Tuesday, October 25, 2011

Hydrostatic Equilibrium and the Virial Theorem

Author: Iryna Butsky
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. 

  • (e) 

1 comment:

  1. (b): That was clever to use dimensional analysis! Another way of looking at it: if a particle starts very very far away (R=infinity) and falls to R_sun, what will its kinetic energy and hence its velocity be?

    (c): A particle in free-fall will definitely be accelerating. If our test particle is accelerating, what effect does that have on your estimate of the timescale for free-fall? How does it affect your argument that we would notice if the sun were in free-fall?

    (e) Awwwww!