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Black Holes
Lecture 20
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Lecture Topics
■
Schwarzchild radius
■
black hole radius
The density of black holes
■ Properties of black holes
■ Falling into a black hole
■
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Making a “Dark Star”
■
Suppose the escape velocity of an object was
equal to the speed of light.
Rs = Schwarzchild radius
Putting in numbers:
Rs = 3M
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Rs in km
M in solar masses
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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:
M, R = mass & distance from center of
Earth
m = mass of rock
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Gravitational Potential Energy
■
The potential energy difference between
two points surrounding a mass is:
GMm GMm
∆ PE =
−
R1
R2
■
R2 > R1
When R2 -> infinity (very large distance)
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Potential ⇒ Kinetic Energy
■
The kinetic energy of the rock when it hits is:
where v = velocity of the rock at impact
m = mass of the rock.
■
This KE comes from the conversion of PE
into KE (by gravity).
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Converting PE to KE
■
All the PE the rock had when it started is
converted to KE at impact.
■
which means
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The Escape Velocity
■
■
■
Reverse the problem:
What is the minimum speed upward the rock
must have to escape the earth.
It’s the same as if you let if fall (only going the
other way)!
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A Black Hole is Born
Black holes bend space
Clip from “Space 1999” – bad 60’s SciFi series
How big are black holes?
Object
Star
Star
Sun
Earth
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Mass (Msun)
10
3
1
3 x 10-6
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Rs
30
9
3
9
km
km
km
mm
How dense are black holes?
■
The average density of a black hole is:
but
More massive black holes are less dense!
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Densities (cont’d)
■
For a black hole with M = Msun.
ρ = 2 x 1016 g/cm3
■
For M = 10 Msun.
ρ = 2 x 1014 g/cm3
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Very Massive Black Holes
■
Suppose we could make a black hole a
big as the solar system, e.g Rs = 40 AU.
Then
and
M = 2 x109 Msun
ρ = 0.005 g/cm3 (!)
- A 2 x109 Msun black hole can not be
formed by a single star.
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Warped Space Time
The Event Horizon
■
■
The event horizon
is located at Rs.
Rs
Anything inside the
event horizon is
gone from sight
forever (nothing can
escape).
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Event
Horizon
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!
■
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Gravitational Redshift
The gravitational redshift gets larger
and larger as objects approach the
event horizon.
■ At the event horizon the redshift
becomes infinite!
■
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Falling in
Falling into a black hole
What happens if you fall in?
Black Hole
Clock
A
“A” falls in while
“B” stays outside.
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B
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Person A falling into BH
Person outside BH sees
1. Photons from A redshifted.
2. Clock A slow down.
3. Person A stretched and
ripped apart by tidal forces.
Black Hole
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Person falling in sees
■
If person a “paused” while falling in then he
would see:
■
■
Clock B is running very fast.
Photons coming from person B and the rest of the
universe are blueshifted.
■
■
Visible photons become X-rays and γ-rays!
The tidal forces will be very bad for the person
falling into the black hole.
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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.
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Tides
People lying in 12 ft.
spaceship near the walls.
Tides pull the two people
towards the outer walls.
Black Hole
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Distance
from BH
5000 km
1000 km
100 km
20 km
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Tidal Force
( g’s )
1.2
144
1.4x105
1.8x107
Circling a Black Hole
Orbiting
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Flying along
event horizon
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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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What’s on the other side?
The singularity
Relevance of Black Holes
■
Black holes can be formed when a
massive star collapses
■
■
(star: M > 15 Msun)
Center of the Milky Way (our Galaxy)
■
■
Mcore > ~ 4 Msun
A 2x106 Msun black hole
Centers of Quasars
■
Black holes up to 2x109 Msun
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In-Class Question
Rs = 3 × M
Rs in km, M in Msun
1) What is the radius of a 1,000,000 Msun black hole (in
km)?
a) 106
b) 30
c) 3x106
d) 3x107
⇑
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In-Class Question
Rs = 3 × M
Rs in km, M in Msun
1) What is the radius of a 1,000,000 Msun black hole (in
km)?
a) 106
b) 30
c) 3x106
d) 3x107
⇑
2) What is the event horizon of a Black Hole?
a) Place where tides rip things apart
⇒ b) Place from which nothing can escape
c) Place to go for a drink
d) Place where photons are emitted
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Nearing a Black Hole
Approaching
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“Entering”
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The impact velocity
■
Putting in some numbers
Mearth = 6 x 1024 kg
Rearth
G
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= 6400 km = 6.4x106 m
= 6.67x10-11 N-m2/kg2 (m3/kg/s2)
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Escape velocity
■
■
■
Escape velocity is the speed an object would
need to escape from a celestial body.
The escape velocity depends on mass.
Examples:
■
■
■
■
■
■
Earth:
Moon:
1 km asteroid:
Sun:
White Dwarf:
11.2 km/sec (25,000 mph)
2.4 km/sec
1.3 m/sec (you could jump off it!)
618 km/sec
6000 km/sec !!
How high can the escape velocity get?
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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.
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Making a “Dark Star”
■
Suppose the escape velocity of an object was
equal to the speed of light.
Rs = Schwarzchild radius
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