Thermonuclear Fusion in the Sun and Other Stars — Question 10

Question 10

Gravitational contraction alone gives a much shorter luminosity timescale

For an ideal-gas star with negligible surface pressure, take gravitational energy $\Omega=-3GM^2/(5R)$ and virial relation $2K+\Omega=0$. Use $M=2.00\times10^{30}$ kg, $R=7.00\times10^8$ m, $G=6.67\times10^{-11}$ SI, luminosity $3.84\times10^{26}$ W and $1\,\mathrm{yr}=3.156\times10^7$ s.

1. Find total energy and energy radiated by contraction from a very large initial radius to R.

2. Estimate the constant-luminosity contraction timescale and compare it with a supplied nuclear-burning timescale of $7.3\times10^9$ yr.

Question 10 - Solution

Gravitational contraction alone gives a much shorter luminosity timescale

(a) Method and calculation

$$K=-\Omega/2\Rightarrow\boxed{E_{\mathrm{total}}=K+\Omega=\Omega/2=-3GM^2/(10R)}.$$$$\boxed{E_{\mathrm{radiated}}=3GM^2/(10R)=1.14343\times10^{41}\,\mathrm J}.$$

(b) Interpretation and further calculation

$$\boxed{t_{\mathrm{KH}}=E_{\mathrm{radiated}}/L=9.43498\times10^6\,\mathrm{yr}}.$$$$\boxed{t_{\mathrm{nuclear}}/t_{\mathrm{KH}}=7.3\times10^9/(9.43498\times10^6)\simeq774}.$$

Physical check

Half the gravitational energy release heats the gas in this virial model. Contraction alone cannot supply the stated luminosity for the nuclear-burning timescale.