Question

Measurements of star motions at the center of the Andromeda Galaxy, also known as M31, show that stars about 3 light-years from the center are orbiting at a speed of 400 km/s. These stars are suspected to be orbiting a supermassive black hole. Use the orbital velocity law from Mathematical Insight 21.2 to estimate the mass of this black hole.

Answer 1

Mass of black hole = 1.5297 × 19^29kg

Step-by-step explanation:

Orbital velocity equation is given by:

V = Sqrt(GM)/ r

Where G= gravitational constant = 6.673×10^-11Nm^2/kg

r = radius of earth = 63800000m

V = orbital velocity= 400000m

Mb= mass of black hole

Making Mb subject of formular

V^2 = GMb/r

Cross multiply

V^2 × r = GMb

(V^2 × r) / G = Mb

(400000^2 × 63800000) /(6.673× 10^-11)

Mb = (1.0208 ×10^19) / ( 6.673 × 10^ -11)

Mb = 1.5297 × 10^29 kg

Answer 2

The mass is 6.81x10^37 kg

Step-by-step explanation:

The orbital velocity is given as

`V=\sqrt{(GM)/(R)}\\`

Rearranging this equation gives

`M=(V^2 R)/(G)`

Here V is given as 400 km/s of 4 x10^5 m/s

R is given as 3 light years so R is given as m

G is the gravitational constant whose value 6.67x10^-11

By putting the values

`M=(V^2 R)/(G)\\M=((4* 10^5)^2* (2.838 * 10^(16)))/(6.67* 10^(-11))\\M=6.81 * 10^(37) kg`

So the mass is 6.81x10^37 kg