Lecture 25: Black Holes
Astronomy 101/103
Terry Herter, Cornell University
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Lecture
Topics
  • Escape velocity
  • Definition of a black hole
  • Schwarzchild radius (black hole radius)
  • The density of black holes
  • Properties of black holes
  • Falling into a black hole

Dark
Stars
  • Rev. John Mitchell - 1783
  • An object more massive than the Sun could have an escape velocity greater than the speed of light!
  • Today we call this object a black hole.
    • An object from which no light can escape.

Dropping
Rocks
  • Suppose we drop a rock from very far out in space. How fast is it going when it hits?
  • The potential energy of the ball is:

Potential
=>
Kinetic
Energy
  • The kinetic energy of the rock when it hits is:

  • This KE comes from the conversion of PE into KE (by gravity).

Converting
PE to KE
  • All the PE the rock had when it started is converted to KE at impact.
  • which means

Impact
Velocity
  • Putting in some numbers


Escape
Velocity
  • Reverse the problem:
  • What is the minimum speed upward the rock must have to escape the earth.
  • It is the same as if you let it fall (only going the other way)!

 

Making a
"Dark Star"
  • Suppose the escape velocity of an object was equal to the speed of light.

Schwarzchild
Radius

  • Rs is the radius which an object must have to become a black hole. Rs is given by:

    Rs = 3 x M (in km)
    • M in solar masses
    • Derived by Karl Schwarzchild using General Relativity.

Size of
Black Holes

Object

Mass (Msun)

Rs

Star

10

30 km

Star

3

9 km

Sun

1

3 km

Earth

3 x 10-6

9 mm


Density of
Black Holes
  • The average density of a black hole is:



    but

More massive black holes are less dense!

  • For a black hole with M = Msun:
  • For M = 10 Msun :

Very massive
Black Holes

  • Suppose we could make a black hole a big as the solar system, e.g. Rs = 40 AU.

  • A 2 x109 Msun black hole cannot be formed by a single star.

The Event
Horizon



  • The event horizon is located at Rs.
  • Anything inside the event horizon is gone from sight forever (nothing can escape).

Time
Dilation

  • Recall that clocks run slower on the surface of the earth than on a mountain top.
  • Viewed from space clocks slow down as they approach the event horizon.
  • At the event horizon, the clock stops!

Gravitational
Redshift

  • The gravitational redshift gets larger and larger as objects approach the event horizon.
  • At the event horizon the redshift becomes infinite!

Falling
Into a
Black Hole


Person A:
Falling into a
Black Hole


View from
Person A

Person falling in sees:

  • Clock B getting faster.
  • Photons coming from person B and from the rest of the universe are blueshifted.
    • Visible photons become X-rays and g-rays!
  • The tidal forces and the rain of high energy photons will be very bad for the person falling into the black hole.

Tides

  • Tidal forces are due to the difference in the gravitational force across an object.
  • Near a black hole gravity changes very rapidly with distance.
    • neutron stars too!
  • Tides pull on the object and stretch it in the direction of the star.

Tides
(cont'd)

Black Holes,
Dangerous?

Are Black Holes Dangerous?

  • BHs don't go around scooping up people and stars.
  • Only if you get very close to one is there a problem.
  • Replacing the sun with a 1 Msun black hole would not change the orbits of the planets!
    • But we'd have a problem keeping warm.
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