A GAS D Y N A M I C MODEL OF THE ROTATING SOLAR

A T M O S P H E R E

O N K A R N A T H

Department of Mathematics, S.C.P.G. College, Ballia, U.P., India

(Received 7 February, 1989; in revised form 29 May, 1991)

Abstract. A model is presented of a solar atmosphere which is heated by the periodic passage of shock waves. The outer atmosphere rotates and is assumed not to affect the strength of the shock waves. This constant shock strength hypothesis is used as the basis of the model of the outer solar atmosphere. From the model it is concluded that the chromospheric temperature rise and flow Mach number are slightly affected by the rotation of the atmosphere.

1. Introduction

Biermann (1948) and Schwarzschild (1948), Schatzmann (1961), Dubov (1960), Weymann (1960), Osterbrock (1961) and Billings (1963) suggested that the heating of the solar chromosphere is due to the energy dissipation of shock waves. The amount of dissipation produced by shock waves depends critically on their strength. The strength of the shock wave is given by the shock Mach number Ms, which is defined as the ratio of the speed of the shock relative to the gas in front of it to the local speed of sound. Courant and Friedrichs (1948) showed that in case of weak shock waves, the dissipation is proportional to the third power of M 7 2.

Due to unsteady processes at photospheric levels there are shock waves propagating in the solar atmosphere. These shocks probably propagate well out into the corona. Assuming the strength of the shock constant, Bird (1964) constructed a dynamical model of the outer solar atmosphere. The equations for the flow Mach number, which is the ratio of the flow velocity to the local velocity of sound, and the temperature gradient in a quasi-steady rotating solar atmosphere have been derived by Ojha and Onkar (1988).

The aim of this paper is to derive and discuss the variation of the flow Mach number and temperature gradient with distance by taking suitable values of the variables for the solar atmosphere.

2. Fundamental Equation

The equation for the flow Mach number in the case of a dynamic atmosphere with an outward mass motion is given by Ojha and Onkar (1988):

- + . ( 1 )

M dr 1 - ~/M 2 2 T dr R T r 2 R T r

Solar Physics 136: 263-268, 1991. �9 1991 Kluwer Academic Publishers. Printed in Belgium.

264

eel,

ONKAR NATH

o 3 0

~F . , . v ,

.55

.~--M-

.3~.

~ ,

2~

26

24

..22

2':::'

16

14

I.~

I'~'

5

0.026 0.206 0.386 0.566 0.746 0.926 r (distance from the solar surface)

Fig. 1. Variation of the temperature (T) as a function of the dimensionless distance from the solar surface

(r) for three different values of the critical Much number M~:, including the rotation of the atmosphere: [] M c = 5 ; + M c = l O ; ~ M c = 1 5 .

2.2

21

P3

]e.

15

15

].4

12 �9

11

0 . 0 2 6 0 . 2 0 6 0 . 3 8 6 0 .566 0 .746 o .~z6 r (distance from the solar surface)

Fig. 2. Same as Figure 1, but excluding the rotation of the atmosphere.

A G A S D Y N A M I C M O D E L O F T H E R O T A T I N G S O L A R A T M O S P H E R E 265

~2.

A~

O

zP.

I g | - 4 . 4 . . + . . . . ~ "," , , . .7".r162

i t -~--4-+"4"++4-+-+-~-f--v-~-4-4~+q__4_f_~ ,~, - - 4 - + - +

r (distance from the solaT ~t~l"faee)

Fig. 3~ Variation of the ttow Mach number (M) as a function of the dimensionless distance from the solar surface (s) for three different values of the critical Math number M~, including the rotation of the atmosphere:

U ] M c = 5 , + M e = f 0 , ~ ' ~ M c = l S .

.<.,w~

o~

o

22

2'~"

~4

t2

"4%.

0.026 0,206 0,386 0.566 0.746 0.926 r (distance from the solar ~urfaee)

Same as Figure 3, but excluding the rotation of the atrnosphere~ Fig. 4.

266 ONKAR NATH

This equation differs slightly from the equation of Bird (1964) for a non-rotating atmosphere. The variables M, T, 7, go, ro, R, r, COo, and m are flow Mach number, temperature, ratio of specific heats, acceleration due to gravity at to, non-dimensional radius, gas constant, reference radius, rotation, and molecular weight, respectively.

The Mach number at which the coefficient of the square bracketed term in an equation of the form (1) becomes infinite is called the critical Mach number. A physically meaningful solution of the equation may be one of an infinite family in which the Mach number is always less than the critical value. These solutions are called the subcritical solutions. If the term in square brackets vanishes at the critical Mach number, the solution of the equation is known as a super-critical solution. The super-critical solution defines the maximum possible mass flow from the atmosphere in the quasi-steady state. As the distance tends to infinity, the super-critical Parker (1965) solution yields a vanishing density and temperature, while the flow velocity tends to a finite limiting value. The sub-critical solution generally tends to an impossibly high value for the density at very large distances, although Chamberlain (1961) has shown that the solution with zero energy at infinity can give any desired small value of the density.

Durney (1971) made a study of stationary coronal models and concluded that at high initial density the Parker solution changes to the Chamberlain breeze solution. A more extended discussion of these solutions has been given by Roberts and Soward (t972). Korevaar (1989) made the first numerical calculation of the complete set of stationary stellar wind solutions in spherical symmetry with boundary conditions specified at the stellar surface and at infinity including the solutions with an interstellar shock or an accretion shock and concluded that if the interstellar density is sufficiently low, the coronal gas expands through the Parker initial point to supersonic velocities. An increase in the interstellar density moves the interstellar shock closer to the star. When it comes closer to the star than the Parker critical point, the flow changes to a breeze solution that is subsonic everywhere. A further increase in the interstellar density reverses the flow with a stationary accretion shock.

The equation for the temperature gradient in a rotating atmosphere derived by Ojha and Onkar (1988) is

1 dT

T dr - 2 [~+ 2 M - (~,~ + 2 0 M 2]-~ [~ + ~ + M ] - ~ x

][{{({ + ~) - 1 + [ [ - 2({ + {)]M} g ~ 1 7 6 x H_ - ? R T r 2

- { ~ { - { M + [ { 2 - 1 - ( 7 - 1)~] M 2 - ( ~ ' - t)~ M3} - -

- {~({ + { ) - 1 + [ ~ - Y({ + {)]M} r6c~176 ,

v R T r Jl (2)

A GAS DYNAMIC MODEL OF THE ROTATING SOLAR ATMOSPHERE 267

where ~ and { are defined (cf. Ojha and Onkar, 1988) by

and

2 M 2 - 1 4 - (3)

y § M,

~_ = {27M~ - (~ - 1)} {2 + (? - 1)M 2 }

(Y + 1) 2Mff (4)

Substituting this equation into (1) we get the following relation for the gradient of flow Mach number:

1 dM

M dr - [ 4 + 2 M - ( 7 4 + ~ ) M Z ] - I [ ~ + [ + M ] -1 x

n- goro m

x I1{(~ + {)(74 + ~.) - 1 + [274 + (7 - 1)~]M} U_ 2 R Tr 2

- {{(2~ + ~) + 2(~ + { ) M - [~2 _ 1 - ( 7 - 1)4{1M2-

_ ( y _ 1)~M3 } 2__ { ( 4 + ~)(74+ ~ ) - 1 + f

+ [ 2 7 4 + ( ? - 1)~]M} y R T r d "

For the super-critical solution the critical Mach number is

(5)

M c

which occurs at

1 + [1 + 4(74+ 2~.)] 1/2 (6)

goro m{~(~ + ~) - 1 + [ ~ - Y(~ + ~)]M} G = +

[O2rom{{(~ + ~) - 1 + [~ - Y(4+ ~)]M}

+ 2RT{4~. - 4?vI+ [~_2 _ i - (2 - 1){~] M2 - (7 - 1){M3}] . (7)

This condition is derived from Equation (2) and it can be shown that it is identical with the corresponding condition derived from (5).

It has been assumed that the shock waves and the quasi-steady flow are spherically symmetric. The term of the form If(M, M,, y)]r - 2 in the equation results from this assumption. In fact, shocks originate at the photosphere and the spherical terms may differ from those assumed. However, the major contributions to these terms arise from the spherical nature of quasi-steady flow, and in any case they are small in comparison with the gravitation term until near the critical point.

268 ONKAR NATH

The constant shock strength hypothesis enables the atmospheric properties to be

found without reference to the energy equation, and leads to a formulation which does

not involve heat conduction and radiation terms. If the conduction and radiation

properties of the gas were known, the energy equation could be solved as an additional

equation and this would give the distribution of heat addition to the atmosphere.

3. Numerical Results and Conclusions

The super-critical solutions of (2.2) and (2.5) have been obtained by numerical inte-

gration for a temperature of 6100 K at a solar radius r C = 1.006 (Van de Hulst, 1953).

The critical Mach number M c is taken to be 5, 10, and 15. The values of the other

quantities for the numerical integration are taken as follows: COo = 10, 7 = 1.4,

go = 0.275 km s -z, R = 8.314 x 103 joule mole 1 The variation of the flow Mach

number and temperature distribution with distance are presented graphically for a

particular shock number M s = 1.1 and different values of Mc with and without rotation.

It is observed from the figures that the rotation has little effect both on the temperature

distribution and on the variation of flow Mach numbers. It is also seen from the

figures that both the temperature and flow Mach number decrease as we move towards

the solar surface.

Acknowledgements

The author acknowledges the continued advise and assistance of Dr S. N. Ohja and the

assistance of Dr K. S. Singh for the numerical computation. The author also thanks the

referee for his valuable suggestions for the improvement of the results.

References

Biermann, L.: 1948, Z. Astrophys. 25, 161. Billings, D. E.: 1963, Astrophys. J. 137, 592. Bird, G. A.: 1964, Astrophys. J. 139, 684. Chamberlain, J. W.: 1961, Astrophys. J. 133, 675. Courant, R. and Friedrichs, K. O.: 1948, Supersonic Flow and Shock Waves, Interscience PubL, New York. Dubov, E. E.: 1960, Izv. Krymsk. Astrofiz. Obs. 22, 101 (translation AFCRL-63-265). Dumey, B. R.: 1971, Astrophys. J. 1(~, 669. Korevaar, P.: 1989, Astron. Astrophys. 226, 209. Ojha, S. N. and Onkar, Nath: 1988, Mod. Sim. Cont. 13, No. 2, 7. Osterbrock, D. E.: 1961, Astrophys. J. 134, 347. Parker, E. N.: 1965, Astrophys. J. 128, 664. Roberts, P. tt. and Soward, A. M.: 1972, Proc. Roy. Astron. Soc. 328, 185. Schatzman, E.: 1961, in R. N. Thomos (ed.), Proc. Fourth Symposium on Cosmical Gas Dynamics, (Suppl.

Nuovo Cimento 22, Ser. 10, 209). Schwarzschild, M.: 1948, Astrophys. J. 107, 1. Van de Hulst, H. C.: 1953, in G- P. Kuiper (ed.), The Sun, University of Chicago Press, Chicago. Weyman, R.: 1960, Astrophys. J. 132, 452.