What is the basic definition and application of the virial theorem?

[Browntown]

TL;DR

Please help me understand virial theorem

Hello,

This term in university I'm taking a second year intro to astrophysics course and my professor talks a lot about different situations and then solves a problem using the virial theorem. The reason I'm confused is because the range of topics that he applies this theorem to vary in many different ways and he never explained the basic definition or how this theorem manages to apply to so many different applications.

If someone could help give me a very basic, low level understanding of what the virial theorem is, it would help me a ton.

Thank you!

 

[Arman777]

Summary: Please help me understand virial theorem

If someone could help give me a very basic, low level understanding of what the virial theorem is, it would help me a ton.

There are ton's of online sources. Maybe if you tell us which part in the theory you did not understand then we might help you better.

https://en.wikipedia.org/wiki/Virial_theoremhttp://hosting.astro.cornell.edu/academics/courses/astro201/vt.htmhttp://astronomy.swin.edu.au/cosmos/V/Virial+Theorem
And there are many more...

 

[Ken G]

The virial theorem is essentially a mathematical trick for expressing the integrated effects of force balance over an entire system. Its most conventional form is for systems with an internal attraction (often gravity) and zero external forces (like pressure), but other forms exist when there are external forces (including one where there is only external pressure and no internal forces on a gas, related to what gets called "the ideal gas law").

So yes, the virial theorem takes many forms, but the "virial" part is just a clever mathematical trick, and the result of the theorem is always just global force balance expressed in energy units, sometimes total energy, sometimes energy per particle, and sometimes energy per volume (as in the ideal gas law).

Often it has the intuitive meaning that for force balance, the total kinetic energy in a system in a steady state must be similar (not necessarily equal) to the energy released in contracting the system from infinity (in the case of an inverse-square force like gravity), or the energy required to expand it from a tiny volume (in the case of a spring force or an external pressure). It's just really useful, so gets used in a lot of contexts, but can be a bit of a pain to derive in detail every time-- so that's probably why your teacher invokes it without really explaining where it comes from.