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Week #10 Notes
Black Holes: The End
of Space and Time
The Formation
of a Stellar-Mass Black Hole
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Astronomers had long assumed that the most massive
stars would somehow lose enough mass to wind up as
white dwarfs.
When the discovery of pulsars (neutron stars) ended this
prejudice, it seemed more reasonable that black holes
could exist.
If more than 2 or 3 times the mass of the Sun remains
after the supernova explosion, the star collapses through
the neutron-star stage.
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We know of no force that can stop the collapse. (In some
cases, the supernova explosion itself may fail, so the
remaining mass can be much larger than 2 or 3 solar
masses.)
The Formation
of a Stellar-Mass Black Hole
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We may then ask what happens to an evolved 5-, 10-, or 50-solarmass star as it collapses, if it retains more than 2 or 3 solar
masses.
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As far as we know, it must keep collapsing, getting denser and
denser.
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Einstein’s general theory of relativity
suggests that the presence of mass or
energy warps (bends) space and even
time in its vicinity (see figure); recall our
discussion of the Sun’s gravity in Section
10.3.
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The greater the density of the material,
the more severe is the warp.
The Formation
of a Stellar-Mass Black Hole
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Eventually, when the mass has been compressed to a certain
size, light from the star can no longer escape into space.
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We say that the star has become a black hole, specifically a
stellar-mass black hole.
Why do we call it a black hole?
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The star has withdrawn from our observable Universe, in that we
can no longer receive radiation from it.
We think of a black surface as a surface that reflects none of the
light that hits it.
Similarly, any radiation that hits a black hole continues into its
interior and is not reflected or transmitted out.
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In this sense, the object is perfectly black.
The Photon Sphere
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As the star collapses, two effects
begin to occur.
Although we on the surface of the
star cannot notice the effects
ourselves, a friend on a planet
revolving around the star could
detect them and radio information
back to us about them.
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For one thing, our friend could see
that our flashlight beam is
redshifted.
Second, our flashlight beam would
be bent by the gravitational field of
the star (see figure, left).
If we shine the beam straight up, it
would continue to go straight up.
But the further we shine it away
from the vertical, the more it would
be bent from the vertical.
When the star reaches a certain
size, a horizontal beam of light
would not escape (see figure, right).
The Photon Sphere
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From this time on, only if the flashlight is
pointed within a certain angle of the vertical
does the light continue outward.
This angle forms a cone, with its apex at the
flashlight, and is called the exit cone (see
figure).
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As the star grows smaller yet, we find that the
flashlight has to be pointed more directly upward
in order for its light to escape.
The exit cone grows smaller as the star shrinks.
When we shine our flashlight in a direction
outside the exit cone, the light is bent
sufficiently that it falls back to the surface of the
star.
When we shine our flashlight exactly along the
side of the exit cone, the light goes into orbit
around the star, neither escaping nor falling
onto the surface.
The Photon Sphere
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The sphere around the star in which light can orbit is
called the photon sphere.
Its radius is calculated theoretically to be 4.5 km for each
solar mass present.
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As the star continues to contract, theory shows that the
exit cone gets narrower and narrower.
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It is thus 45 km in radius for a star of 10 solar masses, for
example.
Light emitted within the exit cone still escapes.
The photon sphere remains at the same height even
though the matter inside it has contracted further, since
the total amount of matter within has not changed.
The Event Horizon
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We might think that the exit cone would simply continue to get narrower
as the star shrinks.
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But Einstein’s general theory of relativity predicts that the cone vanishes
when the star contracts beyond a certain size.
Even light traveling straight up can no longer escape into space, as was
worked out in 1916 by Karl Schwarzschild while solving Einstein’s
equations.
The radius of the star at this time is called the Schwarzschild radius.
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The imaginary surface at that radius is called the
event horizon (see figure). (A horizon on Earth,
similarly, is the limit to which we can see.)
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Its radius is exactly ⅔ times that of the photon
sphere, 3 km for each solar mass.
Formally, the equation for the Schwarzschild radius
is R=2GM/c 2, where M is the star’s mass, G is
Newton’s constant of gravitation, and c is the
speed of light.
A Newtonian Argument
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We can visualize the event horizon in another way, by
considering a classical picture based on the Newtonian
theory of gravitation.
A projectile launched from rest must be given a certain
minimum speed, called the escape velocity, to escape
from the other body, to which it is gravitationally
attracted.
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For example, we would have to launch rockets at 11 km /sec
(40,000 km /hr) or faster in order for them to escape from
the Earth, if they got all their velocity at launch. (From a
more massive body of the same size as Earth, the escape
velocity would be higher.)
A Newtonian Argument
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Note that the escaping object still feels the gravitational
pull of the other body; one cannot “cut off ” or block
gravity.
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Now imagine that this body contracts; we are drawn closer
to the center of the mass.
As this happens, the escape velocity rises.
If the body contracts to half of its former radius, for
instance, the gravitational force on its surface increases by
a factor of four, since gravity follows an inverse-square
law; the corresponding escape velocity therefore also
increases (although not by a factor of four).
A Newtonian Argument
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When all the mass of the body is within its Schwarzschild
radius, the escape velocity becomes equal to the speed of
light.
 Thus, even light cannot escape (see figure).
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If we begin to apply the special
theory of relativity, which deals
with motion at very high speeds,
we might then reason that since
nothing can go faster than the
speed of light, nothing can escape.
Black Holes in General Relativity
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Now let us return to the picture according to the general
theory of relativity, which explains gravity and the effects
caused by large masses.
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A black hole curves space (actually, space-time) to such a
large degree that light cannot escape; it remains trapped
within the event horizon.
The size of the Schwarzschild radius of the event horizon
is directly proportional to the amount of mass that is
collapsing: R=2GM/c 2.
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A star of 3 solar masses, for example, would have a
Schwarzschild radius of 9 km.
A star of 6 solar masses would have a Schwarzschild radius
of 18 km.
Black Holes in General Relativity
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One can calculate the Schwarzschild radii for less massive
stars as well, although the less massive stars would be
held up in the white dwarf or neutron-star stages and not
collapse to their Schwarzschild radii.
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The Sun’s Schwarzschild radius is 3 km.
The Schwarzschild radius for the Earth is only 9 mm; that
is, the Earth would have to be compressed to a sphere
only 9 mm in radius in order to form an event horizon and
be a black hole.
Black Holes in General Relativity
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Anyone or anything on the surface of a star as it
passes its event horizon would not be able to
survive.
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An observer would be stretched out and torn apart
by the tremendous difference in gravity between
his head and feet (see figure).
This resulting force is called a tidal force, since
this kind of difference in gravity also causes tides
on Earth (recall our discussion in Chapter 6).
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The tidal force on an observer would be smaller
near a more massive black hole than near a less
massive black hole: Just outside the event horizon
of a massive black hole, the observer’s feet would
be only a little closer to the center (in comparison
with the Schwarzschild radius) than the head.
14.3b Black Holes in General Relativity
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If the tidal force could be ignored, the observer on the
surface of the star would not notice anything particularly
wrong locally as the star passed its event horizon, except
that the observer’s flashlight signal would never get out.
(On the other hand, the view of space outside the event
horizon would be highly distorted.)
Once the star passes inside its event horizon, it continues
to contract, according to the general theory of relativity.
Nothing can ever stop its contraction.
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In fact, the classical mathematical theory predicts that it will
contract to zero radius—it will reach a singularity.
Quantum effects, however, probably prevent it from
reaching exactly zero radius.
Black Holes in General Relativity
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Even though the mass that causes the black hole has
contracted further, the event horizon doesn’t change.
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It remains at the same radius forever, as long as the amount
of mass inside is constant.
Note that if the Sun were turned into a black hole (by
unknown forces), Earth’s orbit would not be altered: The
masses of the Sun and Earth would remain constant, as
would the distance between them, so the gravitational
force would be unchanged.
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Indeed, the gravitational field would remain the same
everywhere outside the current radius of the Sun.
Only at smaller distances would the force be stronger.
Time Dilation
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According to the general theory of relativity, if you were
far from a black hole and watching a space shuttle
carrying some of your friends fall into it, your friends’
clocks would appear to run progressively slower as they
approached the event horizon. (Eventually they would get
torn apart by tidal forces, as discussed above, but here we
will ignore this complication.)
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From your perspective, time would be slowing down (or
“dilated”) for them; indeed, it would take an infinite amount
of time (as measured by your clock) for them to reach the
horizon.
From your friends’ perspective, in contrast, no such time
dilation occurs; it takes a finite amount of time for them to
reach and cross the event horizon, and shortly thereafter
they hit the singularity.
Time Dilation
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On the other hand, if your friends were to approach the
event horizon and subsequently escape from the vicinity
of the black hole, they would have aged less than you did
(for example, only 3 months instead of 30 years).
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This is a method for jumping into the future while aging very
little!
However, this strategy doesn’t increase longevity—your
friends’ lives would not be extended.
Locally, they would not read more books or see more
movies than you would in the same short time interval (3
months in our case).
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In contrast, while viewing you from the vicinity of the black
hole, they would see you read many books and watch many
movies, signs that you are aging much more than they are
(30 years in our case).
Rotating Black Holes
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Once matter is inside a black hole and reaches the
singularity, it loses its identity in the sense that from
outside a black hole, all we can tell is the mass of the
black hole, the rate at which it is spinning (more precisely,
its angular momentum), and what total electric charge it
has.
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These three quantities are sufficient to completely describe
the black hole.
Rotating Black Holes
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The theoretical calculations about black holes we have
discussed in previous sections are based on the
assumption that black holes do not rotate.
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But this assumption is only a convenience; we think, in fact,
that the rotation of a black hole is one of its important
properties.
It took decades before Einstein’s equations were solved for a
black hole that is rotating. (The realization that the solution
applied to a rotating black hole came after the solution itself
was found.)
In this more general case, an additional special
boundary—the stationary limit —appears, with
somewhat different properties from the original event
horizon.
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Within the stationary limit, no particles can remain at rest
even though they are outside the event horizon.
Rotating Black Holes
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The equator of the stationary limit of a
rotating black hole has the same
diameter as the event horizon of a nonrotating black hole of the same mass.
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But a rotating black hole’s stationary
limit is squashed.
The event horizon touches the
stationary limit at the poles.
Since the event horizon remains a
sphere, it is smaller than the event
horizon of a non-rotating black hole (see
figure).
Rotating Black Holes
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The space between the stationary limit and the event
horizon is the ergosphere.
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This is the region in which particles cannot be at rest.
In principle, we can get energy and matter out of the
ergosphere.
For example, if one sends an object into the ergosphere
along an appropriate trajectory, and part of the object falls
into the black hole at the right moment, the rest of the
object can fly out of the ergosphere with more energy
than the object initially had.
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The extra energy was gained from the rotation of the black
hole, causing it to spin more slowly.
Rotating Black Holes
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A black hole can rotate up to the speed at which a point on the
event horizon’s equator is traveling at the speed of light.
The event horizon’s radius is then half the Schwarzschild radius.
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If a black hole rotated faster than this, its event horizon would
vanish.
Unlike the case of a non-rotating black hole, for which the
singularity is always unreachably hidden within the event
horizon, in this case distant observers could receive signals from
the singularity.
Such a point is called a naked singularity.
Since so much energy might erupt from a naked singularity, we
can conclude from the fact that we do not find signs of them
that there are probably none in our Universe.
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Thus, each black hole cannot exceed its maximum rate of rotation
(or a naked singularity could be seen).
Passageways to Distant Lands?
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In science fiction movies (such as
Contact), it is often claimed that one can
travel through a black hole to a distant
part of our Universe in a very short
amount of time, or perhaps even travel
to other universes.
This misconception arises in part from
diagrams such as the figure for two nonrotating black holes.
One black hole is connected to another
black hole by a tunnel, or wormhole
(officially called an “Einstein-Rosen
bridge”), and it appears possible to
traverse this shortcut.
Passageways to Distant Lands?
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However, this map is misleading; it does not adequately
describe the structure of space–time inside a black hole.
In particular, one would need to travel through space faster
than the speed of light (which nobody can do) to avoid the
singularity and end up in a different region.
Thus, non-rotating black holes definitely seem to be excluded as
passageways to distant lands.
In the case of a rotating black hole, on the other hand, travel
through the wormhole at speeds slower than that of light,
avoiding the singularity, initially seems feasible.
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In fact, it appears as though one could travel back to one’s starting
point in space, possibly arriving at a time prior to departure!
This is quite disturbing, since “causality” could then be violated.
For example, the traveler could affect history in such a way that
he or she would not have been born and could not have made
the journey!
Passageways to Distant Lands?
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More detailed analysis, however, shows that this favorable
geometry of a rotating black hole is only valid for an
idealized black hole into which no material is falling (or
has previously fallen).
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As soon as an object actually tries to traverse the wormhole,
the passageway closes!
One would need to have a very exotic form of matter with
anti-gravitating properties to keep the wormhole open.
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There is no evidence for the existence of such matter, at
least not in the form where it can be gathered. (In Chapter
18, we will introduce the concept of “dark energy,” which
causes the expansion of the Universe to accelerate. But this
energy is uniformly spread throughout space, and cannot be
concentrated into a small volume.)
Detecting a Black Hole
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A star collapsing to become a black hole
would blink out in a fraction of a second,
so the odds are unfavorable that we
would actually see the crucial stage of
star collapse as it approached the event
horizon.
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And a black hole is too small to see
directly.
But all hope is not lost for detecting a
black hole.
Though the black hole disappears, it
leaves its gravity behind.
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It is a bit like the Cheshire Cat from Alice
in Wonderland, which fades away leaving
only its grin behind (see figure).
Cygnus X-1:
The First Plausible Black Hole
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We can determine masses only for certain binary stars.
When we search the position of the x-ray sources, we look for a
single-lined spectroscopic binary Then, if we can show that the
companion is too faint to be a normal, main-sequence star, it
must be a collapsed star.
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If, further, the mass of the unobservable companion is greater than
3 solar masses, it is likely to be a black hole, assuming that the
general theory of relativity is the correct theory of gravity and that
our present understanding of matter is correct.
To definitively show that the black-hole candidate is indeed a
black hole, however, additional evidence is needed, as we will
discuss below.
Cygnus X-1:
The First Plausible Black Hole
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The first and most discussed, though no longer the most persuasive, case
is named Cygnus X-1 (see figure), where X-1 means that it was the first
x-ray source to be discovered in the constellation Cygnus.
A 9th-magnitude star called HDE 226868 has been found at its location.
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This star has the spectrum of a blue supergiant; its mass is uncertain but is
thought to be about 20 times that of the Sun.
Its radial velocity varies with a period of 5.6 days, indicating that the
supergiant and the invisible companion are orbiting each other with that
period.
Cygnus X-1:
The First Plausible Black Hole
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From the orbit, it is deduced that the invisible companion
must probably have a mass greater than 7 solar masses
and less than about 13 solar masses.
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This range makes it very likely that it is a black hole.
But if the visible star is abnormal, and doesn’t really have
a mass of 20 times that of the Sun, the invisible
companion could be less massive, so the case for it being
a black hole is not absolutely conclusive.
Other Black-Hole Candidates
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In 1992, an even more convincing case was found: The dark object in
the x-ray binary star V404 Cygni has a mass of at least 6 Suns, but
probably closer to 12 solar masses (see figure).
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One of the authors (A.F.) has found four additional black-hole candidates with
more than 5 times the Sun’s mass, as well as two somewhat weaker cases for
black holes.
By mid-2005, about two
dozen good or excellent
stellar-mass black-hole
candidates in binary
systems had been
identified and measured
in our Galaxy.
Supermassive Black Holes
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Although the stellar-mass black holes discussed above
provided the best initial evidence for the existence of
black holes, we now have firm observational support for
black holes containing millions or billions of solar masses.
Let us discuss these super-massive black holes briefly
in this black-hole chapter, and say more about them in
Chapter 17, when we consider quasars and active
galaxies.
The more mass involved, the lower the density needed for
a black hole to form.
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For a very massive black hole, one containing hundreds of
millions or even billions of solar masses, the density would
be so low when the event horizon formed that it would be
close to the density of water.
Supermassive Black Holes
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Thus if we were traveling through the Universe in a spaceship,
we couldn’t count on detecting a black hole by noticing a
volume of high density, or by measuring large tidal effects.
We could pass through the event horizon of a high-mass black
hole without being stretched tidally to oblivion, though visual
effects would certainly be bizarre.
We would never be able to get out, but it would be over an hour
before we would notice that we were being drawn into the
center at a rapidly accelerating rate.
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Where could such a supermassive black hole be located?
The center of our Milky Way Galaxy almost certainly contains a
black hole of about 3.7 million solar masses.
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Though we do not observe radiation from the black hole itself, the
gamma rays, xrays, and infrared radiation we detect come from the
gas surrounding the black hole. (See the image opening this
chapter.)
Supermassive Black Holes
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Other galaxies and quasars (see Chapter 17) are also
probable locations for massive black holes.
The Hubble Space Telescope is being used to take images
of galaxies with the highest possible resolution, and is
finding extremely compact, bright cores at the centers of
some of them.
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Many of these cores probably contain black holes that have
a large concentration of stars around them, making their
central regions unusually luminous.
Supermassive Black Holes
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The Hubble Space Telescope has also been able to take spectra
very close to the centers of certain galaxies, though that
particular instrument on the Hubble is currently broken.
The Doppler shifts on opposite sides of the galaxies’ cores,
especially in disks of gas surrounding these cores, show how fast
these points are revolving around the centers of the galaxies (see
figure).
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The speed, in turn, shows how much
mass is present in such a small volume.
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Only a giant black hole can have so
much mass in such a small volume.
Supermassive Black Holes
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It now seems that most galaxies have a
supermassive black hole at their centers.
A majority of these galaxies look relatively
normal, but some are “active” (Chapter 17),
giving off exceptionally large amounts of
electromagnetic radiation from very small
volumes.
The Chandra X-ray Observatory, the XMMNewton Mission, and the Hubble Space
Telescope, working together, are greatly
increasing our knowledge of such supermassive
black holes.
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Chandra has found evidence of a jet in one galaxy
(see figure, top) and confirmed the presence of a
supermassive black hole at the center of the
Andromeda Galaxy (see figure, bottom).
X-ray spectra it took showed features that very
strongly suggested that the objects are black
holes.
Supermassive Black Holes
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Radio techniques that provide very
high resolution can be used to study
the jets of gas emitted from the
vicinity of supermassive black holes.
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The jet from the galaxy M87, the
central galaxy in the Virgo Cluster of
galaxies, has been imaged in various
ways (see figure).
The image has been traced so close
to the central object that it shows a
widening of the jet.
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This widening seems to indicate that
the jet comes from the accretion
disk rather than from the central
black hole itself.
Moderation in All Things
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Until recently, many scientists doubted that there were
black holes with masses intermediate between those of
stars—say, 10 times the Sun’s mass—and those in the
centers of all or most galaxies—perhaps a million to a
billion times the Sun’s mass.
But one of the latest advances in black-hole astrophysics
is the discovery of so-called intermediate-mass black
holes, with masses of “only” 100 –10,000 times the
Sun’s, bridging this gap.
Chandra X-ray images reveal luminous, flickering sources
in some galaxies, suggesting that intermediate-mass black
holes are accreting material sporadically from their
surroundings.
Moderation in All Things
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One possible place for such black holes to form
is in the centers of dense clusters of stars,
perhaps even young globular clusters, which
retain the black holes after they form.
Indeed, in 2002, measurements of the motions
of stars in the central regions of two old globular
clusters seemed to imply the presence of
intermediate-mass black holes, although the
interpretation of the data is still somewhat
controversial.
Moderation in All Things
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Chandra observations of one x-ray source in the galaxy M74 (see
figure) discovered strong variations in its x-ray brightness that
repeated almost periodically about every two hours.
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Because they are almost, but not quite,
periodic, the phenomenon is known as
“quasi-periodic oscillations.”
The object is one of many known as
“ultraluminous x-ray sources,” which are
10 to 1000 times stronger in x-rays than
neutron stars or black holes of stellar
mass.
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These sources are being observed from
both NASA’s Chandra and the European
Space Agency’s XMM-Newton.
Gamma-Ray Bursts:
Birth Cries of Black Holes?
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One of the most exciting astronomical fields in the middle of this first
decade of the new millennium is the study of gamma-ray bursts.
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These events are powerful bursts of extremely short-wavelength radiation,
lasting only about 20 seconds on average.
Each gamma-ray photon, to speak in terms of energies instead of
wavelength, carries a relatively large amount of energy.
Gamma-ray bursts flicker substantially in gamma-ray brightness and each
of them has a unique light curve (see figure).
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The joke so far has been “If you see one gamma-ray burst, you’ve seen one
gamma-ray burst,” a far cry from “you’ve seen them all.”
How Far Away Are
Gamma-Ray Bursts?
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Several other telescopes have also
been automated to make follow-up
observations of gamma-ray bursts.
For example, A robotic telescope at
Lick Observatory that is able to slew
over to the position of a gamma-ray
burst and take relatively deep images
within a few tens of seconds after
the outburst (see figure).
Mini Black Holes
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We have discussed how black holes can form by the
collapse of massive stars.
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But theoretically a black hole should result if a mass
of any amount is sufficiently compressed.
No object containing less than 2 or 3 solar masses
will contract sufficiently under the force of its own
gravity in the course of stellar evolution.
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The density of matter was so high at the time of the
origin of the Universe (see Chapter 19) that smaller
masses may have been sufficiently compressed to
form mini black holes.
Mini Black Holes
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Stephen Hawking (see figure), an English
astrophysicist, has suggested their existence.
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There is no observational evidence for a mini
black hole, but they are theoretically
possible.
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Mini black holes the size of pinheads would
have masses equivalent to those of asteroids.
Hawking has deduced that small black holes
can emit energy in the form of elementary
particles (electrons, neutrinos, and so forth).
The mini black holes would thus evaporate
and eventually disappear, with the final stage
being an explosion of gamma rays.
Mini Black Holes
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Hawking’s idea that black holes radiate may seem to be a contradiction
to the concept that mass can’t escape from a black hole.
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But when we consider effects of quantum physics, the simple picture of a
black hole that we have discussed up to this point is not sufficient.
Physicists already know that “virtual pairs” of particles and antiparticles
can form simultaneously in empty space, though they disappear by
destroying each other a short time later.
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Hawking suggests that a black hole so affects
space near it that the particle or antiparticle
disappears into the black hole, allowing its partner
to escape (see figure).
Photons, which are their own antiparticles,
appear as well.
Mini Black Holes
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As the mass of the mini black hole decreases, the evaporation rate
increases, and the typical energy of emitted particles and photons
increases.
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The final result is an explosion of gamma rays.
Although “gamma-ray bursts” have indeed been detected in the
sky (see the preceding section), their observed properties are not
consistent with the explosions of mini black holes.
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For example, gamma-ray bursts flicker substantially in gamma-ray
brightness, with each one being different (see figure).
Such behavior is not expected of exploding black holes.
14.11 Mini Black Holes
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Instead, as we have seen, gamma-ray bursts may
result from neutron stars merging to form black
holes, or from supernovae in which the collapsing
core forms a black hole.
Gamma-ray bursts are extremely powerful, and
probably have something to do with the formation
of black holes, but unfortunately for Hawking they
are not evidence for exploding mini black holes.