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Help Deriving Hydrostatic Equilibrium and Virial Theorem

  1. Apr 1, 2012 #1
    Hey all,

    Long story short, for my Modern Physics course, we have to do a research paper on a physics topic we didn't cover in class. Since I've always been interested in astronomy and the cosmos, I figured I'd do star formation / life cycle of stars. The paper has to have mathematical and physical reasoning for everything we present.

    Anyway, I found some cool books that have helped me out thus far, but I'm having trouble following an explanation for hydrostatic equilibrium (which more or less directly leads to the virial theorem).

    The book in question "Evolution of Stars and Stellar Populations" by Maurizio Salaris and Santi Cassisi.

    They start the derivation by "finding the equation of motion of a generic infinitesimal cylindrical volume element with axis along the radial direction, located between radii r and r+dr", with a base (perpendicular to the radial direction) of area dA and density ρ.

    They then obtain the mass, dm, contained in the element: dm = ρdrdA.

    "[They] neglect rotation and consider self-gravity and internal pressure as the only forces in action. The mass enclosed within the radius r acts as a gravitational mass located at the center of the star; this generates an inward gravitational acceleration: g(r) = G[itex]m_{r}[/itex] / [itex]r^{2}[/itex]."

    "Due to spherical symmetry, the pressure forces acting on both sides perpendicular to the radial direction are balanced, and only the pressure acting along the radial direction has to be determined. The force acting on the top of the cylinder is P(r+dr)dA, whereas the force acting on the base of the element is P(r)dA. By writing: P(r+dr) = P(r) + [itex]\frac{dP}{dr}[/itex]dr and remembering that drdA = dm/ρ, the equation of motion for the volume element can be written as: [itex]\frac{d^{2}r}{dt^{2}}[/itex]dm = -g(r)dm - [itex]\frac{dP}{dr}\frac{dm}{ρ}[/itex]."


    They go further to get the final equation, but I get lost at this last paragraph. I don't understand where the last term of this equation comes from: P(r+dr) = P(r) + [itex]\frac{dP}{dr}[/itex]dr. Why wouldn't it just be P(r+dr) = P(r)?

    And then once they get that equation, how do they get to this equation: [itex]\frac{d^{2}r}{dt^{2}}[/itex]dm = -g(r)dm - [itex]\frac{dP}{dr}\frac{dm}{ρ}[/itex]? I know they skipped a bunch of steps... Maybe I'm just not thinking right.

    Any help would be appreciated. (BTW, I'm assuming once I understand this, I'll be able to work out the virial theorem on my own, but if not, I'll ask it in a response.)

    Thanks in advance.
  2. jcsd
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