Consider dropping a Professor with an estimated mass of 80 kg from a safe, large distance, onto a SMBH with a mass of 10^7M⊙. The Prof makes an impact at the Schwarzschild radius, whereby he converts all of the gained kinetic energy into radiation with a 100% efficiency. Estimate this amount of energy, and express the result in the units of 1-MT hydrogen bombs; one of those releases about 4×10^22erg if dropped carelessly. Neglect the relativistic effects.
To estimate the amount of energy released when the professor impacts the supermassive black hole (SMBH) at the Schwarzschild radius, we can calculate the gravitational potential energy gained by the professor as he falls from the safe distance to the Schwarzschild radius. This potential energy will then be converted into radiation with 100% efficiency.
First, we need to calculate the gravitational potential energy gained by the professor as he falls towards the SMBH. The gravitational potential energy formula is:
PE = mgh
Where:
PE = Potential Energy
m = mass
g = acceleration due to gravity
h = height
In this case, we can consider the safe distance as the height h. Since the professor's mass is 80 kg, and the acceleration due to gravity remains constant near the surface of the Earth, we can use a value of 9.8 m/s^2 for g.
Now, we need to determine the height from which the professor is dropped. The Schwarzschild radius (rs) of a black hole is given by:
rs = (2 * G * M) / c^2
Where:
G = gravitational constant
M = mass of the SMBH (given as 10^7M☉, where M☉ is the solar mass)
c = speed of light
Using the values G = 6.67430 × 10^-11 m^3 kg^-1 s^-2 and c = 299,792,458 m/s, we can calculate the Schwarzschild radius.
Next, we need to solve for the height h by rearranging the Schwarzschild radius equation:
h = rs - R
Where:
h = height
rs = Schwarzschild radius
R = radius of the SMBH (can be assumed to be negligible compared to the Schwarzschild radius)
Finally, we can calculate the potential energy gained by the professor:
PE = mgh
With all the variables determined, we can proceed with the calculation.
Once we have the potential energy, we can convert it to the energy released in units of 1-MT (megaton) hydrogen bombs. Since it is given that one hydrogen bomb releases about 4 × 10^22 erg when dropped carelessly, we can use this conversion factor.
Now, let's perform the calculations step by step.