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**Astronomy 204 Worksheet: Properties of Black Holes **

Black Holes are compact objects that have zero size – a singularity. They are known for their gravity being so powerful that even light cannot escape. In this worksheet we will use simple concepts from mechanics to approximate some equations for black holes – exact results require advanced concepts from relativity.

1) Remember the concept of escape velocity – how fast does an object of mass m have to be traveling to escape from another (uniform, spherical) object of mass M and radius R. Derive the equation for escape velocity v using the conservation of energy principle (yes, you should lose selfrespect if you peek in your text).

2) Now apply the concept of escape velocity to black holes. Set the escape velocity equal to the speed of light and solve for the size of the region from which light cannot escape (R

Sch

, the

Schwarzchild Radius).

3) In addition, to light not being able to escape, it is also possible for light to go into (circular) orbit around the black hole. This orbit is known as the Photon Sphere. Calculate the radius of the photon sphere R

PS

for a black hole starting with a summation of force equation. (Note the correct value 1.5

R

Sch

).

4) Use the formulae you have derived above to complete the following table.

**Mass M **

**1 kg **

**R**

**Sch**

**R**

**PS**

1 M

**1 M**

**4.5 MM**

5) Convert your answer to part 2 to an equation for R

Sch where mass is in solar units.