Galaxy mass. among the globular clusters orbiting a distant galaxy, one is moving at 420 km/s and is located 11 kpc from the center of the galaxy. assuming the globular cluster is located outside most of the mass of the galaxy, what is the mass of the galaxy? (hint: use the formula for circular velocity . remember to convert units to meters and kilograms.) express your answer in billions of solar masses.

4.50 × 10²

Explanation:

Given:

Velocity of orbiting = 420 km/s = 4.2 × 10⁵ m/s

Distance = 11 kpc = ( 11 × 3.086 × 10¹⁹) m = 3.394 × 10²⁰ m

now,

the cluster is orbiting due to the balancing of the centripetal force and the force due to the gravitational pull by the galaxy

thus,

we have

force due to the centripetal force =   ............(1)

where,

m is the mass of the cluster

v is the velocity of the cluster

r is the distance between the galaxy and the cluster

also,

the force due to the gravitational pull = .......(2)

here,

G is the gravitational force constant =  6.67 × 10⁻¹¹ Nm⁻²kg⁻²

M is the mass of the galaxy

on equating the equation (1) and (2), we have

or

on rearranging and substituting the values, we get

M =

or

M = 8.96 × 10⁴¹ kg

Now,

1 Solar mass = 1.99 × 10³⁰ kg

thus,

1 kg =

Hence,

8.96 × 10⁴¹ kg =  8.96 × 10⁴¹ kg  ×

or

= 4.50 × 10¹¹

or

= ( 4.50 × 10² ) × 10⁹ = 4.50 × 10²