Understanding Virial Theorem: Comparing Equations

[Stratosphere]

In my astrophysics book, it says it's $$\bar{P}=-\frac{1}{3}\frac{E_{gr}}{V}$$

This Wikipedia article has a different equation. http://en.wikipedia.org/wiki/Virial_theorem

Can someone explain the difference?

 

[sylas]

In my astrophysics book, it says it's $$\bar{P}=-\frac{1}{3}\frac{E_{gr}}{V}$$

This Wikipedia article has a different equation. http://en.wikipedia.org/wiki/Virial_theorem

Can someone explain the difference?

Yes; the variables being used are different. Your formula is using pressure and volume. Wikipedia is using kinetic and potential energy, and the "virial". Should work out to be the same, I think.

Cheers -- sylas

 

[sir-pinski]

The virial theorem is a fundamental concept in astrophysics that relates the kinetic and potential energies of a system. It is used to understand the stability and equilibrium of systems, such as stars and galaxies.

The equation for the virial theorem can vary depending on the context and assumptions made. The equation given in the astrophysics book, \bar{P}=-\frac{1}{3}\frac{E_{gr}}{V}, is a simplified version that applies to a system with only gravitational potential energy and no external pressure. This equation is commonly used in the study of stellar evolution and the properties of star clusters.

On the other hand, the Wikipedia article presents a more general form of the virial theorem that includes both gravitational and thermal energies, \langle T \rangle = - \frac{1}{2}\langle V \rangle. This equation is applicable to a wider range of systems, including those with internal pressure and energy sources, such as gas clouds and active galactic nuclei.

It is important to note that both equations are valid and useful in different scenarios. The simpler equation from the astrophysics book is often preferred for its ease of use and application to specific systems, while the more general equation from Wikipedia provides a broader understanding of the virial theorem and its applications. It is also worth mentioning that there may be other variations of the virial theorem that exist in different sources, depending on the specific assumptions and context being considered.

In summary, the difference between the equations for the virial theorem lies in the level of complexity and the type of systems they are applicable to. Both equations are valuable tools in understanding the dynamics and equilibrium of astrophysical systems and should be used accordingly.