The radius R_h, awl mass M_h of a black hole are related by R_h = 2G M_h/c^2, wh

Question

The radius R_h, awl mass M_h of a black hole are related by R_h = 2G M_h/c^2, where c is the speed of light. Assume that the gravitational acceleration a_g of an object at a distance r_0 = 1.001 R_h from the center of a black hole is given by a_g = GM/r^2 _0 (true for large black holes as it is for earth). Take, M_h = 10 solar mass. (a) Calculate Schwarzschild radius, R_h of the black hole. Not even light can escape from the pull of black hole if it gets closer than the Schwarzschild radius. Which means what goes inside this radius, never comes back due to massive pull of the black hole. So, you never see a black hole with telescope. But how can we then detect an black hole? The following answer is courtesy of Yale astrophysicist Kevin Schawinski. "The first way we detect black holes is by their gravitational influence. For example, at the center of the Milky Way, we see an empty spot where all of the stars are circling around as if they were orbiting a really dense mass. That's where the black hole is. The second way is by observing the matter falling into the black hole. As matter falls in, it settles in a disk around the black hole that can get very hot. Some of the energy liberated from falling in is turned into light, which we can then see, for example, in X-rays." (b) In terms of M_h, find a_g at r_o. (c) Does a_g at r_o increase or decrease as M_h, increases? (d) What is a_g at r_o for a very large black hole whose- mass is 1.55 times 10^12 timers the solar mass of 1.99 times 10^30 kg? (c)If an astronaut of height 1.70 in is at r_o with her feet clown, what is the difference in gravitational acceleration between her head and feet? (f) Is the tendency to stretch the astronaut severe?

Explanation / Answer

Rh = 2GM/c2

Mh = 1.99e+31 , 10 times solar mass

a) Rh = 2*6.67e-11*1.99e+31/9.0e+16 = 2.9496 e+4 m

b) ro = 1.001 Rh = 1.001*2*6.67e-11*Mh /9.0e+16 = 1.484e-27 Mh

ag = GM/ro2 = GM/1.0012 * ( c4 /4G2M2)

                       = 81.0e+32 /4*1.0012 *6.67e-11 * Mh

                       = 3.03 e+43/Mh

c) ag decreases as Mh increases and ro increases as Mh increases

d) M = 1.55e+12*1.99e+30 = 3.0845e+42

ro = 1.001*2*6.67e-11*3.0845e+42/9.0e+16 = 4.5765e+15

ag = 3.03 e+43/3.0845e+42 = 9.823 m/s2

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