Answer to Question #90278 in Astronomy | Astrophysics for Shivam Nishad

Question

Estimate the radius of a star in thermal equilibrium of mass 10^30 kg and average internal temperature 10^7 K. It is given that
kB =1.38 x 10^-23 JK^-1, mH - 1.67 x 10^-27 kg and G=6.7 x 10^-11 m³ kg^-1s^-2.

Answer 1

A cloud wtith radius R, mass M, and temperature T will collapse to form a star if the total energy of the cloud is <0, i.e, if the (absolute value) of the potential energy exceeds the thermal energy of the cloud:

$$PE_{grav}=KE_{cloud} (1)$$

We can rewriter (1) as

$$\frac {3}{5} \frac {GM^2}{R}=N \frac {3}{2} kT (2)$$

where N is total number of particles in cloud

The total number of particles in cloud is can written as

$$N=\frac {M}{m_H} (3)$$

Solving for R gives:

$$R=\sqrt {\frac {15}{8\pi} \frac {kT}{G m_H^2 n_H}} (4)$$

Using (4) we get:

R=7×10⁸ m