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Advances in Space Research 49 (2012) 1062–1066

Polytropic star structure analysis under Bonnor–Ebert gassphere astrophysical configuration thorough investigatinganalytical solutions to the related Lane–Emden equation

K. Boubaker a,⇑, A.H. Bhrawy b

a Unite de Physique des dispositifs a Semi-conducteurs, Faculte des Sciences, de Tunis, Universite de Tunis El Manar, 2092 Tunis, Tunisiab Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Received 24 November 2011; received in revised form 10 January 2012; accepted 11 January 2012Available online 20 January 2012

Abstract

In this paper, Bonnor–Ebert gas sphere model of polytropic stars has been investigated through an analytical approach. Two con-firmed and well-established methods have been used: the Enhanced Lagrangian Formulation Method ELFM and the Boubaker Poly-nomials Expansion Scheme BPES. Solutions to the related generalized Lane–Emden equation of the second kind have been expressedand plotted. Results have given evidence to the relevance of the dimensionless Bonnor–Ebert radius, in good agreement with somerecently proposed profiles.� 2012 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Polytropic star; Bonnor–Ebert gas sphere; Lane–Emden equation; Enhanced Lagrangian Formulation Method ELFM; The BoubakerPolynomials Expansion Scheme BPES; Eulerian representation

1. Introduction

Polytropic stars are described by what is now commonlyknown as Bonnor–Ebert gas spheres model (Bonnor, 1956;Ebert, 1955). In this model a star is represented by an iso-thermal gas sphere embedded in a pressurized medium atthe maximum possible mass allowing for hydrostatic equi-librium (Fig. 1). The derivation is based on earlier worksdone by Emden (1907), and hence equation is oftenreferred to as the Lane–Emden equation of the second kind(Momoniat and Harley, 2006; Ramos, 2008; Abbasbandy,2006; Sajid et al., 2007; Liao, 2003). For implementing thismodel (Fig. 1), let us consider Poisson’s equation:

dPðrÞdr¼ �qðrÞG�MðrÞ

r2ð1Þ

and equation for hydrostatic equilibrium given by:

0273-1177/$36.00 � 2012 COSPAR. Published by Elsevier Ltd. All rights rese

doi:10.1016/j.asr.2012.01.007

⇑ Corresponding author. Tel.: +216 71 811418; fax: +216 73 681313.E-mail address: mmbb11112000@yahoo.fr (K. Boubaker).

dMðrÞdr

¼ 4pr2qðrÞ ð2Þ

where P(r) is the pressure at radius r, M(r) is the mass ofthe star at radius r, G is the gravitational constant andq(r) is star r-dependent density. Here the star is assumedto be spherical in shape.

By combining these two equations, it gives:

1

r2

ddr

r2dP ðrÞqðrÞdr

� �¼ �4pGqðrÞ ð3Þ

Eq. (3) could be a linear differential equation if it has not tobe solved both on both P(r) and q(r). The case of polytrop-ic stars gives important simplification to this problem. Infact, polytropes, which are thermodynamic paths similarto adiabatics and isothermals, provide polytropic equationsof state, linking star core pressure P(r) to density q(r):

PðrÞ ¼ W � qðrÞc ð4Þwhere W and c are constants which depend on the thermo-dynamic properties of the isothermal gas sphere. Finally,by setting i.e., c = 1 and introducing the changes:

rved.

Fig. 1. Polytropic star Bonnor–Ebert gas spheres model.

K. Boubaker, A.H. Bhrawy / Advances in Space Research 49 (2012) 1062–1066 1063

yðxÞ ¼ �LnðqðrÞq0Þ

x ¼ r �ffiffiffiffiffiffiffiffiffi4pGq0

W

q0@ ð5Þ

It gives:

y00ðxÞ þ 2

xy0ðxÞ ¼ e�yðxÞ ð6Þ

This equation is a particular case of the Lane–Emden equa-tion of the second kind:

d2ydx2 þ n

xdydxþ f ðyÞ ¼ 0

yð0Þ ¼ 0

y 0ð0Þ ¼ 0

f ðyÞ ¼ e�by

8>>>><>>>>:

ð7Þ

This paper is structured as follows: in the next Section 2, wegive an overview of the Enhanced Lagrangian FormulationMethod ELFM and its relevant properties needed hereaf-ter. Section 3 is devoted to second method: the BoubakerPolynomials Expansion Scheme BPES and its application.Section 4 gathers solutions plots along with comparisonand discussion. Some concluding remarks are given in thefinal Section 5.

2. Enhanced Lagrangian Formulation Method ELFM

2.1. Presentation

In the Eulerian representation of the polytropic starstructure analysis, any specific star constituent dynamic isa function of time t and of its location with respect to afixed point at that instant of time. In this Eulerian repre-sentation, the Bonnor–Ebert mass represents the largestmass that the star can have while still remaining in hydro-static equilibrium and equations governing this hydrostaticequilibrium can be expressed in two-dimensional conserva-tion form as:

y00 ¼ Uðx; y; y0Þ ð8Þ

For solving this equation using the Enhanced LagrangianFormulation Method ELFM, the point type vector fieldis defined:

X ¼ nðx; yÞ @@xþ gðx; tÞ @

@yð9Þ

With, as a first prolongation:

X ½1� ¼ X þ ðgx þ gyy0 � nxy

0 � nyy02Þ @@y

ð10Þ

Lagrangian L(x, y, y0) of Eq. (8) gives the Euler–Lagrange

equivalent equation:

ddx

dL@y 0

� �� dL@y¼ 0 ð11Þ

In this case, X is called: Noether point symmetry generator(Ovsiannikov, 1982; Bluman and Kumei, 1989) corre-sponding to L(x,y,y0) under the condition of existence ofa gauge function B(x,y) which verifies:

@

@xþ y 0

@

@yþ y00

@

@y0þ � � �

� �ðBÞ

¼ X ½1�ðLÞ þ @

@xþ y 0

@

@yþ y 00

@

@y 0þ � � �

� �ðnÞL ð12Þ

A first integral of Eq. (8), associated with the operator X, ishence:

I ¼ nLþ ðg� y0nÞ @L@y� B ð13Þ

If this first integral is associated with the Noether pointsymmetry generator X, it satisfies:

X ½1�I ¼ 0 ð14Þ

The main idea in the Enhanced Lagrangian FormulationMethod ELFM, as a extended form of the already per-formed Variational Iteration Method VIM (Mohyud-Din,2010a,b), is to presume the existence of solutions to Eq.(8) in terms of quadrature. By associating a Noether pointsymmetry generator to the related Lagrangian L(x,y,y0).

2.2. Application of the Enhanced Lagrangian Formulation

Method ELFM

For Eq. (7), the standard Lagrangian is:

L ¼ 1

2xny 02 � xn

Zf ðyÞdy ð15Þ

By substituting this Lagrangian in Eq. (8) and identifyingeach power of y0, one obtains the system:

ny ¼ 0

gy ¼ 12ðnx � nx�1nÞ

xngx ¼ By

�nxn�1nR

f ðyÞdy � xngf ðyÞ � xnnx

Rf ðyÞdy ¼ Bx

8>>><>>>:

ð16Þ

1064 K. Boubaker, A.H. Bhrawy / Advances in Space Research 49 (2012) 1062–1066

This system is consecutively transformed, after straightfor-ward manipulations, to:

n¼ aðxÞg¼¼ 1

2½a0 �nx�1a�yþbðxÞ

B¼ 14xn½a00 �nða=xÞ0

0�y2þb0xnyþ cðxÞ

½�nxn�1a�a0xn�R

f ðyÞdyþ½� 12xna0yþ 1

2nxn�1ay�xnb�f ðyÞ¼

14a000xny2þ 1

2nxn�2a0y2� 1

2nxn�3ay2� 1

4n2y2xn�1ða=xÞ0y2þb0xnyþb0nxn�1yþc0

8>>>>>><>>>>>>:

ð17Þ

2.3. Resolution algorithm

Application of the Enhanced Lagrangian FormulationMethod ELFM starts from assigning the variables in thesystem (18):

n ¼ x

f ðyÞ ¼ aeby ja¼1;b¼1

g ¼ � 2b ¼ �2

B ¼ k

8>>><>>>:

ð18Þ

where k is a constant.

In this case, the second Lie point symmetry generator is:

Y ¼ x ln x@

@x� 2

bð1þ ln xÞ @

@yð19Þ

The obtained solution is finally:

y ¼ x lnux

� �ð20Þ

with:

Zdu

�uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ð1=2Þabu2

p ¼Zðx ln x� xÞdx ð21Þ

Fig. 2. ELFM solution plots for 0 < b < 2.

3. Solution using the Boubaker Polynomials Expansion

Scheme BPES

3.1. Presentation

The Boubaker Polynomials Expansion Scheme BPES(Awojoyogbe and Boubaker; 2008; Belhadj et al.,2009a,b; Tabatabaei et al., 2009; Ghanouchi et al. 2008;Fridjine et al. 2009; Chaouachi et al., 2007; Fridjine andAmlouk, 2009a,b; Ghrib et al., 2008; Guezmir et al.,2009; Oyodum et al., 2009; Slama et al., 2008; 2009a,b;Zhao et al., 2008; Kumar, 2010; Barry and Hennessy,2010; Agida, 2010; Yildirim et al., 2010; Milgram, 2011)is a resolution protocol which has been successfully appliedto several applied-physics and mathematics problems. TheBPES protocol ensures the validity of the related boundaryconditions regardless main equation features. The BPES ismainly based on Boubaker polynomials first derivativesproperties:

PNq¼1

B4qðxÞjx¼0 ¼ �2N–0;

PNq¼1

B4qðxÞjx¼rq¼ 0;

8>>><>>>:

ð22Þ

and:

PNq¼1

dB4qðxÞdx jx¼0 ¼ 0

PNq¼1

dB4qðxÞdx jx¼rq

¼PNq¼1

H q

with : Hn ¼ B04nðrnÞ ¼4rn½2�r2

n��Pn

q¼1

B24qðrnÞ

B4ðnþ1ÞðrnÞ þ 4r3n

0B@

1CA

8>>>>>>>>>>><>>>>>>>>>>>:

ð23Þ

Several solution have been proposed through the BPES inmany fields such as numerical analysis, theoretical physics,mathematical algorithms, heat transfer, homodynamics,material characterization, fuzzy systems modeling andbiology.

3.2. Application

The Boubaker Polynomials Expansion Scheme BPES isapplied to the system (8) through setting the expression:

uðxÞ ¼ 1

2N 0

XN0

k¼1

kk � B4kðxrkÞ ð24Þ

where B4k are the 4k-order Boubaker polynomials, x e

[0, 1] is the normalized variable, rk are B4k minimal positiveroots, N0 is a prefixed integer and kkjk¼1::N0

are unknownpondering real coefficients.

Thanks to the properties expressed by Eqs. (22) and(23), boundary conditions are trivially verified in advanceto resolution process. The system (8) is hence reduced to:

12N0

PN0

k¼1

kk � d2B4kðxrkÞdx2 þ 1

xN 0

PN0

k¼1

kk � dB4kðxrkÞdx ¼

exp � 1N0

PN0

k¼1

kk � B4kðxrkÞ� �

8>>><>>>:

ð25Þ

Fig. 3. BPES solution plots for 0 < b < 2 and different values of N0.

Fig. 4. Polytropic star Bonnor–Ebert gas sphere model density profile.

K. Boubaker, A.H. Bhrawy / Advances in Space Research 49 (2012) 1062–1066 1065

y00ðxÞ þ 2

xy0ðxÞ ¼ e�yðxÞ

The BPES solution is obtained by determining the non-nullset of coefficients ~kkjk¼1...N0

that minimizes the absolutedifference DN0

:

DN0¼ j 1

2N0

PN0

k¼1

~kk�Kk

� �� 1

2N0

PN0

k¼1

~kk�K0k

� �j

with :

Kk ¼ r2k

R 1

0d2B4k

dx2 ðx� rkÞdx

K0k ¼�rk

R 1

02x�

dB4kdx ðx� rkÞ� exp � 1

N0

PN0

k¼1

B4kðx� rkÞ� �� �� �

dx

8>>>>>>>><>>>>>>>>:

ð26Þ

The final solution is hence:

uðxÞ ¼ 1

2N 0

XN0

k¼1

~kk � B4kðxrkÞ ð27Þ

4. Results plots and discussion

Plots of the solutions obtained using the EnhancedLagrangian Formulation Method ELFM are presented inFig. 2 for different values of the parameter b. Fig. 3 pre-sents the solutions obtained using the Boubaker Polynomi-als Expansion Scheme BPES for sampled values of theinteger parameter N0.

While investigating the behaviors of the solutions (Fig. 2and 3), given by the two methods, at the vicinity of the ori-gin, a strict respect of the two initial conditions (y = y0 = 0)could be observed while an exponential parabolic x-depen-dent behavior was recorded for 1 < b < 2 and x > 0.5. Thisresult is confirmed by the results presented in the relevantliterature.

Bonnor–Ebert radius main features, like value, b-depen-dence, and variations, were also concordant with precedentresults, particularly those of detailed by Banerjee and Pud-ritz (2006), Alves et al. (2001), Braithwaite and Spruit(2004) and Andre et al. (2000).

Fig. 3 monitors the effect of increasing values of N0 onthe obtained profile. This figure is an additional proof ofthe rapid convergence of analytical solutions obtainedthrough the Boubaker polynomials expansion scheme(BPES).

Additionally, it had been noticed that the density profilecan be deduced from Eq. (5) using a simple inverse scheme.Fig. 4 monitors the obtained density profile. The presenceof the central core at about 70% from the outer radius,as well as the value of this radius, is in perfect agreementwith the results recorded by Kratter and Matzner (2006).

5. Conclusion

In this work we have tried to give evidence to the effi-ciency of two enhanced protocols: the Enhanced Lagrang-

1066 K. Boubaker, A.H. Bhrawy / Advances in Space Research 49 (2012) 1062–1066

ian Formulation Method ELFM and the Boubaker Poly-nomials expansion Scheme BPES as tools for solvingnon-linear partial differential equations. These schemeshave been successfully applied to the Bonnor–Ebert gassphere model of polytropic stars.

Solutions expressions a graphics have given evidence tothe relevance of density radial profile, dimensionlessBonnor–Ebert radius and cores, whose recorded parameterswere in good agreement with some recently published results.

Obtained results can be a sustainable guide for under-standing star formation/extinction dynamics. Nevertheless,the studied model is seriously contested and hence, manyother modeled stars have to be considered and deeply stud-ied i.e., Protostars, Tauri, Red Giant, White Dwarf, RedDwarf, Neutron and Supergiant Stars.

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