SOLUTIONS TO EXERCISE 20


(a) If light is moving in a stable orbit, then the orbital velocity at that 
distance is the speed of light (c = 3 * 10^8 m/s).  So let's set the mathematical
expression for orbital velocity equal to the speed of light.

  sqrt(GM/r) = c   Now square both sides to get  GM/r = c^2  and rearrange to get   

   r = GM/c^2

So the radius at which light moves in a stable orbit around an object with mass M
is a function of that mass, the gravitational constant, and the speed of light
squared.  (This is called the Schwarzschild radius.)

(b) If M = 6 * 10^24 kg, then 

    r = (6.67 * 10^(-11)) * (6 * 10^24) / (3 * 10^8)^2  = 4.4 * 10^(-3) m

So the diameter of this black hole would be about one centimeter - about the
size of a small marble or gumball.

(c) The number of Earth-mass black holes needed to equal the mass of the solar
system would be

    (2 * 10^30 kg) / (6 * 10^24 kg/black hole) = 333,000 or so.

Each black hole is a sphere with diameter 0.9 cm.  We can thus make a rough
estimate and say that, in a container, each black hole would occupy one cubic
centimeter.  So a container with volume 333,000 cm^3 would hold our solar system's
worth of matter.  A typical bathtub would do - its dimensions would be about
5 feet long, 2.5 feet wide and 1 foot deep, or

   (150 cm long) * (75 cm wide) * (30 cm deep) = 337,500 cm^3.