Question

The U.S.S. Enterprise (NCC‑1701) approaches an unknown system with a Black Hole at the center. One of the planets in this system is observed to have an orbit with a radius of R₁ = 2.6 AU and a period of T₁ = 0.4 years. What is the mass of the Black Hole compared to the mass of the Sun?

The mass 109.81 solar mass

Another planet in the system is observed to have an orbital radius of R₂ = 6.6 AU, how long does it take for this planet to complete one orbit around the Black Hole?

Answer 1

part A)
given
R1 = 2.6 AU = 2.6*1.496*10^11 m
T1 = 0.4 years = 0.4*365*24*60*60 s
a) Let M is the mass of the block Hole

we know, T1 = 2*pi*R1^(3/2)/sqrt(G*M)

T1^2 = 4*pi^2*R1^3/(G*M)

M = 4*pi^2*R1^3/(G*T1^2)

= 4*pi^2*(2.6*1.496*10^11)^3/(6.67*10^-11*(0.4*365*24*60*60)^2)

= 2.189*10^32 kg

= 2.189*10^32/(1.99*10^30)

= 110*M_sun <<<<<<<----------------Answer

part B)
we know,

According to Kepler's third law

square of the time period is proportional to cube of the semi major axis of the planet.

T^2 is proportional to R^3

so, (T2/T1)^2 = (R2/R1)^3

T2/T1 = (R2/R1)^(3/2)

T2 = T1*(R2/R1)^(3/2)

= 0.4*(6.6/2.6)^(3/2)

= 1.62 years <<<<<<<<----------------Answer