Research Article Analytical Results Connecting Stellar Structure … · 2019. 7. 31. · Research...
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Research Article Analytical Results Connecting Stellar Structure Parameters and Extended Reaction Rates Hans J. Haubold 1,2 and Dilip Kumar 2 1 Office for Outer Space Affairs, United Nations, Vienna International Centre, P.O. Box 500, 1400 Vienna, Austria 2 Centre for Mathematical Sciences Pala Campus, Arunapuram P.O., Palai, Kerala 686 574, India Correspondence should be addressed to Hans J. Haubold; [email protected] Received 5 March 2014; Accepted 6 April 2014; Published 12 May 2014 Academic Editor: Milan S. Dimitrijevic Copyright © 2014 H. J. Haubold and D. Kumar. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Possible modification in the velocity distribution in the nonresonant reaction rates leads to an extended reaction rate probability integral. e closed form representation for these thermonuclear functions is used to obtain the stellar luminosity and neutrino emission rates. e composite parameter C that determines the standard nuclear reaction rate through the Maxwell-Boltzmann energy distribution is extended to C ∗ by the extended reaction rates through a more general distribution than the Maxwell- Boltzmann distribution. e new distribution is obtained by the pathway model introduced by Mathai (2005). Simple analytic models considered by various authors are utilized for evaluating stellar luminosity and neutrino emission rates and are obtained in generalized special functions such as Meijer’s G-function and Fox’s H-function. e standard and extended nonresonant thermonuclear functions are compared by plotting them. Behaviour of the new energy distribution, which is more general than the Maxwell-Boltzmann, is also studied. 1. Introduction e mystery behind the distant universe is explored so far by the understanding of the sun, the star near to us. It is the only star whose mass, radius, and luminosity are fairly accurately known. e structural change in the sun is due to the central thermonuclear reactor in it. Solar nuclear energy generation and solar neutrino emission are governed by chains of nuclear reactions in the gravitationally stabilized solar fusion reactor [1, 2]. Qualitative calculations of specific reaction rates require a large amount of experimental inputs and theoretical assumptions. By using the theories from nuclear physics and kinetic theory of gases one can determine the reaction rate for low-energy nonresonant thermonuclear reactions in nondegenerate plasma [3]. e formalization of the calculation of the reaction rate of interacting articles under cosmological or stellar conditions was presented by many authors [4, 5]. For the most common case, a nuclear reaction in which a particle of type 1 strikes a particle of type 2 producing a nucleus 3 and a new particle 4 is symbolically represented as 1+2→3+4+ 12 , (1) where 12 is the energy release given by 12 = ( 1 + 2 − 3 − 4 ) 2 , where , = 1,2,3,4 denote the masses of the particles and denotes the velocity of light. e reaction rate 12 of the interacting particles 1 and 2 is obtained by averaging the reaction cross section over the normalized density function of the relative velocity of the particles [5–7]. Let 1 and 2 denote the number densities of the particles 1 and 2, respectively, and let (V) be the reaction cross section where V is the relative velocity of the particles and (V) is the normalized velocity density; then the reaction rate 12 is given by 12 = (1 − 1 2 12 ) 1 2 ⟨V⟩ 12 = (1 − 1 2 12 ) 1 2 ∫ ∞ 0 V(V)(V) dV = (1 − 1 2 12 ) 1 2 ∫ ∞ 0 () ( 2 ) 1/2 () d, (2) where 12 is the Kronecker delta which is introduced to avoid double counting in the reaction if particles 1 and 2 Hindawi Publishing Corporation Journal of Astrophysics Volume 2014, Article ID 656784, 12 pages http://dx.doi.org/10.1155/2014/656784
Transcript of Research Article Analytical Results Connecting Stellar Structure … · 2019. 7. 31. · Research...
Research ArticleAnalytical Results Connecting Stellar StructureParameters and Extended Reaction Rates
Hans J Haubold12 and Dilip Kumar2
1 Office for Outer Space Affairs United Nations Vienna International Centre PO Box 500 1400 Vienna Austria2 Centre for Mathematical Sciences Pala Campus Arunapuram PO Palai Kerala 686 574 India
Correspondence should be addressed to Hans J Haubold hanshauboldgmailcom
Received 5 March 2014 Accepted 6 April 2014 Published 12 May 2014
Academic Editor Milan S Dimitrijevic
Copyright copy 2014 H J Haubold and D KumarThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Possible modification in the velocity distribution in the nonresonant reaction rates leads to an extended reaction rate probabilityintegral The closed form representation for these thermonuclear functions is used to obtain the stellar luminosity and neutrinoemission rates The composite parameter C that determines the standard nuclear reaction rate through the Maxwell-Boltzmannenergy distribution is extended to Clowast by the extended reaction rates through a more general distribution than the Maxwell-Boltzmann distribution The new distribution is obtained by the pathway model introduced by Mathai (2005) Simple analyticmodels considered by various authors are utilized for evaluating stellar luminosity and neutrino emission rates and are obtainedin generalized special functions such as Meijerrsquos G-function and Foxrsquos H-function The standard and extended nonresonantthermonuclear functions are compared by plotting them Behaviour of the new energy distribution which is more general than theMaxwell-Boltzmann is also studied
1 Introduction
The mystery behind the distant universe is explored so farby the understanding of the sun the star near to us It isthe only star whose mass radius and luminosity are fairlyaccurately known The structural change in the sun is dueto the central thermonuclear reactor in it Solar nuclearenergy generation and solar neutrino emission are governedby chains of nuclear reactions in the gravitationally stabilizedsolar fusion reactor [1 2] Qualitative calculations of specificreaction rates require a large amount of experimental inputsand theoretical assumptions By using the theories fromnuclear physics and kinetic theory of gases one can determinethe reaction rate for low-energy nonresonant thermonuclearreactions in nondegenerate plasma [3] The formalizationof the calculation of the reaction rate of interacting articlesunder cosmological or stellar conditions was presented bymany authors [4 5] For the most common case a nuclearreaction in which a particle of type 1 strikes a particle of type2 producing a nucleus 3 and a new particle 4 is symbolicallyrepresented as
1 + 2 997888rarr 3 + 4 + 11986412 (1)
where 11986412
is the energy release given by 11986412
= (1198981+ 1198982minus
1198983minus 1198984)1198882 where 119898
119894 119894 = 1 2 3 4 denote the masses of
the particles and 119888 denotes the velocity of light The reactionrate 119903
12of the interacting particles 1 and 2 is obtained by
averaging the reaction cross section over the normalizeddensity function of the relative velocity of the particles [5ndash7]Let 1198991and 119899
2denote the number densities of the particles 1
and 2 respectively and let 120590(V) be the reaction cross sectionwhere V is the relative velocity of the particles and 119891(V) is thenormalized velocity density then the reaction rate 119903
12is given
by11990312= (1 minus
1
212057512) 11989911198992⟨120590V⟩12
= (1 minus1
212057512) 11989911198992int
infin
0
V120590 (V) 119891 (V) dV
= (1 minus1
212057512) 11989911198992int
infin
0
120590 (119864) (2119864
120583)
12
119891 (119864) d119864
(2)
where 12057512
is the Kronecker delta which is introduced toavoid double counting in the reaction if particles 1 and 2
Hindawi Publishing CorporationJournal of AstrophysicsVolume 2014 Article ID 656784 12 pageshttpdxdoiorg1011552014656784
2 Journal of Astrophysics
are identical ⟨120590V⟩12is the thermally averaged product which
is in fact the probability per unit time that two particles1 and 2 confined to a unit volume will react with eachother 120583 is the reduced mass of the particles given by 120583 =
(11989811198982)(1198981+ 1198982) 119864 = 120583V22 is the kinetic energy of
the particles in the centre of mass system From literature[4 5 7] it may be noted that all the analytic expressions forastrophysically relevant nuclear reaction rates underline thehypothesis that the distribution of the relative velocities of thereacting particles always remains Maxwell-Boltzmann for anonrelativistic nondegenerate plasma of nuclei in thermody-namic equilibrium The Maxwell-Boltzmann relative kineticenergy distribution can be written as
119891MBD (119864) d119864 = 2120587(1
120587119896119879)
32
exp(minus 119864
119896119879)radic119864d119864 (3)
where 119896 is the Boltzmann constant and 119879 is the temperatureSubstituting (3) in (2) we get
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(1
119896119879)
32
times int
infin
0
119864120590 (119864) exp(minus 119864
119896119879) d119864
(4)
The thermonuclear fusion depends on three physical vari-ables the temperature 119879 the Gamow energy 119864
119866 and the
nuclear fusion factor 119878(119864) If two nuclei of charges 1198851119890 and
1198852119890 collide at low energies below the Coulomb barrier then
the Gamow energy 119864119866is given by [8 9]
119864119866= 2120583(120587120572119885
11198852119888)2
(5)
where 120572 is the electromagnetic fine structure constant givenby
120572 =1198902
ℏ119888 (6)
where 119890 is the quantum of electric charge ℏ is Planckrsquosquantum of action and 120572 is approximately 1137 [9] forour universe Thus the Gamow factor which is determinedby the electromagnetic force and the nuclear fusion factor119878(119864) set the nuclear reaction cross section at low energies fornonresonant charged particles as [7 10]
120590 (119864) =119878 (119864)
119864exp[minus(
119864119866
119864)
12
] (7)
and 119878(119864) is the cross section factor which is often found to beconstant or a slowly varying function of energy over a limitedrange of energy given by [4 5]
119878 (119864) asymp 119878 (0) +d119878 (0)d119864
119864 +1
2
d2119878 (0)d1198642
1198642
=
2
sum
]=0
119878(])(0)
]119864]
(8)
Substituting (7) and (8) in (4) we obtain
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
times (1
119896119879)
32 2
sum
]=0
119878(])(0)
]
times int
infin
0
119864] exp[minus 119864
119896119879minus (
119864119866
119864)
12
] d119864
(9)
This is the nonresonant reaction rate probability integral inthe Maxwell-Boltzmann case The closed form evaluation ofthis integral can be seen in a series of papers by Mathaiand Haubold see for example Haubold and Mathai [6 11]Mathai and Haubold [5] and so forth The main aim of thepresent work is to extend the reaction rate probability integralgiven in (9) by replacing the Maxwell-Boltzmann energydistribution by a more general energy distribution called thepathway energy distribution obtained by using the pathwaymodel of Mathai introduced in 2005
The paper is organized as follows In the next section wediscuss a more general energy distribution than theMaxwell-Boltzmann distribution and obtain the extended reactionrate probability integral in the nonresonant case We takeadvantage of the closed form representation of the extendedthermonuclear reaction rate for finding the luminosity andthe neutrino emission rate of the nonlinear stellar modelunder consideration in Section 3 Section 4 is devoted tofinding the desired connection between stellar structureparameters and the neutrino emission of the stellar model byusing the closed form analytic representation of the extendedreaction rates A comparison of the Maxwell-Boltzmannenergy distribution with the pathway energy distributionis done with the help of graphs in Section 5 Also we tryto discriminate the standard and extended reaction ratesConcluding remarks are included in Section 6
2 Extended Nonresonant ThermonuclearReaction Rate and Its Closed Forms
In recent years possible deviations of the velocity distributionof the plasma particles from the Maxwell-Boltzmann inconnection with the production of neutrinos in the gravi-tationally stabilized solar fusion reactor have been pointedout [2 10 12ndash15] It was initiated by Tsallis the originator ofnonextensive statistical mechanics [16ndash18] who has used 119902-exponential function as the fundamental distribution insteadof the Maxwell-Boltzmann distribution An initial attemptto extend the standard theories of reaction rates to Tsallisstatistics was done by many authors seeMathai and Haubold[19] and Saxena et al [20] In 2005 Mathai introduced thepathway model by which even more general distributionscan be incorporated in the theory of reaction rates [19 21]Initially pathwaymodel was introduced for thematrix variatecase to cover many of the matrix variate statistical densitiesThe scalar case is a particular one there Later Mathai hiscoworkers and others found connection of pathway model
Journal of Astrophysics 3
with the information theory the fractional calculus theMittag-Leffler functions and so forth The pathway modelcan be effectively used in any situation in which we needto switch between three different functional forms namelygeneralized type-1 beta form generalized type-2 beta formand generalized gamma form using the pathway parameter119902 In practical purpose of fitting experimental data pathwaymodel can be utilized to switch between different parametricfamilies with thicker or thinner tail The pathway model forthe real scalar case can be explained as follows
1198911(119909) = 119888
1119909120574minus1
[1 minus 119886 (1 minus 119902) 119909120575
]1(1minus119902)
119886 gt 0 120575 gt 0 1 minus 119886 (1 minus 119902) 119909120575
gt 0 120574 gt 0 119902 lt 1
(10)
is the generalized type-1 beta form of the pathway modelThis is a model with right tail cut-off for 119902 lt 1 The Tsallisstatistics for 119902 lt 1 can be obtained from thismodel by putting120574 = 1 [16ndash18] Other cases available are the regular type-1beta density the Pareto density the power function and thetriangular and related models [22] The generalized type-2beta form of the pathway model is given by
1198912(119909) = 119888
2119909120574minus1
[1 + 119886 (119902 minus 1) 119909120575
]minus1(119902minus1)
0 lt 119909 lt infin 119902 gt 1 119886 gt 0 120574 gt 0 120575 gt 0
(11)
Here also for 120574 = 1 we get the Tsallis statistics for 119902 gt 1 [16ndash18] Other standard distributions coming from this model areregular type-2 beta density 119865-distribution the Levy modeland related models [22] When 119902 rarr 1 119891
1(119909) and 119891
2(119909) will
reduce to the generalized gamma form of the pathway modelgiven by
1198913(119909) = 119888
3119909120574minus1eminus119886119909
120575
119909 gt 0 (12)
This model covers generalized gamma gamma exponen-tial chi-square the Weibull the Maxwell-Boltzmann theRayleigh and related densities 119888
1 1198882 and 119888
3defined in (10)
(11) and (12) respectively are the normalizing constants if weconsider statistical densities
By a suitable modification of the Maxwell-Boltzmanndistribution given in (3) through the pathway model we geta more general energy distribution called the pathway energydistribution given by the density
119891PD (119864) d119864 =2120587(119902 minus 1)
32
(120587119896119879)32
Γ (1 (119902 minus 1))
Γ (1 (119902 minus 1) minus 32)
times radic119864[1 + (119902 minus 1)119864
119896119879]
minus1(119902minus1)
d119864
(13)
for 119902 gt 1 1(119902minus1)minus32 gt 0TheMaxwell-Boltzmann energydistribution can be retrieved from (13) by taking 119902 rarr 1Thus the reaction rate probability integral given in (4) can be
modified by using (13) and we get the extended reaction rateas
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(119902 minus 1
119896119879)
32
timesΓ (1 (119902 minus 1))
Γ (1 (119902 minus 1) minus 32)
2
sum
]=0
119878(])(0)
]
times int
infin
0
119864][1 + (119902 minus 1)
119864
119896119879]
minus1(119902minus1)
times exp[minus(119864119866
119864)
12
] d119864
(14)
for 119902 gt 1 1(119902 minus 1) minus 32 gt 0 Substituting 119910 = 119864119896119879 and 119909 =(119864119866119896119879)12 we obtain the above integral in amore convenient
form as follows
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(119902 minus 1)32
timesΓ (1 (119902 minus 1))
Γ (1 (119902 minus 1) minus 32)
2
sum
]=0(1
119896119879)
minus]+(12)119878(])(0)
]
times int
infin
0
119910][1 + (119902 minus 1) 119910]
minus1(119902minus1)eminus119909119910minus12
d119910
(15)
for 119902 gt 1 1(119902 minus 1) minus 32 gt 0 Here we consider the integralto be evaluated as
1198681119902= int
infin
0
119910][1 + (119902 minus 1) 119910]
minus1(119902minus1)
times eminus119909119910minus12
d119910
(16)
The integral can be evaluated by the techniques in appliedanalysis and can be obtained in closed form via Meijerrsquos 119866-function as [2 23 24]
1198681119902=
(120587)minus12
(119902 minus 1)]+1Γ (1 (119902 minus 1))
times 11986631
13((119902 minus 1) 119909
2
4
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)+]
012]+1)
(17)
which yields the nonresonant reaction rate probability inte-gral in the extended case as
11990312= (1 minus
1
212057512) 11989911198992(8
120583)
12
120587minus1
Γ (1 (119902 minus 1) minus 32)
times
2
sum
]=0(119902 minus 1
119896119879)
minus]+(12)119878(])(0)
]
times 11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)+]
012]+1]
(18)
4 Journal of Astrophysics
Meijerrsquos 119866-function and its properties can be seen in Mathaiand Saxena [25] and Mathai [26] We can obtain seriesexpansions of the 119866-function given in (18) by combining thetheories of residue calculus and generalized special functionssee Kumar and Haubold [24] for series expansions for allpossible values of ] In many cases the nuclear factor 119878(])(0)is approximately constant across the fusion window Taking119878(])(0) = 0 for ] = 1 and ] = 2 and taking 1198780(0) = 119878(0) we
obtain the extended reaction rate probability integral as
11990312= (1 minus
1
212057512) 11989911198992[8 (119902 minus 1)
120583119896119879]
12
120587minus1
Γ (1 (119902 minus 1) minus 32)
times 119878 (0) 11986631
13[(119902 minus 1)119864
119866
4119896119879
10038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
(19)
The series representation for (19) can be obtained as
11990312= (1 minus
1
212057512) 11989911198992[8 (119902 minus 1)
120583119896119879]
12
120587minus1
Γ (1 (119902 minus 1) minus 32)
times 119878 (0) radic120587Γ (1 (119902 minus 1) minus 1)
minus 2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1)119864
119866
4119896119879)
+ (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
times[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(20)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(21)
See the Appendix for detailed evaluation As 119902 rarr 1 in (19)then by using Stirlingrsquos formula for gamma functions givenby
Γ (119911 + 119886) asymp (2120587)12
119911119911+119886minus12eminus119911
|119911| 997888rarr infin 119886 is bounded(22)
we get the reaction rate probability integral in the Maxwell-Boltzmann case as
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(1
119896119879)
32
119878 (0)
times int
infin
0
exp[minus 119864
119896119879minus (
119864119866
119864)
12
] d119864
(23)
= (1 minus1
212057512) 11989911198992(
8
120583119896119879)
12
119878 (0) 120587minus1
times 11986630
03[119864119866
4119896119879
10038161003816100381610038161003816100381610038161003816
minus
0121
]
(24)
which is obtained in a series of papers by Mathai andHaubold see for example Mathai and Haubold [5] Theintegral in (23) is dominated by theminimumvalue of119864119896119879+(119864119866119864)12
= 119892(119864) (say) The minimum value of the function119892(119864) say 119864
0 can be determined as
dd119864
[119864
119896119879+ (
119864119866
119864)
12
]
119864=1198640
=1
119896119879minus1
211986412
119866119864minus32
0= 0 997904rArr 119864
0= 11986412
119866(119896119879
2)
32
(25)
and the function
119892 (1198640) = 3(
119864119866
4119896119879)
13
= 3Θ (26)
where Θ = (1198641198664119896119879)
13 Now by using the Laplace method[27 28] we can obtain an approximate value for (23) as
11990312asymp (1 minus
1
212057512) 11989911198992
8119878 (0)Θ2 exp (minus3Θ)
radic312058712058312057211988511198852119888
(27)
In the next section we will obtain the mass pressure andtemperature for the case of analytic stellar models character-ized by density distribution and corresponding temperaturedistribution suggested by Haubold and Mathai [11]
3 Closed Forms of the Integral over the StellarNuclear Energy Generation Rate
Let us consider the density distribution 984858(119903) considered byHaubold and Mathai [6 11] and Mathai and Haubold [5] inthe form
984858 (119903) = 984858119888[1 minus (
119903
119877)
120575
] 120575 gt 0 (28)
where 984858119888is the central density of the star 119903 is an arbitrary
distance from the center and 119877 is the solar radius Thisdensity function is capable of producing different densitydistributions by choosing the free parameter 120575 Now wedetermine the quantities119872(119903) 119875(119903) and 119879(119903) the mass thepressure and the temperature at 119903
Journal of Astrophysics 5
By the equation of the mass conservation
d119872(119903)
d119903= 4120587119903
2
984858 (119903) (29)
we get
119872(119903) = 4120587984858119888int
119903
0
1199052
[1 minus (119905
119877)
120575
] d119905
=4120587
39848581198881199033
[1 minus3
120575 + 3(119903
119877)
120575
]
(30)
From (30) we get the central density as
984858119888=3 (120575 + 3)
4120587120575
119872 (119877)
1198773
(31)
If an element of a matter at a distance 119903 from the center ofa spherical system is in hydrostatic equilibrium then settingthe sum of the radial forces acting on it to zero we obtain
d119875 (119903)d119903
= minus119866984858 (119903)119872 (119903)
1199032
(32)
where 119866 is the gravitational constant Assuming that thepressure at the center of the sun is 119875
119888and at the surface is
zero we get
119875 (119903) = 119875119888minus 119866int
119903
0
119872(119905) 984858 (119905)
1199052
d119905
=4120587
31198669848582
1198881198772
[120585 minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
(33)
Using the boundary conditions 119875(119877) = 0 we get 119875119888=
(41205873)1198661205859848582
1198881198772 where
120585 =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)+
3
2 (120575 + 1) (120575 + 3) (34)
By the kinetic theory of gases for a perfect gas the pressureis given by
119875 (119903) =119896119873119860
120583984858 (119903) 119879 (119903) (35)
For the temperature of interest for stellar models we neglectthe negligible radiation pressure from the total pressure andobtain from (35) the following
119879 (119903) =120583
119896119873119860
119875 (119903)
984858 (119903)
=4120587
3119896119873119860
1198661205839848581198881198772
[1 minus (119903119877)120575
]
times [120585 minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
(36)
The central temperature 119879119888= (4119896119873
119860)119866120583120585(119872(119877)119877) where
120585 is as defined in (34)Thus we have obtained the mass the pressure and the
temperature throughout the nonlinear stellar model with thedensity distribution defined in (28) Next our aim is to obtainanalytical results for stellar luminosity and neutrino emissionrates for various stellar models
4 Stellar Luminosity and NeutrinoEmission Rate
The energy conservation equation states that the net increasein the rate of energy flux coming out of a spherical shell fromthe inside is the same as the energy produced within the shell[29] If we denote 119871
119903= 119871(119903) as the energy flux through the
sphere of radius 119903 then we have
d119871119903
d119903= 4120587119903
2
984858 (119903) 120576 (119903) (37)
where 120576(119903) is the energy produced per second by nuclearreactions in each gram of stellar matter 120576(119903) depends on thechemical composition in each gram of stellar matter Hereusually 119871
119903is a constant but will be equal to 119871 at the surface of
the star We assume here that the star is chemically homoge-neous (that is a star where chemical composition throughoutis a constant) Also we assume the energy generation rate120576(119903) for one particular nuclear reaction Now if we denote11990312(984858(119903) 119879(119903)) as the extended nonresonant thermonuclear
reaction rate for the particles 1 and 2 defined by (19) thenwe will consider the energy generation rate 120576
12(119903) and it can
be written in terms of the extended reaction rates via
12057612(119903) =
1
984858 (119903)Clowast
9848582
11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
] (38)
where
Clowast
=1198641211990312(984858 (119903) 119879 (119903))
984858211986631
13[(119902 minus 1) 119864
1198664119896119879
1003816100381610038161003816
2minus1(119902minus1)
0121]
(39)
in which 11986412
is the amount of energy given off in a singlereaction It is to be noted that by using the asymptoticbehaviour of 11986631
13((119902 minus 1)119864
1198664119896119879) [26] and as 119902 rarr 1
Clowast rarr C the composite parameter considered by [9] whichis defined as
C =1198641211990312
9848582Θ2
exp (3Θ) (40)
for our universeC asymp 2 times 104 for proton-proton fusion under
typical stellar conditions [9]Then from (37) we have the totalluminosity of the star by integration as follows
119871 (119877) = int
119877
0
41205871199032
984858 (119903) 120576 (119903) d119903 (41)
If we are considering only one specific reaction defined as in(1) then we have
11987112(119877) = int
119877
0
41205871199032
984858 (119903) 12057612(119903) d119903 (42)
6 Journal of Astrophysics
where the energy generation rate is defined in (38) and 984858(119903)is a suitable density distribution explaining the sun Writing(42) in terms of 119903
12(984858(119903) 119879(119903)) we get
11987112(119877) = int
119877
0
41205871199032
Clowast
9848582
11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
] d119903
= int
119877
0
41205871199032
1198641211990312(984858 (119903) 119879 (119903)) d119903
(43)
The number density 119899119894of a particle 119894 for a gas ofmean density
984858(119903) can be expressed as
119899119894(119903) = 984858 (119903)119873
119860
119883119894
119860119894
(44)
where 119873119860stands for Avagadrorsquos constant 119860
119894is the atomic
mass of particle 119894 in atomic mass units and 119883119894is the mass
fraction of particle 119894 such that sum119894119883119894
= 1 Substituting11990312(984858(119903) 119879(119903)) from (19) and using (44) we have
11987112(119877)
= int
119877
0
41205871199032
11986412(1 minus
1
212057512)1198732
1198609848582
(119903)11988311198832
11986011198602
[8 (119902 minus 1)
120583119896119879 (119903)]
12
times120587minus1
Γ (1 (119902 minus 1) minus 32)119878 (0) 119866
31
13
times [
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)(119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
(45)
If we divide the ldquointernal luminosityrdquo 11987112(119877⊙) by the amount
of energy 11986412 then we get the total number of particles per
second11987312liberated in the reaction given by (1) as follows
11987312=11987112(119877)
11986412
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
times int
119877
0
1199032
9848582
(119903)
[119879 (119903)]12
11986631
13
[
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)
times (119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)[
(119902 minus 1) 1205872
120583
2119896(11988511198852e2
ℏ)
2
]
minus119904
times int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903 d119904
(46)
For the density distribution defined in (28) introduced byHaubold and Mathai [5 6] and the corresponding temper-ature distribution (36) we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
119888
times int
119877
0
1199032
[1 minus (119903
119877)
120575
]
52minus119904
times [minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
119904minus12
d119903
(47)
where 120585 is as defined in (34) If we put a substitution 119903 = 119909119877then we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
times int
1
0
1199092
[1 minus 119909120575
]52minus119904
times [120585 minus1
21199092
+120575 + 6
(120575 + 2) (120575 + 3)119909120575+2
minus3
2(120575 + 1)(120575 + 3)1199092120575+2
]
119904minus12
d119909
(48)
Journal of Astrophysics 7
Putting 119909120575 = 119910 and simplifying we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583120585
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
120575
times int
1
0
1199103120575minus1
[1 minus 119910]52minus119904
[1 minus 119906 (119910)]119904minus12d119910
(49)
where 119906(119910) is defined as
119906 (119910) =1199102120575
120585[1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
]
(50)
As 119910 rarr 0 119906(119910) rarr 0 and 119910 rarr 1 119906(119910) rarr 1 If we take
V (119910) =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
(51)
we have V(0) = 12 The minimum value of V(119910) is at 119910 =
(120575 + 1)(120575 + 6)3(120575 + 2) and the value is 12 minus 16((120575 + 6)2(120575 +1)(120575 + 3)(120575 + 2)
2
) Thus the minimum value is nonnegativesince (120575 + 6)2(120575 + 1) (120575 + 3)(120575 + 2)2 decreases steadily from3 to 1 for all 120575 gt 0 Therefore V(119910) le 0 Since 120585 gt 0 for all120575 gt 0 119906(119910) le 0 for all 120575 gt 0 120585 gt 0 Thus [1 minus 119906(119910)]119904minus12 le 0Hence 0 lt 119906(119910) lt 1 for 0 lt 119910 lt 1 and for 120575 gt 0 Thus byusing the binomial expansion we obtain
[1 minus 119906 (119910)]119904minus12
=
infin
sum
119898=0
(12 minus 119904)119898
119898[119906 (119910)]
119898
(52)
where (12 minus 119904)119898is the Pochhammer symbol defined for 119886 isin
C by
(119886)0= 1
(119886)119898=
119886 (119886 + 1) sdot sdot sdot (119886 + 119898 minus 1)
Γ (119886 + 119898)
Γ (119886)
119898 = 1 2 119886 = 0
(53)
whenever Γ(119886) exists Taking 120575 = 2 we get 120585 = 15 and
[1 minus 119906 (119910)]119904minus12
= (1 minus 119910)2119904minus1
(1 minus1
2119910)
119904minus12
(54)
Then from (49) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times int
1
0
11991032minus1
[1 minus 119910]52+119904minus1
(1 minus1
2119910)
119904minus12
d119910
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
(12 minus 119904)119898
119898
1
2119898
times int
1
0
11991032+119898minus1
[1 minus 119910]52+119904minus1d119910
(55)
By using beta integral and using (53) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
Γ (12 minus 119904 + 119898)
119898Γ (12 minus 119904)
1
2119898
Γ (32 + 119898) Γ (52 + 119904)
Γ (4 + 119904 + 119898)
(56)
Now from (46) we obtain the total number of particles persecond liberated in the reaction (1) as follows
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904) Γ (
5
2+ 119904)
timesΓ (1 (119902 minus 1) minus 1 minus 119904) Γ (12 + 119898 minus 119904)
Γ (12 minus 119904) Γ (4 + 119898 + 119904)
times [15120587 (119902 minus 1)119873
119860
81198669848581198881198772
(11988511198852e2
ℏ)
2
]
minus119904
d119904
8 Journal of Astrophysics
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
times 11986642
35
[
[
15120587 (119902 minus 1)119873119860
81198669848581198881198772
times (11988511198852e2
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)12minus1198984+119898
01215212
]
]
(57)
For more details on119866-function and its properties see [2526] Thus we have obtained the the total number of particlesper second liberated in the reaction (1) in terms of the densitydistribution considered by Haubold and Mathai [6 29]
5 Comparison of Pathway Energy Density andthe Maxwell-Boltzmann Energy Density
In Figure 1 it can be obtserved that for the nuclei to reactat energy 119864 they have to borrow an energy 119864 from thethermal environment The probability of such an energy isproportional to theMaxwell-Boltzmann energy exp[minus119864119896119879]The fusion will take place when the nuclei penerate theCoulomb barrier keeping them apart The probability ofpenetration is given by the factor exp[minus(119864
119866119864)12
] Theproduct of these two factors illustrates that fusion mostlyoccurs in the energy window given in the figure
In Figure 2 the pathway energy density is plotted for 119902 =07 09 1 12 14 respectively For different values of 119902we getdifferent energy densities (curves (a) (b) (c) (d) and (e))The nonresonant cross section is also plotted The product ofthe pathway energy density and the nonresonant cross sectionfor different values of 119902 namely 119902 = 07 09 1 12 14 is alsoplotted It is to be noted that as 119902 rarr 1 the pathway energydensity coincides with the Maxwell-Boltzmann energy den-sity and also the fusion window for the Maxwell-Boltzmanncase in Figure 1
The curves in the figure represent pathway density[1 + (119902 minus 1)(119864119896119879)]
minus1(119902minus1) for (a) 119902 = 07 (b) 119902 = 09(c) 119902 = 1 (d) 119902 = 12 and (e) 119902 = 14 (f) repre-sents the nonresonant cross section exp[minus(119864
119866119864)12
] Theproduct [1 + (119902 minus 1)(119864119896119879)]minus1(119902minus1) exp[minus(119864
119866119864)12
] is alsorepresented for (g) 119902 = 07 (h) 119902 = 09 (i) 119902 = 1 (j) 119902 = 12and (k) 119902 = 14
Figure 3(a) shows the pathway energy density defined in(13) for 119902 = 1 12 13 14 and Figure 3(b) shows theMaxwell-Boltzmann energy density defined in (3) As 119902 rarr 1 thepathway energy density reduces to the Maxwell-Boltzmannenergy density The pathway energy density covers manystable and unstable situations as the value of 119902 varies If
Fusion window
0 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
2kT 4kT 6kT 8kT
prop exp[minus E
kT]
prop exp[minus (EGE
)12
]
prop exp[minus E
kTminus (EG
E)12
]
Energy E
Figure 1 Schematic plot of the enrgy-dependent factors for thereaction rate probability integral the Maxwell-Boltzmann energydensity nonresonant nuclear cross section and the product of theMaxwell-Boltzmann density and the nonresonant cross section
Fusion window
0 2kT 4kT 6kT 8kT 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
1 + (q minus 1)E
kT]minus1qminus1
(a)
(b)(c)
(d)(e)
(h)
(i)
(k)
(f)
(j)
(g)
prop[prop exp[minus (EG
E)12
]
Energy E
Figure 2 Schematic plot of the energy-dependent factors for theextended reaction rate probability integral pathway energy densitynonresonant nuclear cross section and the product of the pathwaydensity and the nonresonant cross section
the Maxwell-Boltzmann density is the equilibrium situationmany other nonequilibrium situations are covered by thepathway energy density
Journal of Astrophysics 9
Pathway energy density for k = 1 T = 100K
0004
0003
0002
0001
0
q = 1
q = 12
q = 13
q = 14
0 200 400 600 800 1000
fPD
Pathway energy density for k = 1 T = 100K
q = 1
q = 12
q = 13
q = 14
Energy E
(a)
0004
0003
0002
0001
0
fM
BD
Maxwell-Boltzmann energy densityfor k = 1 T = 100K
0 200 400 600 800 1000
Energy E
(b)
Figure 3 (a) Pathway energy density for 119902 = 1 12 13 14 and for 119896 = 1 119879 = 100119870 (b) The Maxwell-Boltzmann energy density for119896 = 1 119879 = 100119870
6 Concluding Remarks
In this paper we have modified the energy distributionfor a nonresonant reaction rate probability integral Thecomposition parameter C considered by [9] is extended toClowast by the pathway energy density Considering the analyticdensity distributions developed by Haubold and Mathai [629] they are used to obtain the stellar luminosity and the neu-trino emission rates and are obtained in generalized specialfunctions such as Meijerrsquos 119866-function The pathway energydensity considered here covers many density functions andhence the extended reaction rate integral covers a wider classof integral Pathway energy density helps us to obtain variousfusion windows by giving different values to 119902 the pathwayparameter which in turn leads to a new opening in the fusionresearch The graphs plotted here are by using Maple 14 inWindows XP platform
Appendix
Series Representation
The series representation for the right-hand side of (19) canbe obtained through the following procedure Here we applyresidue calculus on the119866-function given in (19) Consider the119866-function as follows
11986631
13((119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
)
=1
2120587119894int
119888+119894infin
119888minus119894infin
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1) 119864119866
4119896119879)
minus119904
d119904
(A1)
The right-hand side is the sum of the residues of the inte-grandThe poles of the gammas in the integral representationin (19) are as follows
poles of Γ(119904) 119904 = 0 minus1 minus2
poles of Γ(12 + 119904) 119904 = minus12 minus32 minus52
poles of Γ(1 + 119904) 119904 = minus1 minus2 minus3
Here the poles of Γ(119904) and Γ(1 + 119904) will coincide with eachother at all points except at 119904 = 0 Note that the pole 119904 = 0 isa pole of order 1 119904 = minus12 minus32 minus52 are each of order 1and 119904 = minus1 minus2 minus3 are each of order 2 We know that
lim119904rarrminus119903
(119904 + 119903) Γ (119904) =(minus1)119903
119903
Γ (119886 minus 119903) =(minus1)119903
Γ (119886)
(1 minus 119886)119903
Γ (119886 + 119898) = Γ (119886) (119886)119898
(A2)
10 Journal of Astrophysics
when Γ(119886) is defined 119903 = 0 1 2 Γ(12) = 12058712
(119886)119903=
119886 (119886 + 1) sdot sdot sdot (119886 + 119903 minus 1) if 119903 ge 1 119886 = 0
1 if 119903 = 0(A3)
The sum of the residues corresponding to the poles 119904 = 0 isgiven by
1198771= radic120587Γ(
1
119902 minus 1minus 1) (A4)
The sum of the residues corresponding to the poles 119904 =
minus12 minus32 minus52 is
1198772=
infin
sum
119903=0
(minus1)119903
119903Γ (minus
1
2minus 119903) Γ (
1
2minus 119903)
times Γ(1
119902 minus 1minus1
2+ 119903) [
(119902 minus 1)119864119866
4119896119879]
minus12+119903
= minus2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1) 119864
119866
4119896119879)
(A5)
where11198652is the hypergeometric function defined by
11198652(119886 119887 119888 119909) =
infin
sum
119903=0
(119886)119903
(119887)119903(119888)119903
119909119903
119903 (A6)
To obtain the sum of the residues corresponding to poles119904 = minus1 minus2 minus3 of order 2 we proceed as follows
1198773
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[(119904 + 1 + 119903)
2
Γ (1 + 119904) Γ (119904) Γ (1
2+ 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1)119864119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[ (Γ2
(2 + 119904 + 119903) Γ (12 + 119904)
times Γ (1 (119902 minus 1) minus 1 minus 119904))
times ((119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904)minus1
times ((119902 minus 1) 119864
119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904Φ (119904)
(A7)
where
Φ (119904) =Γ2
(2 + 119904 + 119903) Γ (12 + 119904) Γ (1 (119902 minus 1) minus 1 minus 119904)
(119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904
times ((119902 minus 1)119864
119866
4119896119879)
minus119904
(A8)
We have
120597
120597119904Φ (119904) = Φ (119904)
120597
120597119904[ln [Φ (119904)]]
lnΦ (119904) = 2 ln [Γ (2 + 119904 + 119903)] + ln [Γ (12+ 119904)]
+ ln [Γ( 1
119902 minus 1minus 1 minus 119904)] minus 119904 ln(
(119902 minus 1) 119864119866
4119896119879)
minus 2 ln (119904 + 119903) minus 2 ln (119904 + 119903 minus 1) minus sdot sdot sdot
minus 2 ln (119904 + 1) minus ln (119904)
120597
120597119904[ln [Φ (119904)]] = 2Ψ (2 + 119904 + 119903) + Ψ(
1
2+ 119904)
+ Ψ(1
119902 minus 1minus 1 minus 119904)
minus ln((119902 minus 1) 119864
119866
4119896119879) minus
2
119904 + 119903
minus2
119904 + 119903 minus 1minus sdot sdot sdot minus
2
119904 + 1minus1
119904
lim119904rarrminus1minus119903
120597
120597119904ln [Φ (119904)] = Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
minus ln((119902 minus 1) 119864
119866
4119896119879)
(A9)
whereΨ(119911) is a Psi function or digamma function (seeMathai[26]) and Ψ(1) = minus120574 120574 = 05772156649 is Eulerrsquosconstant Now
lim119904rarrminus1minus119903
Φ (119904) =(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
times ((119902 minus 1) 119864
119866
4119896119879)
1+119903
(A10)
Journal of Astrophysics 11
Then by using (A7) (A9) and (A10) we get
1198773=
infin
sum
119903=0
(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
((119902 minus 1) 119864
119866
4119896119879)
1+119903
times [Ψ(minus1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903) + Ψ (1 + 119903)
+Ψ (2 + 119903) minus ln((119902 minus 1) 119864
119866
4119896119879)]
= (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(A11)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(A12)
Thus from (A4) (A5) and (A11) we get (20)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the Department of Sci-ence and Technology Government of India New Delhifor the financial assistance for this work under Project noSRS4MS28705 and the Centre for Mathematical Sciencesfor providing all facilities
References
[1] R Davis Jr ldquoNobel Lecture a half-century with solar neutri-nosrdquo Reviews of Modern Physics vol 75 no 3 pp 985ndash9942003
[2] H J Haubold and D Kumar ldquoExtension of thermonuclearfunctions through the pathway model including Maxwell-Boltzmann and Tsallis distributionsrdquo Astroparticle Physics vol29 no 1 pp 70ndash76 2008
[3] H J Haubold and R W John ldquoOn the evaluation of an integralconnected with the thermonuclear reaction rate in closed-formrdquo Astronomische Nachrichten vol 299 no 5 pp 225ndash2321978
[4] W A Fowler G R Caughlan and B A Zimmerman ldquoTher-monuclear rection ratesrdquo Annual Review of Astronomy andAstrophysics vol 5 pp 525ndash570 1967
[5] A M Mathai and H J Haubold Modern Problems in Nuclearand Neutrino Astrophysics Akademie Berlin Germany 1988
[6] H J Haubold and A M Mathai ldquoAnalytic representations ofmodified non-resonant thermonuclear reaction ratesrdquo Journalof Applied Mathematics and Physics vol 37 no 5 pp 685ndash6951986
[7] W A Fowler ldquoExperimental and theoretical nuclear astro-physics the quest for the origin of the elementsrdquo Reviews ofModern Physics vol 56 no 2 pp 149ndash179 1984
[8] A C Phillips The Physics of Stars John Wiley amp SonsChichester UK 2nd edition 1999
[9] F C Adams ldquoStars in other universes stellar structure withdifferent fundamental constantsrdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 8 article 10 2008
[10] M Coraddu G Kaniadakis A Lavagno M Lissia G Mezzo-rani and P Quarati ldquoThermal distributions in stellar plasmasnuclear reactions and solar neutrinosrdquo Brazilian Journal ofPhysics vol 29 no 1 pp 153ndash168 1999
[11] H J Haubold and A M Mathai ldquoOn nuclear reaction ratetheoryrdquo Annalen der Physik vol 41 pp 380ndash396 1984
[12] M Coraddu M Lissia G Mezzorani and P Quarati ldquoSuper-Kamiokande hep neutrino best fit a possible signal of non-Maxwellian solar plasmardquo Physica A Statistical Mechanics andIts Applications vol 326 no 3-4 pp 473ndash481 2003
[13] A Lavagno and P Quarati ldquoClassical and quantum non-extensive statistics effects in nuclear many-body problemsrdquoChaos Solitons and Fractals vol 13 no 3 pp 569ndash580 2002
[14] A Lavagno and P Quarati ldquoMetastability of electron-nuclearastrophysical plasmas motivations signals and conditionsrdquoAstrophysics and Space Science vol 305 no 3 pp 253ndash2592006
[15] M Lissia and P Quarati ldquoNuclear astrophysical plasmas iondistribution functions and fusion ratesrdquo Europhysics News vol36 no 6 pp 211ndash214 2005
[16] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988
[17] C Tsallis Introduction to Non-Extensive Statistical MechanicsSpringer New York NY USA 2009
[18] M Gell-Mann and C Tsallis Eds Nonextensive EntropyInterdisciplinary Applications Oxford University Press NewYork NY USA 2004
[19] A M Mathai and H J Haubold ldquoPathway model superstatis-tics Tsallis statistics and a generalized measure of entropyrdquoPhysica A StatisticalMechanics and Its Applications vol 375 no1 pp 110ndash122 2007
[20] R K Saxena A M Mathai and H J Haubold ldquoAstrophysicalthermonuclear functions for Boltzmann-Gibbs statistics andTsallis statisticsrdquo Physica A Statistical Mechanics and Its Appli-cations vol 344 no 3-4 pp 649ndash656 2004
[21] A MMathai ldquoA pathway tomatrix-variate gamma and normaldensitiesrdquo Linear Algebra and Its Applications vol 396 no 1ndash3pp 317ndash328 2005
[22] A M Mathai and H J Haubold ldquoOn generalized distributionsand pathwaysrdquoPhysics Letters A General Atomic and Solid StatePhysics vol 372 no 12 pp 2109ndash2113 2008
[23] H J Haubold andDKumar ldquoFusion yield Guderleymodel andTsallis statisticsrdquo Journal of Plasma Physics vol 77 no 1 pp 1ndash14 2011
[24] D Kumar and H J Haubold ldquoOn extended thermonuclearfunctions through pathwaymodelrdquoAdvances in Space Researchvol 45 no 5 pp 698ndash708 2010
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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ThermodynamicsJournal of
2 Journal of Astrophysics
are identical ⟨120590V⟩12is the thermally averaged product which
is in fact the probability per unit time that two particles1 and 2 confined to a unit volume will react with eachother 120583 is the reduced mass of the particles given by 120583 =
(11989811198982)(1198981+ 1198982) 119864 = 120583V22 is the kinetic energy of
the particles in the centre of mass system From literature[4 5 7] it may be noted that all the analytic expressions forastrophysically relevant nuclear reaction rates underline thehypothesis that the distribution of the relative velocities of thereacting particles always remains Maxwell-Boltzmann for anonrelativistic nondegenerate plasma of nuclei in thermody-namic equilibrium The Maxwell-Boltzmann relative kineticenergy distribution can be written as
119891MBD (119864) d119864 = 2120587(1
120587119896119879)
32
exp(minus 119864
119896119879)radic119864d119864 (3)
where 119896 is the Boltzmann constant and 119879 is the temperatureSubstituting (3) in (2) we get
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(1
119896119879)
32
times int
infin
0
119864120590 (119864) exp(minus 119864
119896119879) d119864
(4)
The thermonuclear fusion depends on three physical vari-ables the temperature 119879 the Gamow energy 119864
119866 and the
nuclear fusion factor 119878(119864) If two nuclei of charges 1198851119890 and
1198852119890 collide at low energies below the Coulomb barrier then
the Gamow energy 119864119866is given by [8 9]
119864119866= 2120583(120587120572119885
11198852119888)2
(5)
where 120572 is the electromagnetic fine structure constant givenby
120572 =1198902
ℏ119888 (6)
where 119890 is the quantum of electric charge ℏ is Planckrsquosquantum of action and 120572 is approximately 1137 [9] forour universe Thus the Gamow factor which is determinedby the electromagnetic force and the nuclear fusion factor119878(119864) set the nuclear reaction cross section at low energies fornonresonant charged particles as [7 10]
120590 (119864) =119878 (119864)
119864exp[minus(
119864119866
119864)
12
] (7)
and 119878(119864) is the cross section factor which is often found to beconstant or a slowly varying function of energy over a limitedrange of energy given by [4 5]
119878 (119864) asymp 119878 (0) +d119878 (0)d119864
119864 +1
2
d2119878 (0)d1198642
1198642
=
2
sum
]=0
119878(])(0)
]119864]
(8)
Substituting (7) and (8) in (4) we obtain
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
times (1
119896119879)
32 2
sum
]=0
119878(])(0)
]
times int
infin
0
119864] exp[minus 119864
119896119879minus (
119864119866
119864)
12
] d119864
(9)
This is the nonresonant reaction rate probability integral inthe Maxwell-Boltzmann case The closed form evaluation ofthis integral can be seen in a series of papers by Mathaiand Haubold see for example Haubold and Mathai [6 11]Mathai and Haubold [5] and so forth The main aim of thepresent work is to extend the reaction rate probability integralgiven in (9) by replacing the Maxwell-Boltzmann energydistribution by a more general energy distribution called thepathway energy distribution obtained by using the pathwaymodel of Mathai introduced in 2005
The paper is organized as follows In the next section wediscuss a more general energy distribution than theMaxwell-Boltzmann distribution and obtain the extended reactionrate probability integral in the nonresonant case We takeadvantage of the closed form representation of the extendedthermonuclear reaction rate for finding the luminosity andthe neutrino emission rate of the nonlinear stellar modelunder consideration in Section 3 Section 4 is devoted tofinding the desired connection between stellar structureparameters and the neutrino emission of the stellar model byusing the closed form analytic representation of the extendedreaction rates A comparison of the Maxwell-Boltzmannenergy distribution with the pathway energy distributionis done with the help of graphs in Section 5 Also we tryto discriminate the standard and extended reaction ratesConcluding remarks are included in Section 6
2 Extended Nonresonant ThermonuclearReaction Rate and Its Closed Forms
In recent years possible deviations of the velocity distributionof the plasma particles from the Maxwell-Boltzmann inconnection with the production of neutrinos in the gravi-tationally stabilized solar fusion reactor have been pointedout [2 10 12ndash15] It was initiated by Tsallis the originator ofnonextensive statistical mechanics [16ndash18] who has used 119902-exponential function as the fundamental distribution insteadof the Maxwell-Boltzmann distribution An initial attemptto extend the standard theories of reaction rates to Tsallisstatistics was done by many authors seeMathai and Haubold[19] and Saxena et al [20] In 2005 Mathai introduced thepathway model by which even more general distributionscan be incorporated in the theory of reaction rates [19 21]Initially pathwaymodel was introduced for thematrix variatecase to cover many of the matrix variate statistical densitiesThe scalar case is a particular one there Later Mathai hiscoworkers and others found connection of pathway model
Journal of Astrophysics 3
with the information theory the fractional calculus theMittag-Leffler functions and so forth The pathway modelcan be effectively used in any situation in which we needto switch between three different functional forms namelygeneralized type-1 beta form generalized type-2 beta formand generalized gamma form using the pathway parameter119902 In practical purpose of fitting experimental data pathwaymodel can be utilized to switch between different parametricfamilies with thicker or thinner tail The pathway model forthe real scalar case can be explained as follows
1198911(119909) = 119888
1119909120574minus1
[1 minus 119886 (1 minus 119902) 119909120575
]1(1minus119902)
119886 gt 0 120575 gt 0 1 minus 119886 (1 minus 119902) 119909120575
gt 0 120574 gt 0 119902 lt 1
(10)
is the generalized type-1 beta form of the pathway modelThis is a model with right tail cut-off for 119902 lt 1 The Tsallisstatistics for 119902 lt 1 can be obtained from thismodel by putting120574 = 1 [16ndash18] Other cases available are the regular type-1beta density the Pareto density the power function and thetriangular and related models [22] The generalized type-2beta form of the pathway model is given by
1198912(119909) = 119888
2119909120574minus1
[1 + 119886 (119902 minus 1) 119909120575
]minus1(119902minus1)
0 lt 119909 lt infin 119902 gt 1 119886 gt 0 120574 gt 0 120575 gt 0
(11)
Here also for 120574 = 1 we get the Tsallis statistics for 119902 gt 1 [16ndash18] Other standard distributions coming from this model areregular type-2 beta density 119865-distribution the Levy modeland related models [22] When 119902 rarr 1 119891
1(119909) and 119891
2(119909) will
reduce to the generalized gamma form of the pathway modelgiven by
1198913(119909) = 119888
3119909120574minus1eminus119886119909
120575
119909 gt 0 (12)
This model covers generalized gamma gamma exponen-tial chi-square the Weibull the Maxwell-Boltzmann theRayleigh and related densities 119888
1 1198882 and 119888
3defined in (10)
(11) and (12) respectively are the normalizing constants if weconsider statistical densities
By a suitable modification of the Maxwell-Boltzmanndistribution given in (3) through the pathway model we geta more general energy distribution called the pathway energydistribution given by the density
119891PD (119864) d119864 =2120587(119902 minus 1)
32
(120587119896119879)32
Γ (1 (119902 minus 1))
Γ (1 (119902 minus 1) minus 32)
times radic119864[1 + (119902 minus 1)119864
119896119879]
minus1(119902minus1)
d119864
(13)
for 119902 gt 1 1(119902minus1)minus32 gt 0TheMaxwell-Boltzmann energydistribution can be retrieved from (13) by taking 119902 rarr 1Thus the reaction rate probability integral given in (4) can be
modified by using (13) and we get the extended reaction rateas
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(119902 minus 1
119896119879)
32
timesΓ (1 (119902 minus 1))
Γ (1 (119902 minus 1) minus 32)
2
sum
]=0
119878(])(0)
]
times int
infin
0
119864][1 + (119902 minus 1)
119864
119896119879]
minus1(119902minus1)
times exp[minus(119864119866
119864)
12
] d119864
(14)
for 119902 gt 1 1(119902 minus 1) minus 32 gt 0 Substituting 119910 = 119864119896119879 and 119909 =(119864119866119896119879)12 we obtain the above integral in amore convenient
form as follows
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(119902 minus 1)32
timesΓ (1 (119902 minus 1))
Γ (1 (119902 minus 1) minus 32)
2
sum
]=0(1
119896119879)
minus]+(12)119878(])(0)
]
times int
infin
0
119910][1 + (119902 minus 1) 119910]
minus1(119902minus1)eminus119909119910minus12
d119910
(15)
for 119902 gt 1 1(119902 minus 1) minus 32 gt 0 Here we consider the integralto be evaluated as
1198681119902= int
infin
0
119910][1 + (119902 minus 1) 119910]
minus1(119902minus1)
times eminus119909119910minus12
d119910
(16)
The integral can be evaluated by the techniques in appliedanalysis and can be obtained in closed form via Meijerrsquos 119866-function as [2 23 24]
1198681119902=
(120587)minus12
(119902 minus 1)]+1Γ (1 (119902 minus 1))
times 11986631
13((119902 minus 1) 119909
2
4
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)+]
012]+1)
(17)
which yields the nonresonant reaction rate probability inte-gral in the extended case as
11990312= (1 minus
1
212057512) 11989911198992(8
120583)
12
120587minus1
Γ (1 (119902 minus 1) minus 32)
times
2
sum
]=0(119902 minus 1
119896119879)
minus]+(12)119878(])(0)
]
times 11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)+]
012]+1]
(18)
4 Journal of Astrophysics
Meijerrsquos 119866-function and its properties can be seen in Mathaiand Saxena [25] and Mathai [26] We can obtain seriesexpansions of the 119866-function given in (18) by combining thetheories of residue calculus and generalized special functionssee Kumar and Haubold [24] for series expansions for allpossible values of ] In many cases the nuclear factor 119878(])(0)is approximately constant across the fusion window Taking119878(])(0) = 0 for ] = 1 and ] = 2 and taking 1198780(0) = 119878(0) we
obtain the extended reaction rate probability integral as
11990312= (1 minus
1
212057512) 11989911198992[8 (119902 minus 1)
120583119896119879]
12
120587minus1
Γ (1 (119902 minus 1) minus 32)
times 119878 (0) 11986631
13[(119902 minus 1)119864
119866
4119896119879
10038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
(19)
The series representation for (19) can be obtained as
11990312= (1 minus
1
212057512) 11989911198992[8 (119902 minus 1)
120583119896119879]
12
120587minus1
Γ (1 (119902 minus 1) minus 32)
times 119878 (0) radic120587Γ (1 (119902 minus 1) minus 1)
minus 2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1)119864
119866
4119896119879)
+ (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
times[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(20)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(21)
See the Appendix for detailed evaluation As 119902 rarr 1 in (19)then by using Stirlingrsquos formula for gamma functions givenby
Γ (119911 + 119886) asymp (2120587)12
119911119911+119886minus12eminus119911
|119911| 997888rarr infin 119886 is bounded(22)
we get the reaction rate probability integral in the Maxwell-Boltzmann case as
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(1
119896119879)
32
119878 (0)
times int
infin
0
exp[minus 119864
119896119879minus (
119864119866
119864)
12
] d119864
(23)
= (1 minus1
212057512) 11989911198992(
8
120583119896119879)
12
119878 (0) 120587minus1
times 11986630
03[119864119866
4119896119879
10038161003816100381610038161003816100381610038161003816
minus
0121
]
(24)
which is obtained in a series of papers by Mathai andHaubold see for example Mathai and Haubold [5] Theintegral in (23) is dominated by theminimumvalue of119864119896119879+(119864119866119864)12
= 119892(119864) (say) The minimum value of the function119892(119864) say 119864
0 can be determined as
dd119864
[119864
119896119879+ (
119864119866
119864)
12
]
119864=1198640
=1
119896119879minus1
211986412
119866119864minus32
0= 0 997904rArr 119864
0= 11986412
119866(119896119879
2)
32
(25)
and the function
119892 (1198640) = 3(
119864119866
4119896119879)
13
= 3Θ (26)
where Θ = (1198641198664119896119879)
13 Now by using the Laplace method[27 28] we can obtain an approximate value for (23) as
11990312asymp (1 minus
1
212057512) 11989911198992
8119878 (0)Θ2 exp (minus3Θ)
radic312058712058312057211988511198852119888
(27)
In the next section we will obtain the mass pressure andtemperature for the case of analytic stellar models character-ized by density distribution and corresponding temperaturedistribution suggested by Haubold and Mathai [11]
3 Closed Forms of the Integral over the StellarNuclear Energy Generation Rate
Let us consider the density distribution 984858(119903) considered byHaubold and Mathai [6 11] and Mathai and Haubold [5] inthe form
984858 (119903) = 984858119888[1 minus (
119903
119877)
120575
] 120575 gt 0 (28)
where 984858119888is the central density of the star 119903 is an arbitrary
distance from the center and 119877 is the solar radius Thisdensity function is capable of producing different densitydistributions by choosing the free parameter 120575 Now wedetermine the quantities119872(119903) 119875(119903) and 119879(119903) the mass thepressure and the temperature at 119903
Journal of Astrophysics 5
By the equation of the mass conservation
d119872(119903)
d119903= 4120587119903
2
984858 (119903) (29)
we get
119872(119903) = 4120587984858119888int
119903
0
1199052
[1 minus (119905
119877)
120575
] d119905
=4120587
39848581198881199033
[1 minus3
120575 + 3(119903
119877)
120575
]
(30)
From (30) we get the central density as
984858119888=3 (120575 + 3)
4120587120575
119872 (119877)
1198773
(31)
If an element of a matter at a distance 119903 from the center ofa spherical system is in hydrostatic equilibrium then settingthe sum of the radial forces acting on it to zero we obtain
d119875 (119903)d119903
= minus119866984858 (119903)119872 (119903)
1199032
(32)
where 119866 is the gravitational constant Assuming that thepressure at the center of the sun is 119875
119888and at the surface is
zero we get
119875 (119903) = 119875119888minus 119866int
119903
0
119872(119905) 984858 (119905)
1199052
d119905
=4120587
31198669848582
1198881198772
[120585 minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
(33)
Using the boundary conditions 119875(119877) = 0 we get 119875119888=
(41205873)1198661205859848582
1198881198772 where
120585 =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)+
3
2 (120575 + 1) (120575 + 3) (34)
By the kinetic theory of gases for a perfect gas the pressureis given by
119875 (119903) =119896119873119860
120583984858 (119903) 119879 (119903) (35)
For the temperature of interest for stellar models we neglectthe negligible radiation pressure from the total pressure andobtain from (35) the following
119879 (119903) =120583
119896119873119860
119875 (119903)
984858 (119903)
=4120587
3119896119873119860
1198661205839848581198881198772
[1 minus (119903119877)120575
]
times [120585 minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
(36)
The central temperature 119879119888= (4119896119873
119860)119866120583120585(119872(119877)119877) where
120585 is as defined in (34)Thus we have obtained the mass the pressure and the
temperature throughout the nonlinear stellar model with thedensity distribution defined in (28) Next our aim is to obtainanalytical results for stellar luminosity and neutrino emissionrates for various stellar models
4 Stellar Luminosity and NeutrinoEmission Rate
The energy conservation equation states that the net increasein the rate of energy flux coming out of a spherical shell fromthe inside is the same as the energy produced within the shell[29] If we denote 119871
119903= 119871(119903) as the energy flux through the
sphere of radius 119903 then we have
d119871119903
d119903= 4120587119903
2
984858 (119903) 120576 (119903) (37)
where 120576(119903) is the energy produced per second by nuclearreactions in each gram of stellar matter 120576(119903) depends on thechemical composition in each gram of stellar matter Hereusually 119871
119903is a constant but will be equal to 119871 at the surface of
the star We assume here that the star is chemically homoge-neous (that is a star where chemical composition throughoutis a constant) Also we assume the energy generation rate120576(119903) for one particular nuclear reaction Now if we denote11990312(984858(119903) 119879(119903)) as the extended nonresonant thermonuclear
reaction rate for the particles 1 and 2 defined by (19) thenwe will consider the energy generation rate 120576
12(119903) and it can
be written in terms of the extended reaction rates via
12057612(119903) =
1
984858 (119903)Clowast
9848582
11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
] (38)
where
Clowast
=1198641211990312(984858 (119903) 119879 (119903))
984858211986631
13[(119902 minus 1) 119864
1198664119896119879
1003816100381610038161003816
2minus1(119902minus1)
0121]
(39)
in which 11986412
is the amount of energy given off in a singlereaction It is to be noted that by using the asymptoticbehaviour of 11986631
13((119902 minus 1)119864
1198664119896119879) [26] and as 119902 rarr 1
Clowast rarr C the composite parameter considered by [9] whichis defined as
C =1198641211990312
9848582Θ2
exp (3Θ) (40)
for our universeC asymp 2 times 104 for proton-proton fusion under
typical stellar conditions [9]Then from (37) we have the totalluminosity of the star by integration as follows
119871 (119877) = int
119877
0
41205871199032
984858 (119903) 120576 (119903) d119903 (41)
If we are considering only one specific reaction defined as in(1) then we have
11987112(119877) = int
119877
0
41205871199032
984858 (119903) 12057612(119903) d119903 (42)
6 Journal of Astrophysics
where the energy generation rate is defined in (38) and 984858(119903)is a suitable density distribution explaining the sun Writing(42) in terms of 119903
12(984858(119903) 119879(119903)) we get
11987112(119877) = int
119877
0
41205871199032
Clowast
9848582
11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
] d119903
= int
119877
0
41205871199032
1198641211990312(984858 (119903) 119879 (119903)) d119903
(43)
The number density 119899119894of a particle 119894 for a gas ofmean density
984858(119903) can be expressed as
119899119894(119903) = 984858 (119903)119873
119860
119883119894
119860119894
(44)
where 119873119860stands for Avagadrorsquos constant 119860
119894is the atomic
mass of particle 119894 in atomic mass units and 119883119894is the mass
fraction of particle 119894 such that sum119894119883119894
= 1 Substituting11990312(984858(119903) 119879(119903)) from (19) and using (44) we have
11987112(119877)
= int
119877
0
41205871199032
11986412(1 minus
1
212057512)1198732
1198609848582
(119903)11988311198832
11986011198602
[8 (119902 minus 1)
120583119896119879 (119903)]
12
times120587minus1
Γ (1 (119902 minus 1) minus 32)119878 (0) 119866
31
13
times [
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)(119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
(45)
If we divide the ldquointernal luminosityrdquo 11987112(119877⊙) by the amount
of energy 11986412 then we get the total number of particles per
second11987312liberated in the reaction given by (1) as follows
11987312=11987112(119877)
11986412
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
times int
119877
0
1199032
9848582
(119903)
[119879 (119903)]12
11986631
13
[
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)
times (119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)[
(119902 minus 1) 1205872
120583
2119896(11988511198852e2
ℏ)
2
]
minus119904
times int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903 d119904
(46)
For the density distribution defined in (28) introduced byHaubold and Mathai [5 6] and the corresponding temper-ature distribution (36) we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
119888
times int
119877
0
1199032
[1 minus (119903
119877)
120575
]
52minus119904
times [minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
119904minus12
d119903
(47)
where 120585 is as defined in (34) If we put a substitution 119903 = 119909119877then we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
times int
1
0
1199092
[1 minus 119909120575
]52minus119904
times [120585 minus1
21199092
+120575 + 6
(120575 + 2) (120575 + 3)119909120575+2
minus3
2(120575 + 1)(120575 + 3)1199092120575+2
]
119904minus12
d119909
(48)
Journal of Astrophysics 7
Putting 119909120575 = 119910 and simplifying we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583120585
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
120575
times int
1
0
1199103120575minus1
[1 minus 119910]52minus119904
[1 minus 119906 (119910)]119904minus12d119910
(49)
where 119906(119910) is defined as
119906 (119910) =1199102120575
120585[1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
]
(50)
As 119910 rarr 0 119906(119910) rarr 0 and 119910 rarr 1 119906(119910) rarr 1 If we take
V (119910) =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
(51)
we have V(0) = 12 The minimum value of V(119910) is at 119910 =
(120575 + 1)(120575 + 6)3(120575 + 2) and the value is 12 minus 16((120575 + 6)2(120575 +1)(120575 + 3)(120575 + 2)
2
) Thus the minimum value is nonnegativesince (120575 + 6)2(120575 + 1) (120575 + 3)(120575 + 2)2 decreases steadily from3 to 1 for all 120575 gt 0 Therefore V(119910) le 0 Since 120585 gt 0 for all120575 gt 0 119906(119910) le 0 for all 120575 gt 0 120585 gt 0 Thus [1 minus 119906(119910)]119904minus12 le 0Hence 0 lt 119906(119910) lt 1 for 0 lt 119910 lt 1 and for 120575 gt 0 Thus byusing the binomial expansion we obtain
[1 minus 119906 (119910)]119904minus12
=
infin
sum
119898=0
(12 minus 119904)119898
119898[119906 (119910)]
119898
(52)
where (12 minus 119904)119898is the Pochhammer symbol defined for 119886 isin
C by
(119886)0= 1
(119886)119898=
119886 (119886 + 1) sdot sdot sdot (119886 + 119898 minus 1)
Γ (119886 + 119898)
Γ (119886)
119898 = 1 2 119886 = 0
(53)
whenever Γ(119886) exists Taking 120575 = 2 we get 120585 = 15 and
[1 minus 119906 (119910)]119904minus12
= (1 minus 119910)2119904minus1
(1 minus1
2119910)
119904minus12
(54)
Then from (49) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times int
1
0
11991032minus1
[1 minus 119910]52+119904minus1
(1 minus1
2119910)
119904minus12
d119910
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
(12 minus 119904)119898
119898
1
2119898
times int
1
0
11991032+119898minus1
[1 minus 119910]52+119904minus1d119910
(55)
By using beta integral and using (53) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
Γ (12 minus 119904 + 119898)
119898Γ (12 minus 119904)
1
2119898
Γ (32 + 119898) Γ (52 + 119904)
Γ (4 + 119904 + 119898)
(56)
Now from (46) we obtain the total number of particles persecond liberated in the reaction (1) as follows
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904) Γ (
5
2+ 119904)
timesΓ (1 (119902 minus 1) minus 1 minus 119904) Γ (12 + 119898 minus 119904)
Γ (12 minus 119904) Γ (4 + 119898 + 119904)
times [15120587 (119902 minus 1)119873
119860
81198669848581198881198772
(11988511198852e2
ℏ)
2
]
minus119904
d119904
8 Journal of Astrophysics
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
times 11986642
35
[
[
15120587 (119902 minus 1)119873119860
81198669848581198881198772
times (11988511198852e2
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)12minus1198984+119898
01215212
]
]
(57)
For more details on119866-function and its properties see [2526] Thus we have obtained the the total number of particlesper second liberated in the reaction (1) in terms of the densitydistribution considered by Haubold and Mathai [6 29]
5 Comparison of Pathway Energy Density andthe Maxwell-Boltzmann Energy Density
In Figure 1 it can be obtserved that for the nuclei to reactat energy 119864 they have to borrow an energy 119864 from thethermal environment The probability of such an energy isproportional to theMaxwell-Boltzmann energy exp[minus119864119896119879]The fusion will take place when the nuclei penerate theCoulomb barrier keeping them apart The probability ofpenetration is given by the factor exp[minus(119864
119866119864)12
] Theproduct of these two factors illustrates that fusion mostlyoccurs in the energy window given in the figure
In Figure 2 the pathway energy density is plotted for 119902 =07 09 1 12 14 respectively For different values of 119902we getdifferent energy densities (curves (a) (b) (c) (d) and (e))The nonresonant cross section is also plotted The product ofthe pathway energy density and the nonresonant cross sectionfor different values of 119902 namely 119902 = 07 09 1 12 14 is alsoplotted It is to be noted that as 119902 rarr 1 the pathway energydensity coincides with the Maxwell-Boltzmann energy den-sity and also the fusion window for the Maxwell-Boltzmanncase in Figure 1
The curves in the figure represent pathway density[1 + (119902 minus 1)(119864119896119879)]
minus1(119902minus1) for (a) 119902 = 07 (b) 119902 = 09(c) 119902 = 1 (d) 119902 = 12 and (e) 119902 = 14 (f) repre-sents the nonresonant cross section exp[minus(119864
119866119864)12
] Theproduct [1 + (119902 minus 1)(119864119896119879)]minus1(119902minus1) exp[minus(119864
119866119864)12
] is alsorepresented for (g) 119902 = 07 (h) 119902 = 09 (i) 119902 = 1 (j) 119902 = 12and (k) 119902 = 14
Figure 3(a) shows the pathway energy density defined in(13) for 119902 = 1 12 13 14 and Figure 3(b) shows theMaxwell-Boltzmann energy density defined in (3) As 119902 rarr 1 thepathway energy density reduces to the Maxwell-Boltzmannenergy density The pathway energy density covers manystable and unstable situations as the value of 119902 varies If
Fusion window
0 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
2kT 4kT 6kT 8kT
prop exp[minus E
kT]
prop exp[minus (EGE
)12
]
prop exp[minus E
kTminus (EG
E)12
]
Energy E
Figure 1 Schematic plot of the enrgy-dependent factors for thereaction rate probability integral the Maxwell-Boltzmann energydensity nonresonant nuclear cross section and the product of theMaxwell-Boltzmann density and the nonresonant cross section
Fusion window
0 2kT 4kT 6kT 8kT 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
1 + (q minus 1)E
kT]minus1qminus1
(a)
(b)(c)
(d)(e)
(h)
(i)
(k)
(f)
(j)
(g)
prop[prop exp[minus (EG
E)12
]
Energy E
Figure 2 Schematic plot of the energy-dependent factors for theextended reaction rate probability integral pathway energy densitynonresonant nuclear cross section and the product of the pathwaydensity and the nonresonant cross section
the Maxwell-Boltzmann density is the equilibrium situationmany other nonequilibrium situations are covered by thepathway energy density
Journal of Astrophysics 9
Pathway energy density for k = 1 T = 100K
0004
0003
0002
0001
0
q = 1
q = 12
q = 13
q = 14
0 200 400 600 800 1000
fPD
Pathway energy density for k = 1 T = 100K
q = 1
q = 12
q = 13
q = 14
Energy E
(a)
0004
0003
0002
0001
0
fM
BD
Maxwell-Boltzmann energy densityfor k = 1 T = 100K
0 200 400 600 800 1000
Energy E
(b)
Figure 3 (a) Pathway energy density for 119902 = 1 12 13 14 and for 119896 = 1 119879 = 100119870 (b) The Maxwell-Boltzmann energy density for119896 = 1 119879 = 100119870
6 Concluding Remarks
In this paper we have modified the energy distributionfor a nonresonant reaction rate probability integral Thecomposition parameter C considered by [9] is extended toClowast by the pathway energy density Considering the analyticdensity distributions developed by Haubold and Mathai [629] they are used to obtain the stellar luminosity and the neu-trino emission rates and are obtained in generalized specialfunctions such as Meijerrsquos 119866-function The pathway energydensity considered here covers many density functions andhence the extended reaction rate integral covers a wider classof integral Pathway energy density helps us to obtain variousfusion windows by giving different values to 119902 the pathwayparameter which in turn leads to a new opening in the fusionresearch The graphs plotted here are by using Maple 14 inWindows XP platform
Appendix
Series Representation
The series representation for the right-hand side of (19) canbe obtained through the following procedure Here we applyresidue calculus on the119866-function given in (19) Consider the119866-function as follows
11986631
13((119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
)
=1
2120587119894int
119888+119894infin
119888minus119894infin
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1) 119864119866
4119896119879)
minus119904
d119904
(A1)
The right-hand side is the sum of the residues of the inte-grandThe poles of the gammas in the integral representationin (19) are as follows
poles of Γ(119904) 119904 = 0 minus1 minus2
poles of Γ(12 + 119904) 119904 = minus12 minus32 minus52
poles of Γ(1 + 119904) 119904 = minus1 minus2 minus3
Here the poles of Γ(119904) and Γ(1 + 119904) will coincide with eachother at all points except at 119904 = 0 Note that the pole 119904 = 0 isa pole of order 1 119904 = minus12 minus32 minus52 are each of order 1and 119904 = minus1 minus2 minus3 are each of order 2 We know that
lim119904rarrminus119903
(119904 + 119903) Γ (119904) =(minus1)119903
119903
Γ (119886 minus 119903) =(minus1)119903
Γ (119886)
(1 minus 119886)119903
Γ (119886 + 119898) = Γ (119886) (119886)119898
(A2)
10 Journal of Astrophysics
when Γ(119886) is defined 119903 = 0 1 2 Γ(12) = 12058712
(119886)119903=
119886 (119886 + 1) sdot sdot sdot (119886 + 119903 minus 1) if 119903 ge 1 119886 = 0
1 if 119903 = 0(A3)
The sum of the residues corresponding to the poles 119904 = 0 isgiven by
1198771= radic120587Γ(
1
119902 minus 1minus 1) (A4)
The sum of the residues corresponding to the poles 119904 =
minus12 minus32 minus52 is
1198772=
infin
sum
119903=0
(minus1)119903
119903Γ (minus
1
2minus 119903) Γ (
1
2minus 119903)
times Γ(1
119902 minus 1minus1
2+ 119903) [
(119902 minus 1)119864119866
4119896119879]
minus12+119903
= minus2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1) 119864
119866
4119896119879)
(A5)
where11198652is the hypergeometric function defined by
11198652(119886 119887 119888 119909) =
infin
sum
119903=0
(119886)119903
(119887)119903(119888)119903
119909119903
119903 (A6)
To obtain the sum of the residues corresponding to poles119904 = minus1 minus2 minus3 of order 2 we proceed as follows
1198773
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[(119904 + 1 + 119903)
2
Γ (1 + 119904) Γ (119904) Γ (1
2+ 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1)119864119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[ (Γ2
(2 + 119904 + 119903) Γ (12 + 119904)
times Γ (1 (119902 minus 1) minus 1 minus 119904))
times ((119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904)minus1
times ((119902 minus 1) 119864
119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904Φ (119904)
(A7)
where
Φ (119904) =Γ2
(2 + 119904 + 119903) Γ (12 + 119904) Γ (1 (119902 minus 1) minus 1 minus 119904)
(119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904
times ((119902 minus 1)119864
119866
4119896119879)
minus119904
(A8)
We have
120597
120597119904Φ (119904) = Φ (119904)
120597
120597119904[ln [Φ (119904)]]
lnΦ (119904) = 2 ln [Γ (2 + 119904 + 119903)] + ln [Γ (12+ 119904)]
+ ln [Γ( 1
119902 minus 1minus 1 minus 119904)] minus 119904 ln(
(119902 minus 1) 119864119866
4119896119879)
minus 2 ln (119904 + 119903) minus 2 ln (119904 + 119903 minus 1) minus sdot sdot sdot
minus 2 ln (119904 + 1) minus ln (119904)
120597
120597119904[ln [Φ (119904)]] = 2Ψ (2 + 119904 + 119903) + Ψ(
1
2+ 119904)
+ Ψ(1
119902 minus 1minus 1 minus 119904)
minus ln((119902 minus 1) 119864
119866
4119896119879) minus
2
119904 + 119903
minus2
119904 + 119903 minus 1minus sdot sdot sdot minus
2
119904 + 1minus1
119904
lim119904rarrminus1minus119903
120597
120597119904ln [Φ (119904)] = Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
minus ln((119902 minus 1) 119864
119866
4119896119879)
(A9)
whereΨ(119911) is a Psi function or digamma function (seeMathai[26]) and Ψ(1) = minus120574 120574 = 05772156649 is Eulerrsquosconstant Now
lim119904rarrminus1minus119903
Φ (119904) =(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
times ((119902 minus 1) 119864
119866
4119896119879)
1+119903
(A10)
Journal of Astrophysics 11
Then by using (A7) (A9) and (A10) we get
1198773=
infin
sum
119903=0
(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
((119902 minus 1) 119864
119866
4119896119879)
1+119903
times [Ψ(minus1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903) + Ψ (1 + 119903)
+Ψ (2 + 119903) minus ln((119902 minus 1) 119864
119866
4119896119879)]
= (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(A11)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(A12)
Thus from (A4) (A5) and (A11) we get (20)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the Department of Sci-ence and Technology Government of India New Delhifor the financial assistance for this work under Project noSRS4MS28705 and the Centre for Mathematical Sciencesfor providing all facilities
References
[1] R Davis Jr ldquoNobel Lecture a half-century with solar neutri-nosrdquo Reviews of Modern Physics vol 75 no 3 pp 985ndash9942003
[2] H J Haubold and D Kumar ldquoExtension of thermonuclearfunctions through the pathway model including Maxwell-Boltzmann and Tsallis distributionsrdquo Astroparticle Physics vol29 no 1 pp 70ndash76 2008
[3] H J Haubold and R W John ldquoOn the evaluation of an integralconnected with the thermonuclear reaction rate in closed-formrdquo Astronomische Nachrichten vol 299 no 5 pp 225ndash2321978
[4] W A Fowler G R Caughlan and B A Zimmerman ldquoTher-monuclear rection ratesrdquo Annual Review of Astronomy andAstrophysics vol 5 pp 525ndash570 1967
[5] A M Mathai and H J Haubold Modern Problems in Nuclearand Neutrino Astrophysics Akademie Berlin Germany 1988
[6] H J Haubold and A M Mathai ldquoAnalytic representations ofmodified non-resonant thermonuclear reaction ratesrdquo Journalof Applied Mathematics and Physics vol 37 no 5 pp 685ndash6951986
[7] W A Fowler ldquoExperimental and theoretical nuclear astro-physics the quest for the origin of the elementsrdquo Reviews ofModern Physics vol 56 no 2 pp 149ndash179 1984
[8] A C Phillips The Physics of Stars John Wiley amp SonsChichester UK 2nd edition 1999
[9] F C Adams ldquoStars in other universes stellar structure withdifferent fundamental constantsrdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 8 article 10 2008
[10] M Coraddu G Kaniadakis A Lavagno M Lissia G Mezzo-rani and P Quarati ldquoThermal distributions in stellar plasmasnuclear reactions and solar neutrinosrdquo Brazilian Journal ofPhysics vol 29 no 1 pp 153ndash168 1999
[11] H J Haubold and A M Mathai ldquoOn nuclear reaction ratetheoryrdquo Annalen der Physik vol 41 pp 380ndash396 1984
[12] M Coraddu M Lissia G Mezzorani and P Quarati ldquoSuper-Kamiokande hep neutrino best fit a possible signal of non-Maxwellian solar plasmardquo Physica A Statistical Mechanics andIts Applications vol 326 no 3-4 pp 473ndash481 2003
[13] A Lavagno and P Quarati ldquoClassical and quantum non-extensive statistics effects in nuclear many-body problemsrdquoChaos Solitons and Fractals vol 13 no 3 pp 569ndash580 2002
[14] A Lavagno and P Quarati ldquoMetastability of electron-nuclearastrophysical plasmas motivations signals and conditionsrdquoAstrophysics and Space Science vol 305 no 3 pp 253ndash2592006
[15] M Lissia and P Quarati ldquoNuclear astrophysical plasmas iondistribution functions and fusion ratesrdquo Europhysics News vol36 no 6 pp 211ndash214 2005
[16] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988
[17] C Tsallis Introduction to Non-Extensive Statistical MechanicsSpringer New York NY USA 2009
[18] M Gell-Mann and C Tsallis Eds Nonextensive EntropyInterdisciplinary Applications Oxford University Press NewYork NY USA 2004
[19] A M Mathai and H J Haubold ldquoPathway model superstatis-tics Tsallis statistics and a generalized measure of entropyrdquoPhysica A StatisticalMechanics and Its Applications vol 375 no1 pp 110ndash122 2007
[20] R K Saxena A M Mathai and H J Haubold ldquoAstrophysicalthermonuclear functions for Boltzmann-Gibbs statistics andTsallis statisticsrdquo Physica A Statistical Mechanics and Its Appli-cations vol 344 no 3-4 pp 649ndash656 2004
[21] A MMathai ldquoA pathway tomatrix-variate gamma and normaldensitiesrdquo Linear Algebra and Its Applications vol 396 no 1ndash3pp 317ndash328 2005
[22] A M Mathai and H J Haubold ldquoOn generalized distributionsand pathwaysrdquoPhysics Letters A General Atomic and Solid StatePhysics vol 372 no 12 pp 2109ndash2113 2008
[23] H J Haubold andDKumar ldquoFusion yield Guderleymodel andTsallis statisticsrdquo Journal of Plasma Physics vol 77 no 1 pp 1ndash14 2011
[24] D Kumar and H J Haubold ldquoOn extended thermonuclearfunctions through pathwaymodelrdquoAdvances in Space Researchvol 45 no 5 pp 698ndash708 2010
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
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ThermodynamicsJournal of
Journal of Astrophysics 3
with the information theory the fractional calculus theMittag-Leffler functions and so forth The pathway modelcan be effectively used in any situation in which we needto switch between three different functional forms namelygeneralized type-1 beta form generalized type-2 beta formand generalized gamma form using the pathway parameter119902 In practical purpose of fitting experimental data pathwaymodel can be utilized to switch between different parametricfamilies with thicker or thinner tail The pathway model forthe real scalar case can be explained as follows
1198911(119909) = 119888
1119909120574minus1
[1 minus 119886 (1 minus 119902) 119909120575
]1(1minus119902)
119886 gt 0 120575 gt 0 1 minus 119886 (1 minus 119902) 119909120575
gt 0 120574 gt 0 119902 lt 1
(10)
is the generalized type-1 beta form of the pathway modelThis is a model with right tail cut-off for 119902 lt 1 The Tsallisstatistics for 119902 lt 1 can be obtained from thismodel by putting120574 = 1 [16ndash18] Other cases available are the regular type-1beta density the Pareto density the power function and thetriangular and related models [22] The generalized type-2beta form of the pathway model is given by
1198912(119909) = 119888
2119909120574minus1
[1 + 119886 (119902 minus 1) 119909120575
]minus1(119902minus1)
0 lt 119909 lt infin 119902 gt 1 119886 gt 0 120574 gt 0 120575 gt 0
(11)
Here also for 120574 = 1 we get the Tsallis statistics for 119902 gt 1 [16ndash18] Other standard distributions coming from this model areregular type-2 beta density 119865-distribution the Levy modeland related models [22] When 119902 rarr 1 119891
1(119909) and 119891
2(119909) will
reduce to the generalized gamma form of the pathway modelgiven by
1198913(119909) = 119888
3119909120574minus1eminus119886119909
120575
119909 gt 0 (12)
This model covers generalized gamma gamma exponen-tial chi-square the Weibull the Maxwell-Boltzmann theRayleigh and related densities 119888
1 1198882 and 119888
3defined in (10)
(11) and (12) respectively are the normalizing constants if weconsider statistical densities
By a suitable modification of the Maxwell-Boltzmanndistribution given in (3) through the pathway model we geta more general energy distribution called the pathway energydistribution given by the density
119891PD (119864) d119864 =2120587(119902 minus 1)
32
(120587119896119879)32
Γ (1 (119902 minus 1))
Γ (1 (119902 minus 1) minus 32)
times radic119864[1 + (119902 minus 1)119864
119896119879]
minus1(119902minus1)
d119864
(13)
for 119902 gt 1 1(119902minus1)minus32 gt 0TheMaxwell-Boltzmann energydistribution can be retrieved from (13) by taking 119902 rarr 1Thus the reaction rate probability integral given in (4) can be
modified by using (13) and we get the extended reaction rateas
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(119902 minus 1
119896119879)
32
timesΓ (1 (119902 minus 1))
Γ (1 (119902 minus 1) minus 32)
2
sum
]=0
119878(])(0)
]
times int
infin
0
119864][1 + (119902 minus 1)
119864
119896119879]
minus1(119902minus1)
times exp[minus(119864119866
119864)
12
] d119864
(14)
for 119902 gt 1 1(119902 minus 1) minus 32 gt 0 Substituting 119910 = 119864119896119879 and 119909 =(119864119866119896119879)12 we obtain the above integral in amore convenient
form as follows
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(119902 minus 1)32
timesΓ (1 (119902 minus 1))
Γ (1 (119902 minus 1) minus 32)
2
sum
]=0(1
119896119879)
minus]+(12)119878(])(0)
]
times int
infin
0
119910][1 + (119902 minus 1) 119910]
minus1(119902minus1)eminus119909119910minus12
d119910
(15)
for 119902 gt 1 1(119902 minus 1) minus 32 gt 0 Here we consider the integralto be evaluated as
1198681119902= int
infin
0
119910][1 + (119902 minus 1) 119910]
minus1(119902minus1)
times eminus119909119910minus12
d119910
(16)
The integral can be evaluated by the techniques in appliedanalysis and can be obtained in closed form via Meijerrsquos 119866-function as [2 23 24]
1198681119902=
(120587)minus12
(119902 minus 1)]+1Γ (1 (119902 minus 1))
times 11986631
13((119902 minus 1) 119909
2
4
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)+]
012]+1)
(17)
which yields the nonresonant reaction rate probability inte-gral in the extended case as
11990312= (1 minus
1
212057512) 11989911198992(8
120583)
12
120587minus1
Γ (1 (119902 minus 1) minus 32)
times
2
sum
]=0(119902 minus 1
119896119879)
minus]+(12)119878(])(0)
]
times 11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)+]
012]+1]
(18)
4 Journal of Astrophysics
Meijerrsquos 119866-function and its properties can be seen in Mathaiand Saxena [25] and Mathai [26] We can obtain seriesexpansions of the 119866-function given in (18) by combining thetheories of residue calculus and generalized special functionssee Kumar and Haubold [24] for series expansions for allpossible values of ] In many cases the nuclear factor 119878(])(0)is approximately constant across the fusion window Taking119878(])(0) = 0 for ] = 1 and ] = 2 and taking 1198780(0) = 119878(0) we
obtain the extended reaction rate probability integral as
11990312= (1 minus
1
212057512) 11989911198992[8 (119902 minus 1)
120583119896119879]
12
120587minus1
Γ (1 (119902 minus 1) minus 32)
times 119878 (0) 11986631
13[(119902 minus 1)119864
119866
4119896119879
10038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
(19)
The series representation for (19) can be obtained as
11990312= (1 minus
1
212057512) 11989911198992[8 (119902 minus 1)
120583119896119879]
12
120587minus1
Γ (1 (119902 minus 1) minus 32)
times 119878 (0) radic120587Γ (1 (119902 minus 1) minus 1)
minus 2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1)119864
119866
4119896119879)
+ (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
times[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(20)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(21)
See the Appendix for detailed evaluation As 119902 rarr 1 in (19)then by using Stirlingrsquos formula for gamma functions givenby
Γ (119911 + 119886) asymp (2120587)12
119911119911+119886minus12eminus119911
|119911| 997888rarr infin 119886 is bounded(22)
we get the reaction rate probability integral in the Maxwell-Boltzmann case as
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(1
119896119879)
32
119878 (0)
times int
infin
0
exp[minus 119864
119896119879minus (
119864119866
119864)
12
] d119864
(23)
= (1 minus1
212057512) 11989911198992(
8
120583119896119879)
12
119878 (0) 120587minus1
times 11986630
03[119864119866
4119896119879
10038161003816100381610038161003816100381610038161003816
minus
0121
]
(24)
which is obtained in a series of papers by Mathai andHaubold see for example Mathai and Haubold [5] Theintegral in (23) is dominated by theminimumvalue of119864119896119879+(119864119866119864)12
= 119892(119864) (say) The minimum value of the function119892(119864) say 119864
0 can be determined as
dd119864
[119864
119896119879+ (
119864119866
119864)
12
]
119864=1198640
=1
119896119879minus1
211986412
119866119864minus32
0= 0 997904rArr 119864
0= 11986412
119866(119896119879
2)
32
(25)
and the function
119892 (1198640) = 3(
119864119866
4119896119879)
13
= 3Θ (26)
where Θ = (1198641198664119896119879)
13 Now by using the Laplace method[27 28] we can obtain an approximate value for (23) as
11990312asymp (1 minus
1
212057512) 11989911198992
8119878 (0)Θ2 exp (minus3Θ)
radic312058712058312057211988511198852119888
(27)
In the next section we will obtain the mass pressure andtemperature for the case of analytic stellar models character-ized by density distribution and corresponding temperaturedistribution suggested by Haubold and Mathai [11]
3 Closed Forms of the Integral over the StellarNuclear Energy Generation Rate
Let us consider the density distribution 984858(119903) considered byHaubold and Mathai [6 11] and Mathai and Haubold [5] inthe form
984858 (119903) = 984858119888[1 minus (
119903
119877)
120575
] 120575 gt 0 (28)
where 984858119888is the central density of the star 119903 is an arbitrary
distance from the center and 119877 is the solar radius Thisdensity function is capable of producing different densitydistributions by choosing the free parameter 120575 Now wedetermine the quantities119872(119903) 119875(119903) and 119879(119903) the mass thepressure and the temperature at 119903
Journal of Astrophysics 5
By the equation of the mass conservation
d119872(119903)
d119903= 4120587119903
2
984858 (119903) (29)
we get
119872(119903) = 4120587984858119888int
119903
0
1199052
[1 minus (119905
119877)
120575
] d119905
=4120587
39848581198881199033
[1 minus3
120575 + 3(119903
119877)
120575
]
(30)
From (30) we get the central density as
984858119888=3 (120575 + 3)
4120587120575
119872 (119877)
1198773
(31)
If an element of a matter at a distance 119903 from the center ofa spherical system is in hydrostatic equilibrium then settingthe sum of the radial forces acting on it to zero we obtain
d119875 (119903)d119903
= minus119866984858 (119903)119872 (119903)
1199032
(32)
where 119866 is the gravitational constant Assuming that thepressure at the center of the sun is 119875
119888and at the surface is
zero we get
119875 (119903) = 119875119888minus 119866int
119903
0
119872(119905) 984858 (119905)
1199052
d119905
=4120587
31198669848582
1198881198772
[120585 minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
(33)
Using the boundary conditions 119875(119877) = 0 we get 119875119888=
(41205873)1198661205859848582
1198881198772 where
120585 =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)+
3
2 (120575 + 1) (120575 + 3) (34)
By the kinetic theory of gases for a perfect gas the pressureis given by
119875 (119903) =119896119873119860
120583984858 (119903) 119879 (119903) (35)
For the temperature of interest for stellar models we neglectthe negligible radiation pressure from the total pressure andobtain from (35) the following
119879 (119903) =120583
119896119873119860
119875 (119903)
984858 (119903)
=4120587
3119896119873119860
1198661205839848581198881198772
[1 minus (119903119877)120575
]
times [120585 minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
(36)
The central temperature 119879119888= (4119896119873
119860)119866120583120585(119872(119877)119877) where
120585 is as defined in (34)Thus we have obtained the mass the pressure and the
temperature throughout the nonlinear stellar model with thedensity distribution defined in (28) Next our aim is to obtainanalytical results for stellar luminosity and neutrino emissionrates for various stellar models
4 Stellar Luminosity and NeutrinoEmission Rate
The energy conservation equation states that the net increasein the rate of energy flux coming out of a spherical shell fromthe inside is the same as the energy produced within the shell[29] If we denote 119871
119903= 119871(119903) as the energy flux through the
sphere of radius 119903 then we have
d119871119903
d119903= 4120587119903
2
984858 (119903) 120576 (119903) (37)
where 120576(119903) is the energy produced per second by nuclearreactions in each gram of stellar matter 120576(119903) depends on thechemical composition in each gram of stellar matter Hereusually 119871
119903is a constant but will be equal to 119871 at the surface of
the star We assume here that the star is chemically homoge-neous (that is a star where chemical composition throughoutis a constant) Also we assume the energy generation rate120576(119903) for one particular nuclear reaction Now if we denote11990312(984858(119903) 119879(119903)) as the extended nonresonant thermonuclear
reaction rate for the particles 1 and 2 defined by (19) thenwe will consider the energy generation rate 120576
12(119903) and it can
be written in terms of the extended reaction rates via
12057612(119903) =
1
984858 (119903)Clowast
9848582
11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
] (38)
where
Clowast
=1198641211990312(984858 (119903) 119879 (119903))
984858211986631
13[(119902 minus 1) 119864
1198664119896119879
1003816100381610038161003816
2minus1(119902minus1)
0121]
(39)
in which 11986412
is the amount of energy given off in a singlereaction It is to be noted that by using the asymptoticbehaviour of 11986631
13((119902 minus 1)119864
1198664119896119879) [26] and as 119902 rarr 1
Clowast rarr C the composite parameter considered by [9] whichis defined as
C =1198641211990312
9848582Θ2
exp (3Θ) (40)
for our universeC asymp 2 times 104 for proton-proton fusion under
typical stellar conditions [9]Then from (37) we have the totalluminosity of the star by integration as follows
119871 (119877) = int
119877
0
41205871199032
984858 (119903) 120576 (119903) d119903 (41)
If we are considering only one specific reaction defined as in(1) then we have
11987112(119877) = int
119877
0
41205871199032
984858 (119903) 12057612(119903) d119903 (42)
6 Journal of Astrophysics
where the energy generation rate is defined in (38) and 984858(119903)is a suitable density distribution explaining the sun Writing(42) in terms of 119903
12(984858(119903) 119879(119903)) we get
11987112(119877) = int
119877
0
41205871199032
Clowast
9848582
11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
] d119903
= int
119877
0
41205871199032
1198641211990312(984858 (119903) 119879 (119903)) d119903
(43)
The number density 119899119894of a particle 119894 for a gas ofmean density
984858(119903) can be expressed as
119899119894(119903) = 984858 (119903)119873
119860
119883119894
119860119894
(44)
where 119873119860stands for Avagadrorsquos constant 119860
119894is the atomic
mass of particle 119894 in atomic mass units and 119883119894is the mass
fraction of particle 119894 such that sum119894119883119894
= 1 Substituting11990312(984858(119903) 119879(119903)) from (19) and using (44) we have
11987112(119877)
= int
119877
0
41205871199032
11986412(1 minus
1
212057512)1198732
1198609848582
(119903)11988311198832
11986011198602
[8 (119902 minus 1)
120583119896119879 (119903)]
12
times120587minus1
Γ (1 (119902 minus 1) minus 32)119878 (0) 119866
31
13
times [
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)(119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
(45)
If we divide the ldquointernal luminosityrdquo 11987112(119877⊙) by the amount
of energy 11986412 then we get the total number of particles per
second11987312liberated in the reaction given by (1) as follows
11987312=11987112(119877)
11986412
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
times int
119877
0
1199032
9848582
(119903)
[119879 (119903)]12
11986631
13
[
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)
times (119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)[
(119902 minus 1) 1205872
120583
2119896(11988511198852e2
ℏ)
2
]
minus119904
times int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903 d119904
(46)
For the density distribution defined in (28) introduced byHaubold and Mathai [5 6] and the corresponding temper-ature distribution (36) we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
119888
times int
119877
0
1199032
[1 minus (119903
119877)
120575
]
52minus119904
times [minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
119904minus12
d119903
(47)
where 120585 is as defined in (34) If we put a substitution 119903 = 119909119877then we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
times int
1
0
1199092
[1 minus 119909120575
]52minus119904
times [120585 minus1
21199092
+120575 + 6
(120575 + 2) (120575 + 3)119909120575+2
minus3
2(120575 + 1)(120575 + 3)1199092120575+2
]
119904minus12
d119909
(48)
Journal of Astrophysics 7
Putting 119909120575 = 119910 and simplifying we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583120585
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
120575
times int
1
0
1199103120575minus1
[1 minus 119910]52minus119904
[1 minus 119906 (119910)]119904minus12d119910
(49)
where 119906(119910) is defined as
119906 (119910) =1199102120575
120585[1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
]
(50)
As 119910 rarr 0 119906(119910) rarr 0 and 119910 rarr 1 119906(119910) rarr 1 If we take
V (119910) =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
(51)
we have V(0) = 12 The minimum value of V(119910) is at 119910 =
(120575 + 1)(120575 + 6)3(120575 + 2) and the value is 12 minus 16((120575 + 6)2(120575 +1)(120575 + 3)(120575 + 2)
2
) Thus the minimum value is nonnegativesince (120575 + 6)2(120575 + 1) (120575 + 3)(120575 + 2)2 decreases steadily from3 to 1 for all 120575 gt 0 Therefore V(119910) le 0 Since 120585 gt 0 for all120575 gt 0 119906(119910) le 0 for all 120575 gt 0 120585 gt 0 Thus [1 minus 119906(119910)]119904minus12 le 0Hence 0 lt 119906(119910) lt 1 for 0 lt 119910 lt 1 and for 120575 gt 0 Thus byusing the binomial expansion we obtain
[1 minus 119906 (119910)]119904minus12
=
infin
sum
119898=0
(12 minus 119904)119898
119898[119906 (119910)]
119898
(52)
where (12 minus 119904)119898is the Pochhammer symbol defined for 119886 isin
C by
(119886)0= 1
(119886)119898=
119886 (119886 + 1) sdot sdot sdot (119886 + 119898 minus 1)
Γ (119886 + 119898)
Γ (119886)
119898 = 1 2 119886 = 0
(53)
whenever Γ(119886) exists Taking 120575 = 2 we get 120585 = 15 and
[1 minus 119906 (119910)]119904minus12
= (1 minus 119910)2119904minus1
(1 minus1
2119910)
119904minus12
(54)
Then from (49) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times int
1
0
11991032minus1
[1 minus 119910]52+119904minus1
(1 minus1
2119910)
119904minus12
d119910
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
(12 minus 119904)119898
119898
1
2119898
times int
1
0
11991032+119898minus1
[1 minus 119910]52+119904minus1d119910
(55)
By using beta integral and using (53) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
Γ (12 minus 119904 + 119898)
119898Γ (12 minus 119904)
1
2119898
Γ (32 + 119898) Γ (52 + 119904)
Γ (4 + 119904 + 119898)
(56)
Now from (46) we obtain the total number of particles persecond liberated in the reaction (1) as follows
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904) Γ (
5
2+ 119904)
timesΓ (1 (119902 minus 1) minus 1 minus 119904) Γ (12 + 119898 minus 119904)
Γ (12 minus 119904) Γ (4 + 119898 + 119904)
times [15120587 (119902 minus 1)119873
119860
81198669848581198881198772
(11988511198852e2
ℏ)
2
]
minus119904
d119904
8 Journal of Astrophysics
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
times 11986642
35
[
[
15120587 (119902 minus 1)119873119860
81198669848581198881198772
times (11988511198852e2
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)12minus1198984+119898
01215212
]
]
(57)
For more details on119866-function and its properties see [2526] Thus we have obtained the the total number of particlesper second liberated in the reaction (1) in terms of the densitydistribution considered by Haubold and Mathai [6 29]
5 Comparison of Pathway Energy Density andthe Maxwell-Boltzmann Energy Density
In Figure 1 it can be obtserved that for the nuclei to reactat energy 119864 they have to borrow an energy 119864 from thethermal environment The probability of such an energy isproportional to theMaxwell-Boltzmann energy exp[minus119864119896119879]The fusion will take place when the nuclei penerate theCoulomb barrier keeping them apart The probability ofpenetration is given by the factor exp[minus(119864
119866119864)12
] Theproduct of these two factors illustrates that fusion mostlyoccurs in the energy window given in the figure
In Figure 2 the pathway energy density is plotted for 119902 =07 09 1 12 14 respectively For different values of 119902we getdifferent energy densities (curves (a) (b) (c) (d) and (e))The nonresonant cross section is also plotted The product ofthe pathway energy density and the nonresonant cross sectionfor different values of 119902 namely 119902 = 07 09 1 12 14 is alsoplotted It is to be noted that as 119902 rarr 1 the pathway energydensity coincides with the Maxwell-Boltzmann energy den-sity and also the fusion window for the Maxwell-Boltzmanncase in Figure 1
The curves in the figure represent pathway density[1 + (119902 minus 1)(119864119896119879)]
minus1(119902minus1) for (a) 119902 = 07 (b) 119902 = 09(c) 119902 = 1 (d) 119902 = 12 and (e) 119902 = 14 (f) repre-sents the nonresonant cross section exp[minus(119864
119866119864)12
] Theproduct [1 + (119902 minus 1)(119864119896119879)]minus1(119902minus1) exp[minus(119864
119866119864)12
] is alsorepresented for (g) 119902 = 07 (h) 119902 = 09 (i) 119902 = 1 (j) 119902 = 12and (k) 119902 = 14
Figure 3(a) shows the pathway energy density defined in(13) for 119902 = 1 12 13 14 and Figure 3(b) shows theMaxwell-Boltzmann energy density defined in (3) As 119902 rarr 1 thepathway energy density reduces to the Maxwell-Boltzmannenergy density The pathway energy density covers manystable and unstable situations as the value of 119902 varies If
Fusion window
0 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
2kT 4kT 6kT 8kT
prop exp[minus E
kT]
prop exp[minus (EGE
)12
]
prop exp[minus E
kTminus (EG
E)12
]
Energy E
Figure 1 Schematic plot of the enrgy-dependent factors for thereaction rate probability integral the Maxwell-Boltzmann energydensity nonresonant nuclear cross section and the product of theMaxwell-Boltzmann density and the nonresonant cross section
Fusion window
0 2kT 4kT 6kT 8kT 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
1 + (q minus 1)E
kT]minus1qminus1
(a)
(b)(c)
(d)(e)
(h)
(i)
(k)
(f)
(j)
(g)
prop[prop exp[minus (EG
E)12
]
Energy E
Figure 2 Schematic plot of the energy-dependent factors for theextended reaction rate probability integral pathway energy densitynonresonant nuclear cross section and the product of the pathwaydensity and the nonresonant cross section
the Maxwell-Boltzmann density is the equilibrium situationmany other nonequilibrium situations are covered by thepathway energy density
Journal of Astrophysics 9
Pathway energy density for k = 1 T = 100K
0004
0003
0002
0001
0
q = 1
q = 12
q = 13
q = 14
0 200 400 600 800 1000
fPD
Pathway energy density for k = 1 T = 100K
q = 1
q = 12
q = 13
q = 14
Energy E
(a)
0004
0003
0002
0001
0
fM
BD
Maxwell-Boltzmann energy densityfor k = 1 T = 100K
0 200 400 600 800 1000
Energy E
(b)
Figure 3 (a) Pathway energy density for 119902 = 1 12 13 14 and for 119896 = 1 119879 = 100119870 (b) The Maxwell-Boltzmann energy density for119896 = 1 119879 = 100119870
6 Concluding Remarks
In this paper we have modified the energy distributionfor a nonresonant reaction rate probability integral Thecomposition parameter C considered by [9] is extended toClowast by the pathway energy density Considering the analyticdensity distributions developed by Haubold and Mathai [629] they are used to obtain the stellar luminosity and the neu-trino emission rates and are obtained in generalized specialfunctions such as Meijerrsquos 119866-function The pathway energydensity considered here covers many density functions andhence the extended reaction rate integral covers a wider classof integral Pathway energy density helps us to obtain variousfusion windows by giving different values to 119902 the pathwayparameter which in turn leads to a new opening in the fusionresearch The graphs plotted here are by using Maple 14 inWindows XP platform
Appendix
Series Representation
The series representation for the right-hand side of (19) canbe obtained through the following procedure Here we applyresidue calculus on the119866-function given in (19) Consider the119866-function as follows
11986631
13((119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
)
=1
2120587119894int
119888+119894infin
119888minus119894infin
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1) 119864119866
4119896119879)
minus119904
d119904
(A1)
The right-hand side is the sum of the residues of the inte-grandThe poles of the gammas in the integral representationin (19) are as follows
poles of Γ(119904) 119904 = 0 minus1 minus2
poles of Γ(12 + 119904) 119904 = minus12 minus32 minus52
poles of Γ(1 + 119904) 119904 = minus1 minus2 minus3
Here the poles of Γ(119904) and Γ(1 + 119904) will coincide with eachother at all points except at 119904 = 0 Note that the pole 119904 = 0 isa pole of order 1 119904 = minus12 minus32 minus52 are each of order 1and 119904 = minus1 minus2 minus3 are each of order 2 We know that
lim119904rarrminus119903
(119904 + 119903) Γ (119904) =(minus1)119903
119903
Γ (119886 minus 119903) =(minus1)119903
Γ (119886)
(1 minus 119886)119903
Γ (119886 + 119898) = Γ (119886) (119886)119898
(A2)
10 Journal of Astrophysics
when Γ(119886) is defined 119903 = 0 1 2 Γ(12) = 12058712
(119886)119903=
119886 (119886 + 1) sdot sdot sdot (119886 + 119903 minus 1) if 119903 ge 1 119886 = 0
1 if 119903 = 0(A3)
The sum of the residues corresponding to the poles 119904 = 0 isgiven by
1198771= radic120587Γ(
1
119902 minus 1minus 1) (A4)
The sum of the residues corresponding to the poles 119904 =
minus12 minus32 minus52 is
1198772=
infin
sum
119903=0
(minus1)119903
119903Γ (minus
1
2minus 119903) Γ (
1
2minus 119903)
times Γ(1
119902 minus 1minus1
2+ 119903) [
(119902 minus 1)119864119866
4119896119879]
minus12+119903
= minus2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1) 119864
119866
4119896119879)
(A5)
where11198652is the hypergeometric function defined by
11198652(119886 119887 119888 119909) =
infin
sum
119903=0
(119886)119903
(119887)119903(119888)119903
119909119903
119903 (A6)
To obtain the sum of the residues corresponding to poles119904 = minus1 minus2 minus3 of order 2 we proceed as follows
1198773
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[(119904 + 1 + 119903)
2
Γ (1 + 119904) Γ (119904) Γ (1
2+ 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1)119864119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[ (Γ2
(2 + 119904 + 119903) Γ (12 + 119904)
times Γ (1 (119902 minus 1) minus 1 minus 119904))
times ((119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904)minus1
times ((119902 minus 1) 119864
119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904Φ (119904)
(A7)
where
Φ (119904) =Γ2
(2 + 119904 + 119903) Γ (12 + 119904) Γ (1 (119902 minus 1) minus 1 minus 119904)
(119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904
times ((119902 minus 1)119864
119866
4119896119879)
minus119904
(A8)
We have
120597
120597119904Φ (119904) = Φ (119904)
120597
120597119904[ln [Φ (119904)]]
lnΦ (119904) = 2 ln [Γ (2 + 119904 + 119903)] + ln [Γ (12+ 119904)]
+ ln [Γ( 1
119902 minus 1minus 1 minus 119904)] minus 119904 ln(
(119902 minus 1) 119864119866
4119896119879)
minus 2 ln (119904 + 119903) minus 2 ln (119904 + 119903 minus 1) minus sdot sdot sdot
minus 2 ln (119904 + 1) minus ln (119904)
120597
120597119904[ln [Φ (119904)]] = 2Ψ (2 + 119904 + 119903) + Ψ(
1
2+ 119904)
+ Ψ(1
119902 minus 1minus 1 minus 119904)
minus ln((119902 minus 1) 119864
119866
4119896119879) minus
2
119904 + 119903
minus2
119904 + 119903 minus 1minus sdot sdot sdot minus
2
119904 + 1minus1
119904
lim119904rarrminus1minus119903
120597
120597119904ln [Φ (119904)] = Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
minus ln((119902 minus 1) 119864
119866
4119896119879)
(A9)
whereΨ(119911) is a Psi function or digamma function (seeMathai[26]) and Ψ(1) = minus120574 120574 = 05772156649 is Eulerrsquosconstant Now
lim119904rarrminus1minus119903
Φ (119904) =(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
times ((119902 minus 1) 119864
119866
4119896119879)
1+119903
(A10)
Journal of Astrophysics 11
Then by using (A7) (A9) and (A10) we get
1198773=
infin
sum
119903=0
(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
((119902 minus 1) 119864
119866
4119896119879)
1+119903
times [Ψ(minus1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903) + Ψ (1 + 119903)
+Ψ (2 + 119903) minus ln((119902 minus 1) 119864
119866
4119896119879)]
= (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(A11)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(A12)
Thus from (A4) (A5) and (A11) we get (20)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the Department of Sci-ence and Technology Government of India New Delhifor the financial assistance for this work under Project noSRS4MS28705 and the Centre for Mathematical Sciencesfor providing all facilities
References
[1] R Davis Jr ldquoNobel Lecture a half-century with solar neutri-nosrdquo Reviews of Modern Physics vol 75 no 3 pp 985ndash9942003
[2] H J Haubold and D Kumar ldquoExtension of thermonuclearfunctions through the pathway model including Maxwell-Boltzmann and Tsallis distributionsrdquo Astroparticle Physics vol29 no 1 pp 70ndash76 2008
[3] H J Haubold and R W John ldquoOn the evaluation of an integralconnected with the thermonuclear reaction rate in closed-formrdquo Astronomische Nachrichten vol 299 no 5 pp 225ndash2321978
[4] W A Fowler G R Caughlan and B A Zimmerman ldquoTher-monuclear rection ratesrdquo Annual Review of Astronomy andAstrophysics vol 5 pp 525ndash570 1967
[5] A M Mathai and H J Haubold Modern Problems in Nuclearand Neutrino Astrophysics Akademie Berlin Germany 1988
[6] H J Haubold and A M Mathai ldquoAnalytic representations ofmodified non-resonant thermonuclear reaction ratesrdquo Journalof Applied Mathematics and Physics vol 37 no 5 pp 685ndash6951986
[7] W A Fowler ldquoExperimental and theoretical nuclear astro-physics the quest for the origin of the elementsrdquo Reviews ofModern Physics vol 56 no 2 pp 149ndash179 1984
[8] A C Phillips The Physics of Stars John Wiley amp SonsChichester UK 2nd edition 1999
[9] F C Adams ldquoStars in other universes stellar structure withdifferent fundamental constantsrdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 8 article 10 2008
[10] M Coraddu G Kaniadakis A Lavagno M Lissia G Mezzo-rani and P Quarati ldquoThermal distributions in stellar plasmasnuclear reactions and solar neutrinosrdquo Brazilian Journal ofPhysics vol 29 no 1 pp 153ndash168 1999
[11] H J Haubold and A M Mathai ldquoOn nuclear reaction ratetheoryrdquo Annalen der Physik vol 41 pp 380ndash396 1984
[12] M Coraddu M Lissia G Mezzorani and P Quarati ldquoSuper-Kamiokande hep neutrino best fit a possible signal of non-Maxwellian solar plasmardquo Physica A Statistical Mechanics andIts Applications vol 326 no 3-4 pp 473ndash481 2003
[13] A Lavagno and P Quarati ldquoClassical and quantum non-extensive statistics effects in nuclear many-body problemsrdquoChaos Solitons and Fractals vol 13 no 3 pp 569ndash580 2002
[14] A Lavagno and P Quarati ldquoMetastability of electron-nuclearastrophysical plasmas motivations signals and conditionsrdquoAstrophysics and Space Science vol 305 no 3 pp 253ndash2592006
[15] M Lissia and P Quarati ldquoNuclear astrophysical plasmas iondistribution functions and fusion ratesrdquo Europhysics News vol36 no 6 pp 211ndash214 2005
[16] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988
[17] C Tsallis Introduction to Non-Extensive Statistical MechanicsSpringer New York NY USA 2009
[18] M Gell-Mann and C Tsallis Eds Nonextensive EntropyInterdisciplinary Applications Oxford University Press NewYork NY USA 2004
[19] A M Mathai and H J Haubold ldquoPathway model superstatis-tics Tsallis statistics and a generalized measure of entropyrdquoPhysica A StatisticalMechanics and Its Applications vol 375 no1 pp 110ndash122 2007
[20] R K Saxena A M Mathai and H J Haubold ldquoAstrophysicalthermonuclear functions for Boltzmann-Gibbs statistics andTsallis statisticsrdquo Physica A Statistical Mechanics and Its Appli-cations vol 344 no 3-4 pp 649ndash656 2004
[21] A MMathai ldquoA pathway tomatrix-variate gamma and normaldensitiesrdquo Linear Algebra and Its Applications vol 396 no 1ndash3pp 317ndash328 2005
[22] A M Mathai and H J Haubold ldquoOn generalized distributionsand pathwaysrdquoPhysics Letters A General Atomic and Solid StatePhysics vol 372 no 12 pp 2109ndash2113 2008
[23] H J Haubold andDKumar ldquoFusion yield Guderleymodel andTsallis statisticsrdquo Journal of Plasma Physics vol 77 no 1 pp 1ndash14 2011
[24] D Kumar and H J Haubold ldquoOn extended thermonuclearfunctions through pathwaymodelrdquoAdvances in Space Researchvol 45 no 5 pp 698ndash708 2010
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Superconductivity
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Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
4 Journal of Astrophysics
Meijerrsquos 119866-function and its properties can be seen in Mathaiand Saxena [25] and Mathai [26] We can obtain seriesexpansions of the 119866-function given in (18) by combining thetheories of residue calculus and generalized special functionssee Kumar and Haubold [24] for series expansions for allpossible values of ] In many cases the nuclear factor 119878(])(0)is approximately constant across the fusion window Taking119878(])(0) = 0 for ] = 1 and ] = 2 and taking 1198780(0) = 119878(0) we
obtain the extended reaction rate probability integral as
11990312= (1 minus
1
212057512) 11989911198992[8 (119902 minus 1)
120583119896119879]
12
120587minus1
Γ (1 (119902 minus 1) minus 32)
times 119878 (0) 11986631
13[(119902 minus 1)119864
119866
4119896119879
10038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
(19)
The series representation for (19) can be obtained as
11990312= (1 minus
1
212057512) 11989911198992[8 (119902 minus 1)
120583119896119879]
12
120587minus1
Γ (1 (119902 minus 1) minus 32)
times 119878 (0) radic120587Γ (1 (119902 minus 1) minus 1)
minus 2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1)119864
119866
4119896119879)
+ (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
times[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(20)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(21)
See the Appendix for detailed evaluation As 119902 rarr 1 in (19)then by using Stirlingrsquos formula for gamma functions givenby
Γ (119911 + 119886) asymp (2120587)12
119911119911+119886minus12eminus119911
|119911| 997888rarr infin 119886 is bounded(22)
we get the reaction rate probability integral in the Maxwell-Boltzmann case as
11990312= (1 minus
1
212057512) 11989911198992(8
120587120583)
12
(1
119896119879)
32
119878 (0)
times int
infin
0
exp[minus 119864
119896119879minus (
119864119866
119864)
12
] d119864
(23)
= (1 minus1
212057512) 11989911198992(
8
120583119896119879)
12
119878 (0) 120587minus1
times 11986630
03[119864119866
4119896119879
10038161003816100381610038161003816100381610038161003816
minus
0121
]
(24)
which is obtained in a series of papers by Mathai andHaubold see for example Mathai and Haubold [5] Theintegral in (23) is dominated by theminimumvalue of119864119896119879+(119864119866119864)12
= 119892(119864) (say) The minimum value of the function119892(119864) say 119864
0 can be determined as
dd119864
[119864
119896119879+ (
119864119866
119864)
12
]
119864=1198640
=1
119896119879minus1
211986412
119866119864minus32
0= 0 997904rArr 119864
0= 11986412
119866(119896119879
2)
32
(25)
and the function
119892 (1198640) = 3(
119864119866
4119896119879)
13
= 3Θ (26)
where Θ = (1198641198664119896119879)
13 Now by using the Laplace method[27 28] we can obtain an approximate value for (23) as
11990312asymp (1 minus
1
212057512) 11989911198992
8119878 (0)Θ2 exp (minus3Θ)
radic312058712058312057211988511198852119888
(27)
In the next section we will obtain the mass pressure andtemperature for the case of analytic stellar models character-ized by density distribution and corresponding temperaturedistribution suggested by Haubold and Mathai [11]
3 Closed Forms of the Integral over the StellarNuclear Energy Generation Rate
Let us consider the density distribution 984858(119903) considered byHaubold and Mathai [6 11] and Mathai and Haubold [5] inthe form
984858 (119903) = 984858119888[1 minus (
119903
119877)
120575
] 120575 gt 0 (28)
where 984858119888is the central density of the star 119903 is an arbitrary
distance from the center and 119877 is the solar radius Thisdensity function is capable of producing different densitydistributions by choosing the free parameter 120575 Now wedetermine the quantities119872(119903) 119875(119903) and 119879(119903) the mass thepressure and the temperature at 119903
Journal of Astrophysics 5
By the equation of the mass conservation
d119872(119903)
d119903= 4120587119903
2
984858 (119903) (29)
we get
119872(119903) = 4120587984858119888int
119903
0
1199052
[1 minus (119905
119877)
120575
] d119905
=4120587
39848581198881199033
[1 minus3
120575 + 3(119903
119877)
120575
]
(30)
From (30) we get the central density as
984858119888=3 (120575 + 3)
4120587120575
119872 (119877)
1198773
(31)
If an element of a matter at a distance 119903 from the center ofa spherical system is in hydrostatic equilibrium then settingthe sum of the radial forces acting on it to zero we obtain
d119875 (119903)d119903
= minus119866984858 (119903)119872 (119903)
1199032
(32)
where 119866 is the gravitational constant Assuming that thepressure at the center of the sun is 119875
119888and at the surface is
zero we get
119875 (119903) = 119875119888minus 119866int
119903
0
119872(119905) 984858 (119905)
1199052
d119905
=4120587
31198669848582
1198881198772
[120585 minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
(33)
Using the boundary conditions 119875(119877) = 0 we get 119875119888=
(41205873)1198661205859848582
1198881198772 where
120585 =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)+
3
2 (120575 + 1) (120575 + 3) (34)
By the kinetic theory of gases for a perfect gas the pressureis given by
119875 (119903) =119896119873119860
120583984858 (119903) 119879 (119903) (35)
For the temperature of interest for stellar models we neglectthe negligible radiation pressure from the total pressure andobtain from (35) the following
119879 (119903) =120583
119896119873119860
119875 (119903)
984858 (119903)
=4120587
3119896119873119860
1198661205839848581198881198772
[1 minus (119903119877)120575
]
times [120585 minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
(36)
The central temperature 119879119888= (4119896119873
119860)119866120583120585(119872(119877)119877) where
120585 is as defined in (34)Thus we have obtained the mass the pressure and the
temperature throughout the nonlinear stellar model with thedensity distribution defined in (28) Next our aim is to obtainanalytical results for stellar luminosity and neutrino emissionrates for various stellar models
4 Stellar Luminosity and NeutrinoEmission Rate
The energy conservation equation states that the net increasein the rate of energy flux coming out of a spherical shell fromthe inside is the same as the energy produced within the shell[29] If we denote 119871
119903= 119871(119903) as the energy flux through the
sphere of radius 119903 then we have
d119871119903
d119903= 4120587119903
2
984858 (119903) 120576 (119903) (37)
where 120576(119903) is the energy produced per second by nuclearreactions in each gram of stellar matter 120576(119903) depends on thechemical composition in each gram of stellar matter Hereusually 119871
119903is a constant but will be equal to 119871 at the surface of
the star We assume here that the star is chemically homoge-neous (that is a star where chemical composition throughoutis a constant) Also we assume the energy generation rate120576(119903) for one particular nuclear reaction Now if we denote11990312(984858(119903) 119879(119903)) as the extended nonresonant thermonuclear
reaction rate for the particles 1 and 2 defined by (19) thenwe will consider the energy generation rate 120576
12(119903) and it can
be written in terms of the extended reaction rates via
12057612(119903) =
1
984858 (119903)Clowast
9848582
11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
] (38)
where
Clowast
=1198641211990312(984858 (119903) 119879 (119903))
984858211986631
13[(119902 minus 1) 119864
1198664119896119879
1003816100381610038161003816
2minus1(119902minus1)
0121]
(39)
in which 11986412
is the amount of energy given off in a singlereaction It is to be noted that by using the asymptoticbehaviour of 11986631
13((119902 minus 1)119864
1198664119896119879) [26] and as 119902 rarr 1
Clowast rarr C the composite parameter considered by [9] whichis defined as
C =1198641211990312
9848582Θ2
exp (3Θ) (40)
for our universeC asymp 2 times 104 for proton-proton fusion under
typical stellar conditions [9]Then from (37) we have the totalluminosity of the star by integration as follows
119871 (119877) = int
119877
0
41205871199032
984858 (119903) 120576 (119903) d119903 (41)
If we are considering only one specific reaction defined as in(1) then we have
11987112(119877) = int
119877
0
41205871199032
984858 (119903) 12057612(119903) d119903 (42)
6 Journal of Astrophysics
where the energy generation rate is defined in (38) and 984858(119903)is a suitable density distribution explaining the sun Writing(42) in terms of 119903
12(984858(119903) 119879(119903)) we get
11987112(119877) = int
119877
0
41205871199032
Clowast
9848582
11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
] d119903
= int
119877
0
41205871199032
1198641211990312(984858 (119903) 119879 (119903)) d119903
(43)
The number density 119899119894of a particle 119894 for a gas ofmean density
984858(119903) can be expressed as
119899119894(119903) = 984858 (119903)119873
119860
119883119894
119860119894
(44)
where 119873119860stands for Avagadrorsquos constant 119860
119894is the atomic
mass of particle 119894 in atomic mass units and 119883119894is the mass
fraction of particle 119894 such that sum119894119883119894
= 1 Substituting11990312(984858(119903) 119879(119903)) from (19) and using (44) we have
11987112(119877)
= int
119877
0
41205871199032
11986412(1 minus
1
212057512)1198732
1198609848582
(119903)11988311198832
11986011198602
[8 (119902 minus 1)
120583119896119879 (119903)]
12
times120587minus1
Γ (1 (119902 minus 1) minus 32)119878 (0) 119866
31
13
times [
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)(119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
(45)
If we divide the ldquointernal luminosityrdquo 11987112(119877⊙) by the amount
of energy 11986412 then we get the total number of particles per
second11987312liberated in the reaction given by (1) as follows
11987312=11987112(119877)
11986412
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
times int
119877
0
1199032
9848582
(119903)
[119879 (119903)]12
11986631
13
[
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)
times (119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)[
(119902 minus 1) 1205872
120583
2119896(11988511198852e2
ℏ)
2
]
minus119904
times int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903 d119904
(46)
For the density distribution defined in (28) introduced byHaubold and Mathai [5 6] and the corresponding temper-ature distribution (36) we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
119888
times int
119877
0
1199032
[1 minus (119903
119877)
120575
]
52minus119904
times [minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
119904minus12
d119903
(47)
where 120585 is as defined in (34) If we put a substitution 119903 = 119909119877then we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
times int
1
0
1199092
[1 minus 119909120575
]52minus119904
times [120585 minus1
21199092
+120575 + 6
(120575 + 2) (120575 + 3)119909120575+2
minus3
2(120575 + 1)(120575 + 3)1199092120575+2
]
119904minus12
d119909
(48)
Journal of Astrophysics 7
Putting 119909120575 = 119910 and simplifying we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583120585
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
120575
times int
1
0
1199103120575minus1
[1 minus 119910]52minus119904
[1 minus 119906 (119910)]119904minus12d119910
(49)
where 119906(119910) is defined as
119906 (119910) =1199102120575
120585[1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
]
(50)
As 119910 rarr 0 119906(119910) rarr 0 and 119910 rarr 1 119906(119910) rarr 1 If we take
V (119910) =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
(51)
we have V(0) = 12 The minimum value of V(119910) is at 119910 =
(120575 + 1)(120575 + 6)3(120575 + 2) and the value is 12 minus 16((120575 + 6)2(120575 +1)(120575 + 3)(120575 + 2)
2
) Thus the minimum value is nonnegativesince (120575 + 6)2(120575 + 1) (120575 + 3)(120575 + 2)2 decreases steadily from3 to 1 for all 120575 gt 0 Therefore V(119910) le 0 Since 120585 gt 0 for all120575 gt 0 119906(119910) le 0 for all 120575 gt 0 120585 gt 0 Thus [1 minus 119906(119910)]119904minus12 le 0Hence 0 lt 119906(119910) lt 1 for 0 lt 119910 lt 1 and for 120575 gt 0 Thus byusing the binomial expansion we obtain
[1 minus 119906 (119910)]119904minus12
=
infin
sum
119898=0
(12 minus 119904)119898
119898[119906 (119910)]
119898
(52)
where (12 minus 119904)119898is the Pochhammer symbol defined for 119886 isin
C by
(119886)0= 1
(119886)119898=
119886 (119886 + 1) sdot sdot sdot (119886 + 119898 minus 1)
Γ (119886 + 119898)
Γ (119886)
119898 = 1 2 119886 = 0
(53)
whenever Γ(119886) exists Taking 120575 = 2 we get 120585 = 15 and
[1 minus 119906 (119910)]119904minus12
= (1 minus 119910)2119904minus1
(1 minus1
2119910)
119904minus12
(54)
Then from (49) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times int
1
0
11991032minus1
[1 minus 119910]52+119904minus1
(1 minus1
2119910)
119904minus12
d119910
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
(12 minus 119904)119898
119898
1
2119898
times int
1
0
11991032+119898minus1
[1 minus 119910]52+119904minus1d119910
(55)
By using beta integral and using (53) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
Γ (12 minus 119904 + 119898)
119898Γ (12 minus 119904)
1
2119898
Γ (32 + 119898) Γ (52 + 119904)
Γ (4 + 119904 + 119898)
(56)
Now from (46) we obtain the total number of particles persecond liberated in the reaction (1) as follows
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904) Γ (
5
2+ 119904)
timesΓ (1 (119902 minus 1) minus 1 minus 119904) Γ (12 + 119898 minus 119904)
Γ (12 minus 119904) Γ (4 + 119898 + 119904)
times [15120587 (119902 minus 1)119873
119860
81198669848581198881198772
(11988511198852e2
ℏ)
2
]
minus119904
d119904
8 Journal of Astrophysics
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
times 11986642
35
[
[
15120587 (119902 minus 1)119873119860
81198669848581198881198772
times (11988511198852e2
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)12minus1198984+119898
01215212
]
]
(57)
For more details on119866-function and its properties see [2526] Thus we have obtained the the total number of particlesper second liberated in the reaction (1) in terms of the densitydistribution considered by Haubold and Mathai [6 29]
5 Comparison of Pathway Energy Density andthe Maxwell-Boltzmann Energy Density
In Figure 1 it can be obtserved that for the nuclei to reactat energy 119864 they have to borrow an energy 119864 from thethermal environment The probability of such an energy isproportional to theMaxwell-Boltzmann energy exp[minus119864119896119879]The fusion will take place when the nuclei penerate theCoulomb barrier keeping them apart The probability ofpenetration is given by the factor exp[minus(119864
119866119864)12
] Theproduct of these two factors illustrates that fusion mostlyoccurs in the energy window given in the figure
In Figure 2 the pathway energy density is plotted for 119902 =07 09 1 12 14 respectively For different values of 119902we getdifferent energy densities (curves (a) (b) (c) (d) and (e))The nonresonant cross section is also plotted The product ofthe pathway energy density and the nonresonant cross sectionfor different values of 119902 namely 119902 = 07 09 1 12 14 is alsoplotted It is to be noted that as 119902 rarr 1 the pathway energydensity coincides with the Maxwell-Boltzmann energy den-sity and also the fusion window for the Maxwell-Boltzmanncase in Figure 1
The curves in the figure represent pathway density[1 + (119902 minus 1)(119864119896119879)]
minus1(119902minus1) for (a) 119902 = 07 (b) 119902 = 09(c) 119902 = 1 (d) 119902 = 12 and (e) 119902 = 14 (f) repre-sents the nonresonant cross section exp[minus(119864
119866119864)12
] Theproduct [1 + (119902 minus 1)(119864119896119879)]minus1(119902minus1) exp[minus(119864
119866119864)12
] is alsorepresented for (g) 119902 = 07 (h) 119902 = 09 (i) 119902 = 1 (j) 119902 = 12and (k) 119902 = 14
Figure 3(a) shows the pathway energy density defined in(13) for 119902 = 1 12 13 14 and Figure 3(b) shows theMaxwell-Boltzmann energy density defined in (3) As 119902 rarr 1 thepathway energy density reduces to the Maxwell-Boltzmannenergy density The pathway energy density covers manystable and unstable situations as the value of 119902 varies If
Fusion window
0 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
2kT 4kT 6kT 8kT
prop exp[minus E
kT]
prop exp[minus (EGE
)12
]
prop exp[minus E
kTminus (EG
E)12
]
Energy E
Figure 1 Schematic plot of the enrgy-dependent factors for thereaction rate probability integral the Maxwell-Boltzmann energydensity nonresonant nuclear cross section and the product of theMaxwell-Boltzmann density and the nonresonant cross section
Fusion window
0 2kT 4kT 6kT 8kT 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
1 + (q minus 1)E
kT]minus1qminus1
(a)
(b)(c)
(d)(e)
(h)
(i)
(k)
(f)
(j)
(g)
prop[prop exp[minus (EG
E)12
]
Energy E
Figure 2 Schematic plot of the energy-dependent factors for theextended reaction rate probability integral pathway energy densitynonresonant nuclear cross section and the product of the pathwaydensity and the nonresonant cross section
the Maxwell-Boltzmann density is the equilibrium situationmany other nonequilibrium situations are covered by thepathway energy density
Journal of Astrophysics 9
Pathway energy density for k = 1 T = 100K
0004
0003
0002
0001
0
q = 1
q = 12
q = 13
q = 14
0 200 400 600 800 1000
fPD
Pathway energy density for k = 1 T = 100K
q = 1
q = 12
q = 13
q = 14
Energy E
(a)
0004
0003
0002
0001
0
fM
BD
Maxwell-Boltzmann energy densityfor k = 1 T = 100K
0 200 400 600 800 1000
Energy E
(b)
Figure 3 (a) Pathway energy density for 119902 = 1 12 13 14 and for 119896 = 1 119879 = 100119870 (b) The Maxwell-Boltzmann energy density for119896 = 1 119879 = 100119870
6 Concluding Remarks
In this paper we have modified the energy distributionfor a nonresonant reaction rate probability integral Thecomposition parameter C considered by [9] is extended toClowast by the pathway energy density Considering the analyticdensity distributions developed by Haubold and Mathai [629] they are used to obtain the stellar luminosity and the neu-trino emission rates and are obtained in generalized specialfunctions such as Meijerrsquos 119866-function The pathway energydensity considered here covers many density functions andhence the extended reaction rate integral covers a wider classof integral Pathway energy density helps us to obtain variousfusion windows by giving different values to 119902 the pathwayparameter which in turn leads to a new opening in the fusionresearch The graphs plotted here are by using Maple 14 inWindows XP platform
Appendix
Series Representation
The series representation for the right-hand side of (19) canbe obtained through the following procedure Here we applyresidue calculus on the119866-function given in (19) Consider the119866-function as follows
11986631
13((119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
)
=1
2120587119894int
119888+119894infin
119888minus119894infin
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1) 119864119866
4119896119879)
minus119904
d119904
(A1)
The right-hand side is the sum of the residues of the inte-grandThe poles of the gammas in the integral representationin (19) are as follows
poles of Γ(119904) 119904 = 0 minus1 minus2
poles of Γ(12 + 119904) 119904 = minus12 minus32 minus52
poles of Γ(1 + 119904) 119904 = minus1 minus2 minus3
Here the poles of Γ(119904) and Γ(1 + 119904) will coincide with eachother at all points except at 119904 = 0 Note that the pole 119904 = 0 isa pole of order 1 119904 = minus12 minus32 minus52 are each of order 1and 119904 = minus1 minus2 minus3 are each of order 2 We know that
lim119904rarrminus119903
(119904 + 119903) Γ (119904) =(minus1)119903
119903
Γ (119886 minus 119903) =(minus1)119903
Γ (119886)
(1 minus 119886)119903
Γ (119886 + 119898) = Γ (119886) (119886)119898
(A2)
10 Journal of Astrophysics
when Γ(119886) is defined 119903 = 0 1 2 Γ(12) = 12058712
(119886)119903=
119886 (119886 + 1) sdot sdot sdot (119886 + 119903 minus 1) if 119903 ge 1 119886 = 0
1 if 119903 = 0(A3)
The sum of the residues corresponding to the poles 119904 = 0 isgiven by
1198771= radic120587Γ(
1
119902 minus 1minus 1) (A4)
The sum of the residues corresponding to the poles 119904 =
minus12 minus32 minus52 is
1198772=
infin
sum
119903=0
(minus1)119903
119903Γ (minus
1
2minus 119903) Γ (
1
2minus 119903)
times Γ(1
119902 minus 1minus1
2+ 119903) [
(119902 minus 1)119864119866
4119896119879]
minus12+119903
= minus2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1) 119864
119866
4119896119879)
(A5)
where11198652is the hypergeometric function defined by
11198652(119886 119887 119888 119909) =
infin
sum
119903=0
(119886)119903
(119887)119903(119888)119903
119909119903
119903 (A6)
To obtain the sum of the residues corresponding to poles119904 = minus1 minus2 minus3 of order 2 we proceed as follows
1198773
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[(119904 + 1 + 119903)
2
Γ (1 + 119904) Γ (119904) Γ (1
2+ 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1)119864119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[ (Γ2
(2 + 119904 + 119903) Γ (12 + 119904)
times Γ (1 (119902 minus 1) minus 1 minus 119904))
times ((119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904)minus1
times ((119902 minus 1) 119864
119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904Φ (119904)
(A7)
where
Φ (119904) =Γ2
(2 + 119904 + 119903) Γ (12 + 119904) Γ (1 (119902 minus 1) minus 1 minus 119904)
(119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904
times ((119902 minus 1)119864
119866
4119896119879)
minus119904
(A8)
We have
120597
120597119904Φ (119904) = Φ (119904)
120597
120597119904[ln [Φ (119904)]]
lnΦ (119904) = 2 ln [Γ (2 + 119904 + 119903)] + ln [Γ (12+ 119904)]
+ ln [Γ( 1
119902 minus 1minus 1 minus 119904)] minus 119904 ln(
(119902 minus 1) 119864119866
4119896119879)
minus 2 ln (119904 + 119903) minus 2 ln (119904 + 119903 minus 1) minus sdot sdot sdot
minus 2 ln (119904 + 1) minus ln (119904)
120597
120597119904[ln [Φ (119904)]] = 2Ψ (2 + 119904 + 119903) + Ψ(
1
2+ 119904)
+ Ψ(1
119902 minus 1minus 1 minus 119904)
minus ln((119902 minus 1) 119864
119866
4119896119879) minus
2
119904 + 119903
minus2
119904 + 119903 minus 1minus sdot sdot sdot minus
2
119904 + 1minus1
119904
lim119904rarrminus1minus119903
120597
120597119904ln [Φ (119904)] = Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
minus ln((119902 minus 1) 119864
119866
4119896119879)
(A9)
whereΨ(119911) is a Psi function or digamma function (seeMathai[26]) and Ψ(1) = minus120574 120574 = 05772156649 is Eulerrsquosconstant Now
lim119904rarrminus1minus119903
Φ (119904) =(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
times ((119902 minus 1) 119864
119866
4119896119879)
1+119903
(A10)
Journal of Astrophysics 11
Then by using (A7) (A9) and (A10) we get
1198773=
infin
sum
119903=0
(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
((119902 minus 1) 119864
119866
4119896119879)
1+119903
times [Ψ(minus1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903) + Ψ (1 + 119903)
+Ψ (2 + 119903) minus ln((119902 minus 1) 119864
119866
4119896119879)]
= (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(A11)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(A12)
Thus from (A4) (A5) and (A11) we get (20)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the Department of Sci-ence and Technology Government of India New Delhifor the financial assistance for this work under Project noSRS4MS28705 and the Centre for Mathematical Sciencesfor providing all facilities
References
[1] R Davis Jr ldquoNobel Lecture a half-century with solar neutri-nosrdquo Reviews of Modern Physics vol 75 no 3 pp 985ndash9942003
[2] H J Haubold and D Kumar ldquoExtension of thermonuclearfunctions through the pathway model including Maxwell-Boltzmann and Tsallis distributionsrdquo Astroparticle Physics vol29 no 1 pp 70ndash76 2008
[3] H J Haubold and R W John ldquoOn the evaluation of an integralconnected with the thermonuclear reaction rate in closed-formrdquo Astronomische Nachrichten vol 299 no 5 pp 225ndash2321978
[4] W A Fowler G R Caughlan and B A Zimmerman ldquoTher-monuclear rection ratesrdquo Annual Review of Astronomy andAstrophysics vol 5 pp 525ndash570 1967
[5] A M Mathai and H J Haubold Modern Problems in Nuclearand Neutrino Astrophysics Akademie Berlin Germany 1988
[6] H J Haubold and A M Mathai ldquoAnalytic representations ofmodified non-resonant thermonuclear reaction ratesrdquo Journalof Applied Mathematics and Physics vol 37 no 5 pp 685ndash6951986
[7] W A Fowler ldquoExperimental and theoretical nuclear astro-physics the quest for the origin of the elementsrdquo Reviews ofModern Physics vol 56 no 2 pp 149ndash179 1984
[8] A C Phillips The Physics of Stars John Wiley amp SonsChichester UK 2nd edition 1999
[9] F C Adams ldquoStars in other universes stellar structure withdifferent fundamental constantsrdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 8 article 10 2008
[10] M Coraddu G Kaniadakis A Lavagno M Lissia G Mezzo-rani and P Quarati ldquoThermal distributions in stellar plasmasnuclear reactions and solar neutrinosrdquo Brazilian Journal ofPhysics vol 29 no 1 pp 153ndash168 1999
[11] H J Haubold and A M Mathai ldquoOn nuclear reaction ratetheoryrdquo Annalen der Physik vol 41 pp 380ndash396 1984
[12] M Coraddu M Lissia G Mezzorani and P Quarati ldquoSuper-Kamiokande hep neutrino best fit a possible signal of non-Maxwellian solar plasmardquo Physica A Statistical Mechanics andIts Applications vol 326 no 3-4 pp 473ndash481 2003
[13] A Lavagno and P Quarati ldquoClassical and quantum non-extensive statistics effects in nuclear many-body problemsrdquoChaos Solitons and Fractals vol 13 no 3 pp 569ndash580 2002
[14] A Lavagno and P Quarati ldquoMetastability of electron-nuclearastrophysical plasmas motivations signals and conditionsrdquoAstrophysics and Space Science vol 305 no 3 pp 253ndash2592006
[15] M Lissia and P Quarati ldquoNuclear astrophysical plasmas iondistribution functions and fusion ratesrdquo Europhysics News vol36 no 6 pp 211ndash214 2005
[16] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988
[17] C Tsallis Introduction to Non-Extensive Statistical MechanicsSpringer New York NY USA 2009
[18] M Gell-Mann and C Tsallis Eds Nonextensive EntropyInterdisciplinary Applications Oxford University Press NewYork NY USA 2004
[19] A M Mathai and H J Haubold ldquoPathway model superstatis-tics Tsallis statistics and a generalized measure of entropyrdquoPhysica A StatisticalMechanics and Its Applications vol 375 no1 pp 110ndash122 2007
[20] R K Saxena A M Mathai and H J Haubold ldquoAstrophysicalthermonuclear functions for Boltzmann-Gibbs statistics andTsallis statisticsrdquo Physica A Statistical Mechanics and Its Appli-cations vol 344 no 3-4 pp 649ndash656 2004
[21] A MMathai ldquoA pathway tomatrix-variate gamma and normaldensitiesrdquo Linear Algebra and Its Applications vol 396 no 1ndash3pp 317ndash328 2005
[22] A M Mathai and H J Haubold ldquoOn generalized distributionsand pathwaysrdquoPhysics Letters A General Atomic and Solid StatePhysics vol 372 no 12 pp 2109ndash2113 2008
[23] H J Haubold andDKumar ldquoFusion yield Guderleymodel andTsallis statisticsrdquo Journal of Plasma Physics vol 77 no 1 pp 1ndash14 2011
[24] D Kumar and H J Haubold ldquoOn extended thermonuclearfunctions through pathwaymodelrdquoAdvances in Space Researchvol 45 no 5 pp 698ndash708 2010
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Journal of Astrophysics 5
By the equation of the mass conservation
d119872(119903)
d119903= 4120587119903
2
984858 (119903) (29)
we get
119872(119903) = 4120587984858119888int
119903
0
1199052
[1 minus (119905
119877)
120575
] d119905
=4120587
39848581198881199033
[1 minus3
120575 + 3(119903
119877)
120575
]
(30)
From (30) we get the central density as
984858119888=3 (120575 + 3)
4120587120575
119872 (119877)
1198773
(31)
If an element of a matter at a distance 119903 from the center ofa spherical system is in hydrostatic equilibrium then settingthe sum of the radial forces acting on it to zero we obtain
d119875 (119903)d119903
= minus119866984858 (119903)119872 (119903)
1199032
(32)
where 119866 is the gravitational constant Assuming that thepressure at the center of the sun is 119875
119888and at the surface is
zero we get
119875 (119903) = 119875119888minus 119866int
119903
0
119872(119905) 984858 (119905)
1199052
d119905
=4120587
31198669848582
1198881198772
[120585 minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
(33)
Using the boundary conditions 119875(119877) = 0 we get 119875119888=
(41205873)1198661205859848582
1198881198772 where
120585 =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)+
3
2 (120575 + 1) (120575 + 3) (34)
By the kinetic theory of gases for a perfect gas the pressureis given by
119875 (119903) =119896119873119860
120583984858 (119903) 119879 (119903) (35)
For the temperature of interest for stellar models we neglectthe negligible radiation pressure from the total pressure andobtain from (35) the following
119879 (119903) =120583
119896119873119860
119875 (119903)
984858 (119903)
=4120587
3119896119873119860
1198661205839848581198881198772
[1 minus (119903119877)120575
]
times [120585 minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
(36)
The central temperature 119879119888= (4119896119873
119860)119866120583120585(119872(119877)119877) where
120585 is as defined in (34)Thus we have obtained the mass the pressure and the
temperature throughout the nonlinear stellar model with thedensity distribution defined in (28) Next our aim is to obtainanalytical results for stellar luminosity and neutrino emissionrates for various stellar models
4 Stellar Luminosity and NeutrinoEmission Rate
The energy conservation equation states that the net increasein the rate of energy flux coming out of a spherical shell fromthe inside is the same as the energy produced within the shell[29] If we denote 119871
119903= 119871(119903) as the energy flux through the
sphere of radius 119903 then we have
d119871119903
d119903= 4120587119903
2
984858 (119903) 120576 (119903) (37)
where 120576(119903) is the energy produced per second by nuclearreactions in each gram of stellar matter 120576(119903) depends on thechemical composition in each gram of stellar matter Hereusually 119871
119903is a constant but will be equal to 119871 at the surface of
the star We assume here that the star is chemically homoge-neous (that is a star where chemical composition throughoutis a constant) Also we assume the energy generation rate120576(119903) for one particular nuclear reaction Now if we denote11990312(984858(119903) 119879(119903)) as the extended nonresonant thermonuclear
reaction rate for the particles 1 and 2 defined by (19) thenwe will consider the energy generation rate 120576
12(119903) and it can
be written in terms of the extended reaction rates via
12057612(119903) =
1
984858 (119903)Clowast
9848582
11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
] (38)
where
Clowast
=1198641211990312(984858 (119903) 119879 (119903))
984858211986631
13[(119902 minus 1) 119864
1198664119896119879
1003816100381610038161003816
2minus1(119902minus1)
0121]
(39)
in which 11986412
is the amount of energy given off in a singlereaction It is to be noted that by using the asymptoticbehaviour of 11986631
13((119902 minus 1)119864
1198664119896119879) [26] and as 119902 rarr 1
Clowast rarr C the composite parameter considered by [9] whichis defined as
C =1198641211990312
9848582Θ2
exp (3Θ) (40)
for our universeC asymp 2 times 104 for proton-proton fusion under
typical stellar conditions [9]Then from (37) we have the totalluminosity of the star by integration as follows
119871 (119877) = int
119877
0
41205871199032
984858 (119903) 120576 (119903) d119903 (41)
If we are considering only one specific reaction defined as in(1) then we have
11987112(119877) = int
119877
0
41205871199032
984858 (119903) 12057612(119903) d119903 (42)
6 Journal of Astrophysics
where the energy generation rate is defined in (38) and 984858(119903)is a suitable density distribution explaining the sun Writing(42) in terms of 119903
12(984858(119903) 119879(119903)) we get
11987112(119877) = int
119877
0
41205871199032
Clowast
9848582
11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
] d119903
= int
119877
0
41205871199032
1198641211990312(984858 (119903) 119879 (119903)) d119903
(43)
The number density 119899119894of a particle 119894 for a gas ofmean density
984858(119903) can be expressed as
119899119894(119903) = 984858 (119903)119873
119860
119883119894
119860119894
(44)
where 119873119860stands for Avagadrorsquos constant 119860
119894is the atomic
mass of particle 119894 in atomic mass units and 119883119894is the mass
fraction of particle 119894 such that sum119894119883119894
= 1 Substituting11990312(984858(119903) 119879(119903)) from (19) and using (44) we have
11987112(119877)
= int
119877
0
41205871199032
11986412(1 minus
1
212057512)1198732
1198609848582
(119903)11988311198832
11986011198602
[8 (119902 minus 1)
120583119896119879 (119903)]
12
times120587minus1
Γ (1 (119902 minus 1) minus 32)119878 (0) 119866
31
13
times [
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)(119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
(45)
If we divide the ldquointernal luminosityrdquo 11987112(119877⊙) by the amount
of energy 11986412 then we get the total number of particles per
second11987312liberated in the reaction given by (1) as follows
11987312=11987112(119877)
11986412
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
times int
119877
0
1199032
9848582
(119903)
[119879 (119903)]12
11986631
13
[
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)
times (119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)[
(119902 minus 1) 1205872
120583
2119896(11988511198852e2
ℏ)
2
]
minus119904
times int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903 d119904
(46)
For the density distribution defined in (28) introduced byHaubold and Mathai [5 6] and the corresponding temper-ature distribution (36) we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
119888
times int
119877
0
1199032
[1 minus (119903
119877)
120575
]
52minus119904
times [minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
119904minus12
d119903
(47)
where 120585 is as defined in (34) If we put a substitution 119903 = 119909119877then we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
times int
1
0
1199092
[1 minus 119909120575
]52minus119904
times [120585 minus1
21199092
+120575 + 6
(120575 + 2) (120575 + 3)119909120575+2
minus3
2(120575 + 1)(120575 + 3)1199092120575+2
]
119904minus12
d119909
(48)
Journal of Astrophysics 7
Putting 119909120575 = 119910 and simplifying we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583120585
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
120575
times int
1
0
1199103120575minus1
[1 minus 119910]52minus119904
[1 minus 119906 (119910)]119904minus12d119910
(49)
where 119906(119910) is defined as
119906 (119910) =1199102120575
120585[1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
]
(50)
As 119910 rarr 0 119906(119910) rarr 0 and 119910 rarr 1 119906(119910) rarr 1 If we take
V (119910) =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
(51)
we have V(0) = 12 The minimum value of V(119910) is at 119910 =
(120575 + 1)(120575 + 6)3(120575 + 2) and the value is 12 minus 16((120575 + 6)2(120575 +1)(120575 + 3)(120575 + 2)
2
) Thus the minimum value is nonnegativesince (120575 + 6)2(120575 + 1) (120575 + 3)(120575 + 2)2 decreases steadily from3 to 1 for all 120575 gt 0 Therefore V(119910) le 0 Since 120585 gt 0 for all120575 gt 0 119906(119910) le 0 for all 120575 gt 0 120585 gt 0 Thus [1 minus 119906(119910)]119904minus12 le 0Hence 0 lt 119906(119910) lt 1 for 0 lt 119910 lt 1 and for 120575 gt 0 Thus byusing the binomial expansion we obtain
[1 minus 119906 (119910)]119904minus12
=
infin
sum
119898=0
(12 minus 119904)119898
119898[119906 (119910)]
119898
(52)
where (12 minus 119904)119898is the Pochhammer symbol defined for 119886 isin
C by
(119886)0= 1
(119886)119898=
119886 (119886 + 1) sdot sdot sdot (119886 + 119898 minus 1)
Γ (119886 + 119898)
Γ (119886)
119898 = 1 2 119886 = 0
(53)
whenever Γ(119886) exists Taking 120575 = 2 we get 120585 = 15 and
[1 minus 119906 (119910)]119904minus12
= (1 minus 119910)2119904minus1
(1 minus1
2119910)
119904minus12
(54)
Then from (49) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times int
1
0
11991032minus1
[1 minus 119910]52+119904minus1
(1 minus1
2119910)
119904minus12
d119910
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
(12 minus 119904)119898
119898
1
2119898
times int
1
0
11991032+119898minus1
[1 minus 119910]52+119904minus1d119910
(55)
By using beta integral and using (53) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
Γ (12 minus 119904 + 119898)
119898Γ (12 minus 119904)
1
2119898
Γ (32 + 119898) Γ (52 + 119904)
Γ (4 + 119904 + 119898)
(56)
Now from (46) we obtain the total number of particles persecond liberated in the reaction (1) as follows
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904) Γ (
5
2+ 119904)
timesΓ (1 (119902 minus 1) minus 1 minus 119904) Γ (12 + 119898 minus 119904)
Γ (12 minus 119904) Γ (4 + 119898 + 119904)
times [15120587 (119902 minus 1)119873
119860
81198669848581198881198772
(11988511198852e2
ℏ)
2
]
minus119904
d119904
8 Journal of Astrophysics
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
times 11986642
35
[
[
15120587 (119902 minus 1)119873119860
81198669848581198881198772
times (11988511198852e2
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)12minus1198984+119898
01215212
]
]
(57)
For more details on119866-function and its properties see [2526] Thus we have obtained the the total number of particlesper second liberated in the reaction (1) in terms of the densitydistribution considered by Haubold and Mathai [6 29]
5 Comparison of Pathway Energy Density andthe Maxwell-Boltzmann Energy Density
In Figure 1 it can be obtserved that for the nuclei to reactat energy 119864 they have to borrow an energy 119864 from thethermal environment The probability of such an energy isproportional to theMaxwell-Boltzmann energy exp[minus119864119896119879]The fusion will take place when the nuclei penerate theCoulomb barrier keeping them apart The probability ofpenetration is given by the factor exp[minus(119864
119866119864)12
] Theproduct of these two factors illustrates that fusion mostlyoccurs in the energy window given in the figure
In Figure 2 the pathway energy density is plotted for 119902 =07 09 1 12 14 respectively For different values of 119902we getdifferent energy densities (curves (a) (b) (c) (d) and (e))The nonresonant cross section is also plotted The product ofthe pathway energy density and the nonresonant cross sectionfor different values of 119902 namely 119902 = 07 09 1 12 14 is alsoplotted It is to be noted that as 119902 rarr 1 the pathway energydensity coincides with the Maxwell-Boltzmann energy den-sity and also the fusion window for the Maxwell-Boltzmanncase in Figure 1
The curves in the figure represent pathway density[1 + (119902 minus 1)(119864119896119879)]
minus1(119902minus1) for (a) 119902 = 07 (b) 119902 = 09(c) 119902 = 1 (d) 119902 = 12 and (e) 119902 = 14 (f) repre-sents the nonresonant cross section exp[minus(119864
119866119864)12
] Theproduct [1 + (119902 minus 1)(119864119896119879)]minus1(119902minus1) exp[minus(119864
119866119864)12
] is alsorepresented for (g) 119902 = 07 (h) 119902 = 09 (i) 119902 = 1 (j) 119902 = 12and (k) 119902 = 14
Figure 3(a) shows the pathway energy density defined in(13) for 119902 = 1 12 13 14 and Figure 3(b) shows theMaxwell-Boltzmann energy density defined in (3) As 119902 rarr 1 thepathway energy density reduces to the Maxwell-Boltzmannenergy density The pathway energy density covers manystable and unstable situations as the value of 119902 varies If
Fusion window
0 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
2kT 4kT 6kT 8kT
prop exp[minus E
kT]
prop exp[minus (EGE
)12
]
prop exp[minus E
kTminus (EG
E)12
]
Energy E
Figure 1 Schematic plot of the enrgy-dependent factors for thereaction rate probability integral the Maxwell-Boltzmann energydensity nonresonant nuclear cross section and the product of theMaxwell-Boltzmann density and the nonresonant cross section
Fusion window
0 2kT 4kT 6kT 8kT 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
1 + (q minus 1)E
kT]minus1qminus1
(a)
(b)(c)
(d)(e)
(h)
(i)
(k)
(f)
(j)
(g)
prop[prop exp[minus (EG
E)12
]
Energy E
Figure 2 Schematic plot of the energy-dependent factors for theextended reaction rate probability integral pathway energy densitynonresonant nuclear cross section and the product of the pathwaydensity and the nonresonant cross section
the Maxwell-Boltzmann density is the equilibrium situationmany other nonequilibrium situations are covered by thepathway energy density
Journal of Astrophysics 9
Pathway energy density for k = 1 T = 100K
0004
0003
0002
0001
0
q = 1
q = 12
q = 13
q = 14
0 200 400 600 800 1000
fPD
Pathway energy density for k = 1 T = 100K
q = 1
q = 12
q = 13
q = 14
Energy E
(a)
0004
0003
0002
0001
0
fM
BD
Maxwell-Boltzmann energy densityfor k = 1 T = 100K
0 200 400 600 800 1000
Energy E
(b)
Figure 3 (a) Pathway energy density for 119902 = 1 12 13 14 and for 119896 = 1 119879 = 100119870 (b) The Maxwell-Boltzmann energy density for119896 = 1 119879 = 100119870
6 Concluding Remarks
In this paper we have modified the energy distributionfor a nonresonant reaction rate probability integral Thecomposition parameter C considered by [9] is extended toClowast by the pathway energy density Considering the analyticdensity distributions developed by Haubold and Mathai [629] they are used to obtain the stellar luminosity and the neu-trino emission rates and are obtained in generalized specialfunctions such as Meijerrsquos 119866-function The pathway energydensity considered here covers many density functions andhence the extended reaction rate integral covers a wider classof integral Pathway energy density helps us to obtain variousfusion windows by giving different values to 119902 the pathwayparameter which in turn leads to a new opening in the fusionresearch The graphs plotted here are by using Maple 14 inWindows XP platform
Appendix
Series Representation
The series representation for the right-hand side of (19) canbe obtained through the following procedure Here we applyresidue calculus on the119866-function given in (19) Consider the119866-function as follows
11986631
13((119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
)
=1
2120587119894int
119888+119894infin
119888minus119894infin
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1) 119864119866
4119896119879)
minus119904
d119904
(A1)
The right-hand side is the sum of the residues of the inte-grandThe poles of the gammas in the integral representationin (19) are as follows
poles of Γ(119904) 119904 = 0 minus1 minus2
poles of Γ(12 + 119904) 119904 = minus12 minus32 minus52
poles of Γ(1 + 119904) 119904 = minus1 minus2 minus3
Here the poles of Γ(119904) and Γ(1 + 119904) will coincide with eachother at all points except at 119904 = 0 Note that the pole 119904 = 0 isa pole of order 1 119904 = minus12 minus32 minus52 are each of order 1and 119904 = minus1 minus2 minus3 are each of order 2 We know that
lim119904rarrminus119903
(119904 + 119903) Γ (119904) =(minus1)119903
119903
Γ (119886 minus 119903) =(minus1)119903
Γ (119886)
(1 minus 119886)119903
Γ (119886 + 119898) = Γ (119886) (119886)119898
(A2)
10 Journal of Astrophysics
when Γ(119886) is defined 119903 = 0 1 2 Γ(12) = 12058712
(119886)119903=
119886 (119886 + 1) sdot sdot sdot (119886 + 119903 minus 1) if 119903 ge 1 119886 = 0
1 if 119903 = 0(A3)
The sum of the residues corresponding to the poles 119904 = 0 isgiven by
1198771= radic120587Γ(
1
119902 minus 1minus 1) (A4)
The sum of the residues corresponding to the poles 119904 =
minus12 minus32 minus52 is
1198772=
infin
sum
119903=0
(minus1)119903
119903Γ (minus
1
2minus 119903) Γ (
1
2minus 119903)
times Γ(1
119902 minus 1minus1
2+ 119903) [
(119902 minus 1)119864119866
4119896119879]
minus12+119903
= minus2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1) 119864
119866
4119896119879)
(A5)
where11198652is the hypergeometric function defined by
11198652(119886 119887 119888 119909) =
infin
sum
119903=0
(119886)119903
(119887)119903(119888)119903
119909119903
119903 (A6)
To obtain the sum of the residues corresponding to poles119904 = minus1 minus2 minus3 of order 2 we proceed as follows
1198773
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[(119904 + 1 + 119903)
2
Γ (1 + 119904) Γ (119904) Γ (1
2+ 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1)119864119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[ (Γ2
(2 + 119904 + 119903) Γ (12 + 119904)
times Γ (1 (119902 minus 1) minus 1 minus 119904))
times ((119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904)minus1
times ((119902 minus 1) 119864
119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904Φ (119904)
(A7)
where
Φ (119904) =Γ2
(2 + 119904 + 119903) Γ (12 + 119904) Γ (1 (119902 minus 1) minus 1 minus 119904)
(119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904
times ((119902 minus 1)119864
119866
4119896119879)
minus119904
(A8)
We have
120597
120597119904Φ (119904) = Φ (119904)
120597
120597119904[ln [Φ (119904)]]
lnΦ (119904) = 2 ln [Γ (2 + 119904 + 119903)] + ln [Γ (12+ 119904)]
+ ln [Γ( 1
119902 minus 1minus 1 minus 119904)] minus 119904 ln(
(119902 minus 1) 119864119866
4119896119879)
minus 2 ln (119904 + 119903) minus 2 ln (119904 + 119903 minus 1) minus sdot sdot sdot
minus 2 ln (119904 + 1) minus ln (119904)
120597
120597119904[ln [Φ (119904)]] = 2Ψ (2 + 119904 + 119903) + Ψ(
1
2+ 119904)
+ Ψ(1
119902 minus 1minus 1 minus 119904)
minus ln((119902 minus 1) 119864
119866
4119896119879) minus
2
119904 + 119903
minus2
119904 + 119903 minus 1minus sdot sdot sdot minus
2
119904 + 1minus1
119904
lim119904rarrminus1minus119903
120597
120597119904ln [Φ (119904)] = Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
minus ln((119902 minus 1) 119864
119866
4119896119879)
(A9)
whereΨ(119911) is a Psi function or digamma function (seeMathai[26]) and Ψ(1) = minus120574 120574 = 05772156649 is Eulerrsquosconstant Now
lim119904rarrminus1minus119903
Φ (119904) =(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
times ((119902 minus 1) 119864
119866
4119896119879)
1+119903
(A10)
Journal of Astrophysics 11
Then by using (A7) (A9) and (A10) we get
1198773=
infin
sum
119903=0
(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
((119902 minus 1) 119864
119866
4119896119879)
1+119903
times [Ψ(minus1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903) + Ψ (1 + 119903)
+Ψ (2 + 119903) minus ln((119902 minus 1) 119864
119866
4119896119879)]
= (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(A11)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(A12)
Thus from (A4) (A5) and (A11) we get (20)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the Department of Sci-ence and Technology Government of India New Delhifor the financial assistance for this work under Project noSRS4MS28705 and the Centre for Mathematical Sciencesfor providing all facilities
References
[1] R Davis Jr ldquoNobel Lecture a half-century with solar neutri-nosrdquo Reviews of Modern Physics vol 75 no 3 pp 985ndash9942003
[2] H J Haubold and D Kumar ldquoExtension of thermonuclearfunctions through the pathway model including Maxwell-Boltzmann and Tsallis distributionsrdquo Astroparticle Physics vol29 no 1 pp 70ndash76 2008
[3] H J Haubold and R W John ldquoOn the evaluation of an integralconnected with the thermonuclear reaction rate in closed-formrdquo Astronomische Nachrichten vol 299 no 5 pp 225ndash2321978
[4] W A Fowler G R Caughlan and B A Zimmerman ldquoTher-monuclear rection ratesrdquo Annual Review of Astronomy andAstrophysics vol 5 pp 525ndash570 1967
[5] A M Mathai and H J Haubold Modern Problems in Nuclearand Neutrino Astrophysics Akademie Berlin Germany 1988
[6] H J Haubold and A M Mathai ldquoAnalytic representations ofmodified non-resonant thermonuclear reaction ratesrdquo Journalof Applied Mathematics and Physics vol 37 no 5 pp 685ndash6951986
[7] W A Fowler ldquoExperimental and theoretical nuclear astro-physics the quest for the origin of the elementsrdquo Reviews ofModern Physics vol 56 no 2 pp 149ndash179 1984
[8] A C Phillips The Physics of Stars John Wiley amp SonsChichester UK 2nd edition 1999
[9] F C Adams ldquoStars in other universes stellar structure withdifferent fundamental constantsrdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 8 article 10 2008
[10] M Coraddu G Kaniadakis A Lavagno M Lissia G Mezzo-rani and P Quarati ldquoThermal distributions in stellar plasmasnuclear reactions and solar neutrinosrdquo Brazilian Journal ofPhysics vol 29 no 1 pp 153ndash168 1999
[11] H J Haubold and A M Mathai ldquoOn nuclear reaction ratetheoryrdquo Annalen der Physik vol 41 pp 380ndash396 1984
[12] M Coraddu M Lissia G Mezzorani and P Quarati ldquoSuper-Kamiokande hep neutrino best fit a possible signal of non-Maxwellian solar plasmardquo Physica A Statistical Mechanics andIts Applications vol 326 no 3-4 pp 473ndash481 2003
[13] A Lavagno and P Quarati ldquoClassical and quantum non-extensive statistics effects in nuclear many-body problemsrdquoChaos Solitons and Fractals vol 13 no 3 pp 569ndash580 2002
[14] A Lavagno and P Quarati ldquoMetastability of electron-nuclearastrophysical plasmas motivations signals and conditionsrdquoAstrophysics and Space Science vol 305 no 3 pp 253ndash2592006
[15] M Lissia and P Quarati ldquoNuclear astrophysical plasmas iondistribution functions and fusion ratesrdquo Europhysics News vol36 no 6 pp 211ndash214 2005
[16] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988
[17] C Tsallis Introduction to Non-Extensive Statistical MechanicsSpringer New York NY USA 2009
[18] M Gell-Mann and C Tsallis Eds Nonextensive EntropyInterdisciplinary Applications Oxford University Press NewYork NY USA 2004
[19] A M Mathai and H J Haubold ldquoPathway model superstatis-tics Tsallis statistics and a generalized measure of entropyrdquoPhysica A StatisticalMechanics and Its Applications vol 375 no1 pp 110ndash122 2007
[20] R K Saxena A M Mathai and H J Haubold ldquoAstrophysicalthermonuclear functions for Boltzmann-Gibbs statistics andTsallis statisticsrdquo Physica A Statistical Mechanics and Its Appli-cations vol 344 no 3-4 pp 649ndash656 2004
[21] A MMathai ldquoA pathway tomatrix-variate gamma and normaldensitiesrdquo Linear Algebra and Its Applications vol 396 no 1ndash3pp 317ndash328 2005
[22] A M Mathai and H J Haubold ldquoOn generalized distributionsand pathwaysrdquoPhysics Letters A General Atomic and Solid StatePhysics vol 372 no 12 pp 2109ndash2113 2008
[23] H J Haubold andDKumar ldquoFusion yield Guderleymodel andTsallis statisticsrdquo Journal of Plasma Physics vol 77 no 1 pp 1ndash14 2011
[24] D Kumar and H J Haubold ldquoOn extended thermonuclearfunctions through pathwaymodelrdquoAdvances in Space Researchvol 45 no 5 pp 698ndash708 2010
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Superconductivity
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PhotonicsJournal of
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Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
6 Journal of Astrophysics
where the energy generation rate is defined in (38) and 984858(119903)is a suitable density distribution explaining the sun Writing(42) in terms of 119903
12(984858(119903) 119879(119903)) we get
11987112(119877) = int
119877
0
41205871199032
Clowast
9848582
11986631
13[(119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
] d119903
= int
119877
0
41205871199032
1198641211990312(984858 (119903) 119879 (119903)) d119903
(43)
The number density 119899119894of a particle 119894 for a gas ofmean density
984858(119903) can be expressed as
119899119894(119903) = 984858 (119903)119873
119860
119883119894
119860119894
(44)
where 119873119860stands for Avagadrorsquos constant 119860
119894is the atomic
mass of particle 119894 in atomic mass units and 119883119894is the mass
fraction of particle 119894 such that sum119894119883119894
= 1 Substituting11990312(984858(119903) 119879(119903)) from (19) and using (44) we have
11987112(119877)
= int
119877
0
41205871199032
11986412(1 minus
1
212057512)1198732
1198609848582
(119903)11988311198832
11986011198602
[8 (119902 minus 1)
120583119896119879 (119903)]
12
times120587minus1
Γ (1 (119902 minus 1) minus 32)119878 (0) 119866
31
13
times [
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)(119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
(45)
If we divide the ldquointernal luminosityrdquo 11987112(119877⊙) by the amount
of energy 11986412 then we get the total number of particles per
second11987312liberated in the reaction given by (1) as follows
11987312=11987112(119877)
11986412
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
times int
119877
0
1199032
9848582
(119903)
[119879 (119903)]12
11986631
13
[
[
(119902 minus 1) 1205872
120583
2119896119879 (119903)
times (119885111988521198902
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
]
]
d119903
= 4 (1 minus1
212057512)1198732
119860
11988311198832
11986011198602
[8(119902 minus 1)
120583119896]
12
times1
Γ (1 (119902 minus 1) minus 32)119878 (0)
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)[
(119902 minus 1) 1205872
120583
2119896(11988511198852e2
ℏ)
2
]
minus119904
times int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903 d119904
(46)
For the density distribution defined in (28) introduced byHaubold and Mathai [5 6] and the corresponding temper-ature distribution (36) we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
119888
times int
119877
0
1199032
[1 minus (119903
119877)
120575
]
52minus119904
times [minus1
2(119903
119877)
2
+120575 + 6
(120575 + 2) (120575 + 3)(119903
119877)
120575+2
minus3
2 (120575 + 1) (120575 + 3)(119903
119877)
2120575+2
]
119904minus12
d119903
(47)
where 120585 is as defined in (34) If we put a substitution 119903 = 119909119877then we get
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
times int
1
0
1199092
[1 minus 119909120575
]52minus119904
times [120585 minus1
21199092
+120575 + 6
(120575 + 2) (120575 + 3)119909120575+2
minus3
2(120575 + 1)(120575 + 3)1199092120575+2
]
119904minus12
d119909
(48)
Journal of Astrophysics 7
Putting 119909120575 = 119910 and simplifying we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583120585
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
120575
times int
1
0
1199103120575minus1
[1 minus 119910]52minus119904
[1 minus 119906 (119910)]119904minus12d119910
(49)
where 119906(119910) is defined as
119906 (119910) =1199102120575
120585[1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
]
(50)
As 119910 rarr 0 119906(119910) rarr 0 and 119910 rarr 1 119906(119910) rarr 1 If we take
V (119910) =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
(51)
we have V(0) = 12 The minimum value of V(119910) is at 119910 =
(120575 + 1)(120575 + 6)3(120575 + 2) and the value is 12 minus 16((120575 + 6)2(120575 +1)(120575 + 3)(120575 + 2)
2
) Thus the minimum value is nonnegativesince (120575 + 6)2(120575 + 1) (120575 + 3)(120575 + 2)2 decreases steadily from3 to 1 for all 120575 gt 0 Therefore V(119910) le 0 Since 120585 gt 0 for all120575 gt 0 119906(119910) le 0 for all 120575 gt 0 120585 gt 0 Thus [1 minus 119906(119910)]119904minus12 le 0Hence 0 lt 119906(119910) lt 1 for 0 lt 119910 lt 1 and for 120575 gt 0 Thus byusing the binomial expansion we obtain
[1 minus 119906 (119910)]119904minus12
=
infin
sum
119898=0
(12 minus 119904)119898
119898[119906 (119910)]
119898
(52)
where (12 minus 119904)119898is the Pochhammer symbol defined for 119886 isin
C by
(119886)0= 1
(119886)119898=
119886 (119886 + 1) sdot sdot sdot (119886 + 119898 minus 1)
Γ (119886 + 119898)
Γ (119886)
119898 = 1 2 119886 = 0
(53)
whenever Γ(119886) exists Taking 120575 = 2 we get 120585 = 15 and
[1 minus 119906 (119910)]119904minus12
= (1 minus 119910)2119904minus1
(1 minus1
2119910)
119904minus12
(54)
Then from (49) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times int
1
0
11991032minus1
[1 minus 119910]52+119904minus1
(1 minus1
2119910)
119904minus12
d119910
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
(12 minus 119904)119898
119898
1
2119898
times int
1
0
11991032+119898minus1
[1 minus 119910]52+119904minus1d119910
(55)
By using beta integral and using (53) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
Γ (12 minus 119904 + 119898)
119898Γ (12 minus 119904)
1
2119898
Γ (32 + 119898) Γ (52 + 119904)
Γ (4 + 119904 + 119898)
(56)
Now from (46) we obtain the total number of particles persecond liberated in the reaction (1) as follows
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904) Γ (
5
2+ 119904)
timesΓ (1 (119902 minus 1) minus 1 minus 119904) Γ (12 + 119898 minus 119904)
Γ (12 minus 119904) Γ (4 + 119898 + 119904)
times [15120587 (119902 minus 1)119873
119860
81198669848581198881198772
(11988511198852e2
ℏ)
2
]
minus119904
d119904
8 Journal of Astrophysics
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
times 11986642
35
[
[
15120587 (119902 minus 1)119873119860
81198669848581198881198772
times (11988511198852e2
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)12minus1198984+119898
01215212
]
]
(57)
For more details on119866-function and its properties see [2526] Thus we have obtained the the total number of particlesper second liberated in the reaction (1) in terms of the densitydistribution considered by Haubold and Mathai [6 29]
5 Comparison of Pathway Energy Density andthe Maxwell-Boltzmann Energy Density
In Figure 1 it can be obtserved that for the nuclei to reactat energy 119864 they have to borrow an energy 119864 from thethermal environment The probability of such an energy isproportional to theMaxwell-Boltzmann energy exp[minus119864119896119879]The fusion will take place when the nuclei penerate theCoulomb barrier keeping them apart The probability ofpenetration is given by the factor exp[minus(119864
119866119864)12
] Theproduct of these two factors illustrates that fusion mostlyoccurs in the energy window given in the figure
In Figure 2 the pathway energy density is plotted for 119902 =07 09 1 12 14 respectively For different values of 119902we getdifferent energy densities (curves (a) (b) (c) (d) and (e))The nonresonant cross section is also plotted The product ofthe pathway energy density and the nonresonant cross sectionfor different values of 119902 namely 119902 = 07 09 1 12 14 is alsoplotted It is to be noted that as 119902 rarr 1 the pathway energydensity coincides with the Maxwell-Boltzmann energy den-sity and also the fusion window for the Maxwell-Boltzmanncase in Figure 1
The curves in the figure represent pathway density[1 + (119902 minus 1)(119864119896119879)]
minus1(119902minus1) for (a) 119902 = 07 (b) 119902 = 09(c) 119902 = 1 (d) 119902 = 12 and (e) 119902 = 14 (f) repre-sents the nonresonant cross section exp[minus(119864
119866119864)12
] Theproduct [1 + (119902 minus 1)(119864119896119879)]minus1(119902minus1) exp[minus(119864
119866119864)12
] is alsorepresented for (g) 119902 = 07 (h) 119902 = 09 (i) 119902 = 1 (j) 119902 = 12and (k) 119902 = 14
Figure 3(a) shows the pathway energy density defined in(13) for 119902 = 1 12 13 14 and Figure 3(b) shows theMaxwell-Boltzmann energy density defined in (3) As 119902 rarr 1 thepathway energy density reduces to the Maxwell-Boltzmannenergy density The pathway energy density covers manystable and unstable situations as the value of 119902 varies If
Fusion window
0 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
2kT 4kT 6kT 8kT
prop exp[minus E
kT]
prop exp[minus (EGE
)12
]
prop exp[minus E
kTminus (EG
E)12
]
Energy E
Figure 1 Schematic plot of the enrgy-dependent factors for thereaction rate probability integral the Maxwell-Boltzmann energydensity nonresonant nuclear cross section and the product of theMaxwell-Boltzmann density and the nonresonant cross section
Fusion window
0 2kT 4kT 6kT 8kT 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
1 + (q minus 1)E
kT]minus1qminus1
(a)
(b)(c)
(d)(e)
(h)
(i)
(k)
(f)
(j)
(g)
prop[prop exp[minus (EG
E)12
]
Energy E
Figure 2 Schematic plot of the energy-dependent factors for theextended reaction rate probability integral pathway energy densitynonresonant nuclear cross section and the product of the pathwaydensity and the nonresonant cross section
the Maxwell-Boltzmann density is the equilibrium situationmany other nonequilibrium situations are covered by thepathway energy density
Journal of Astrophysics 9
Pathway energy density for k = 1 T = 100K
0004
0003
0002
0001
0
q = 1
q = 12
q = 13
q = 14
0 200 400 600 800 1000
fPD
Pathway energy density for k = 1 T = 100K
q = 1
q = 12
q = 13
q = 14
Energy E
(a)
0004
0003
0002
0001
0
fM
BD
Maxwell-Boltzmann energy densityfor k = 1 T = 100K
0 200 400 600 800 1000
Energy E
(b)
Figure 3 (a) Pathway energy density for 119902 = 1 12 13 14 and for 119896 = 1 119879 = 100119870 (b) The Maxwell-Boltzmann energy density for119896 = 1 119879 = 100119870
6 Concluding Remarks
In this paper we have modified the energy distributionfor a nonresonant reaction rate probability integral Thecomposition parameter C considered by [9] is extended toClowast by the pathway energy density Considering the analyticdensity distributions developed by Haubold and Mathai [629] they are used to obtain the stellar luminosity and the neu-trino emission rates and are obtained in generalized specialfunctions such as Meijerrsquos 119866-function The pathway energydensity considered here covers many density functions andhence the extended reaction rate integral covers a wider classof integral Pathway energy density helps us to obtain variousfusion windows by giving different values to 119902 the pathwayparameter which in turn leads to a new opening in the fusionresearch The graphs plotted here are by using Maple 14 inWindows XP platform
Appendix
Series Representation
The series representation for the right-hand side of (19) canbe obtained through the following procedure Here we applyresidue calculus on the119866-function given in (19) Consider the119866-function as follows
11986631
13((119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
)
=1
2120587119894int
119888+119894infin
119888minus119894infin
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1) 119864119866
4119896119879)
minus119904
d119904
(A1)
The right-hand side is the sum of the residues of the inte-grandThe poles of the gammas in the integral representationin (19) are as follows
poles of Γ(119904) 119904 = 0 minus1 minus2
poles of Γ(12 + 119904) 119904 = minus12 minus32 minus52
poles of Γ(1 + 119904) 119904 = minus1 minus2 minus3
Here the poles of Γ(119904) and Γ(1 + 119904) will coincide with eachother at all points except at 119904 = 0 Note that the pole 119904 = 0 isa pole of order 1 119904 = minus12 minus32 minus52 are each of order 1and 119904 = minus1 minus2 minus3 are each of order 2 We know that
lim119904rarrminus119903
(119904 + 119903) Γ (119904) =(minus1)119903
119903
Γ (119886 minus 119903) =(minus1)119903
Γ (119886)
(1 minus 119886)119903
Γ (119886 + 119898) = Γ (119886) (119886)119898
(A2)
10 Journal of Astrophysics
when Γ(119886) is defined 119903 = 0 1 2 Γ(12) = 12058712
(119886)119903=
119886 (119886 + 1) sdot sdot sdot (119886 + 119903 minus 1) if 119903 ge 1 119886 = 0
1 if 119903 = 0(A3)
The sum of the residues corresponding to the poles 119904 = 0 isgiven by
1198771= radic120587Γ(
1
119902 minus 1minus 1) (A4)
The sum of the residues corresponding to the poles 119904 =
minus12 minus32 minus52 is
1198772=
infin
sum
119903=0
(minus1)119903
119903Γ (minus
1
2minus 119903) Γ (
1
2minus 119903)
times Γ(1
119902 minus 1minus1
2+ 119903) [
(119902 minus 1)119864119866
4119896119879]
minus12+119903
= minus2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1) 119864
119866
4119896119879)
(A5)
where11198652is the hypergeometric function defined by
11198652(119886 119887 119888 119909) =
infin
sum
119903=0
(119886)119903
(119887)119903(119888)119903
119909119903
119903 (A6)
To obtain the sum of the residues corresponding to poles119904 = minus1 minus2 minus3 of order 2 we proceed as follows
1198773
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[(119904 + 1 + 119903)
2
Γ (1 + 119904) Γ (119904) Γ (1
2+ 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1)119864119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[ (Γ2
(2 + 119904 + 119903) Γ (12 + 119904)
times Γ (1 (119902 minus 1) minus 1 minus 119904))
times ((119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904)minus1
times ((119902 minus 1) 119864
119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904Φ (119904)
(A7)
where
Φ (119904) =Γ2
(2 + 119904 + 119903) Γ (12 + 119904) Γ (1 (119902 minus 1) minus 1 minus 119904)
(119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904
times ((119902 minus 1)119864
119866
4119896119879)
minus119904
(A8)
We have
120597
120597119904Φ (119904) = Φ (119904)
120597
120597119904[ln [Φ (119904)]]
lnΦ (119904) = 2 ln [Γ (2 + 119904 + 119903)] + ln [Γ (12+ 119904)]
+ ln [Γ( 1
119902 minus 1minus 1 minus 119904)] minus 119904 ln(
(119902 minus 1) 119864119866
4119896119879)
minus 2 ln (119904 + 119903) minus 2 ln (119904 + 119903 minus 1) minus sdot sdot sdot
minus 2 ln (119904 + 1) minus ln (119904)
120597
120597119904[ln [Φ (119904)]] = 2Ψ (2 + 119904 + 119903) + Ψ(
1
2+ 119904)
+ Ψ(1
119902 minus 1minus 1 minus 119904)
minus ln((119902 minus 1) 119864
119866
4119896119879) minus
2
119904 + 119903
minus2
119904 + 119903 minus 1minus sdot sdot sdot minus
2
119904 + 1minus1
119904
lim119904rarrminus1minus119903
120597
120597119904ln [Φ (119904)] = Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
minus ln((119902 minus 1) 119864
119866
4119896119879)
(A9)
whereΨ(119911) is a Psi function or digamma function (seeMathai[26]) and Ψ(1) = minus120574 120574 = 05772156649 is Eulerrsquosconstant Now
lim119904rarrminus1minus119903
Φ (119904) =(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
times ((119902 minus 1) 119864
119866
4119896119879)
1+119903
(A10)
Journal of Astrophysics 11
Then by using (A7) (A9) and (A10) we get
1198773=
infin
sum
119903=0
(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
((119902 minus 1) 119864
119866
4119896119879)
1+119903
times [Ψ(minus1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903) + Ψ (1 + 119903)
+Ψ (2 + 119903) minus ln((119902 minus 1) 119864
119866
4119896119879)]
= (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(A11)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(A12)
Thus from (A4) (A5) and (A11) we get (20)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the Department of Sci-ence and Technology Government of India New Delhifor the financial assistance for this work under Project noSRS4MS28705 and the Centre for Mathematical Sciencesfor providing all facilities
References
[1] R Davis Jr ldquoNobel Lecture a half-century with solar neutri-nosrdquo Reviews of Modern Physics vol 75 no 3 pp 985ndash9942003
[2] H J Haubold and D Kumar ldquoExtension of thermonuclearfunctions through the pathway model including Maxwell-Boltzmann and Tsallis distributionsrdquo Astroparticle Physics vol29 no 1 pp 70ndash76 2008
[3] H J Haubold and R W John ldquoOn the evaluation of an integralconnected with the thermonuclear reaction rate in closed-formrdquo Astronomische Nachrichten vol 299 no 5 pp 225ndash2321978
[4] W A Fowler G R Caughlan and B A Zimmerman ldquoTher-monuclear rection ratesrdquo Annual Review of Astronomy andAstrophysics vol 5 pp 525ndash570 1967
[5] A M Mathai and H J Haubold Modern Problems in Nuclearand Neutrino Astrophysics Akademie Berlin Germany 1988
[6] H J Haubold and A M Mathai ldquoAnalytic representations ofmodified non-resonant thermonuclear reaction ratesrdquo Journalof Applied Mathematics and Physics vol 37 no 5 pp 685ndash6951986
[7] W A Fowler ldquoExperimental and theoretical nuclear astro-physics the quest for the origin of the elementsrdquo Reviews ofModern Physics vol 56 no 2 pp 149ndash179 1984
[8] A C Phillips The Physics of Stars John Wiley amp SonsChichester UK 2nd edition 1999
[9] F C Adams ldquoStars in other universes stellar structure withdifferent fundamental constantsrdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 8 article 10 2008
[10] M Coraddu G Kaniadakis A Lavagno M Lissia G Mezzo-rani and P Quarati ldquoThermal distributions in stellar plasmasnuclear reactions and solar neutrinosrdquo Brazilian Journal ofPhysics vol 29 no 1 pp 153ndash168 1999
[11] H J Haubold and A M Mathai ldquoOn nuclear reaction ratetheoryrdquo Annalen der Physik vol 41 pp 380ndash396 1984
[12] M Coraddu M Lissia G Mezzorani and P Quarati ldquoSuper-Kamiokande hep neutrino best fit a possible signal of non-Maxwellian solar plasmardquo Physica A Statistical Mechanics andIts Applications vol 326 no 3-4 pp 473ndash481 2003
[13] A Lavagno and P Quarati ldquoClassical and quantum non-extensive statistics effects in nuclear many-body problemsrdquoChaos Solitons and Fractals vol 13 no 3 pp 569ndash580 2002
[14] A Lavagno and P Quarati ldquoMetastability of electron-nuclearastrophysical plasmas motivations signals and conditionsrdquoAstrophysics and Space Science vol 305 no 3 pp 253ndash2592006
[15] M Lissia and P Quarati ldquoNuclear astrophysical plasmas iondistribution functions and fusion ratesrdquo Europhysics News vol36 no 6 pp 211ndash214 2005
[16] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988
[17] C Tsallis Introduction to Non-Extensive Statistical MechanicsSpringer New York NY USA 2009
[18] M Gell-Mann and C Tsallis Eds Nonextensive EntropyInterdisciplinary Applications Oxford University Press NewYork NY USA 2004
[19] A M Mathai and H J Haubold ldquoPathway model superstatis-tics Tsallis statistics and a generalized measure of entropyrdquoPhysica A StatisticalMechanics and Its Applications vol 375 no1 pp 110ndash122 2007
[20] R K Saxena A M Mathai and H J Haubold ldquoAstrophysicalthermonuclear functions for Boltzmann-Gibbs statistics andTsallis statisticsrdquo Physica A Statistical Mechanics and Its Appli-cations vol 344 no 3-4 pp 649ndash656 2004
[21] A MMathai ldquoA pathway tomatrix-variate gamma and normaldensitiesrdquo Linear Algebra and Its Applications vol 396 no 1ndash3pp 317ndash328 2005
[22] A M Mathai and H J Haubold ldquoOn generalized distributionsand pathwaysrdquoPhysics Letters A General Atomic and Solid StatePhysics vol 372 no 12 pp 2109ndash2113 2008
[23] H J Haubold andDKumar ldquoFusion yield Guderleymodel andTsallis statisticsrdquo Journal of Plasma Physics vol 77 no 1 pp 1ndash14 2011
[24] D Kumar and H J Haubold ldquoOn extended thermonuclearfunctions through pathwaymodelrdquoAdvances in Space Researchvol 45 no 5 pp 698ndash708 2010
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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ThermodynamicsJournal of
Journal of Astrophysics 7
Putting 119909120575 = 119910 and simplifying we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583120585
3119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
120575
times int
1
0
1199103120575minus1
[1 minus 119910]52minus119904
[1 minus 119906 (119910)]119904minus12d119910
(49)
where 119906(119910) is defined as
119906 (119910) =1199102120575
120585[1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
]
(50)
As 119910 rarr 0 119906(119910) rarr 0 and 119910 rarr 1 119906(119910) rarr 1 If we take
V (119910) =1
2minus
120575 + 6
(120575 + 2) (120575 + 3)119910 +
3
2 (120575 + 1) (120575 + 3)1199102
(51)
we have V(0) = 12 The minimum value of V(119910) is at 119910 =
(120575 + 1)(120575 + 6)3(120575 + 2) and the value is 12 minus 16((120575 + 6)2(120575 +1)(120575 + 3)(120575 + 2)
2
) Thus the minimum value is nonnegativesince (120575 + 6)2(120575 + 1) (120575 + 3)(120575 + 2)2 decreases steadily from3 to 1 for all 120575 gt 0 Therefore V(119910) le 0 Since 120585 gt 0 for all120575 gt 0 119906(119910) le 0 for all 120575 gt 0 120585 gt 0 Thus [1 minus 119906(119910)]119904minus12 le 0Hence 0 lt 119906(119910) lt 1 for 0 lt 119910 lt 1 and for 120575 gt 0 Thus byusing the binomial expansion we obtain
[1 minus 119906 (119910)]119904minus12
=
infin
sum
119898=0
(12 minus 119904)119898
119898[119906 (119910)]
119898
(52)
where (12 minus 119904)119898is the Pochhammer symbol defined for 119886 isin
C by
(119886)0= 1
(119886)119898=
119886 (119886 + 1) sdot sdot sdot (119886 + 119898 minus 1)
Γ (119886 + 119898)
Γ (119886)
119898 = 1 2 119886 = 0
(53)
whenever Γ(119886) exists Taking 120575 = 2 we get 120585 = 15 and
[1 minus 119906 (119910)]119904minus12
= (1 minus 119910)2119904minus1
(1 minus1
2119910)
119904minus12
(54)
Then from (49) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times int
1
0
11991032minus1
[1 minus 119910]52+119904minus1
(1 minus1
2119910)
119904minus12
d119910
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
(12 minus 119904)119898
119898
1
2119898
times int
1
0
11991032+119898minus1
[1 minus 119910]52+119904minus1d119910
(55)
By using beta integral and using (53) we obtain
int
119877
0
1199032
9848582
(119903) [119879 (119903)]minus12+119904d119903
= [4120587119866120583
15119896119873119860
9848581198881198772
]
119904minus12
9848582
1198881198773
2
times
infin
sum
119898=0
Γ (12 minus 119904 + 119898)
119898Γ (12 minus 119904)
1
2119898
Γ (32 + 119898) Γ (52 + 119904)
Γ (4 + 119904 + 119898)
(56)
Now from (46) we obtain the total number of particles persecond liberated in the reaction (1) as follows
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
1
2120587119894
times int
119871
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904) Γ (
5
2+ 119904)
timesΓ (1 (119902 minus 1) minus 1 minus 119904) Γ (12 + 119898 minus 119904)
Γ (12 minus 119904) Γ (4 + 119898 + 119904)
times [15120587 (119902 minus 1)119873
119860
81198669848581198881198772
(11988511198852e2
ℏ)
2
]
minus119904
d119904
8 Journal of Astrophysics
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
times 11986642
35
[
[
15120587 (119902 minus 1)119873119860
81198669848581198881198772
times (11988511198852e2
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)12minus1198984+119898
01215212
]
]
(57)
For more details on119866-function and its properties see [2526] Thus we have obtained the the total number of particlesper second liberated in the reaction (1) in terms of the densitydistribution considered by Haubold and Mathai [6 29]
5 Comparison of Pathway Energy Density andthe Maxwell-Boltzmann Energy Density
In Figure 1 it can be obtserved that for the nuclei to reactat energy 119864 they have to borrow an energy 119864 from thethermal environment The probability of such an energy isproportional to theMaxwell-Boltzmann energy exp[minus119864119896119879]The fusion will take place when the nuclei penerate theCoulomb barrier keeping them apart The probability ofpenetration is given by the factor exp[minus(119864
119866119864)12
] Theproduct of these two factors illustrates that fusion mostlyoccurs in the energy window given in the figure
In Figure 2 the pathway energy density is plotted for 119902 =07 09 1 12 14 respectively For different values of 119902we getdifferent energy densities (curves (a) (b) (c) (d) and (e))The nonresonant cross section is also plotted The product ofthe pathway energy density and the nonresonant cross sectionfor different values of 119902 namely 119902 = 07 09 1 12 14 is alsoplotted It is to be noted that as 119902 rarr 1 the pathway energydensity coincides with the Maxwell-Boltzmann energy den-sity and also the fusion window for the Maxwell-Boltzmanncase in Figure 1
The curves in the figure represent pathway density[1 + (119902 minus 1)(119864119896119879)]
minus1(119902minus1) for (a) 119902 = 07 (b) 119902 = 09(c) 119902 = 1 (d) 119902 = 12 and (e) 119902 = 14 (f) repre-sents the nonresonant cross section exp[minus(119864
119866119864)12
] Theproduct [1 + (119902 minus 1)(119864119896119879)]minus1(119902minus1) exp[minus(119864
119866119864)12
] is alsorepresented for (g) 119902 = 07 (h) 119902 = 09 (i) 119902 = 1 (j) 119902 = 12and (k) 119902 = 14
Figure 3(a) shows the pathway energy density defined in(13) for 119902 = 1 12 13 14 and Figure 3(b) shows theMaxwell-Boltzmann energy density defined in (3) As 119902 rarr 1 thepathway energy density reduces to the Maxwell-Boltzmannenergy density The pathway energy density covers manystable and unstable situations as the value of 119902 varies If
Fusion window
0 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
2kT 4kT 6kT 8kT
prop exp[minus E
kT]
prop exp[minus (EGE
)12
]
prop exp[minus E
kTminus (EG
E)12
]
Energy E
Figure 1 Schematic plot of the enrgy-dependent factors for thereaction rate probability integral the Maxwell-Boltzmann energydensity nonresonant nuclear cross section and the product of theMaxwell-Boltzmann density and the nonresonant cross section
Fusion window
0 2kT 4kT 6kT 8kT 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
1 + (q minus 1)E
kT]minus1qminus1
(a)
(b)(c)
(d)(e)
(h)
(i)
(k)
(f)
(j)
(g)
prop[prop exp[minus (EG
E)12
]
Energy E
Figure 2 Schematic plot of the energy-dependent factors for theextended reaction rate probability integral pathway energy densitynonresonant nuclear cross section and the product of the pathwaydensity and the nonresonant cross section
the Maxwell-Boltzmann density is the equilibrium situationmany other nonequilibrium situations are covered by thepathway energy density
Journal of Astrophysics 9
Pathway energy density for k = 1 T = 100K
0004
0003
0002
0001
0
q = 1
q = 12
q = 13
q = 14
0 200 400 600 800 1000
fPD
Pathway energy density for k = 1 T = 100K
q = 1
q = 12
q = 13
q = 14
Energy E
(a)
0004
0003
0002
0001
0
fM
BD
Maxwell-Boltzmann energy densityfor k = 1 T = 100K
0 200 400 600 800 1000
Energy E
(b)
Figure 3 (a) Pathway energy density for 119902 = 1 12 13 14 and for 119896 = 1 119879 = 100119870 (b) The Maxwell-Boltzmann energy density for119896 = 1 119879 = 100119870
6 Concluding Remarks
In this paper we have modified the energy distributionfor a nonresonant reaction rate probability integral Thecomposition parameter C considered by [9] is extended toClowast by the pathway energy density Considering the analyticdensity distributions developed by Haubold and Mathai [629] they are used to obtain the stellar luminosity and the neu-trino emission rates and are obtained in generalized specialfunctions such as Meijerrsquos 119866-function The pathway energydensity considered here covers many density functions andhence the extended reaction rate integral covers a wider classof integral Pathway energy density helps us to obtain variousfusion windows by giving different values to 119902 the pathwayparameter which in turn leads to a new opening in the fusionresearch The graphs plotted here are by using Maple 14 inWindows XP platform
Appendix
Series Representation
The series representation for the right-hand side of (19) canbe obtained through the following procedure Here we applyresidue calculus on the119866-function given in (19) Consider the119866-function as follows
11986631
13((119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
)
=1
2120587119894int
119888+119894infin
119888minus119894infin
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1) 119864119866
4119896119879)
minus119904
d119904
(A1)
The right-hand side is the sum of the residues of the inte-grandThe poles of the gammas in the integral representationin (19) are as follows
poles of Γ(119904) 119904 = 0 minus1 minus2
poles of Γ(12 + 119904) 119904 = minus12 minus32 minus52
poles of Γ(1 + 119904) 119904 = minus1 minus2 minus3
Here the poles of Γ(119904) and Γ(1 + 119904) will coincide with eachother at all points except at 119904 = 0 Note that the pole 119904 = 0 isa pole of order 1 119904 = minus12 minus32 minus52 are each of order 1and 119904 = minus1 minus2 minus3 are each of order 2 We know that
lim119904rarrminus119903
(119904 + 119903) Γ (119904) =(minus1)119903
119903
Γ (119886 minus 119903) =(minus1)119903
Γ (119886)
(1 minus 119886)119903
Γ (119886 + 119898) = Γ (119886) (119886)119898
(A2)
10 Journal of Astrophysics
when Γ(119886) is defined 119903 = 0 1 2 Γ(12) = 12058712
(119886)119903=
119886 (119886 + 1) sdot sdot sdot (119886 + 119903 minus 1) if 119903 ge 1 119886 = 0
1 if 119903 = 0(A3)
The sum of the residues corresponding to the poles 119904 = 0 isgiven by
1198771= radic120587Γ(
1
119902 minus 1minus 1) (A4)
The sum of the residues corresponding to the poles 119904 =
minus12 minus32 minus52 is
1198772=
infin
sum
119903=0
(minus1)119903
119903Γ (minus
1
2minus 119903) Γ (
1
2minus 119903)
times Γ(1
119902 minus 1minus1
2+ 119903) [
(119902 minus 1)119864119866
4119896119879]
minus12+119903
= minus2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1) 119864
119866
4119896119879)
(A5)
where11198652is the hypergeometric function defined by
11198652(119886 119887 119888 119909) =
infin
sum
119903=0
(119886)119903
(119887)119903(119888)119903
119909119903
119903 (A6)
To obtain the sum of the residues corresponding to poles119904 = minus1 minus2 minus3 of order 2 we proceed as follows
1198773
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[(119904 + 1 + 119903)
2
Γ (1 + 119904) Γ (119904) Γ (1
2+ 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1)119864119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[ (Γ2
(2 + 119904 + 119903) Γ (12 + 119904)
times Γ (1 (119902 minus 1) minus 1 minus 119904))
times ((119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904)minus1
times ((119902 minus 1) 119864
119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904Φ (119904)
(A7)
where
Φ (119904) =Γ2
(2 + 119904 + 119903) Γ (12 + 119904) Γ (1 (119902 minus 1) minus 1 minus 119904)
(119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904
times ((119902 minus 1)119864
119866
4119896119879)
minus119904
(A8)
We have
120597
120597119904Φ (119904) = Φ (119904)
120597
120597119904[ln [Φ (119904)]]
lnΦ (119904) = 2 ln [Γ (2 + 119904 + 119903)] + ln [Γ (12+ 119904)]
+ ln [Γ( 1
119902 minus 1minus 1 minus 119904)] minus 119904 ln(
(119902 minus 1) 119864119866
4119896119879)
minus 2 ln (119904 + 119903) minus 2 ln (119904 + 119903 minus 1) minus sdot sdot sdot
minus 2 ln (119904 + 1) minus ln (119904)
120597
120597119904[ln [Φ (119904)]] = 2Ψ (2 + 119904 + 119903) + Ψ(
1
2+ 119904)
+ Ψ(1
119902 minus 1minus 1 minus 119904)
minus ln((119902 minus 1) 119864
119866
4119896119879) minus
2
119904 + 119903
minus2
119904 + 119903 minus 1minus sdot sdot sdot minus
2
119904 + 1minus1
119904
lim119904rarrminus1minus119903
120597
120597119904ln [Φ (119904)] = Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
minus ln((119902 minus 1) 119864
119866
4119896119879)
(A9)
whereΨ(119911) is a Psi function or digamma function (seeMathai[26]) and Ψ(1) = minus120574 120574 = 05772156649 is Eulerrsquosconstant Now
lim119904rarrminus1minus119903
Φ (119904) =(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
times ((119902 minus 1) 119864
119866
4119896119879)
1+119903
(A10)
Journal of Astrophysics 11
Then by using (A7) (A9) and (A10) we get
1198773=
infin
sum
119903=0
(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
((119902 minus 1) 119864
119866
4119896119879)
1+119903
times [Ψ(minus1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903) + Ψ (1 + 119903)
+Ψ (2 + 119903) minus ln((119902 minus 1) 119864
119866
4119896119879)]
= (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(A11)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(A12)
Thus from (A4) (A5) and (A11) we get (20)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the Department of Sci-ence and Technology Government of India New Delhifor the financial assistance for this work under Project noSRS4MS28705 and the Centre for Mathematical Sciencesfor providing all facilities
References
[1] R Davis Jr ldquoNobel Lecture a half-century with solar neutri-nosrdquo Reviews of Modern Physics vol 75 no 3 pp 985ndash9942003
[2] H J Haubold and D Kumar ldquoExtension of thermonuclearfunctions through the pathway model including Maxwell-Boltzmann and Tsallis distributionsrdquo Astroparticle Physics vol29 no 1 pp 70ndash76 2008
[3] H J Haubold and R W John ldquoOn the evaluation of an integralconnected with the thermonuclear reaction rate in closed-formrdquo Astronomische Nachrichten vol 299 no 5 pp 225ndash2321978
[4] W A Fowler G R Caughlan and B A Zimmerman ldquoTher-monuclear rection ratesrdquo Annual Review of Astronomy andAstrophysics vol 5 pp 525ndash570 1967
[5] A M Mathai and H J Haubold Modern Problems in Nuclearand Neutrino Astrophysics Akademie Berlin Germany 1988
[6] H J Haubold and A M Mathai ldquoAnalytic representations ofmodified non-resonant thermonuclear reaction ratesrdquo Journalof Applied Mathematics and Physics vol 37 no 5 pp 685ndash6951986
[7] W A Fowler ldquoExperimental and theoretical nuclear astro-physics the quest for the origin of the elementsrdquo Reviews ofModern Physics vol 56 no 2 pp 149ndash179 1984
[8] A C Phillips The Physics of Stars John Wiley amp SonsChichester UK 2nd edition 1999
[9] F C Adams ldquoStars in other universes stellar structure withdifferent fundamental constantsrdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 8 article 10 2008
[10] M Coraddu G Kaniadakis A Lavagno M Lissia G Mezzo-rani and P Quarati ldquoThermal distributions in stellar plasmasnuclear reactions and solar neutrinosrdquo Brazilian Journal ofPhysics vol 29 no 1 pp 153ndash168 1999
[11] H J Haubold and A M Mathai ldquoOn nuclear reaction ratetheoryrdquo Annalen der Physik vol 41 pp 380ndash396 1984
[12] M Coraddu M Lissia G Mezzorani and P Quarati ldquoSuper-Kamiokande hep neutrino best fit a possible signal of non-Maxwellian solar plasmardquo Physica A Statistical Mechanics andIts Applications vol 326 no 3-4 pp 473ndash481 2003
[13] A Lavagno and P Quarati ldquoClassical and quantum non-extensive statistics effects in nuclear many-body problemsrdquoChaos Solitons and Fractals vol 13 no 3 pp 569ndash580 2002
[14] A Lavagno and P Quarati ldquoMetastability of electron-nuclearastrophysical plasmas motivations signals and conditionsrdquoAstrophysics and Space Science vol 305 no 3 pp 253ndash2592006
[15] M Lissia and P Quarati ldquoNuclear astrophysical plasmas iondistribution functions and fusion ratesrdquo Europhysics News vol36 no 6 pp 211ndash214 2005
[16] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988
[17] C Tsallis Introduction to Non-Extensive Statistical MechanicsSpringer New York NY USA 2009
[18] M Gell-Mann and C Tsallis Eds Nonextensive EntropyInterdisciplinary Applications Oxford University Press NewYork NY USA 2004
[19] A M Mathai and H J Haubold ldquoPathway model superstatis-tics Tsallis statistics and a generalized measure of entropyrdquoPhysica A StatisticalMechanics and Its Applications vol 375 no1 pp 110ndash122 2007
[20] R K Saxena A M Mathai and H J Haubold ldquoAstrophysicalthermonuclear functions for Boltzmann-Gibbs statistics andTsallis statisticsrdquo Physica A Statistical Mechanics and Its Appli-cations vol 344 no 3-4 pp 649ndash656 2004
[21] A MMathai ldquoA pathway tomatrix-variate gamma and normaldensitiesrdquo Linear Algebra and Its Applications vol 396 no 1ndash3pp 317ndash328 2005
[22] A M Mathai and H J Haubold ldquoOn generalized distributionsand pathwaysrdquoPhysics Letters A General Atomic and Solid StatePhysics vol 372 no 12 pp 2109ndash2113 2008
[23] H J Haubold andDKumar ldquoFusion yield Guderleymodel andTsallis statisticsrdquo Journal of Plasma Physics vol 77 no 1 pp 1ndash14 2011
[24] D Kumar and H J Haubold ldquoOn extended thermonuclearfunctions through pathwaymodelrdquoAdvances in Space Researchvol 45 no 5 pp 698ndash708 2010
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
8 Journal of Astrophysics
11987312=21198732
1198609848582
1198881198772
120583(1 minus
1
212057512)11988311198832
11986011198602
[30 (119902 minus 1)119873
119860
120587119866984858119888
]
12
times 119878 (0)1
Γ (1 (119902 minus 1) minus 32)
times
infin
sum
119898=0
1
2119898
Γ (32 + 119898)
119898
times 11986642
35
[
[
15120587 (119902 minus 1)119873119860
81198669848581198881198772
times (11988511198852e2
ℏ)
21003816100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)12minus1198984+119898
01215212
]
]
(57)
For more details on119866-function and its properties see [2526] Thus we have obtained the the total number of particlesper second liberated in the reaction (1) in terms of the densitydistribution considered by Haubold and Mathai [6 29]
5 Comparison of Pathway Energy Density andthe Maxwell-Boltzmann Energy Density
In Figure 1 it can be obtserved that for the nuclei to reactat energy 119864 they have to borrow an energy 119864 from thethermal environment The probability of such an energy isproportional to theMaxwell-Boltzmann energy exp[minus119864119896119879]The fusion will take place when the nuclei penerate theCoulomb barrier keeping them apart The probability ofpenetration is given by the factor exp[minus(119864
119866119864)12
] Theproduct of these two factors illustrates that fusion mostlyoccurs in the energy window given in the figure
In Figure 2 the pathway energy density is plotted for 119902 =07 09 1 12 14 respectively For different values of 119902we getdifferent energy densities (curves (a) (b) (c) (d) and (e))The nonresonant cross section is also plotted The product ofthe pathway energy density and the nonresonant cross sectionfor different values of 119902 namely 119902 = 07 09 1 12 14 is alsoplotted It is to be noted that as 119902 rarr 1 the pathway energydensity coincides with the Maxwell-Boltzmann energy den-sity and also the fusion window for the Maxwell-Boltzmanncase in Figure 1
The curves in the figure represent pathway density[1 + (119902 minus 1)(119864119896119879)]
minus1(119902minus1) for (a) 119902 = 07 (b) 119902 = 09(c) 119902 = 1 (d) 119902 = 12 and (e) 119902 = 14 (f) repre-sents the nonresonant cross section exp[minus(119864
119866119864)12
] Theproduct [1 + (119902 minus 1)(119864119896119879)]minus1(119902minus1) exp[minus(119864
119866119864)12
] is alsorepresented for (g) 119902 = 07 (h) 119902 = 09 (i) 119902 = 1 (j) 119902 = 12and (k) 119902 = 14
Figure 3(a) shows the pathway energy density defined in(13) for 119902 = 1 12 13 14 and Figure 3(b) shows theMaxwell-Boltzmann energy density defined in (3) As 119902 rarr 1 thepathway energy density reduces to the Maxwell-Boltzmannenergy density The pathway energy density covers manystable and unstable situations as the value of 119902 varies If
Fusion window
0 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
2kT 4kT 6kT 8kT
prop exp[minus E
kT]
prop exp[minus (EGE
)12
]
prop exp[minus E
kTminus (EG
E)12
]
Energy E
Figure 1 Schematic plot of the enrgy-dependent factors for thereaction rate probability integral the Maxwell-Boltzmann energydensity nonresonant nuclear cross section and the product of theMaxwell-Boltzmann density and the nonresonant cross section
Fusion window
0 2kT 4kT 6kT 8kT 10
1
08
06
04
02
0
Fusio
n pr
obab
ility
1 + (q minus 1)E
kT]minus1qminus1
(a)
(b)(c)
(d)(e)
(h)
(i)
(k)
(f)
(j)
(g)
prop[prop exp[minus (EG
E)12
]
Energy E
Figure 2 Schematic plot of the energy-dependent factors for theextended reaction rate probability integral pathway energy densitynonresonant nuclear cross section and the product of the pathwaydensity and the nonresonant cross section
the Maxwell-Boltzmann density is the equilibrium situationmany other nonequilibrium situations are covered by thepathway energy density
Journal of Astrophysics 9
Pathway energy density for k = 1 T = 100K
0004
0003
0002
0001
0
q = 1
q = 12
q = 13
q = 14
0 200 400 600 800 1000
fPD
Pathway energy density for k = 1 T = 100K
q = 1
q = 12
q = 13
q = 14
Energy E
(a)
0004
0003
0002
0001
0
fM
BD
Maxwell-Boltzmann energy densityfor k = 1 T = 100K
0 200 400 600 800 1000
Energy E
(b)
Figure 3 (a) Pathway energy density for 119902 = 1 12 13 14 and for 119896 = 1 119879 = 100119870 (b) The Maxwell-Boltzmann energy density for119896 = 1 119879 = 100119870
6 Concluding Remarks
In this paper we have modified the energy distributionfor a nonresonant reaction rate probability integral Thecomposition parameter C considered by [9] is extended toClowast by the pathway energy density Considering the analyticdensity distributions developed by Haubold and Mathai [629] they are used to obtain the stellar luminosity and the neu-trino emission rates and are obtained in generalized specialfunctions such as Meijerrsquos 119866-function The pathway energydensity considered here covers many density functions andhence the extended reaction rate integral covers a wider classof integral Pathway energy density helps us to obtain variousfusion windows by giving different values to 119902 the pathwayparameter which in turn leads to a new opening in the fusionresearch The graphs plotted here are by using Maple 14 inWindows XP platform
Appendix
Series Representation
The series representation for the right-hand side of (19) canbe obtained through the following procedure Here we applyresidue calculus on the119866-function given in (19) Consider the119866-function as follows
11986631
13((119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
)
=1
2120587119894int
119888+119894infin
119888minus119894infin
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1) 119864119866
4119896119879)
minus119904
d119904
(A1)
The right-hand side is the sum of the residues of the inte-grandThe poles of the gammas in the integral representationin (19) are as follows
poles of Γ(119904) 119904 = 0 minus1 minus2
poles of Γ(12 + 119904) 119904 = minus12 minus32 minus52
poles of Γ(1 + 119904) 119904 = minus1 minus2 minus3
Here the poles of Γ(119904) and Γ(1 + 119904) will coincide with eachother at all points except at 119904 = 0 Note that the pole 119904 = 0 isa pole of order 1 119904 = minus12 minus32 minus52 are each of order 1and 119904 = minus1 minus2 minus3 are each of order 2 We know that
lim119904rarrminus119903
(119904 + 119903) Γ (119904) =(minus1)119903
119903
Γ (119886 minus 119903) =(minus1)119903
Γ (119886)
(1 minus 119886)119903
Γ (119886 + 119898) = Γ (119886) (119886)119898
(A2)
10 Journal of Astrophysics
when Γ(119886) is defined 119903 = 0 1 2 Γ(12) = 12058712
(119886)119903=
119886 (119886 + 1) sdot sdot sdot (119886 + 119903 minus 1) if 119903 ge 1 119886 = 0
1 if 119903 = 0(A3)
The sum of the residues corresponding to the poles 119904 = 0 isgiven by
1198771= radic120587Γ(
1
119902 minus 1minus 1) (A4)
The sum of the residues corresponding to the poles 119904 =
minus12 minus32 minus52 is
1198772=
infin
sum
119903=0
(minus1)119903
119903Γ (minus
1
2minus 119903) Γ (
1
2minus 119903)
times Γ(1
119902 minus 1minus1
2+ 119903) [
(119902 minus 1)119864119866
4119896119879]
minus12+119903
= minus2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1) 119864
119866
4119896119879)
(A5)
where11198652is the hypergeometric function defined by
11198652(119886 119887 119888 119909) =
infin
sum
119903=0
(119886)119903
(119887)119903(119888)119903
119909119903
119903 (A6)
To obtain the sum of the residues corresponding to poles119904 = minus1 minus2 minus3 of order 2 we proceed as follows
1198773
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[(119904 + 1 + 119903)
2
Γ (1 + 119904) Γ (119904) Γ (1
2+ 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1)119864119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[ (Γ2
(2 + 119904 + 119903) Γ (12 + 119904)
times Γ (1 (119902 minus 1) minus 1 minus 119904))
times ((119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904)minus1
times ((119902 minus 1) 119864
119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904Φ (119904)
(A7)
where
Φ (119904) =Γ2
(2 + 119904 + 119903) Γ (12 + 119904) Γ (1 (119902 minus 1) minus 1 minus 119904)
(119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904
times ((119902 minus 1)119864
119866
4119896119879)
minus119904
(A8)
We have
120597
120597119904Φ (119904) = Φ (119904)
120597
120597119904[ln [Φ (119904)]]
lnΦ (119904) = 2 ln [Γ (2 + 119904 + 119903)] + ln [Γ (12+ 119904)]
+ ln [Γ( 1
119902 minus 1minus 1 minus 119904)] minus 119904 ln(
(119902 minus 1) 119864119866
4119896119879)
minus 2 ln (119904 + 119903) minus 2 ln (119904 + 119903 minus 1) minus sdot sdot sdot
minus 2 ln (119904 + 1) minus ln (119904)
120597
120597119904[ln [Φ (119904)]] = 2Ψ (2 + 119904 + 119903) + Ψ(
1
2+ 119904)
+ Ψ(1
119902 minus 1minus 1 minus 119904)
minus ln((119902 minus 1) 119864
119866
4119896119879) minus
2
119904 + 119903
minus2
119904 + 119903 minus 1minus sdot sdot sdot minus
2
119904 + 1minus1
119904
lim119904rarrminus1minus119903
120597
120597119904ln [Φ (119904)] = Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
minus ln((119902 minus 1) 119864
119866
4119896119879)
(A9)
whereΨ(119911) is a Psi function or digamma function (seeMathai[26]) and Ψ(1) = minus120574 120574 = 05772156649 is Eulerrsquosconstant Now
lim119904rarrminus1minus119903
Φ (119904) =(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
times ((119902 minus 1) 119864
119866
4119896119879)
1+119903
(A10)
Journal of Astrophysics 11
Then by using (A7) (A9) and (A10) we get
1198773=
infin
sum
119903=0
(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
((119902 minus 1) 119864
119866
4119896119879)
1+119903
times [Ψ(minus1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903) + Ψ (1 + 119903)
+Ψ (2 + 119903) minus ln((119902 minus 1) 119864
119866
4119896119879)]
= (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(A11)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(A12)
Thus from (A4) (A5) and (A11) we get (20)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the Department of Sci-ence and Technology Government of India New Delhifor the financial assistance for this work under Project noSRS4MS28705 and the Centre for Mathematical Sciencesfor providing all facilities
References
[1] R Davis Jr ldquoNobel Lecture a half-century with solar neutri-nosrdquo Reviews of Modern Physics vol 75 no 3 pp 985ndash9942003
[2] H J Haubold and D Kumar ldquoExtension of thermonuclearfunctions through the pathway model including Maxwell-Boltzmann and Tsallis distributionsrdquo Astroparticle Physics vol29 no 1 pp 70ndash76 2008
[3] H J Haubold and R W John ldquoOn the evaluation of an integralconnected with the thermonuclear reaction rate in closed-formrdquo Astronomische Nachrichten vol 299 no 5 pp 225ndash2321978
[4] W A Fowler G R Caughlan and B A Zimmerman ldquoTher-monuclear rection ratesrdquo Annual Review of Astronomy andAstrophysics vol 5 pp 525ndash570 1967
[5] A M Mathai and H J Haubold Modern Problems in Nuclearand Neutrino Astrophysics Akademie Berlin Germany 1988
[6] H J Haubold and A M Mathai ldquoAnalytic representations ofmodified non-resonant thermonuclear reaction ratesrdquo Journalof Applied Mathematics and Physics vol 37 no 5 pp 685ndash6951986
[7] W A Fowler ldquoExperimental and theoretical nuclear astro-physics the quest for the origin of the elementsrdquo Reviews ofModern Physics vol 56 no 2 pp 149ndash179 1984
[8] A C Phillips The Physics of Stars John Wiley amp SonsChichester UK 2nd edition 1999
[9] F C Adams ldquoStars in other universes stellar structure withdifferent fundamental constantsrdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 8 article 10 2008
[10] M Coraddu G Kaniadakis A Lavagno M Lissia G Mezzo-rani and P Quarati ldquoThermal distributions in stellar plasmasnuclear reactions and solar neutrinosrdquo Brazilian Journal ofPhysics vol 29 no 1 pp 153ndash168 1999
[11] H J Haubold and A M Mathai ldquoOn nuclear reaction ratetheoryrdquo Annalen der Physik vol 41 pp 380ndash396 1984
[12] M Coraddu M Lissia G Mezzorani and P Quarati ldquoSuper-Kamiokande hep neutrino best fit a possible signal of non-Maxwellian solar plasmardquo Physica A Statistical Mechanics andIts Applications vol 326 no 3-4 pp 473ndash481 2003
[13] A Lavagno and P Quarati ldquoClassical and quantum non-extensive statistics effects in nuclear many-body problemsrdquoChaos Solitons and Fractals vol 13 no 3 pp 569ndash580 2002
[14] A Lavagno and P Quarati ldquoMetastability of electron-nuclearastrophysical plasmas motivations signals and conditionsrdquoAstrophysics and Space Science vol 305 no 3 pp 253ndash2592006
[15] M Lissia and P Quarati ldquoNuclear astrophysical plasmas iondistribution functions and fusion ratesrdquo Europhysics News vol36 no 6 pp 211ndash214 2005
[16] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988
[17] C Tsallis Introduction to Non-Extensive Statistical MechanicsSpringer New York NY USA 2009
[18] M Gell-Mann and C Tsallis Eds Nonextensive EntropyInterdisciplinary Applications Oxford University Press NewYork NY USA 2004
[19] A M Mathai and H J Haubold ldquoPathway model superstatis-tics Tsallis statistics and a generalized measure of entropyrdquoPhysica A StatisticalMechanics and Its Applications vol 375 no1 pp 110ndash122 2007
[20] R K Saxena A M Mathai and H J Haubold ldquoAstrophysicalthermonuclear functions for Boltzmann-Gibbs statistics andTsallis statisticsrdquo Physica A Statistical Mechanics and Its Appli-cations vol 344 no 3-4 pp 649ndash656 2004
[21] A MMathai ldquoA pathway tomatrix-variate gamma and normaldensitiesrdquo Linear Algebra and Its Applications vol 396 no 1ndash3pp 317ndash328 2005
[22] A M Mathai and H J Haubold ldquoOn generalized distributionsand pathwaysrdquoPhysics Letters A General Atomic and Solid StatePhysics vol 372 no 12 pp 2109ndash2113 2008
[23] H J Haubold andDKumar ldquoFusion yield Guderleymodel andTsallis statisticsrdquo Journal of Plasma Physics vol 77 no 1 pp 1ndash14 2011
[24] D Kumar and H J Haubold ldquoOn extended thermonuclearfunctions through pathwaymodelrdquoAdvances in Space Researchvol 45 no 5 pp 698ndash708 2010
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Journal of Astrophysics 9
Pathway energy density for k = 1 T = 100K
0004
0003
0002
0001
0
q = 1
q = 12
q = 13
q = 14
0 200 400 600 800 1000
fPD
Pathway energy density for k = 1 T = 100K
q = 1
q = 12
q = 13
q = 14
Energy E
(a)
0004
0003
0002
0001
0
fM
BD
Maxwell-Boltzmann energy densityfor k = 1 T = 100K
0 200 400 600 800 1000
Energy E
(b)
Figure 3 (a) Pathway energy density for 119902 = 1 12 13 14 and for 119896 = 1 119879 = 100119870 (b) The Maxwell-Boltzmann energy density for119896 = 1 119879 = 100119870
6 Concluding Remarks
In this paper we have modified the energy distributionfor a nonresonant reaction rate probability integral Thecomposition parameter C considered by [9] is extended toClowast by the pathway energy density Considering the analyticdensity distributions developed by Haubold and Mathai [629] they are used to obtain the stellar luminosity and the neu-trino emission rates and are obtained in generalized specialfunctions such as Meijerrsquos 119866-function The pathway energydensity considered here covers many density functions andhence the extended reaction rate integral covers a wider classof integral Pathway energy density helps us to obtain variousfusion windows by giving different values to 119902 the pathwayparameter which in turn leads to a new opening in the fusionresearch The graphs plotted here are by using Maple 14 inWindows XP platform
Appendix
Series Representation
The series representation for the right-hand side of (19) canbe obtained through the following procedure Here we applyresidue calculus on the119866-function given in (19) Consider the119866-function as follows
11986631
13((119902 minus 1) 119864
119866
4119896119879
100381610038161003816100381610038161003816100381610038161003816
2minus1(119902minus1)
0121
)
=1
2120587119894int
119888+119894infin
119888minus119894infin
Γ (119904) Γ (1
2+ 119904) Γ (1 + 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1) 119864119866
4119896119879)
minus119904
d119904
(A1)
The right-hand side is the sum of the residues of the inte-grandThe poles of the gammas in the integral representationin (19) are as follows
poles of Γ(119904) 119904 = 0 minus1 minus2
poles of Γ(12 + 119904) 119904 = minus12 minus32 minus52
poles of Γ(1 + 119904) 119904 = minus1 minus2 minus3
Here the poles of Γ(119904) and Γ(1 + 119904) will coincide with eachother at all points except at 119904 = 0 Note that the pole 119904 = 0 isa pole of order 1 119904 = minus12 minus32 minus52 are each of order 1and 119904 = minus1 minus2 minus3 are each of order 2 We know that
lim119904rarrminus119903
(119904 + 119903) Γ (119904) =(minus1)119903
119903
Γ (119886 minus 119903) =(minus1)119903
Γ (119886)
(1 minus 119886)119903
Γ (119886 + 119898) = Γ (119886) (119886)119898
(A2)
10 Journal of Astrophysics
when Γ(119886) is defined 119903 = 0 1 2 Γ(12) = 12058712
(119886)119903=
119886 (119886 + 1) sdot sdot sdot (119886 + 119903 minus 1) if 119903 ge 1 119886 = 0
1 if 119903 = 0(A3)
The sum of the residues corresponding to the poles 119904 = 0 isgiven by
1198771= radic120587Γ(
1
119902 minus 1minus 1) (A4)
The sum of the residues corresponding to the poles 119904 =
minus12 minus32 minus52 is
1198772=
infin
sum
119903=0
(minus1)119903
119903Γ (minus
1
2minus 119903) Γ (
1
2minus 119903)
times Γ(1
119902 minus 1minus1
2+ 119903) [
(119902 minus 1)119864119866
4119896119879]
minus12+119903
= minus2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1) 119864
119866
4119896119879)
(A5)
where11198652is the hypergeometric function defined by
11198652(119886 119887 119888 119909) =
infin
sum
119903=0
(119886)119903
(119887)119903(119888)119903
119909119903
119903 (A6)
To obtain the sum of the residues corresponding to poles119904 = minus1 minus2 minus3 of order 2 we proceed as follows
1198773
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[(119904 + 1 + 119903)
2
Γ (1 + 119904) Γ (119904) Γ (1
2+ 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1)119864119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[ (Γ2
(2 + 119904 + 119903) Γ (12 + 119904)
times Γ (1 (119902 minus 1) minus 1 minus 119904))
times ((119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904)minus1
times ((119902 minus 1) 119864
119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904Φ (119904)
(A7)
where
Φ (119904) =Γ2
(2 + 119904 + 119903) Γ (12 + 119904) Γ (1 (119902 minus 1) minus 1 minus 119904)
(119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904
times ((119902 minus 1)119864
119866
4119896119879)
minus119904
(A8)
We have
120597
120597119904Φ (119904) = Φ (119904)
120597
120597119904[ln [Φ (119904)]]
lnΦ (119904) = 2 ln [Γ (2 + 119904 + 119903)] + ln [Γ (12+ 119904)]
+ ln [Γ( 1
119902 minus 1minus 1 minus 119904)] minus 119904 ln(
(119902 minus 1) 119864119866
4119896119879)
minus 2 ln (119904 + 119903) minus 2 ln (119904 + 119903 minus 1) minus sdot sdot sdot
minus 2 ln (119904 + 1) minus ln (119904)
120597
120597119904[ln [Φ (119904)]] = 2Ψ (2 + 119904 + 119903) + Ψ(
1
2+ 119904)
+ Ψ(1
119902 minus 1minus 1 minus 119904)
minus ln((119902 minus 1) 119864
119866
4119896119879) minus
2
119904 + 119903
minus2
119904 + 119903 minus 1minus sdot sdot sdot minus
2
119904 + 1minus1
119904
lim119904rarrminus1minus119903
120597
120597119904ln [Φ (119904)] = Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
minus ln((119902 minus 1) 119864
119866
4119896119879)
(A9)
whereΨ(119911) is a Psi function or digamma function (seeMathai[26]) and Ψ(1) = minus120574 120574 = 05772156649 is Eulerrsquosconstant Now
lim119904rarrminus1minus119903
Φ (119904) =(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
times ((119902 minus 1) 119864
119866
4119896119879)
1+119903
(A10)
Journal of Astrophysics 11
Then by using (A7) (A9) and (A10) we get
1198773=
infin
sum
119903=0
(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
((119902 minus 1) 119864
119866
4119896119879)
1+119903
times [Ψ(minus1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903) + Ψ (1 + 119903)
+Ψ (2 + 119903) minus ln((119902 minus 1) 119864
119866
4119896119879)]
= (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(A11)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(A12)
Thus from (A4) (A5) and (A11) we get (20)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the Department of Sci-ence and Technology Government of India New Delhifor the financial assistance for this work under Project noSRS4MS28705 and the Centre for Mathematical Sciencesfor providing all facilities
References
[1] R Davis Jr ldquoNobel Lecture a half-century with solar neutri-nosrdquo Reviews of Modern Physics vol 75 no 3 pp 985ndash9942003
[2] H J Haubold and D Kumar ldquoExtension of thermonuclearfunctions through the pathway model including Maxwell-Boltzmann and Tsallis distributionsrdquo Astroparticle Physics vol29 no 1 pp 70ndash76 2008
[3] H J Haubold and R W John ldquoOn the evaluation of an integralconnected with the thermonuclear reaction rate in closed-formrdquo Astronomische Nachrichten vol 299 no 5 pp 225ndash2321978
[4] W A Fowler G R Caughlan and B A Zimmerman ldquoTher-monuclear rection ratesrdquo Annual Review of Astronomy andAstrophysics vol 5 pp 525ndash570 1967
[5] A M Mathai and H J Haubold Modern Problems in Nuclearand Neutrino Astrophysics Akademie Berlin Germany 1988
[6] H J Haubold and A M Mathai ldquoAnalytic representations ofmodified non-resonant thermonuclear reaction ratesrdquo Journalof Applied Mathematics and Physics vol 37 no 5 pp 685ndash6951986
[7] W A Fowler ldquoExperimental and theoretical nuclear astro-physics the quest for the origin of the elementsrdquo Reviews ofModern Physics vol 56 no 2 pp 149ndash179 1984
[8] A C Phillips The Physics of Stars John Wiley amp SonsChichester UK 2nd edition 1999
[9] F C Adams ldquoStars in other universes stellar structure withdifferent fundamental constantsrdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 8 article 10 2008
[10] M Coraddu G Kaniadakis A Lavagno M Lissia G Mezzo-rani and P Quarati ldquoThermal distributions in stellar plasmasnuclear reactions and solar neutrinosrdquo Brazilian Journal ofPhysics vol 29 no 1 pp 153ndash168 1999
[11] H J Haubold and A M Mathai ldquoOn nuclear reaction ratetheoryrdquo Annalen der Physik vol 41 pp 380ndash396 1984
[12] M Coraddu M Lissia G Mezzorani and P Quarati ldquoSuper-Kamiokande hep neutrino best fit a possible signal of non-Maxwellian solar plasmardquo Physica A Statistical Mechanics andIts Applications vol 326 no 3-4 pp 473ndash481 2003
[13] A Lavagno and P Quarati ldquoClassical and quantum non-extensive statistics effects in nuclear many-body problemsrdquoChaos Solitons and Fractals vol 13 no 3 pp 569ndash580 2002
[14] A Lavagno and P Quarati ldquoMetastability of electron-nuclearastrophysical plasmas motivations signals and conditionsrdquoAstrophysics and Space Science vol 305 no 3 pp 253ndash2592006
[15] M Lissia and P Quarati ldquoNuclear astrophysical plasmas iondistribution functions and fusion ratesrdquo Europhysics News vol36 no 6 pp 211ndash214 2005
[16] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988
[17] C Tsallis Introduction to Non-Extensive Statistical MechanicsSpringer New York NY USA 2009
[18] M Gell-Mann and C Tsallis Eds Nonextensive EntropyInterdisciplinary Applications Oxford University Press NewYork NY USA 2004
[19] A M Mathai and H J Haubold ldquoPathway model superstatis-tics Tsallis statistics and a generalized measure of entropyrdquoPhysica A StatisticalMechanics and Its Applications vol 375 no1 pp 110ndash122 2007
[20] R K Saxena A M Mathai and H J Haubold ldquoAstrophysicalthermonuclear functions for Boltzmann-Gibbs statistics andTsallis statisticsrdquo Physica A Statistical Mechanics and Its Appli-cations vol 344 no 3-4 pp 649ndash656 2004
[21] A MMathai ldquoA pathway tomatrix-variate gamma and normaldensitiesrdquo Linear Algebra and Its Applications vol 396 no 1ndash3pp 317ndash328 2005
[22] A M Mathai and H J Haubold ldquoOn generalized distributionsand pathwaysrdquoPhysics Letters A General Atomic and Solid StatePhysics vol 372 no 12 pp 2109ndash2113 2008
[23] H J Haubold andDKumar ldquoFusion yield Guderleymodel andTsallis statisticsrdquo Journal of Plasma Physics vol 77 no 1 pp 1ndash14 2011
[24] D Kumar and H J Haubold ldquoOn extended thermonuclearfunctions through pathwaymodelrdquoAdvances in Space Researchvol 45 no 5 pp 698ndash708 2010
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
10 Journal of Astrophysics
when Γ(119886) is defined 119903 = 0 1 2 Γ(12) = 12058712
(119886)119903=
119886 (119886 + 1) sdot sdot sdot (119886 + 119903 minus 1) if 119903 ge 1 119886 = 0
1 if 119903 = 0(A3)
The sum of the residues corresponding to the poles 119904 = 0 isgiven by
1198771= radic120587Γ(
1
119902 minus 1minus 1) (A4)
The sum of the residues corresponding to the poles 119904 =
minus12 minus32 minus52 is
1198772=
infin
sum
119903=0
(minus1)119903
119903Γ (minus
1
2minus 119903) Γ (
1
2minus 119903)
times Γ(1
119902 minus 1minus1
2+ 119903) [
(119902 minus 1)119864119866
4119896119879]
minus12+119903
= minus2120587Γ(1
119902 minus 1minus1
2) [
(119902 minus 1)119864119866
4119896119879]
12
times11198652(
1
119902 minus 1minus1
23
21
2 minus(119902 minus 1) 119864
119866
4119896119879)
(A5)
where11198652is the hypergeometric function defined by
11198652(119886 119887 119888 119909) =
infin
sum
119903=0
(119886)119903
(119887)119903(119888)119903
119909119903
119903 (A6)
To obtain the sum of the residues corresponding to poles119904 = minus1 minus2 minus3 of order 2 we proceed as follows
1198773
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[(119904 + 1 + 119903)
2
Γ (1 + 119904) Γ (119904) Γ (1
2+ 119904)
times Γ(1
119902 minus 1minus 1 minus 119904)(
(119902 minus 1)119864119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904[ (Γ2
(2 + 119904 + 119903) Γ (12 + 119904)
times Γ (1 (119902 minus 1) minus 1 minus 119904))
times ((119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904)minus1
times ((119902 minus 1) 119864
119866
4119896119879)
minus119904
]
=
infin
sum
119903=0
lim119904rarrminus1minus119903
120597
120597119904Φ (119904)
(A7)
where
Φ (119904) =Γ2
(2 + 119904 + 119903) Γ (12 + 119904) Γ (1 (119902 minus 1) minus 1 minus 119904)
(119904 + 119903)2
(119904 + 119903 minus 1)2
sdot sdot sdot (119904 + 1)2
119904
times ((119902 minus 1)119864
119866
4119896119879)
minus119904
(A8)
We have
120597
120597119904Φ (119904) = Φ (119904)
120597
120597119904[ln [Φ (119904)]]
lnΦ (119904) = 2 ln [Γ (2 + 119904 + 119903)] + ln [Γ (12+ 119904)]
+ ln [Γ( 1
119902 minus 1minus 1 minus 119904)] minus 119904 ln(
(119902 minus 1) 119864119866
4119896119879)
minus 2 ln (119904 + 119903) minus 2 ln (119904 + 119903 minus 1) minus sdot sdot sdot
minus 2 ln (119904 + 1) minus ln (119904)
120597
120597119904[ln [Φ (119904)]] = 2Ψ (2 + 119904 + 119903) + Ψ(
1
2+ 119904)
+ Ψ(1
119902 minus 1minus 1 minus 119904)
minus ln((119902 minus 1) 119864
119866
4119896119879) minus
2
119904 + 119903
minus2
119904 + 119903 minus 1minus sdot sdot sdot minus
2
119904 + 1minus1
119904
lim119904rarrminus1minus119903
120597
120597119904ln [Φ (119904)] = Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
minus ln((119902 minus 1) 119864
119866
4119896119879)
(A9)
whereΨ(119911) is a Psi function or digamma function (seeMathai[26]) and Ψ(1) = minus120574 120574 = 05772156649 is Eulerrsquosconstant Now
lim119904rarrminus1minus119903
Φ (119904) =(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
times ((119902 minus 1) 119864
119866
4119896119879)
1+119903
(A10)
Journal of Astrophysics 11
Then by using (A7) (A9) and (A10) we get
1198773=
infin
sum
119903=0
(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
((119902 minus 1) 119864
119866
4119896119879)
1+119903
times [Ψ(minus1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903) + Ψ (1 + 119903)
+Ψ (2 + 119903) minus ln((119902 minus 1) 119864
119866
4119896119879)]
= (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(A11)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(A12)
Thus from (A4) (A5) and (A11) we get (20)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the Department of Sci-ence and Technology Government of India New Delhifor the financial assistance for this work under Project noSRS4MS28705 and the Centre for Mathematical Sciencesfor providing all facilities
References
[1] R Davis Jr ldquoNobel Lecture a half-century with solar neutri-nosrdquo Reviews of Modern Physics vol 75 no 3 pp 985ndash9942003
[2] H J Haubold and D Kumar ldquoExtension of thermonuclearfunctions through the pathway model including Maxwell-Boltzmann and Tsallis distributionsrdquo Astroparticle Physics vol29 no 1 pp 70ndash76 2008
[3] H J Haubold and R W John ldquoOn the evaluation of an integralconnected with the thermonuclear reaction rate in closed-formrdquo Astronomische Nachrichten vol 299 no 5 pp 225ndash2321978
[4] W A Fowler G R Caughlan and B A Zimmerman ldquoTher-monuclear rection ratesrdquo Annual Review of Astronomy andAstrophysics vol 5 pp 525ndash570 1967
[5] A M Mathai and H J Haubold Modern Problems in Nuclearand Neutrino Astrophysics Akademie Berlin Germany 1988
[6] H J Haubold and A M Mathai ldquoAnalytic representations ofmodified non-resonant thermonuclear reaction ratesrdquo Journalof Applied Mathematics and Physics vol 37 no 5 pp 685ndash6951986
[7] W A Fowler ldquoExperimental and theoretical nuclear astro-physics the quest for the origin of the elementsrdquo Reviews ofModern Physics vol 56 no 2 pp 149ndash179 1984
[8] A C Phillips The Physics of Stars John Wiley amp SonsChichester UK 2nd edition 1999
[9] F C Adams ldquoStars in other universes stellar structure withdifferent fundamental constantsrdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 8 article 10 2008
[10] M Coraddu G Kaniadakis A Lavagno M Lissia G Mezzo-rani and P Quarati ldquoThermal distributions in stellar plasmasnuclear reactions and solar neutrinosrdquo Brazilian Journal ofPhysics vol 29 no 1 pp 153ndash168 1999
[11] H J Haubold and A M Mathai ldquoOn nuclear reaction ratetheoryrdquo Annalen der Physik vol 41 pp 380ndash396 1984
[12] M Coraddu M Lissia G Mezzorani and P Quarati ldquoSuper-Kamiokande hep neutrino best fit a possible signal of non-Maxwellian solar plasmardquo Physica A Statistical Mechanics andIts Applications vol 326 no 3-4 pp 473ndash481 2003
[13] A Lavagno and P Quarati ldquoClassical and quantum non-extensive statistics effects in nuclear many-body problemsrdquoChaos Solitons and Fractals vol 13 no 3 pp 569ndash580 2002
[14] A Lavagno and P Quarati ldquoMetastability of electron-nuclearastrophysical plasmas motivations signals and conditionsrdquoAstrophysics and Space Science vol 305 no 3 pp 253ndash2592006
[15] M Lissia and P Quarati ldquoNuclear astrophysical plasmas iondistribution functions and fusion ratesrdquo Europhysics News vol36 no 6 pp 211ndash214 2005
[16] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988
[17] C Tsallis Introduction to Non-Extensive Statistical MechanicsSpringer New York NY USA 2009
[18] M Gell-Mann and C Tsallis Eds Nonextensive EntropyInterdisciplinary Applications Oxford University Press NewYork NY USA 2004
[19] A M Mathai and H J Haubold ldquoPathway model superstatis-tics Tsallis statistics and a generalized measure of entropyrdquoPhysica A StatisticalMechanics and Its Applications vol 375 no1 pp 110ndash122 2007
[20] R K Saxena A M Mathai and H J Haubold ldquoAstrophysicalthermonuclear functions for Boltzmann-Gibbs statistics andTsallis statisticsrdquo Physica A Statistical Mechanics and Its Appli-cations vol 344 no 3-4 pp 649ndash656 2004
[21] A MMathai ldquoA pathway tomatrix-variate gamma and normaldensitiesrdquo Linear Algebra and Its Applications vol 396 no 1ndash3pp 317ndash328 2005
[22] A M Mathai and H J Haubold ldquoOn generalized distributionsand pathwaysrdquoPhysics Letters A General Atomic and Solid StatePhysics vol 372 no 12 pp 2109ndash2113 2008
[23] H J Haubold andDKumar ldquoFusion yield Guderleymodel andTsallis statisticsrdquo Journal of Plasma Physics vol 77 no 1 pp 1ndash14 2011
[24] D Kumar and H J Haubold ldquoOn extended thermonuclearfunctions through pathwaymodelrdquoAdvances in Space Researchvol 45 no 5 pp 698ndash708 2010
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Journal of Astrophysics 11
Then by using (A7) (A9) and (A10) we get
1198773=
infin
sum
119903=0
(minus1)1+119903
2radic120587Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
((119902 minus 1) 119864
119866
4119896119879)
1+119903
times [Ψ(minus1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903) + Ψ (1 + 119903)
+Ψ (2 + 119903) minus ln((119902 minus 1) 119864
119866
4119896119879)]
= (2radic120587 (119902 minus 1) 119864
119866
4119896119879)
times
infin
sum
119903=0
((119902 minus 1) 119864
119866
4119896119879)
119903
[119860119903minus ln(
(119902 minus 1) 119864119866
4119896119879)]119861119903
(A11)
where
119860119903= Ψ(minus
1
2minus 119903) + Ψ(
1
119902 minus 1+ 119903)
+ Ψ (1 + 119903) + Ψ (2 + 119903)
119861119903=(minus1)119903
Γ (1 (119902 minus 1) + 119903)
(32)119903119903 (1 + 119903)
(A12)
Thus from (A4) (A5) and (A11) we get (20)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the Department of Sci-ence and Technology Government of India New Delhifor the financial assistance for this work under Project noSRS4MS28705 and the Centre for Mathematical Sciencesfor providing all facilities
References
[1] R Davis Jr ldquoNobel Lecture a half-century with solar neutri-nosrdquo Reviews of Modern Physics vol 75 no 3 pp 985ndash9942003
[2] H J Haubold and D Kumar ldquoExtension of thermonuclearfunctions through the pathway model including Maxwell-Boltzmann and Tsallis distributionsrdquo Astroparticle Physics vol29 no 1 pp 70ndash76 2008
[3] H J Haubold and R W John ldquoOn the evaluation of an integralconnected with the thermonuclear reaction rate in closed-formrdquo Astronomische Nachrichten vol 299 no 5 pp 225ndash2321978
[4] W A Fowler G R Caughlan and B A Zimmerman ldquoTher-monuclear rection ratesrdquo Annual Review of Astronomy andAstrophysics vol 5 pp 525ndash570 1967
[5] A M Mathai and H J Haubold Modern Problems in Nuclearand Neutrino Astrophysics Akademie Berlin Germany 1988
[6] H J Haubold and A M Mathai ldquoAnalytic representations ofmodified non-resonant thermonuclear reaction ratesrdquo Journalof Applied Mathematics and Physics vol 37 no 5 pp 685ndash6951986
[7] W A Fowler ldquoExperimental and theoretical nuclear astro-physics the quest for the origin of the elementsrdquo Reviews ofModern Physics vol 56 no 2 pp 149ndash179 1984
[8] A C Phillips The Physics of Stars John Wiley amp SonsChichester UK 2nd edition 1999
[9] F C Adams ldquoStars in other universes stellar structure withdifferent fundamental constantsrdquo Journal of Cosmology andAstroparticle Physics vol 2008 no 8 article 10 2008
[10] M Coraddu G Kaniadakis A Lavagno M Lissia G Mezzo-rani and P Quarati ldquoThermal distributions in stellar plasmasnuclear reactions and solar neutrinosrdquo Brazilian Journal ofPhysics vol 29 no 1 pp 153ndash168 1999
[11] H J Haubold and A M Mathai ldquoOn nuclear reaction ratetheoryrdquo Annalen der Physik vol 41 pp 380ndash396 1984
[12] M Coraddu M Lissia G Mezzorani and P Quarati ldquoSuper-Kamiokande hep neutrino best fit a possible signal of non-Maxwellian solar plasmardquo Physica A Statistical Mechanics andIts Applications vol 326 no 3-4 pp 473ndash481 2003
[13] A Lavagno and P Quarati ldquoClassical and quantum non-extensive statistics effects in nuclear many-body problemsrdquoChaos Solitons and Fractals vol 13 no 3 pp 569ndash580 2002
[14] A Lavagno and P Quarati ldquoMetastability of electron-nuclearastrophysical plasmas motivations signals and conditionsrdquoAstrophysics and Space Science vol 305 no 3 pp 253ndash2592006
[15] M Lissia and P Quarati ldquoNuclear astrophysical plasmas iondistribution functions and fusion ratesrdquo Europhysics News vol36 no 6 pp 211ndash214 2005
[16] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988
[17] C Tsallis Introduction to Non-Extensive Statistical MechanicsSpringer New York NY USA 2009
[18] M Gell-Mann and C Tsallis Eds Nonextensive EntropyInterdisciplinary Applications Oxford University Press NewYork NY USA 2004
[19] A M Mathai and H J Haubold ldquoPathway model superstatis-tics Tsallis statistics and a generalized measure of entropyrdquoPhysica A StatisticalMechanics and Its Applications vol 375 no1 pp 110ndash122 2007
[20] R K Saxena A M Mathai and H J Haubold ldquoAstrophysicalthermonuclear functions for Boltzmann-Gibbs statistics andTsallis statisticsrdquo Physica A Statistical Mechanics and Its Appli-cations vol 344 no 3-4 pp 649ndash656 2004
[21] A MMathai ldquoA pathway tomatrix-variate gamma and normaldensitiesrdquo Linear Algebra and Its Applications vol 396 no 1ndash3pp 317ndash328 2005
[22] A M Mathai and H J Haubold ldquoOn generalized distributionsand pathwaysrdquoPhysics Letters A General Atomic and Solid StatePhysics vol 372 no 12 pp 2109ndash2113 2008
[23] H J Haubold andDKumar ldquoFusion yield Guderleymodel andTsallis statisticsrdquo Journal of Plasma Physics vol 77 no 1 pp 1ndash14 2011
[24] D Kumar and H J Haubold ldquoOn extended thermonuclearfunctions through pathwaymodelrdquoAdvances in Space Researchvol 45 no 5 pp 698ndash708 2010
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
12 Journal of Astrophysics
[25] A M Mathai and R K Saxena Generalized HypergeometricFunctions with Applications in Statistics and Physical Sciencesvol 348 of Lecture Notes in Mathematics Springer New YorkNY USA 1973
[26] A M Mathai A Handbook of Generalized Special Functions forStatistics and Physical Sciences Clarendo Press Oxford UK1993
[27] A Erdelyi Asymptotic Expansions Dover New York NY USA1956
[28] F W J Olver Asymptotics and Spcecial Functions AcademicPress New York NY USA 1974
[29] H J Haubold and A M Mathai ldquoAnalytical results connectingstellar structure parameters and neutrino uxesrdquo Annalen derPhysik vol 44 no 2 pp 103ndash116 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
