Thermonuclear Fusion in the Sun and Other Stars — Question 9

Question 9

Hydrostatic pressure supplies a central temperature scale

For a uniform-density, fully ionized hydrogen star, use $P_c=3GM^2/(8\pi R^4)$, $\rho=3M/(4\pi R^3)$ and $P=2\rho kT/m_p$. Take $M=2.00\times10^{30}$ kg, $R=7.00\times10^8$ m, $G=6.67\times10^{-11}$ SI, $m_p=1.67\times10^{-27}$ kg and $k=1.38\times10^{-23}$ J/K.

1. Derive a central temperature expression and evaluate it.

2. At fixed mass, infer how this temperature scale changes if the radius contracts by 10 percent.

Question 9 - Solution

Hydrostatic pressure supplies a central temperature scale

(a) Method and calculation

$$T_c=\frac{m_p}{2k}\frac{P_c}{\rho}=\frac{m_p}{2k}\frac{GM}{2R}.$$$$\boxed{T_c=\frac{GMm_p}{4Rk}=5.76548\times10^6\,\mathrm K}.$$

(b) Interpretation and further calculation

$$T_c\propto R^{-1}\Rightarrow\boxed{T_c\prime/T_c=1/.900=1.11111}.$$$$\boxed{\Delta T_c/T_c=.11111}\quad\text{an 11.111 percent increase}.$$

Physical check

The factor two counts ions and electrons. Uniform density is crude; realistic profiles, composition and radiation pressure change central temperature.