black hole

1308 Black Hole

Black holes, the Schwarzschild Radius, and the Event Horizon

If a nonrotating spherical mass distribution with total mass M has a radius less than its Schwarzschild radius RSR_S, it is called a black hole. The gravitational interaction prevents anything, including light, from escaping from within a sphere with radius RSR_S.

RS=2GMc2\begin{aligned} R_S = \frac{2GM}{c^2} \end{aligned}

RSR_S - Schwarzschild radius of a black hole.
GG - Gravitational constant.
MM - Mass of black hole.
cc - Speed of light in vacuum

Exercises

38, 39, 40

(13.38) Mini Black Holes. Cosmologists have speculated that black holes the size of a proton could have formed during the early days of the Big Bang when the universe began. If we take the diameter of a proton to be 1.0×10−15m1.0 \times 10^{-15}m, what would be the mass of a mini black hole?

Solution

RS=2GMc2⇒M=RSc22G=0.5⋅10−15⋅(3.0⋅108)22G=3.37⋅1011kg\begin{aligned} R_S &= \frac{2GM}{c^2}\\ \To M &= \frac{R_S c^2}{2G}\\ &= \frac{0.5\cdot 10^{-15} \cdot (3.0 \cdot 10^8)^2}{2G}\\ &= 3.37 \cdot 10^{11}kg \end{aligned}

(13.39) At the Galaxy's Core. Astronomers have observed a small, massive object at the center of our Milky Way galaxy (see Section 13.8). A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about 200 km/s.
(a) Determine the mass of the object at the center of the Milky Way galaxy. Give your answer both in kilograms and in solar masses (one solar mass is the mass of the sun).
(b) Observations of stars, as well as theories of the structure of stars, suggest that it is impossible for a single star to have a mass of more than about 50 solar masses. Can this massive object be a single, ordinary star?
(c) Many astronomers believe that the massive object at the center of the Milky Way galaxy is a black hole. If so, what must the Schwarzschild radius of this black hole be? Would a black hole of this size fit inside the earth's orbit around the sun?

Solution

Let the radius of the material orbiting the massive object be rr and the mass of the massive object be MM.

r=7.5⋅365⋅24⋅3600⋅c=7.1⋅1016mv=GMr⇒M=v2rG=(2⋅105)2⋅7.1⋅1016G=4.3⋅1037kg=(2.1⋅107)MS\begin{aligned} r &= 7.5 \cdot 365 \cdot 24 \cdot 3600\cdot c = 7.1 \cdot 10^{16}m\\ v &= \sqrt{\frac{GM}{r}}\\ \To M &= \frac{v^2 r}{G}\\ &= \frac{(2\cdot 10^5)^2 \cdot 7.1 \cdot 10^{16}}{G}\\ &= 4.3 \cdot 10^{37}kg = (2.1\cdot 10^7)M_S \end{aligned}

If the massive object is a black hole, the radius is

RS=2GMc2=2G⋅4.3⋅1037c2=6.37⋅1010m\begin{aligned} R_S &= \frac{2GM}{c^2}\\ &= \frac{2G \cdot 4.3 \cdot 10^{37}}{c^2}\\ &= 6.37 \cdot 10^{10}m \end{aligned}

(13.40) In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,00030,000 km/s.
(a) How far are these clumps from the center of the black hole?
(b) What is the mass of this black hole, assuming circular orbits? Express your answer in kilograms and as a multiple of our sun's mass.
(c) What is the radius of its event horizon?

Solution

Todo