====== Virial theorem ======

The **virial theorem** provides a fundamental link between the internal pressure (or kinetic energy) and the gravitational binding energy of a self-gravitating system.  
For a star, galaxy, or molecular cloud in steady equilibrium, it can be derived directly from the condition of **hydrostatic equilibrium**.

{{:courses:ast301:virial.webp?nolink&650|}}

Starting from hydrostatic equilibrium,
$$
\frac{dP}{dr}=-\rho(r)\frac{G\,M(r)}{r^{2}},
$$
multiply by the spherical volume element \(4\pi r^{3}dr\) and integrate from center (\(r=0\)) to surface (\(r=R\)):
$$
\int_{0}^{R}4\pi r^{3}\frac{dP}{dr}\,dr=-\int_{0}^{R}4\pi r^{3}\rho(r)\frac{G\,M(r)}{r^{2}}\,dr.
$$

**Left-hand side (pressure term).** Integrate by parts:
$$
\int_{0}^{R}4\pi r^{3}dP=\Big[4\pi r^{3}P\Big]_{0}^{R}-\int_{0}^{R}12\pi r^{2}P\,dr
\approx-3\!\int_{0}^{R}4\pi r^{2}P\,dr=-3\!\int P\,dV,
$$
where \(P(R)\simeq 0\).

**Right-hand side (gravity term).** Using \(dM=4\pi r^{2}\rho\,dr\),
$$
-\int_{0}^{R}4\pi r^{3}\rho\,\frac{G\,M}{r^{2}}\,dr
= -\int_{0}^{R}\frac{G\,M(r)}{r}\,dM(r)
\equiv \Omega ,
$$
which is the (negative) **gravitational potential energy**.

Equating both sides gives the pressure–gravity balance:
\begin{equation}
3\!\int P\,dV+\Omega=0.
\end{equation}

For an **ideal gas** the thermal kinetic energy is \(K=\tfrac{3}{2}\!\int P\,dV\), hence
\begin{equation}
2K+\Omega=0,
\end{equation}
the virial theorem in its standard energy form.

===== Satellites and stars =====

A satellite of mass \(m\) in a circular orbit around a mass \(M\) at radius \(r\) obeys
$$
K=\tfrac12 m v^{2}=\tfrac12\,\frac{G M m}{r},\qquad
\Omega=-\frac{G M m}{r},
$$
so that
\begin{equation}
K=-\tfrac12\,\Omega ,
\end{equation}
consistent with \(2K+\Omega=0\).

{{https://upload.wikimedia.org/wikipedia/commons/thumb/8/83/The_Sun_in_white_light.jpg/1024px-The_Sun_in_white_light.jpg?nolink&500}}

A **star without an internal nuclear source** behaves analogously: as it radiates, its total energy \(E=K+\Omega\) decreases; to re-establish virial balance the star **contracts**, making \(\Omega\) more negative and increasing \(K\) (heating the gas).  
As the star shrinks:
  * \(\Omega\) becomes more negative (stronger binding),
  * \(K\) increases (higher temperature).

Thus a self-gravitating gas has **negative specific heat**: losing energy can raise its temperature.  
During stellar birth the contracting cloud spins up and heats until pressure and rotation counter gravity; during late stages, core contraction again heats the gas and ignites progressively heavier fuels.  
When nuclear power equals surface losses, the star reaches **true equilibrium**—hydrostatic balance sustained by the virial relation.

===== Clusters of galaxies =====
{{https://upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Webb%27s_First_Deep_Field_%28adjusted%29.jpg/1007px-Webb%27s_First_Deep_Field_%28adjusted%29.jpg?nolink&600}}

The virial theorem also yields the **total mass** of a galaxy cluster from member velocities.  
For \(N\) identical galaxies (mass \(m\)),
$$
2\sum_i \tfrac12 m v_i^{2}-\sum_{i\neq j}\frac{G m^{2}}{r_{ij}}=0.
$$
Rearranging,
$$
m\sum_i v_i^{2}-G m^{2}\sum_{i\neq j}\frac{1}{r_{ij}}=0.
$$
Multiplying the first term by \(N/N\) and the second by \(N^{2}/N^{2}\),
$$
N m\!\left[\frac{1}{N}\sum_i v_i^{2}\right]
-\frac{G(Nm)^{2}}{2}\!\left[\frac{1}{N(N-1)/2}\sum_{i\neq j}\frac{1}{r_{ij}}\right]=0.
$$
With \(M\equiv Nm\) (total mass) and defining \(\langle \cdot \rangle\) as the average over members, we obtain
\begin{equation}
M\,\langle v^{2}\rangle-\frac{G M^{2}}{2}\,\langle r_{ij}^{-1}\rangle=0,
\end{equation}
hence the **virial mass**
\begin{equation}
M=\frac{2\,\langle v^{2}\rangle}{G\,\langle r_{ij}^{-1}\rangle}.
\end{equation}

Only the **line-of-sight** component \(v_{\text{los}}\) is observed.  
Assuming isotropy,
$$
\langle v^{2}\rangle=3\,\langle v_{\text{los}}^{2}\rangle .
$$
Thus the virial theorem converts measured velocity dispersions and projected separations into the **total (including dark) mass** of a cluster.

===== Insights =====

The virial theorem states that for any **self-gravitating system in equilibrium**, the total kinetic energy equals half the magnitude of the gravitational binding energy.
  * In stars: \(3\int P\,dV+\Omega=0\) links pressure support to binding energy.
  * In clusters: velocity dispersion \(\langle v^{2}\rangle\) maps to total mass through \(M=2\langle v^{2}\rangle/(G\langle r_{ij}^{-1}\rangle)\).
  * The negative specific heat of bound gravitating systems explains why contraction heats stars and clouds.

===== Inquiries =====

  - Starting from \( \frac{dP}{dr}=-\rho\,G M(r)/r^{2} \), derive \(3\int P\,dV+\Omega=0\) (show the parts integration and the \(dM\) substitution).  
  - Explain physically how \(2K+\Omega=0\) implies **negative heat capacity** for self-gravitating gas.  
  - Using the cluster derivation, show how \( \langle v_{\text{los}}^{2}\rangle \) and projected \( r_{ij} \) yield \( M=2\langle v^{2}\rangle/(G\langle r_{ij}^{-1}\rangle) \); state assumptions and corrections (isotropy, projection).