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 Incoming links: Constant-opacity homologous contraction to CNO ignition

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 71 / 1 / Solution Created 2026-10-03 Updated 2026-10-14

Use

R

for the stellar gas constant, reserving

R

for the stellar radius. The specific internal energy of a monatomic gas obeying the ideal gas law is

e = P / [(γ - 1) ρ] = 3 P / (2 ρ)

. At fixed enclosed mass, the first law of thermodynamics gives the heat available to the outgoing luminosity as

ϵ = - \partial_{t} e - P \partial_{t} (1/ ρ)

when there is no stellar nuclear fusion heating. Therefore

ϵ = - \frac{3}{2 ρ} (\frac{\partial P}{\partial t})_{m} + \frac{5 P}{2 ρ ^{2}} (\frac{\partial ρ}{\partial t})_{m} .

The second coefficient includes both the mass density dependence of

e

and pressure work. This is gravitational energy generation in a homologously contracting ideal-gas star.

Write the characteristic scales as

ρ_{0} = \frac{M}{4 π R ^{3}}, P_{0} = \frac{G M ^{2}}{4 π R ^{4}}, T_{0} = \frac{μ GM}{R R} .

Then

ρ = ρ_{0} b

,

P = P_{0} p

,

T = T_{0} p / b

,

m = Mq

and

L_{r} = L l

. Since

d / d r = R^{- 1} d / d x

, direct substitution in the stellar hydrostatic equation and mass conservation gives

\frac{d p}{d x} = - \frac{b q}{x ^{2}}, \frac{d q}{d x} = x^{2} b .

For radiative diffusion in a star, dividing the temperature equation by

T_{0} / R

gives

\frac{d}{d x} (\frac{p}{b}) = - \frac{3 κ _{0} ρ _{0} L}{16 πa c R T _{0}^{4}} \frac{b l}{x ^{2} ( p / b ) ^{3}} = - D \frac{b ^{4} l}{x ^{2} p ^{3}}, D = \frac{3 κ _{0} R ^{4} L}{64 π ^{2} a c μ ^{4} G ^{4} M ^{3}} .

In particular the fourth power here is of

R

, not of the radius.

Because

q (x)

is fixed and

M

is constant, a fixed mass label has fixed

x

. Thus stellar homology gives

\partial_{t} ρ = - 3 ρ \dot{R} / R

and

\partial_{t} P = - 4 P \dot{R} / R

. The heating rate reduces to

ϵ = - 3 P \dot{R} / (2 ρR)

, positive during Kelvin-Helmholtz contraction. Substitution in

d L_{r} / d r = 4 π r^{2} ρ ϵ

now yields

\frac{d l}{d x} = - \frac{6 π R ^{2} R ˙}{L} x^{2} P = E x^{2} p, E = - \frac{3 G M ^{2} R ˙}{2 R ^{2} L} > 0.

The dimensionless profiles fix

D

and

E

. In a common stellar homology family with fixed stellar composition and opacity normalization, the expression for

D

implies

L = \frac{64 π ^{2} a c μ ^{4} G ^{4} D}{3 κ _{0} R ^{4}} M^{3} \propto M^{3} .

It is independent of radius and constant in time for a particular fixed-mass star. The formula for

E

gives

\dot{R} = - 2 E L R^{2} / (3 G M^{2})

, which integrates to

\frac{1}{R ( t )} = \frac{1}{R _{0}} + \frac{2 E L t}{3 G M ^{2}} .

When the initial-radius term is negligible,

\frac{R L t}{G M ^{2}} = \frac{3}{2 E} .

This is the Kelvin-Helmholtz contraction time law in the prescribed stellar homology model. Infinite initial radius is a limiting idealization; a finite initial radius retains the first term.

At main sequence arrival, CNO cycle heating with fixed stellar composition has luminosity

L_{nuc} = ϵ_{0} M ρ_{0} T_{0}^{16} \int_{0}^{1} b (q) (\frac{p ( q )}{b ( q )})^{16} d q \propto \frac{M ^{18}}{R ^{19}} .

The integral is dimensionless and fixed within the stellar homology family. The constant-opacity radiative scaling still gives

L \propto M^{3}

. Equating the stellar nuclear fusion and transported luminosities gives

R_{MS} \propto M^{15/19} .

Finally the Kelvin-Helmholtz contraction law evaluated at

R_{MS}

gives

t_{MS} \propto M^{2} / (L R_{MS}) \propto M^{- 34/19}

. In a coeval star cluster, the mass just arriving at the main sequence consequently satisfies

M_{arrival} \propto t^{- 19/34} .

These results comprise constant-opacity homologous contraction to CNO ignition; their mass exponents compare fixed-composition models with common dimensionless profiles.

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