Transcript from Epic of Evolution: Life, the Earth and the Cosmos (BEP 210A)
February 14, 2000 - Lecture by Claude Bernard
Okay, let’s get started. I’m going to change a little bit the topics for this week from what’s on
the syllabus. In the first week, I didn’t get to the formation of galaxies and the formation of stars
and how all our matter got made in the center of stars. I was planning to do it one time fast
through on the first week and then slower later on, but I’ll just start now with formation of
galaxies and then formation of stars and do it once. I’ll probably end up spending most of the
week doing that. So we’ll be talking about the life cycle of stars and how our elements got made
and how galaxies got made.
So last time we were about up to here [transparency #2]. The last thing I talked about in the first
week was decoupling. Now I’m going to talk about the formation of galaxies and the formation
of stars. Just to remind you: at the period of decoupling the temperature was cool enough that
atoms could form and the electrons wouldn’t be boiled off back out into space. The electrons
would stay with the nuclei: For the first time we had atoms, which are neutral.
At the point when neutral atoms are formed, the photons no longer interact much with the
particles. Photons are electromagnetic radiation. Electromagnetic things interact with charges,
i.e., plus charges and minus charges. But when the plus charges and minus charges are
combined into a neutral atom, the interaction with the photons is much smaller than it was
before. I’ll talk a little bit more about that in a few minutes. And once the atoms were formed
and the photons stopped interacting significantly with them, then the matter was free to clump
together in big clumps due to the force of gravity. Before that, the photons, bouncing back and
forth between the particles at the speed of light, created a pressure that pushed the particles apart
and opposed the collapsing effect of gravity. But once atoms were formed, the photons just
streamed right through and barely ever hit an atom or interacted with an atom. So the only thing
left was the force of gravity. Gravity gradually pulled together big clumps of atoms making the
huge conglomerations that are galaxies. Gravity also pulled together smaller clumps that made
stars, possibly with planets. (We don’t really know what fraction of stars have planets around
them. Presumably a lot of them have planets. It’s unlikely that they all do.)
Okay, so that’s what’s happening after the period of decoupling. The matter is clumping
together and making galaxies; inside galaxies there is further clumping making stars. The
photons are just continuing to move along and almost all of them don’t hit anything anymore. A
few of them may every once in awhile hit an atom, but most of them have just been moving in a
straight line ever since that time. So if we look out in space we may see one of those left-over
photons arriving. That’s the cosmic microwave background that I talked about at the very end of
the last lecture. The fact that the cosmic microwave background exists is crucial evidence in
favor of the Big Bang picture. We’re not just making it up. We have the evidence of the cosmic
microwave background with just the right temperature. The photons are just the right
temperature (or energy) now to correspond to the expected temperature at decoupling (300,000
years after the Big Bang). From then till now the photons haven’t done anything except, as
always, to stretch because the universe is expanding. As the photons stretch their wavelength
increases, so their frequency, and hence their energy, decreases. Remember, that a photon’s
energy is proportional to the frequency of the electromagnetic wave that makes it up.
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Okay, I want to describe today how galaxies formed. Probably we won’t get to stars till
Wednesday. But before I start, I want to back up and discuss in a more formal way the
properties of the four forces that we know about between particles. These are the four
fundamental forces in the universe. Every other force that you might identify is merely some
manifestation of these four fundamental forces, as least as far as we know. There could be other
ones that we haven’t discovered yet, but as far as we know these are the only ones.
Okay, so what are the four forces? I’m going to list them from strongest to weakest. You have
to be a little careful in stating what you mean by strong or weak, because the force between two
objects will depend on what types of objects they are and how far apart they are. The way I’ll
define strong and weak is by looking at the forces between two protons very close together --almost touching. So here are two protons and they have various forces between them. They
have an electromagnetic force. As you know, two positive charges repel each other. They also
have a strong nuclear force between them that is attractive. They even have a gravitational force
between them because every two objects in the universe attract each other with a force of
gravity. But we can list the forces in order of their decreasing strengths. I’ll talk about the
strongest force first, then the next, the next, etc.
The strongest force --- not surprisingly --- is called the “Strong Force”. And what does the
Strong Force do? I mentioned some of its properties before. It binds quarks into protons and
neutrons. It also affects the protons and neutrons themselves. It binds protons and neutrons (I’ll
abbreviate them p and n as before) into the nucleus of atoms. So the Strong Force is what holds
the nucleus of atoms together. It’s the strongest force of the four forces, but on the other hand it
has what we call a short range. If you take two protons that are as close to each other as I’ve
drawn them, the Strong Force is enormous. But if you pull them apart so that they are 10 times
the diameter of a proton apart (or even five times), the Strong Force between them gets very
small. Saying it has a short range means that the force falls very rapidly with distance. So
although the Strong Force is very strong when two protons are almost touching, if the two
protons are separated by a significant number of proton diameters, the Strong Force is almost
entirely negligible. So it’s a very rapidly decreasing force.
The next force is the electromagnetic force. It’s one of the first forces you learned about,
probably the second after gravity. You learn about gravity when you’re a toddler and are trying
to walk without falling down --- so you already have a feeling for gravity in your bones. You
probably learned something about the electromagnetic force in high school, or maybe even
elementary school. And the key fact about the electromagnetic force is that it affects objects
with electric charge. And charge comes in two varieties, which we happen to call plus and
minus (which are arbitrary names for these charges). As I’m sure most of you know, opposite
electric charges attract (a plus and a minus attract each other), but two charges of the same type
repel (two pluses or two minuses repel). So the electromagnetic force can be attractive. That’s
what holds atoms together: the negatively charged electrons are attracted to the positively
charged nucleus. But the electromagnetic force can also be repulsive if the charges are the same.
Electromagnetism is what we call a long-range force. Like all the forces, it decreases with
distance (i.e., if the distance is bigger, the force is less). But electromagnetism decreases very
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slowly with distance, at least in comparison with the Strong Force. Increasing the distance
between our two protons by a factor of 10 will reduce the electromagnetic force by a factor of
100. But it will decrease the Strong Force by a factor of at least 20,000 --- and possibly by a lot
more, if the two protons were actually “touching” to begin with.
The fact that the electromagnetic force acts on both plus and minus charges means that in many
cases the electrical force can cancel. For example, as we said, an atom is neutral. An atom has
plus charges in the center (the protons in the nucleus) and an equal number of minus charges (the
electrons) orbiting around the outside. If you were to look at an atom from a distance, it would
be difficult to tell that the pluses and minuses are in separate locations: the atom would just
appear to be a totally neutral object. I guess the easiest analogy is to a black and white
newspaper photo. If you’ve ever looked at it closely (especially with a magnifying glass), you
have seen that the gray in the black and white photo is made out of separate little dots of white
and black. However, if you look at the photo from a distance you can no longer resolve the
separate dots. All you see is gray, the combination of black and white. An atom works in just
the same way. If you “look” at it closely (say if you’re another nearby atom), you could
recognize that the minuses and pluses are separated from each other, not sitting right on top of
each other. But if you “look” at an atom from a distance many times the diameter of the atom,
the fact that the minuses and pluses are not on top of each other no longer is important. What
you just see is sort of a “gray,” a neutral gray. It’s neutral because there are as many minuses as
pluses. So if you’re far enough away from an atom that you can’t “see” the minuses and pluses
separately, the atom will act like a completely neutral object --- with no charges of any type. In
other words, a neutral atom has no electrical effect on objects far away from it. The only way an
atom can have electrical effect on something is if that object is close enough to be able to “see”
the minuses and pluses of the atom separately.
Thus neutral atoms can have electrical forces, but only on other objects that are very close to
them. And when atoms are close to each other they have all kinds of electrical forces. Those
electrical forces are what make chemistry work: all the complicated interactions between atoms
and molecules that Ursula was talking about last week. Electrical forces between nearby atoms
allow them to combine to make molecules, including the complex ones such as proteins and
DNA that are important for life. When the atoms are close the electrical forces can allow them
for example to transfer one electron from one atom to the other (ionic bond). It can also allow
them to share electrons among themselves (covalent bond). These effects produce the chemical
bonds between the atoms. So chemical bonds are just electrical forces in a particular situation:
electrical forces between atoms that are close to one another other.
When atoms are far apart you get the opposite effect. Then the fact that they’re neutral is crucial.
You can’t resolve that the minuses and pluses aren’t really on top of each other. From a distance
it all looks gray, neutral gray. You see a net charge of the atom of zero (equal amounts of pluses
and minuses). That means two atoms that are far apart from each other will have zero force
between them. (A purist would say that the force isn’t exactly zero, but just extremely tiny --but for our purposes we can just say that there is zero force between them.) If they’re far apart--many diameters of an atom ---- the forces between atoms are utterly negligible.
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Now, maybe I shouldn’t belabor the point, but it’s really interesting that the electrical force
between neutral atoms decreases very rapidly with distance, despite the fact that electrical forces
themselves are long range. The force between one proton and another proton falls off slowly
with distance because electric force is long range. But the force between two atoms, which are
neutral, falls off very rapidly with distance. What’s happening is a cancellation of the attractions
(between pluses in one atom and minuses in the other atom) and repulsions (between pluses in
one atom and pluses in the other atom, or between the minuses in one atom and the minuses in
the other). So between atoms the force gets very small very rapidly with distance. Between
single charges the force falls off much more slowly with distance. Any questions on electrical
forces?
Okay, the next one is the Weak Force. It isn’t the weakest of the four of them, so this name isn’t
the best in the world. The Weak Force is similar to the Strong Force in that it’s a short-range
force: the Weak Force falls off very rapidly with distance. On the other hand, it’s much weaker
than the Strong Force. So you might say “well, if it’s like the Strong Force but much weaker
what good is it?” And the answer is that, as we’ve already seen, the Weak Force can do certain
things that no other force can accomplish. It allows certain kinds of processes to take place that
are impossible with any other kind of force. One thing that the Weak Force does that the other
forces don’t is that it treats particles and antiparticles slightly differently. And this allows for the
process I called baryogenesis. Because it treats quarks and antiquarks differently, the Weak
Force makes possible the generation of a few more quarks than antiquarks in the early universe.
That’s why our universe ends up with particles and not antiparticles.
Another thing that the Weak Force does is to make possible a change in particle type. (I briefly
mentioned this before and actually put a little bit more into the transcripts of the lectures.) What
do I mean by “change in particle type?” Well, for example, it allows for a neutron to decay into
a proton plus an electron plus an “antineutrino.” [We’ll may talk more about neutrinos later on,
but for the moment it’s just some other particle which I haven’t described much.] A neutrino
doesn’t have any electrical charge. You can hear the “neutral” in its name. [The neutrino in this
process is actually an antineutrino but it doesn’t really matter to us.] This process (n -> p + e +
antineutrino) can take place because of the Weak Force. It’s the reason that neutrons by
themselves are unstable --- they break apart. A single neutron will decay after around 10
minutes. If you had a box full of neutrons, then after 10 minutes about half of them would be
gone, turned into protons, electrons and antineutrinos. And the reason this decay is possible is
because the mass of the neutron is more than the sum of the other three. So it’s possible to make
the other three and even have a little bit of energy left over (and mass is energy, remember).
Therefore neutrons by themselves don’t live very long.
You might wonder: “If neutrons don’t live long, why were there all these neutrons around in the
period of nucleosynthesis in the Big Bang. We talked about neutrons meeting protons and
making helium and deuterium and other things. Why weren’t the neutrons gone?” Well, that
was too early in the universe for the neutrons to have a chance to decay. Nucleosynthesis occurs
only about 1 minute to 3 minutes after the Big Bang, so most of the neutrons are still there. They
haven’t had a chance to decay yet. If you waited 10 minutes after the Big Bang only half the
neutrons would still be there. And after a few hours, the number of neutrons would be miniscule,
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so there’d be no way to have processes like proton plus neutron makes deuterium. Okay, any
questions on that?
And this process, as I said, is the decay of a neutron. Another process due to the Weak Force
takes place in supernovas, which are exploding stars. That process is just the reverse of neutron
decay. Like all processes these things can go backwards. If you take a proton plus an electron
plus enough energy (in other words, the proton and the electron are moving fast enough), the
energy can be converted back into mass, and you can actually make a neutron plus a neutrino.
And that’s a process that happens in supernova. In fact, what ends up happening in supernova is
all the atoms are crushed. The electrons are pushed on top of the protons in the nucleus, and the
protons are turned into neutrons. And then a neutrino is sent off with lots of energy.
This process is called “reverse neutron decay” or “inverse neutron decay.”
The process of reverse neutron decay is crucial to our existence. It’s not just interesting as a
sidelight. Exploding stars are the way that the elements that are made inside stars get thrown
back out into the clouds of gas and dust in space. When those clouds condense later on (for
example, in the early solar system) the planets will have heavy elements in them. (“Heavy” here
means anything heavier than helium --- i.e., any atom with more than 4 protons + neutrons in its
nucleus.) Our carbon, oxygen, nitrogen, calcium, iron, etc. all come from exploding stars, and a
large fraction of those explosions are of the particularly violent type called supernovas. So
supernova explosions are crucial to us being here. Without supernovas, even if planets had
formed they would just have mainly light elements, in particular hydrogen and helium. It’s hard
to imagine making life out of just hydrogen and helium. Helium doesn’t combine with anything.
Hydrogen combining with itself just makes hydrogen molecules (H2) and they’re not very
interesting.
There’s also another Weak process, which I don’t want to talk about now but is very similar to
the ones I’ve talked about already, that occurs in the center of our sun. That process keeps the
sun burning at a nice steady rate. Without these Weak processes, which change types of
particles, we certainly wouldn’t be here. Of course that’s true of all these forces. If one of them
was absent everything would be different and we wouldn’t be here. Michael?
[Michael: What keeps the matter stable? Or is there some other something that keeps the
neutrons from decaying? Or is there a process by which new neutrons are continuously made to
replace the ones that decay?]
Once they’re combined in the nucleus, neutrons are no longer unstable. It’s sort of subtle so
maybe I’ll avoid that question. [Note added to the transcript: After thinking about it, I realize
that it’s not hard to explain this, but it requires information that I will not talk about until the next
lecture. So I will add an explanation to the transcript of that lecture.]
[Michael: So my body’s not going to fly apart . . .]
No. Well, it’s a good point because free neutrons will decay in 10 minutes but a neutron bound
inside a normal nucleus will stay there forever as a neutron. A nucleus like the nucleus of a
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helium atom is not unstable at all. It has two protons and two neutrons, and the neutrons there do
not decay. Once they’re bound into a nucleus they’re very happy just to stay there.
So now we’re up to the weakest force (for two protons sitting next to each other). That force is
gravity. Yet, as I said, gravity is the force you learn about first in your life. Why is that? Well,
gravity has two very important features. First of all, like electromagnetism it’s long range so it
doesn’t decrease rapidly with distance. And in fact the way electromagnetism and gravity
decrease with distance is exactly the same mathematically. But although electromagnetism is
long range, once you start getting atoms formed electromagnetism doesn’t affect things that are
far apart. Atoms are neutral, so the electromagnetic force between widely separated atoms
cancels. On the other hand, gravity doesn’t have any pluses and minuses. Everything attracts
everything else. All matter attracts all other matter. Even the distinction between matter and
antimatter is irrelevant to gravity. Antimatter also attracts matter (or other antimatter)
gravitationally. It’s “anti” in various senses, but gravity doesn’t care. [Note: like all rules, the
one that says everything gravitationally attracts everything else may actually have an exception -- there may be a repulsive gravitational effect. But that’s not important to us now, and the
relevance of the idea is in any case still somewhat controversial. I just mention it because I don’t
want there to be wrong statements on the transcript.] Sam?
[Student: Is there inconsistency between the fact that all matter attracts other matter and the
entropy of everything moving apart from one another? Does that mean everything is opposing
gravity all the time? We talked about all these things after Big Bang moving apart.]
Well, yeah, everything, after the Big Bang we have things that are moving apart and gravity
pulling them back together.
[Student: Are things accelerating or are they slowing down because of gravity?]
Well, this gets into the very question I was trying to avoid here! If we just had ordinary matter in
the universe and by that I’m even including antimatter, then because of the gravitational force
between the matter and all other matter it would make the expansion of the universe slow down.
It’s the same as when you throw a rock up in the air: it slows down as it goes up. And the same
thing would happen here. The initial velocity that the particles had going out would gradually
decrease and slow down. However, measurements in the last couple of years seem to show that
the universe is actually speeding up in its expansion. That would mean something else --different from ordinary matter attracting other matter --- is going on. And I’ll talk a bit about
that at the end of the course, but I’ll leave it for the moment. And I don’t think I really dealt with
your question about entropy, but why don’t we talk about that after or some other time because
it’s a little bit subtle. [Note added about this subtle question: The expansion of universe gives
particles more space to rattle around in. That allows entropy (disorder) to increase --- or at least
stay the same --- even though the average temperature is decreasing as the photons get stretched
and cool off. Normally, increasing entropy would mean increasing temperature. Later, when
gravity starts pulling clouds of gas together, the particles in the cloud have less room to rattle
around, but their entropy still increases because of their increase in temperature. You are not
responsible for the above explanation about entropy; it’s just added for completeness.] Okay,
other questions?
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Since all matter attracts all other matter gravitationally, gravity becomes more and more
important as you look at bigger and bigger clumps of matter. And from the time of decoupling
until the time when the first stars turn on (about 1 billion years after the Big Bang), gravity is the
only game in town. The electromagnetic force is no longer important because atoms have
formed and they’re neutral. Of course there’s a possibility of having some molecules like those
of hydrogen gas, H2, which is formed from two atoms of hydrogen. So in that sense electrical
forces can have some effect, but that’s only for things right next to each other. If you take the
big view of things, electrical forces are irrelevant because atoms are neutral. The Weak and
Strong Forces are irrelevant because things are spread out, and the Weak and Strong Forces are
very short range. They only affect things that are very close to each other. Of course it is true
again there are helium nuclei, which have been made in the Big Bang, and they’re still being
held together by the Strong Force. But those helium atoms don’t have any Strong Force
attraction for other helium atoms. They’re far away from each other and the Strong Force has no
effect. Similarly, the Weak Force has no effect, so all you have is gravity.
Here’s what gravity does after the time of decoupling: Wherever there was a slight irregularity,
where the matter was a little bit closer together, the matter would have slightly stronger
gravitational effect on the stuff around it. That would tend to pull in more nearby stuff, and you
get bigger and bigger clumps forming. And the biggest clumps we’ll talk about are
protogalaxies, the beginnings of galaxies. Now remember, gravity is the last one of the four
forces. It’s quite weak. It only matters when you have lots and lots of matter is a big region.
But then it’s much more important than the other forces. So gravity gradually pulls things
together. Between the time of decoupling (300,000 years), and about 1 billion years after the Big
Bang, we have clumping gradually going on. [Actually clumping begins at the time of matter
domination (10,000 years), but it accelerates after decoupling, when electromagnetism stops
having any effect at all. The origin of the initial irregularities --- “seeds” around which the
gravitational clumping takes place --- is uncertain, although most cosmologists think the basic
idea is understood. I hope to talk more about that at the end of the course.]
At about one billion years after the Big Bang, you start having the first recognizable galaxies.
Within those galaxies, smaller clumps have also made the first recognizable stars, and possibly
some planetary systems. But note that those planets wouldn’t be anything like Earth because
they would only be made out of hydrogen and helium. Those are the only elements we have at
this time. So there could be gaseous planets like Jupiter but nothing like Earth. So life --- at
least life resembling in any way life on earth --- would be impossible. Questions?
Okay, so how do galaxies form? Well, the process of galaxies forming due to gravitational
attraction is very similar to something you’ve already seen. Michael’s talked about the clumping
of the pre-solar nebula. First it was a big clump, a big sphere more or less, of gas and dust. Now
by random fluctuations, the clump is likely to have some small amount of rotational motion
around some axis or another. Then as the clump collapses or condenses due to gravity, it’s the
same deal as the skater pulling in her arms. The clump will turn faster and faster. Then the very
turning makes the clump spread out along its “equator” due to centrifugal force. So you get a
disk-like structure forming, with a little bit extra in the middle where the centrifugal force isn’t
so big. If you’ve ever put a kid on a merry-go-round, you know that they like to ride on the
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outside because it’s more fun; it’s more interesting. In the middle you’re going slowly, and you
don’t feel much different from normal. In fact, you can walk around in the middle of a merrygo-round without noticing much. But if you’re on the outside you’re going much faster: it’s
more interesting, it’s more fun. So far away you’re really affected by the rotation and it gets
very disk-like. In the middle it can be more spherical so you get a cross-section like that.
The process of galaxy formation is exactly analogous to the formation of a solar system (where
the spherical part is the sun and the disk-like part condenses into planets). But here we’re talking
about 100 billion times more matter: a galaxy has maybe 100 billion stars. And just as within the
solar nebula there was clumping out here making individual planets, in this case there’ll be
clumping out in the disk making individual stars and possibly planets around those stars. But the
process is basically the same. As Michael said, it’s all about angular momentum, which is the
momentum something has when it’s spinning around. And angular momentum explains why a
skater spins faster as she pulls in her arms.
I’m going to talk more about galaxy formation, but I since I keep bringing in this TV and never
end up showing you the clip from the movie, I’ll do it now. This is taken from a movie we’ve
looked at before called the “The Cosmic Voyage: An IMAX Film.” This part is their description
of the Big Bang. A lot of the processes we’ve already talked about. But it not only has galaxies
forming into disks, but also shows spirals in the disks. I’ll talk about where the spirals come
later. And even sometimes two spiral galaxies can collide with one another and they make
something called an elliptical galaxy. The wreckage of two colliding spiral galaxies is an
elliptical galaxy. That’s where we’re headed, since our spiral galaxy (the Milky Way) is going to
collide in a few billion years with the Andromeda Galaxy. What’s left will probably be a big
glob, called an elliptical galaxy, instead of a nice pretty spiral.
So this is the movie. The guy you see at the beginning is actually a famous cosmologist named
Rocky Kolb. I think he’s actually kind of a cool guy but he looks totally stilted in this movie.
He had his script and he went with it. Kolb is leading a tour through Fermilab. Fermilab, which
I’ve mentioned before, is a particle accelerator laboratory outside of Chicago. Experiments at
Fermilab have been some of the most important in validating the current theories of elementary
particles.
[movie clip]
Kolb was a little more daring than me. He was willing to trace back the observable universe to a
size smaller than a ping-pong ball. I only did a beach ball!
All right, so there’s your Big Bang. I sent to the Web page today a description of a link to a site
in Germany. They’ve done simulations of particles interacting due to their mutual gravitational
forces and condensing to make galaxies. They zoom in on one part, and you can actually see a
spiral galaxy forming. There are two simulations I really like (some of the others are a little
boring). There’s one where you get a very nice spiral galaxy. In the other one it makes two
spiral galaxies, which then collide with each other and make an elliptical galaxy. If you get a
chance you might want to get on the Web and take a look at that. You need to be able to run an
mpeg video on your computer to see it.
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Okay, I have a little bit more time so let me talk about what happens in the center of a galaxy as
it condenses. Remember, it’s analogous to what happened as our solar system condensed. In the
solar system you’ve got the sun, which is much more massive than any of the planets going
around the outside. In a galaxy a similar thing happens. In the center you get a collapse of an
enormous amounts of matter. We think that there’s usually enough mass in the collapsing center
to make a black hole. I haven’t yet talked about what black holes are; let me say a little but now.
(I will talk more about black holes later on in the course.) A black hole is created when matter
gets enormously compressed. The black holes in centers of galaxies may something like 10 to
500 million times the mass of the sun compressed into a region smaller than the size of the solar
system. Because of the enormous compression, a black hole has such enormous gravity at its
surface that nothing can leave it, not even light. Even if a photon is shot out it from the surface,
it will be pulled back and crash back down into the black hole. Nothing can get out of a black
hole, hence the name. They’re black; no light can get out.
So you might think that the center of a galaxy would be dark because of the black hole. But
that’s actually just the opposite of what happens. Near the black hole, but still outside it, there are
all kinds of gas and stars, which are pulled into the hole. It’s like an enormous vacuum cleaner
sucking in everything from the neighborhood. The stuff falls into the black hole with a
tremendous amount of energy, and that energy is converted into heat. So the gas & stars falling
in gets very hot and emits light. The light can get out because it’s not yet in the black hole. You
get light and all kinds of other electromagnetic radiation --- gamma rays, X-rays --- coming out.
So there is a lot of radiation coming from the center of the galaxy. And in a young galaxy, this is
tremendously bright. It’s what’s called a quasar. So a quasar is black hole at a center of a young
galaxy, sucking in all kinds of matter and emitting lots of light. As the galaxy gets older, the
black hole has already sucked in most of the stuff around it, so the center is no longer very
bright. Only young galaxies have quasars in the center. In older galaxies, there’s still a black
hole in the center, but the area becomes darker. It’s not emitting much anymore because it’s
already sucked in everything in the neighborhood.
[end of lecture]
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