IGNOU PHE 15 2014 - Derive the Virial theorem. Use the theorem to estimate the mass of a s

Question

Derive the Virial theorem. Use the theorem to estimate the mass of a star cluster with a radius of 10¹⁸m. It is given that the stars have random
velocities of the order of 10⁵ ms^-1.

26 Jun 2018

Answer :

Word Count : 572

The Virial Theorem is a fundamental principle in astrophysics that relates the kinetic and potential energies of a system in a stable, quasi-equilibrium state. It is particularly useful in understanding the dynamics and stability of astronomical objects such as star clusters. Let's derive the Virial Theorem and then use it to estimate the mass of a star cluster.

### Derivation of the Virial Theorem:

The Virial Theorem can be derived from the equations of motion and the assumptions of a stable, quasi-equilibrium state. Consider a system of N point masses, such as stars in a cluster, with masses \(m_i\) and positions \(\mathbf{r}_i\) in three-dimensional space. The total kinetic energy \(T\) of the system is given by:

\[ T = \frac{1}{2}\sum_{i=1}^{N}m_i|\mathbf{v}_i|^2 \]

where \(\mathbf{v}_i\) is the velocity

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