The Boundary Conditions of Relativistic Polytropic Fluid Spheres

THE BOUNDARY CONDITIONS FOR RELATIVISTIC POLYTROPIC

FLUID SPHERES

T. G. DALLAS and V. S. GEROYANNIS Astronomy Laboratory, Dept. of Physics, Univ. of Patras, Greece

(Received: 25 June, 1992)

AbstracL The differential equations goveming relativistic polytropic fluid spheres have been integrated numerically for polytropic index n = 0.0 (0.1) 4.9 and relativity parameter « = 0.0 (0.1) 0.9, and the resulting boundary conditions for ~~, Tl, v(~1) and other related quantiües are presented in this paper.

1. The Relativistic Polytropic Model

In out current investigations concerning polytropes on the basis of zatrikean pre- geometry (Geroyannis, 1992), we need the boundary conditions of relativistic polytropic fluid spheres with satisfactory accuracy. Since these results may be of wider interest, we present them in the present paper.

The relativistic polytropes (Tooper, 1964) obey the set of differential equations:

«2dO 1 - 2a(n + 1)v/« + v + cr«O dv d---~ 1 + crO -~ = O,

(1)

dt, _ ~ 2 0 n d-~: -- (2)

where, n is the polytropic index and cr is the ratio of the pressure to the energy density at the centre of the sphere. The above equations reduce to the classical Emden equation for cr = 0. The quantity ~, in the above equations, however, is not the wdue measured by an outside observer; a quantity which represents the true distance from the centre of the polytrope, is defined by

~== [1 - 2cr(n + 1)v/«]-U2d« => d--~ = 0

VI - 2«(n + 1)v/«" (3)

The above system of equations satisfies the initial conditions

0(o) = 1, v ( o ) = o, ~ (o ) = o. (4)

The boundary of the sphere is represented, as in the classical theory, by the first zero ~1 of 0, that is

Astrophysics and Space Science 201:249-271, 1993. @ 1993 KluwerAcademic PubIishers. Printed in Belgium.

250 T.G. DALLAS AND V. S. GEROYANNIS

0(«1) = 0 , (5)

and the total mass of the sphere is proportional to the va lue V(~l) at that point. The mass-radius relationship is

G M a ( n + l) v(«l) ¢2R - - ~1 ' (6)

while the ratio of the central to the mean density is

po «~ m (7)

The phase velocity of sound at the centre of the sphere is

B n + l v = c (8)

n

and the inequality v / c < 1 sets the likely maximum a for every n.

2. The Numerical Treatment The system of differential equations (1)-(3) obeying the initial conditions (4) can be integrated numerically.

However, when ~ goes beyond ~1,0(~) becomes negative; as a result 0 n becomes indefinite in the real axis and integration cannot continue beyond ~1. A simple way to overcome this difficulty is to solve the equations in the complex plane; this method has been proved very efficient in the past (Geroyannis, 1988). Thus, we can integrate beyond ~1, and check for the zero value of 0(() by using a simple root-finding algorithm.

The differential equations (1)-(3) have been solved by use of a variable order Adams-Bashford-Moulton method (Shampine and Gordon, 1975) modified to work in the complex plane. The root finding has been performed by the classical Secant method, using the real part of the complex numbers provided by the integration algorithm. The relative error requested has been set to 1 × 10 -15. The absolute error has the same value initially, but it has then been increased by multiplying with ln(~) whenever it proved too rough for the integration method. This occurred often for n > 4.0, and occasionally for n > 3.0 and large a. The root ~1 was the point at which the real part of 0(~) has become less than 1 x 10 -15.

The numerical method has been checked for a=0 against accurate boundary conditions for classical polytropes (Jabbar, 1983). The results are in agreement within 7 to 10 digits for n < 4; the accuracy declines for larger n down to a minimum of 5 digits for n > 4.5, possibly due to the looser tolerances on the absolute error. We believe that the displayed 8 digits are accurate for n < 3 and 6 digits are still accurate for up to n < 4.5. For larger values of n and high cr there is

RELATIV1STIC POLYTROPIC FLUID SPHERES 251

a nuraefical difficulty because 0(~) converges to zero very slowly. For instance, at the extreme case n = 4.9, cr = 0.9, 0(~) is less than 7 × 10 -14 for~ = 2 × 1028, but 0(~) becomes less than our tolerance at ~ = 9.7 × 1028. Possibly, at these extremes no more than two digits are correct. The original results of Tooper proved to be correct to four (occasionally three) digits - the two-digit accuracy for ~1, only at n = 1.5, cr = 0.1 must be a printing error.

The results of our calculations are presented in Table I. They are sorted by increasing n and er. The 8 columns contain (from left to right) n, ~r, ~1, 71, v(~l), GM/'c2R, po/-fi and v/c. A 3.5" MS-DOS disk containing ~, 7, v(~), 0(~), at the boundary plus at 100 points inside each polytropic configuration is available from the authors upon request.

The dimensionless coordinate radius ~1, decreases monotonically with ~r for n < 1.5. For 1.5 < n < 3.0 the coordinate radius first decreases and then increases° For n > 3.0 we notice fluctuations of the coordinate radius with er. The dimensionless invariant radius Tl, follows similar behaviour, but it is not analogous to ~1.

Generally, as cr increases the matter is concentrated towards the centre of the polytrope. This i~s demonstrated by the increase of the ratio Po/-fi when a increases, but again there are fluctuations of this ratio for n > 3.0. This marked behaviour for n > 3.0 and large « can be attributed to the large instability of the particular configurations.

References

Geroyannis, V. S.: 1988, Astrophys. J. 327, 273. Geroyannis, V. S.: 1992, Astrophys. Space Sci. 199, 1. Jabbar, J. R., 1984: Astrophys. Space Sci. 100, 447. Shamping L. E and Gordon M. K., 1975: Computer Solution of Ordhmry Differential Equations,

W. H. Freeman and Company. Tooper, R. E, 1964: Astrophys. J. 140, 434.

252 T.G. DALLAS AND V. S. GEROYANN1S

r"

ù~

0

.~

'~.

2~

0

0 r..)

0

d ~ £ ~ M M N N N d d b ~ ~ ~ ~

~ + + + + + + + + + + + + + + + + + + + + + + + + +

~I~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

+ + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + +

d d d d d d d d d d d d d d d d d d d d d d d d d

~ d d d d d d d d d d d d d d d d d d d d d d d d d

RELATIVISTIC POLYTROPIC FLUID SPHERES 253

.1

I~ ° ° ° ° ~ o~~oooooooo o o o o ~ ~ ~ ~ ° ~ ~ ~ ~ oooooooo~**÷÷****

~ J l l t t t t l l l l l ß t t t t l l l I l l ~ t

+ + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + +

b d d d d d d d d d d d d d d d d & d d d d d d d d

~~ d d d d d d d d d d d d d d & d d d d d d d d d d


"~,

~ O O O O O O O O ~ O O O O O + + + + + + + + + + + + + + ~ ~ ~ ~

Õ 0 © © 0 0 0 0 0 0 © © © 0 0 © © 0 © 0 0 ~ © 0 0 0 0 0 I ~ + + + + + + + + + + + + + + + + + + + + + + + + + + + 1

& ~ ~ O ~ ~ ~ ~ ~ ~ b ~ ~ h ~ ~ O ~ O ~ ~ ~ ~ ~ ~ O ~ ~ ~ l

M ~ N N N N N d ~ ~ M N M N N M N d ~ ~ N N N M M N N

O O O O O O O O O O O O ~ O O O O O O O O O O O O O O O ~ O O O O O O O O O O O O O O O O O O O O O O O O O O O + + + + + + + + + + + + + + + + + + + + + + + + + + +

liooooo~ooooooooooooooooooooo + + + + + + + + + + + + + + + + + + + + + + + + + + +

. ~ ~ . . . . . ~ ~ ~ . . . . . . . ~ ~ ~ . . . . ~ ~ ~

d d d S d d d d d d o d d d d d d s d d d d d d d d 5

RELATIV1ST1C POLYTROPIC FLU1D SPHERES 255

o

~5

oo I ~~- L~

~q

0

0 0 o g õ g g g g g g g o õ ~ g g õ g o o g g ° 0 0

t l l l l l l l t t ~ l l l l l l l ~ ~ t ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ õ ~ ~ ~ ~

0 0 ~ 0 ~ 0 0 0 ~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0


ü

»1

+ ~ + + ~ + + + + + + +

O

+ + + + + + + + + + + + + + + + + + + + + + + +

N d M M M M M N M N N N N M M M M N N M M M N M

+ + + + + + + + + + + + + + + + + + + + + + + +

b d d d d d d d d d d d d d d d d d d d d d d d d

RE L AT IVIST IC POLYTROP1C F L U I D SPHERES 257

~ t'-.I ~ ,3 ' , ~ - ~ ~ ~ . ~ - ",~- ~ t--.I ~ ~ ,~ ~"5 , ~ ~ t ' ~

+ + + + + + + + + + 4- + + + + + + + + + + 4- + + +

. . . . . . . . . . ~ ~ ~ ~ ~

M ~ ~ ~ ~ ~ m m ~ M d ~ ~ ~ M ~ d ~ M N N ~ d ~ M 4

~ + + + + + + + + + + + + + + + + + + + + + + + + +


~ ~ , ~.1 ~ _

+ + + + .2. ' .2. ' + + + + ' ' ' ' ' ' + + + +

+ + + + + + + + + + + + + + + + + + + + + + + +

9 9 9 m m N

ù-2 ~ ,--:

+ + + + + + + + + + + + + + + + + + + + + + + +

RELATIVIST1C POLYTROPIC FLU1D SPHERES 259

ü

M m ¢

I1~ ~ ~ ~ ~ ~ ~ ~ ~ ~ O ~ ~ ~ ~ ~ ~ ~ ~O O O O O O O O O O O O O O O O O O O O O O O O + + + + + + + + + + + + + + + + + + + + + + + +

o o o o o o ~ o ~ ~ ~ o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ o ~ , , , , , , , ,

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 ~ 0 0 0 0 ~ 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 + + + + + + + + + + + + + + + + + + + + + + + +

~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + + + + + + + + + + + + + + + + + + + + + + + +

~ ~ d d d d d d d d d d d d d & & d d d d d d d d d d

' ~ ~ ~ ~ ~ ~ ~ . . . . . . . . . . . . . . . . .

260 T.G. DALLAS AND V. S. GEROYANN1S

ù~ ~o

c~


ù~

7~

«q

O O O O O O O O O O O O O O O O O O O O O O O O + + + + + + + + + + + + + + + + + + + + + + + +

O O O O O O O O

õ~o~~~

e 9 9 9 9 9 9 ~ ~ ~ e 9 9 e 9 9 9 9 ~ ~ o e 9 o

~ M ~ ~ ~ M M M ~ ~ ~ M ~ M ~ M M M N ~ ~ M ~ d

+ + + + + + + + + + + + + + + + + + + + + + + +

~ O O O O O O O O O O O O O O O O O ~ O O O O O O O O O O O O O O O O O O O O O O ~ O O O O O O O + + + + + + + + + + + + + + + + + + + + + + + +

t ~ O O O O O O O O O O O O O O O O O ~ O ~ O O O O O

~~ M M N N M N M N N N N N N N N N N M N N ~ N N N ~

262 T, G. DALLAS AND V. S. GEROYANNIS

~ + + + + + + + + + + + + + + + + + + + + + + + +

M M M M ~ ~ ~ ~ d d d M N M £ N M ~ d d d d d M

+ + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + +

b d d d d d d d d d d d d d d d d ~ d d d d d d ~

M M M M M M M N M N M M M ~ M M N M M M M M M M


~ ~ 0 ~ ~ ~ ~ ~

~:'~ I t̀ ̀

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ + + + + + + + + + + + + + + + + + + + + + + + +

o o o o o o o o o ~ o ~ o ~

m m m m m ~ m ~ m ~ ~ m ~ m ~ m m m ~ m ~ m m m

~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + + + + + + + + + + + + + + + + + + + + + + + +

d d d d d d d d d d d d d d d d d d d d d d d d d


e~ T~

I~lõ

i

n l n u l l a ~ ~ ~ l n

~ ~ ~ ~ ~ o ~ ~ ~ o ~ ~ ~ ~ ~ ~ ~ ~~~ ~ ~ ~ ' ~ ~ ~ ~ ~ ~ ~ ~ ~ o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

~ ô ô õ ~ ~ ~ ô ô ~ õ õ ô õ ~ ~ ~ ~ ô ~ ~ ~ ~ õ ~~~5~_ ~ ~ ~ ~ ~ ~ ~ ~ ~ 5 ~~~~~~~5~

0 0 0 ~ 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0 + + + + + + + + + + + + + + + + + + + + + + + +

RELAT1VISTIC POLYTROPtC FLUID SPHERES 265

I ~, -- -- -- ~. -- -- o o o . . . . . . .. o o o -- ~. -- -- .. -- ~. ! o

I

- + + + + + + + + + + + + + + + + + + + + + + + + + + + +

I ~~~~mm~.mo~~~~mmmmmõ~~m~mmmmm

=,~oo~oooooooo~oooooo, o, ~°õõ°o°°

+ + + + + + + + + + + + + + + + + + + + + + + + + + + +

, ~ 0o ~ ~ ~ ~ ~I- o0 oo 0o ~ ~ 0o 0o ~I" ~¢~ ~ ~ 0o ~ ~ ~ oo ~ ~ ~ ~

~ + + + + + + + + + + + + + + + + + + + + + + + + + + + +

266 T . G . D A L L A S A N D V. S. G E R O Y A N N I S

.1

o

oi

, o o o o o o o o ~ ~ ~ ~ ~ ~ ~ O O O O O ~ O O O

~ ~ o o o o o o o o o ~ o o o o o o o o o

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 + + + + + + + + + + + + + + + + + + + + + + + + +

d & d d d d d d d d d d d d d d d d d d d d d d d

d ~ ~ d d d M d d d M d ~ ~ d d d ~ ~ ~ ~ ~ d d ~

RELAT1V1STIC POLYTROPIC FLUID SPHERES 267

~<

+ + + + + + + + + + + + + + + + + + + + + + + +

Q ~ M ~ M ~ ~ £ d ~ M d d ~ ~ Œ d 2 ~ 2 2 Œ d N N ~ d

+ + + + + + + + + + + + + + + + + + + + + + + + +

~ + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + +


.3

+ + + + + + + + + + + + + + + + + + + + + + + +

~ ~ m ~ m m ~ m m m g

+ + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + +

b d d d d d d d d d d d d d d d d d d d d d d d d d

RELATIV1STIC POLYTROPIC FLUID SPHERES 269

I ~~- eL~ r~ ù1-

~-~

b

~~~~~~~~~°°~~~~~~-°-~~~~- ° ° ° ° 9 - ~ - - 9 ~ 9 ~ ~ ~ ~ ~ ~

2 d ~ d £ ~ m 2 £ M 2 d 2 d ~ £ d 2 N 2 ~ 2 ~ d

. . . . . ~ . . ~ ~ ~ ~ . ~ ~ . ~ . . . . ~ . .

d d d d d d d d d d & d & d d d d d d d d d d d d


I + + + + + + + + + + + + + + + + + + + + + + + + + I ~ 1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ 1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

9 o . . . . . . . . . . . . . ~ Z - Z . . . . . . ~ ,

. . . . . . . . ~ ~ ~ q ~ ~ N N ~ ~ ~ m ~ ~ ~

+ + + + + + + + + + + + + + + + + + + + + + + +

~ + + + + + + + + + + + + + + + + + + + + + + + +

~ + + + + + + + + + + + + + + + + + + + + + + + +


~3 7~ .1

qz~i

e-i

~~~~ooooo~o~

+ + + + + + + + + + + +

+ + + + + + + + + + + + +

+ + + + + + + + + + + + +

The Boundary Conditions of Relativistic Polytropic Fluid Spheres

Documents

Transcript of The Boundary Conditions of Relativistic Polytropic Fluid Spheres