USAAAO 2022, I кръг — Задача 9

• D

In order to estimate the mass of the globular cluster, we use the Virial theorem, which relates the time-averaged values of the kinetic and potential energy of a gravitationally bound system:

Note that $\sigma^2 = \sigma_x^2 + \sigma_y^2 + \sigma_z^2$ and $\sigma_r^2 = \sigma_x^2 = \sigma_y^2 = \sigma_z^2$ (assuming velocity dispersions are randomly distributed in a globular cluster). Thus, $\sigma^2 = 3\sigma_r^2$.

Additionally, we use the expression

for the gravitational potential energy of a uniform sphere.

Then,

We also note that:

,

where theta needs to be converted from arcminutes to radians before being substituted.

Solving for $M$, replacing $R$ with the expression from above, and substituting numerical values, we find that:

Answer: D