Homework Set-V for PHY-305A: Physics of the Universe

Date: 23rd October, 2017

Instructor: Kaushik Bhattarcharya1
Office: FB-387, Phone: 7306

1. Let us consider a spherical star whose mass density and pressure at position r is ρ(r) and
P (r).

(a) Consider an infinitesimally thin spherical shell whose thickness is dr. (It is the shell
between two spherical shells of radii r and r + dr.) Find the mass dm(r) of this shell
and show that
dm
= 4πr2 ρ , (1)
dr
where the mass of the star, up to radius r, is
Z r
m(r) = dm .
0

(b) Next consider a cylinder, whose two circular faces, of area dS, are on the two con-
centric spheres of radii r and r + dr (obviously the height of the cylinder is dr).
Suppose the pressure on the surface dS situated on the outward (from the centre)
spherical shell is P (r + dr) and the pressure on surface dS situated on the inward
spherical shell id P (r). From this show that the force acting radially outward on the
cylinder is
dP
Fp = P (r)dS − P (r + dr)dS = − drdS ,
dr
where you require the taylor expansion of P (r + dr) to get to the above result.
Gravitational force is also acting on this cylinder of area dS and height dr whose
mass is drdSρ. In our notation the mass of the sphere whose radius is r is given by
m(r). Consequently the graviational force on the cylinder acting radially out is

Gmρ
Fg = − drdS .
r2
If the cylinder is in hydrostatic equilibrium then Fg + Fp = 0. From this show that

dP Gmρ
=− 2 . (2)
dr r

(c) Replace the expression of m(r), from Eq. (2), into Eq. (1) to get the following
equation,
!
1 d r2 dP
= −4πGρ . (3)
r2 dr ρ dr
1
Email: kaushikb@iitk.ac.in

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(d) Suppose the pressure and mass density of the star are related by the following relation

P (r) = κρ(r)γ , (4)

where κ and γ are constants. This equation defines the equation of state of a
polytrope (the gas which forms the star in this case is the polytrope) and the above
equation of state is called a polytropic equation of state. Show that one can rewrite
Eq. (3) in terms of ρ by using the polytropic equation of state as:
!
1 d r2 dρ
κγργ−1 = −4πGρ . (5)
r2 dr ρ dr
Now define a function θ(r) and a constant quantity γ as follows:
n+1
ρ(r) ≡ λθn (r) , γ ≡ ,
n
where λ is a constant. Show that in terms of these new quantities one can write
Eq. (5) as:
" # !
(n + 1) (1−n) 1 d 2 dθ
κλ n r = −θn . (6)
4πG r2 dr dr

(e) Now define the quantities ξ(r) and α as follows:

" #1/2
r (n + 1) (1−n)
ξ(r) ≡ , α ≡ κλ n ,
α 4πG
where α is again another constant. Show that in terms of ξ one can write Eq. (6)
as:
!
1 d dθ
2
ξ2 = −θn . (7)
ξ dξ dξ
This is the standard equation of a polytropic star. Solution of this equation
gives how θ (related to the mass desnsity via the relation (ρ/λ)1/n ) varies with ξ
(proportional to r as r = ξα). The above equation is the celebrated Lane-Emden
Equation for polytropic stars.
(f) As this is a second order non-linear differential equation, it requires proper boundary
conditions. In the present case the boundary conditions are chosen in such a way
that one sets

ρ(ξ = 0) = λ ≡ ρc , (8)

where ρc is the central density of the star. In this case you get θn (ξ = 0) = 1.
More over at r = 0, one gets from Eq. (2), dp/dr = 0 as m(0) = 0. Show that one
can use these two conditions to write down the boundary conditions as:

dθ 
θ(ξ = 0) = 1 , = 0. (9)
dξ ξ=0


From the definitions of ξ and θ, show that both of them are dimensionless.

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(g) If ξ1 is the first zero of the function θ(ξ), then one naturally defines the radius of the
star to be
R ≡ αξ1 . (10)
Can you justify the above definition? With the above definition one can write the
mass of the polytropic star as
Z R
M= 4πr2 ρ(r)dr .
0

Show that this relation yields

" #
dθ
M = 4πα3 ρc −ξ 2 . (11)
dξ ξ=ξ1

Here you will require to use the Lane-Emden equation to get the above form.
(h) If the average density of the star is defined as
M
ρ̄ = 4 , (12)
3
πR3
then using Eq. (11) show that the relation between the central density to the average
density of the star is given by:
ρc = Dn ρ̄ , (13)
where " −1
#
3 dθ
Dn ≡  −  .
ξ dξ ξ=ξ1

(i) From the definition of R and the boundary conditions show that you can write
" #1/2
(n + 1) κ 1−n
R= ρc 2n ξ1 . (14)
4π G
Using this equation and the form of M given in Eq. (11) show that the relationship
between mass and radius of a polytropic star with polytropic index n is given by
3−n n−1 κ
R n M n = , (15)
GNn
where the constant Nn is
" #  1−n
(4π)1/n 
n
dθ n−3
Nn = −ξ 2  ξ1 n .
n+1 dξ ξ=ξ1

(j) From Eq. (15) show that the general relationship of the mass and radius of a poly-
tropic star with polytropic index n is
1
R3−n ∝ . (16)
M n−1
Show explicitely from Eq. (15) that for n = 1 the radius becomes independent of the
mass. Find the radius in this case. Also show that for the case of n = 3 the mass of
the star becomes independent of its radius. Find the mass of the star in this case.