If we found a star cluster at the edge of the disk of the Milky Way at a distance of 10 kpc from the center,which orbited with a velocity of 300 km/s,what would be the total mass enclosed within this radius?

Question

Quizplus · Accepted Answer

This question requires the use of the virial theorem, which relates the total kinetic energy of a system to its potential energy and enables us to estimate the mass of the system. The virial theorem states that 2K +U =0, where K is the total kinetic energy of the system and U is its potential energy.
The kinetic energy of the star cluster can be estimated using its velocity and distance from the center of the Milky Way:
K = 0.5mv^2
where m is the mass of the star cluster and v is its velocity.
Using units of kpc and km/s, we have:
K = 0.5mv^2 = 0.5×m×(300 km/s)^2 = 45m
The potential energy of the star cluster is related to the total mass enclosed within its radius, M(<10 kpc), and the gravitational constant, G:
U = - GmM(<10 kpc) / R
where R is the radius of the star cluster and is assumed to be equal to 10 kpc for this question.
Rearranging this equation and substituting for U and K from the virial theorem gives:
M(<10 kpc) = - Rv^2 / (2G) = 2.25×10^11 M⊙
Thus, the total mass enclosed within the radius of the star cluster is approximately 2 × 10^11 solar masses, which corresponds to answer choice C.

If We Found a Star Cluster at the Edge of the Disk