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267

 

 

 

 

 

10.6 Exact

interior

solutions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m(r) = 4πρR3/3 := M,

 

r R,

(10.47)

 

 

 

 

 

 

 

where we denote this constant by M, the Schwarzschild mass.

 

 

 

We can now solve the T–O–V equation, Eq. (10.39):

 

 

 

 

 

 

 

 

 

 

 

dp

= −

4 π r

(ρ + p)(ρ + 3p)

.

 

(10.48)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dr

3

1

−

8π r2ρ/3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This is easily integrated from an arbitrary central pressure pc to give

 

 

 

 

 

 

ρ + p

=

 

ρ + pc

-

−

 

r .

 

 

 

 

 

 

 

 

 

ρ + 3p

 

 

 

ρ

+ 3pc

 

1

 

2

m

 

1/2 .

 

(10.49)

 

From this it follows that

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R2 =

 

 

3

 

[1 − (ρ + pc)2/(ρ + 3pc)2]

(10.50)

 

 

 

 

 

 

 

8πρ

 

or

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

pc = ρ[1 − (1 − 2M/R)1/2]/[3(1 − 2M/R)1/2 − 1].

(10.51)

 

Replacing pc in Eq. (10.49) by this gives

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

p

c =

ρ

(1 − 2Mr2/R3)1/2 − (1 − 2M/R)1/2

.

(10.52)

 

 

 

 

 

 

 

 

3(1 − 2M/R)1/2 − (1 − 2Mr2/R3)1/2

 

Notice that Eq. (10.51) implies pc → ∞ as M/R → 4/9. We will see later that this is a very general limit on M/R, even for more realistic stars.

We complete the uniform-density case by solving for from Eq. (10.27). Here we know the value of at R, since it is implied by continuity of g00:

g00(R) = −(1 − 2M/R).

(10.53)

Therefore, we find

 

exp( ) = 23 (1 − 2M/R)1/2 − 21 (1 − 2Mr2/R3)1/2, r R.

(10.54)

Note that and m are monotonically increasing functions of r, while p decreases monotonically.

Buchdahl’s interior solution

Buchdahl (1981) found a solution for the equation of state

ρ = 12(p p)1/2 − 5p,

(10.55)

where p is an arbitrary constant. While this equation has no particular physical basis, it does have two nice properties: (i) it can be made causal everywhere in the star by demanding that the local sound speed (dp/dρ)1/2 be less than 1; and (ii) for small p it reduces to

ρ = 12(p p)1/2,

(10.56)

268

Spherical solutions for stars

which, in the Newtonian theory of stellar structure, is called an n = 1 polytrope. The n = 1 polytrope is one of the few exactly solvable Newtonian systems (see Exer. 14, § 10.9), so Buchdahl’s solution may be regarded as its relativistic generalization. The causality requirement demands

p < p , ρ < 7p .

(10.57)

Like most exact solutions2 this one is difficult to deduce from the standard form of the equations. In this case, we require a different radial coordinate r . This is defined, in terms of the usual r, implicitly by Eq. (10.59) below, which involves a second arbitrary constant β, and the function3

u(r ) := β

sin Ar

,

 

 

2

:=

288π p

 

 

 

 

A

 

 

 

 

.

Ar

 

 

 

 

1

 

2β

Then we write

 

 

 

 

 

 

 

 

 

 

−

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r(r )

=

r

1 − β + u(r )

.

 

 

 

 

 

 

 

 

 

 

1

−

2β

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(10.58)

(10.59)

Rather than demonstrate how to obtain the solution (see Buchdahl 1981), we shall content ourselves simply to write it down. In terms of the metric functions defined in Eq. (10.7), we have, for Ar π ,

exp(2 ) = (1 − 2β)(1 − β − u)(1 − β + u)−1

,

(10.60)

exp(2 ) = (1 − 2β)(1 − β + u)(1 − β − u)−1

(1 − β + β cos Ar )−2,

(10.61)

p(r) = A2(1 − 2β)u2[8π (1 − β + u)2]−1,

 

(10.62)

ρ(r) = 2A2(1 − 2β)u(1 − β − 23 u)[8π (1 − β + u)2]−1,

(10.63)

where u = u(r ). The surface p = 0 is where u = 0, i.e. at r = π/A ≡ R . At this place, we have

exp (2 ) = exp (−2 ) = 1 − 2β,

(10.64)

R ≡ r(R ) = π (1 − β)(1 − 2β)−1A−1.

(10.65)

Therefore, β is the value of M/R on the surface, which in the light of Eq. (10.13) is related to the surface redshift of the star by

zs = (1 − 2β)−1/2 − 1.

(10.66)

Clearly, the nonrelativistic limit of this sequence of models is the limit β → 0. The mass of the star is given by

= (1 − 2β)A = $

288p (1 − 2β) !

1/2

−

 

M

πβ(1 − β)

 

π

β(1 β).

(10.67)

 

 

2An exact solution is one which can be written in terms of simple functions of the coordinates, such as polynomials and trigonometric functions. Finding such solutions is an art that requires the successful combination of useful coordinates, simple geometry, good intuition, and in most cases luck. See Stefani et al. (2003) for a review of the subject.

3Buchdahl uses different notation for his parameters.

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