A black hole is an object with mass, but no spatial extent. It truly is a particle. A black hole may form from a dead star. Such a black hole has a mass several times the mass of the Sun. Imagine a black hole whose mass is fifty times the mass of the Sun.
(a) Would you expect the period of an object orbiting the black hole with a semimajor axis of 1 AU to have a period greater than, less than, or equal to 1 yr?
(b) Use the equation
T^2 = (4𝜋^2/GM) a^3 to calculate this period.
To answer these questions, we can use the equation you mentioned: T^2 = (4𝜋^2/GM) a^3, where T is the period of the object's orbit, G is the gravitational constant, M is the mass of the black hole, and a is the semimajor axis of the object's orbit.
(a) If we consider a black hole with a mass fifty times that of the Sun, we can substitute M = 50 M_sun into the equation. However, we need to compare the period to 1 year, which is equivalent to approximately 31,556,952 seconds.
We can rearrange the equation to solve for T:
T^2 = (4𝜋^2/GM) a^3
T = √((4𝜋^2/GM) a^3)
Now we can plug in the values:
T = √((4𝜋^2 / G(50 M_sun)) a^3)
(b) To calculate the period, we need to know the value of G, the gravitational constant. G is approximately equal to 6.6743 × 10^-11 m^3 kg^-1 s^-2.
To calculate the period, we also need to know the value of a, the semimajor axis of the object's orbit. In this case, a = 1 AU, which is equivalent to approximately 1.496 × 10^11 meters.
Now we can substitute the values into the equation and calculate the period:
T = √((4𝜋^2 / (6.6743 × 10^-11)(50 M_sun)) (1.496 × 10^11)^3)
After plugging in the values and performing the calculations, we can determine the period of the object's orbit around the black hole.