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

Answer:

The mass is 6.81x10^37 kg

Explanation:

The orbital velocity is given as

$V=\sqrt{\dfrac{GM}{R}}$

Rearranging this equation gives

$M=\dfrac{V^2 R}{G}$

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

$M=\dfrac{V^2 R}{G}\\M=\dfrac{(4\times 10^5)^2* (2.838 \times 10^{16})}{6.67\times 10^{-11}}\\M=6.81 \times 10^{37} kg$

So the mass is 6.81x10^37 kg

Answer 2

Answer:

Mass of black hole = 1.5297 × 19^29kg

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