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 determine the orbit radius of the blob, we can use Kepler's Third Law of Planetary Motion, which relates the orbital period and radius of a celestial object to the mass of the central object.
Kepler's Third Law is given by the equation:
T^2 = (4π^2/GM)r^3,
where T is the orbital period, G is the gravitational constant, M is the mass of the central object, and r is the orbit radius.
In this case, the orbital period (T) of the blob is given as 7.84 ms = 7.84 × 10^-3 seconds.
The mass of the black hole (M) is 13.5 times the mass of the Sun, which is 13.5 * 1.991 × 10^30 kg.
The gravitational constant (G) is 6.67259 × 10^-11 N · m^2/kg^2.
We can rearrange the equation to solve for the orbit radius (r):
r^3 = (T^2 * GM) / (4π^2)
Substituting the given values and solving for r:
r^3 = [(7.84 × 10^-3)^2 * (6.67259 × 10^-11) * (13.5 * 1.991 × 10^30)] / (4π^2)
r^3 = (6.161856 × 10^-17 * 1.79922 × 10^32) / (4π^2)
r^3 = 1.1073894352 × 10^15 / (4π^2)
r^3 ≈ 8.85058885055 × 10^13
Taking the cube root of both sides to solve for r:
r ≈ ∛(8.85058885055 × 10^13)
r ≈ 4.313 × 10^4 m
To convert meters to kilometers, divide by 1000:
r ≈ 4.313 × 10^4 / 1000 km
r ≈ 43.13 km
Therefore, the orbit radius of the blob around the black hole is approximately 43.13 km.