Co-authors: Juliette Becker and Monica He
This is a write up of question 1 of the Hydrostatic Equilibrium and the Virial Theorem worksheet.
- (a) We're looking for the total potential energy contained in the sun. We know that potential energy is expressed by U = - GMm/r. More specifically, in this case where M_shell is the mass and M_int is the mass of the sphere inside that shell. We also have and .The change in potential energy, dU: therefore, after combining the terms we have that . Keeping in mind that we can simplify the expression to
- (b) Potential energy is related to force in the following expression: By doing dimensional analysis, we know that U has units of cm^3g^-1s^2 and dr/dt has units of cm/s. Therefore the velocity of a free-falling element would have the following relation to potential energy:
- (c) We can replace dr/dt in the equation above with . We're assuming that the free-fall velocity will be roughly constant. Therefore, if we define tau to be the time it takes a particle to fall from the surface of the star to the center and R_solar as the radius of the Sun, then we can make this substitution to an order of magnitude.
- (d) Plugging in our values into this expression we find that . This means that if the Sun were truly undergoing free fall collapse, it would collapse in 34 minutes, which is surely a time-scale that we would notice.