Black Holes - Cornell Universityhosting.astro.cornell.edu/.../20BlackHoles.pdf · Lec 20: Black...

Lec 20: Black Holes 1

Black Holes

Lecture 20


Lecture Topics■ Schwarzchild radius

■ black hole radius

■ The density of black holes■ Properties of black holes■ Falling into a black hole


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


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


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


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).


Converting PE to KE■ All the PE the rock had when it started is

converted to KE at impact.

■ which means


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


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


How dense are black holes?■ The average density of a black hole is:

but

More massive black holes are less dense!


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


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


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


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 in


Falling into a black holeBlack Hole

A

“A” falls in while“B” stays outside.

What happens if you fall in?

B

Clock


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.


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.


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

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


Circling a Black Hole

Orbiting Flying along event horizon


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


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


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


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


Nearing a Black Hole

Approaching “Entering”


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)


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?


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.


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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