Question

1) Use the gravitational potential energy and energy conservation arguments to write an expression for the Kelvin-Helmholtz time, t_KH, for stars. Calculate t_KH (in years) for stars of mass 10 Mo, 1 Mo, and 0.1 Mo. Assume they have luminosities and radii like main sequence stars with these masses

1) Use the gravitational potential energy and energy conservation arguments to write an expression for the Kelvin-Helmholtz time, th, for stars. Calculate tkh (in years) for stars of mass 10 M., 1 M, and 0.1 M.. Assume they have luminosities and radii like main sequence stars with these masses (use data in Appendix G of your textbook).

Answer 1

Kelvin-Helmholtz Time is the time needed to radiate away a significant fraction of the thermal energy of a star. For today's Sun, this timescale is about 30 million years.

It is the ratio of total energy divided by the energy lost

Gravitational potential energy of star is given by E= GM²/R

Suppose the star is not in equilibrium: there were no forces opposing gravitational force.  In order for the collapse to proceed, the pressure supporting the star against collapse would have to be lost. As the star radiates its energy away, pressure support would decrease, and it would contract.

The Star would slowly contract, radiating away all of its gravitational potential energy. If it does so at its present luminosity, the time it takes to radiate away its energy is given by the total energy divided by the rate the energy is lost (which is the luminosity). This time,

t_KH = GM²/RL

tkh of 0.1 M star = GM²/RL

= 6.7*10^-11 * (0.1*1.9891*10^30)²/(107280*0.00006)

=4.118*10^47