Astronomy 101 Lecture 21, Apr. 9, 2003

Black Holes– (Chapter 22.5 – 22.8 in text)

In the gravitational collapse preceeding a supernova Type II, the core condensed into a ball of neutrons. After blowing off the outer layers in the supernova, neutron star is left ‘naked’. The gravity squeezing the neutron star is immense due to its tiny size and large mass. For neutron stars with mass below about 3 solar masses, the degenerate neutron pressure is enough to withstand gravity and an equilibrium results.

What happens if the neutron star is above 3 solar masses?

In this case, the gravitational force is larger than even degenerate neutron pressure can withstand. Gravity wins the final battle and the star collapses into a

Black Hole

Stars that are originally above about 25 solar masses typically leave remnants after supernova explosion of more than 3 solar masses and become black holes. The theory of Relativity is needed to describe Black Holes.

Special Relativity:In 1905, Einstein published a paper establishing special relativity, built on a seemingly simple and innocuous propositions:

The laws of physics are the same in all inertial frames of reference -- and the speed of light is the same in all frames too.A frame of reference is simply the x, y, z

axes that define the space, for example in this room, x is to right, y is up and z is from the blackboard towards you.

An inertial frame is one that is moving at a constant velocity, relative say to our laboratory frame. (non-accelerating frame)

blackboard/screen

x

y

z (lab) frame of reference

vObserver on train sees ball rise and fall with same laws of motion and same acceleration as observer on the ground. Both are in inertial frames.

The assumption that light moves with the same speed in all reference frames is actually pretty odd:

Shoot a bullet at 1000 km/hr from a car moving at 100 km/hr; observer on the ground will see bullet moving at 1100 km/hr in his frame.Einstein’s special relativity says that if you shoot a laser beam (light travelling at speed c = 3x105 km/s) from a space ship moving at 0.5c (1/2 times speed of light), observer sees the light moving at c, not 1.5c ! This is counterintuitive, but true!

In fact, nothing can move faster than the speed of light in any frame of reference.

Other oddities from special relativity for fast moving objects: clock on moving object, seen by observer, runs slow; length of object seen by observer is shrunk.

General relativity:

Einstein noted that we can’t tell the difference between acceleration of a frame of reference, and the presence of a gravitational force.

Imagine two persons in closed elevators. One elevator is stationary in the vicinity of the earth and the person feels a ‘force’ that pushes him down. The other elevator is accelerating upward (with acceleration g). The person in this one feels a force that pushes him toward the floor.

The two feel the same thing – so conclude gravitational force is associated with acceleration.

In general relativity, instead of talking about masses attracting other masses by gravity, we think of a mass warping the space around it, so that objects move on curved trajectories in the curved space, rather than in straight lines as in Newton’s First Law - in the absence of other forces.

In General Relativity, Einstein said that instead of there being gravitational forces, the very fabric of space is ‘warped’ by the presence of masses nearby. Space is curved due to the presence of mass. Objects in this warped space do not follow straight lines in the absence of a force as Newton’s First Law said, but follow ‘geodesics’ in the curved space. The geodesic path is exactly what would be predicted from Newton’s law of gravity for masses that are not too large. For large masses, there are differences.

mass

More mass, more warp (equivalent to stronger attraction)

Warping of two-dimensional space due to the presence of a mass at the center.

A 2-dimensional analogy: A marble on such a warped space will roll toward the central mass – in exactly the same way that it would in flat space under Newton’s Law of Gravity.

Light, as well as matter particles follow the curved geodesics of warped space!

Escaping from the vicinity of a large mass.

Consider a rocket leaving the earth’s surface. If we shoot it up with a small velocity, it rises to a maximum height and returns to earth. If we increase the velocity, we reach a critical ESCAPE VELOCITY, vesc at which the rocket just leaves the earth’s pull and drifts outward to have zero velocity infinitely far from earth.

For a rocket leaving the earth’s surface, vesc = 11 km/s.

For objects escaping from planets with different mass and radius, the escape velocity varies as vesc ~ √(M/R) , so for larger masses or smaller radii, the escape velocity grows. If earth shrunk to 1/100 of its current size and kept the same mass M, vesc would increase to 110 km/s.

If Earth shrunk to about 1 cm, the escape velocity would reach c = speed of light!

Since nothing can travel faster than the speed of light, an earth mass contained within 1 cm could not emit anything at all – no particles, no light, no nothing !

M

R

A mass which is contained within a radius smaller than that from which light – or anything else - can escape is a BLACK HOLE (BH).

How big is the ball of matter within the black hole and how is it configured? Those are rather meaningless questions, since nothing can emerge from the BH, so we can never observe details inside. The only things that we can know about a BH are its Mass, its electric charge, and its spin. We think however that the collapse of the matter continues until all mass is located at a point – the BH singularity.

What does have meaning is the imaginary sphere centered on the BH within which nothing escapes and outside of which communication to the outside world is possible. This sphere is called the EVENT HORIZON and the radius of the event horizon is called the SCHWARZCHILD RADIUS.

In General relativity, we say the Black Hole warps the space surrounding it so severely that things can only fall into it, never escape.

An analogy: People live on a rubber sheet. If they try all to congregate at one place, the sheet deforms.

When enough of them get to the spot, the sheet forms a pocket from which no one can escape.

Tests of general relativity

The laws of general relativity modify the way objects move near large masses (relative to their Newtonian motion). For example, Mercury’s orbit close to the sun is changed so that its perihelion ‘precesses’ with time.

The precession of Mercury’s orbit due to general relativity is only 43 arc seconds per century, but observations confirm it. (There are additional sources of precession)

Starlight passing near the sun is bent (general relativity says that the sun warps the space, so the geodesic for light is a curve). This effect is observed (only visible when the moon eclipses the sun and we can see a distant star going behind the sun).

Does a black hole suck up everything near it? NO: far from the black hole, the mass causes objects (other stars, planets …) to orbit around it in the same way as any mass does in Newtonian gravity.

But when an object comes close to the event horizon, general relativity modifies the Newtonian orbits. And near the black hole, the tidal forces (e.g. differences in force on the head and foot of a person trying to stand near a BH) will rip the object apart. The motions of the electrons and protons ripped out of the object become very rapid as the object falls toward the BH and can emit X-rays that we can observe.

Light emitted from just outside a BH is gravitationally red-shifted.

The light emitted has to climb ‘uphill’ in escaping from the deep gravitational well. Unlike the rocket trying to escape the earth, light cannot lose energy by slowing down [light travels always at c ! Special relativity] So the light loses energy the only way it can – by reducing its frequency (increasing its wavelenth). Remember: E = hf (f = c)

Forming Black Holes:

The collapsing neutron core of a star about to undergo a supernova explosion turns into a black hole if its mass exceeds about 3 solar masses.

A 1.4 solar mass neutron star has a radius of about 10 km and a Schwarzchild radius of 4.2 km, so it does not become a black hole.

If however, mass is added to this neutron star, its radius decreases slightly (more compression from the weight) and the Schwarzchild radius grows until it is equal to its actual radius. This neutron star then collapses into a black hole.

Light from the event horizon is red shifted to infinite wavelength, which is equivalent to having no light at all. (see next slide)

Also, if we were to drop a clock into a black hole, it would slow down and as it approached the event horizon, its time would stand still. Thus from the outside, we would never actually see an object fall into a BH, though a person riding into the black hole (assuming she were not ripped apart by the tidal force) would not experience anything special on passing through the Schwarzchild radius.

Wavelength of light is shifted to red (longer ) as it leaves the vicinity of a black hole.

How do we observe a Black Hole?

By definition, we can’t see light from one. But we can observe the effect of a black hole on matter outside the BH.

For example, in binary star pairs where one has evolved into a BH, the effects on the companion can be violent. We noted that if material from the companion accretes onto a neutron star (or BH), it is ripped apart into electrons and protons which emit X-rays.

Cygnus X-1 is such a binary, and we see very bright X-rays emitted at a location where nothing is visible, near to a blue B-type supergiant.

In Cygnus X-1, the companion supergiant is known from its position on the HR diagram to have about 25 solar masses.

The motion of the companion in the spectroscopic binary shows that the unseen companion a period of 5.6 days, and a mass of about 10 solar masses – well above the limit of 3 solar masses for a BH to form.

The Doppler shift of spectral lines shows that matter is streaming from the supergiant to the unseen companion.

Strong X-ray radiation is observed. The time variations of the X-ray bursts occur very quickly, indicating that the size of the emission region is less than a few hundred km (a large object emitting from across its size will wash out any rapid local time variations).

It looks like a BH, it walks like a BH, it quacks like a BH – it must BE a Black Hole !

We not only find Black Holes as the end result of stellar evolution – we believe that there is black hole at the center of our galaxy (and many others).

The evidence is based on Kepler’s laws. We see stars near the center of the galaxy that are orbiting very rapidly around a central point. Knowing the distance to them and the period, we can use

P2 = a3/(Mhole + mstar)

to estimate the mass of the black hole candidate. Knowing the size of the orbit also tells us the upper limit on the size of the black hole candidate.

There is nothing besides a black hole that could be so massive and so small.

Mass of the supermassive black hole at the center of the galaxy is around 3 million solar masses.

Do black holes really exist?

We have good very good observational evidence for a very compact objects that do not radiate by themselves at all, and have Black Hole sized masses.

The laws of general relativity predict black holes, and general relativity is reasonably well tested in other ways, so some confidence that it is correct.

So, while we never can prove something in Nature is absolutely true (can only prove something is not true), scientists are confident that Black Holes exist.

Searches in labs on earth are underway to show the existence of Black Holes.

New experiments on earth are starting to detect gravitational waves that would be emitted during the collapse of a supernova core into a black hole.

LIGO interferometer in Hanford Washington; two 4 km lasers sense deformations due to gravity waves. Another similar interferometer in Louisiana to allow simultaneous detection and reduce background noise.

Plans now to build a larger interferometer in space.

Big chunk of matter (maybe another star) spiralling into Black Hole would generate a gravitational wave ‘chirp’ that may be detectable in LIGO.

Gravity wave intensity

time

Mt. Blanc

Lake Geneva

8.6 km

Large Hadron Collider near Geneva Switzerland, where particle physicists could concievably make mini black holes in the laboratory.

Black holes were predicted by Stephen Hawking to ‘evaporate’ electrons and positrons, making them, at least theoretically, observable. Very high energy accelerator experiments could produce mini black holes that would evaporate quickly into a burst of observable particles.

Geneva city

Simulation of a mini-black hole production and evaporation in a high energy accelerator experiment.