X-ray pulses from Cygnus X-1, a celestial x-
ray source, have been recorded during high-
altitude rocket flights. The signals can be
interpreted as originating when a blob of ion-
ized matter orbits a black hole with a period
of 7.84 ms.
If the blob were in a circular orbit about
a black hole whose mass is 13.5 times the
mass of the Sun, what is the orbit radius?
The value of the gravitational constant is
6.67259 × 10−11 N · m2/kg2 and mass of the
Sun is 1.991 × 1030 kg.
Answer in units of km.
To find the orbit radius of the blob around the black hole, we can use the law of universal gravitation and the period of the orbit.
We can start by rearranging the formula for the period of an orbit to solve for the radius of the orbit.
T = 2π√(r^3/GM)
Where:
T is the period of the orbit (given as 7.84 ms = 0.00784 s)
r is the orbit radius (to be determined)
G is the gravitational constant (given as 6.67259 × 10^−11 N · m^2/kg^2)
M is the mass of the black hole (given as 13.5 times the mass of the Sun = 13.5 * 1.991 × 10^30 kg)
Now, let's plug in the known values and solve for r:
0.00784 s = 2π√(r^3/(6.67259 × 10^−11 N · m^2/kg^2 * 13.5 * 1.991 × 10^30 kg))
To solve for r, we need to isolate it on one side. We can square both sides and divide by (2π)^2:
(0.00784 s)^2 / (4π^2) = r^3 / (6.67259 × 10^−11 N · m^2/kg^2 * 13.5 * 1.991 × 10^30 kg)
Now, let's calculate the right side of the equation:
r^3 = (0.00784 s)^2 * (6.67259 × 10^−11 N · m^2/kg^2 * 13.5 * 1.991 × 10^30 kg) / (4π^2)
r^3 = 0.3850597178 × 10^16 m^3/kg · N
To find r, we can take the cube root of both sides:
r = ∛(0.3850597178 × 10^16 m^3/kg · N)
r ≈ 3061055.09 m
Finally, let's convert the answer to kilometers:
r ≈ 3061055.09 m / 1000 = 3061.05509 km
Therefore, the orbit radius of the blob around the black hole is approximately 3061.05509 km.
Assistance needed.
Please type your subject (not your school name) in the School Subject box. Any other words are likely to delay responses from a teacher who knows that subject well.