# Harvard Astronomy 201b

## CHAPTER: The Virial Theorem

In Book Chapter on March 12, 2013 at 3:21 pm

See Draine pp 395-396 and appendix J for more details.

The Virial Theorem provides insight about how a volume of gas subject to many forces will evolve. Lets start with virial equilibrium. For a surface S,

$0 = \frac12 \frac{\mathrm D^2I}{\mathrm Dt^2} = 2\Gamma + 3\Pi + \mathscr M + W + \frac1{4\pi}\int_S(\mathbf r \cdot \mathbf B_ \mathbf B \cdot \mathrm d \mathbf s - \int_S \left(p+\frac{B^2}{8\pi}\right)\mathbf r \cdot \mathrm d\mathbf s,$

see Spitzer pp.~217–218. Here I is the moment of inertia:

$I = \int \varrho r^2 \mathrm dV$

$\Gamma$ is the bulk kinetic energy of the fluid (macroscopic kinetic energy):

$\Gamma = \frac12 \int \varrho v^2 \mathrm dV,$

$\Pi$ is $\frac23$ of the random kinetic energy of thermal particles (molecular motion), or $\frac13$ of random kinetic energy of relativistic particles (microscopic kinetic energy):

$\Pi = \int p \mathrm dV,$

$\mathscr M$ is the magnetic energy within S:

$\mathscr M = \frac1{8\pi} \int B^2 \mathrm dV$

and W is the total gravitational energy of the system if masses outside S don’t contribute to the potential:

$W = - \int \varrho \mathbf r \cdot \nabla \Phi \mathrm dV.$

Among all these terms, the most used ones are $\Gamma$, $\mathscr M$ and $W$. But most often the equation is just quoted as $2\Gamma+W=0$. Note that the virial theorem always holds, inapplicability is only a problem when important terms are omitted.

This kind of simple analysis is often used to determine how bound a system is, and predict its future, e.g. collapse, expansion or evaporation. Specific examples will show up later in the course, including instability analyses.

The virial theorem as Chandrasekhar and Fermi formulated it in 1953 is the following:

$\underbrace {2T_m} _{2\Gamma} + \underbrace {2T_k} _{3\Pi} + \underbrace {\Omega} _{W} + \mathscr M = \underbrace {0} _{\frac {\mathrm D^2 I} {\mathrm D t^2}}.$

This uses a different notation but expresses the same idea, which is very useful in terms of the ISM.