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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:MR 3s = 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:
21 RGMm
RGMmPE −=∆
■ When R2 -> infinity (very large distance)
R2 > R1
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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)!
A Black Hole is Born
Black holes bend space
Clip from “Space 1999” – bad 60’s SciFi series
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How big are black holes?Object Mass (Msun) Rs
Star 10 30 kmStar 3 9 kmSun 1 3 kmEarth 3 x 10-6 9 mm
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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 M = 2 x109 Msun
and ρ = 0.005 g/cm3 (!)
- A 2 x109 Msun black hole can not be formed by a single star.
Warped Space Time
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The Event Horizon■ The event horizon
is located at Rs.■ Anything inside the
event horizon is gone from sight forever (nothing can escape).
Rs
Event Horizon
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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!
Falling in
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Falling into a black holeBlack Hole
A
“A” falls in while“B” stays outside.
What happens if you fall in?
B
Clock
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Black Hole
Person A falling into BH
Person outside BH sees1. Photons from A redshifted.2. Clock A slow down.3. Person A stretched and ripped apart by tidal forces.
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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
Black Hole
Tides pull the two peopletowards the outer walls.
People lying in 12 ft. spaceship near the walls.
Distance Tidal Forcefrom BH ( g’s )5000 km 1.21000 km 144 100 km 1.4x105
20 km 1.8x107
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Circling a Black Hole
Orbiting 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.
What’s on the other side?
The singularity
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Relevance of Black Holes■ Black holes can be formed when a
massive star collapses■ Mcore > ~ 4 Msun (star: M > 15 Msun)
■ Center of the Milky Way (our Galaxy)■ A 2x106 Msun black hole
■ Centers of Quasars■ Black holes up to 2x109 Msun
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In-Class Question
1) What is the radius of a 1,000,000 Msun black hole (in km)?
a) 106 b) 30 c) 3x106 d) 3x107 ⇑
MRs ×= 3
Rs in km, M in Msun
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In-Class Question
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 apartb) Place from which nothing can escapec) Place to go for a drink d) Place where photons are emitted
⇒
MRs ×= 3
Rs in km, M in Msun
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Nearing a Black Hole
Approaching “Entering”
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The impact velocity■ Putting in some numbers
Mearth = 6 x 1024 kg Rearth = 6400 km = 6.4x106 m G = 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: 11.2 km/sec (25,000 mph)■ Moon: 2.4 km/sec■ 1 km asteroid: 1.3 m/sec (you could jump off it!)■ Sun: 618 km/sec■ White Dwarf: 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