# Understanding Virial Theorem: Comparing Equations

• Stratosphere
In summary, The conversation discussed the differences between the \bar{P}=-\frac{1}{3}\frac{E_{gr}}{V} equation found in an astrophysics book and the kinetic and potential energy equation in the Wikipedia article on Virial Theorem. While they use different variables, they should ultimately result in the same value.
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?

Stratosphere said:
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

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.

## 1. What is the Virial Theorem?

The Virial Theorem is a principle in physics that relates the average kinetic energy of a system to its average potential energy. It states that for a stable, isolated system in equilibrium, the average kinetic energy is equal to the negative of half the average potential energy.

## 2. Why is the Virial Theorem important?

The Virial Theorem is important because it allows scientists to understand the behavior and stability of systems, such as galaxies, stars, and molecules. It also has applications in various fields, such as astrophysics, thermodynamics, and quantum mechanics.

## 3. How is the Virial Theorem used in comparing equations?

The Virial Theorem is used in comparing equations by providing a way to relate different physical quantities in a system. For example, it can be used to compare the kinetic and potential energies of a system, or to relate the temperature and pressure of a gas.

## 4. What are the assumptions of the Virial Theorem?

The Virial Theorem assumes that the system is stable, isolated, and in equilibrium. It also assumes that the system follows classical mechanics and that the potential energy can be described by a power law potential.

## 5. Can the Virial Theorem be applied to all systems?

No, the Virial Theorem can only be applied to systems that meet its assumptions. For example, it cannot be used for systems that are not in equilibrium or that do not follow classical mechanics. Additionally, it may not be applicable to systems with complex or non-power law potentials.

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