Transcript lecture21

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)
v
blackboard/screen
y
x
z
(lab) frame
of reference
Observer 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.
Warping of two-dimensional
space due to the presence of a
mass at the center.
mass
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.
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!
More mass, more warp
(equivalent to stronger
attraction)
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!
M
R
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 !
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
(lf = c)
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.
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.
Wavelength of light is shifted to red (longer l) 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
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.
Mt. Blanc
Lake Geneva
Geneva city
Large Hadron Collider near Geneva Switzerland, where particle physicists
could concievably make mini black holes in the laboratory.
Simulation of a mini-black hole production and evaporation in a high
energy accelerator experiment.