The Virial Theorem and Dark Matter

The Virial Theorem and Dark Matter

For a gravitationally bound system in dynamical equilibrium, the virial theorem states:

$2\langle K \rangle + \langle U \rangle = 0$

The total energy is $E = K + U = -K = U/2$, meaning the system is bound with $E < 0$.

Measuring Cluster Masses

For a galaxy cluster of $N$ galaxies with line-of-sight velocity dispersion $\sigma$ and characteristic radius $R$, the virial mass is:

$M_{\text{virial}} = \frac{5\,\sigma^2\,R}{G}$

This lets astronomers weigh galaxy clusters using only the velocities of member galaxies — no knowledge of the underlying mass distribution is required.

Zwicky and Dark Matter

In 1933, Fritz Zwicky applied the virial theorem to the Coma Cluster ($\sigma \approx 1000$ km/s, $R \approx 2$ Mpc) and found a virial mass orders of magnitude larger than the luminous mass. He called this missing mass dunkle Materie — dark matter. For the Coma Cluster:

$M_{\text{virial}} = \frac{5 \times (10^6\text{ m/s})^2 \times (2 \times 3.086 \times 10^{22}\text{ m})}{6.674 \times 10^{-11}} \approx 2.3 \times 10^{15}\,M_\odot$

Your Task

Implement three functions. All constants must be defined inside each function.

• virial_mass_kg(sigma_m_s, R_m) — returns $M = 5\sigma^2 R / G$ in kg
• virial_mass_solar(sigma_km_s, R_Mpc) — returns virial mass in solar masses (convenience units: $\sigma$ in km/s, $R$ in Mpc where $1\text{ Mpc} = 3.086 \times 10^{22}$ m)
• kinetic_energy_J(M_kg, sigma_m_s) — returns $K = \frac{1}{2} M \sigma^2$ in joules

Use $G = 6.674 \times 10^{-11}$ N m² kg⁻², $M_\odot = 1.989 \times 10^{30}$ kg.