Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 14B (Cosmology)

Table of contents
- a 14B
  - Solution a
- b 14B
  - Solution b
- c 14B
  - Solution c
- d 14B
  - Solution d

a

Solution

At the centre, spherical symmetry and regularity require a finite central density

ρ (0) = ρ_{c}

and

\frac{d P}{d r}_{r = 0} = 0.

At the stellar surface

r = R

, the pressure vanishes:

P (R) = 0.

For a stellar polytrope these become the dimensionless conditions

θ (0) = 1

,

θ^{'} (0) = 0

, and the surface is the first zero

θ (ξ_{1}) = 0

; specifying

ρ_{c}

fixes the overall scale.

b

Solution

With the polytropic equation of state and substitutions

P = K ρ_{c}^{1 + 1/ n} θ^{n + 1}, ρ = ρ_{c} θ^{n}, r = a ξ,

we have

\frac{1}{ρ} \frac{d P}{d r} = (n + 1) K ρ_{c}^{1/ n} \frac{d θ}{d r} .

The pressure-support equation therefore becomes

(n + 1) K ρ_{c}^{1/ n} \frac{1}{r ^{2}} \frac{d}{d r} (r^{2} \frac{d θ}{d r}) = - 4 π G ρ_{c} θ^{n} .

Choose

a^{2} = \frac{( n + 1 ) K}{4 π G} ρ_{c}^{(1 - n) / n} .

Changing from

r

to

ξ

then yields the Lane-Emden equation

\frac{1}{ξ ^{2}} \frac{d}{d ξ} (ξ^{2} \frac{d θ}{d ξ}) = - θ^{n} .

These are the Lane-Emden variables for a stellar polytrope.

c

Solution

Using the enclosed mass integral,

R = a ξ_{1}

, and the Lane-Emden equation,

M = 4 π \int_{0}^{R} ρ (r) r^{2} d r = 4 π a^{3} ρ_{c} \int_{0}^{ξ_{1}} θ^{n} ξ^{2} d ξ = 4 π a^{3} ρ_{c} [- ξ^{2} θ^{'} (ξ)]_{ξ = ξ_{1}} = 4 π a^{3} ρ_{c} Y_{n} .

Substituting the value of

a

from part (b) gives the Lane-Emden mass formula

M = \frac{1}{2 π} (\frac{( n + 1 ) K}{G})^{3/2} ρ_{c}^{(3 - n) / (2 n)} Y_{n}, Y_{n} = - ξ_{1}^{2} θ^{'} (ξ_{1}) .

d

Solution

Since

R = a ξ_{1}

, the scaling in part (b) gives

ρ_{c}^{(1 - n) / n} = \frac{4 π G}{( n + 1 ) K} \frac{R ^{2}}{ξ _{1}^{2}} .

Substitute this into

M = 4 π (R / ξ_{1})^{3} ρ_{c} Y_{n}

. After collecting powers,

M = A_{n} K^{n / (n - 1)} R^{(3 - n) / (1 - n)},

where

A_{n} = 4 π Y_{n} (\frac{n + 1}{4 π G})^{n / (n - 1)} ξ_{1}^{(3 - n) / (n - 1)} .

This is the polytropic mass-radius relation.

An

n = 3

polytrope models a white dwarf whose electrons are an ultrarelativistic degenerate gas. For

n = 3

, the radius exponent vanishes and the mass is independent of both

R

and

ρ_{c}

. This fixed limiting value is the Chandrasekhar mass: increasing the central density can no longer support a larger mass, so a cold nonrotating white dwarf above the limit cannot remain in hydrostatic equilibrium.

 Ancestors (9)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version