Today, I want to talk about the motions of

stars in the Milky Way. I believe that this diagram is exactly the one from your text.

The three parts of the Milky Way — they also differ in the way their stars move. First

of all, if we look at the disk stars, the starts and the disk move along circular, slightly

undulated orbits where the stars — they stay close to the disk, but they move up and down.

Overall, their orbit is circular with this slight undulation, up and down. They also all move in essentially the same

direction, just like all the planets in the solar system revolve around the Sun in the

same direction, which is counterclockwise, as one would observe from far above the Earth’s

North Pole, so the disk stars all move in the same direction. Our Sun moves at the speed of about 220 kilometers

per second around the galactic center. You can ask, “How do we know that?” We are revolving

around the Sun, how possibly can we find the speed with which the Sun is moving relative

to the galactic center. Basically, you determine that from the Doppler shifts of the light

that we receive from neighboring stars and different parts of the galaxy. By collecting all this data, you can actually

figure out the speed of the Sun and us, moving around the Sun, moving around the galactic

center. We can determine the orbital speed. We can also determine the distance of the

Sun and us from the galactic center, which is about 30,000 light years. We know the circumference

of the Sun’s orbit around the galactic center. We know the speed, and from that, we can find

out how long it takes the Sun to orbit around the galactic center once. You simply take the length of the orbit, the

circumference of the circle, whose radius is about 30,000 light years. You divide it

with orbital speed, and then you get the time. It turns out that it takes the Sun about 230

million years to revolve once around the galactic center. Using scientific notation, that will

be 2.3 million is 10^6. Then if I write this, 230 is 2.3*10^2. Altogether I have 2.3*10^8

years. To put this number into some kind of perspective,

most of you have heard or learned that there was a cataclysmic collision between the asteroid

and the Earth some 65 million years ago that wiped out, among other things, dinosaurs.

Until that time, they were the most dominant species on the Earth. But then there was this

catastrophic event that channeled the evolution in the favor of mammals because they were

small animals living underground, in burrows covered with fur. That was advantageous because so much dust

was kicked up as a result of that collision that sunlight was blocked. The conditions

of so-called Nuclear Winter were created. No sun could get to the plants. They died.

Dinosaurs didn’t have anything to eat. They died out, and also there was severe conditions

that…Not usual rain was falling, but acid rain. It burned everything. Mammals were able

to survive living underground in burrows. Eventually, through further evolution, became,

through us, the most dominant species on Earth. Evolution is not something that happens slowly,

on and on forever. Every now and then you have these e catastrophic events that can

channel the evolution in a particular direction. That happened 65 million years ago. In that,

it takes the Sun 230 million years to complete one full orbit. Since then, since that catastrophic

event, the Earth has traveled only between one-third and one-quarter of its way around

the galactic center. Then you can appreciate how long that period of time is. Also, knowing that the Sun was formed 4.6

billion years, we can calculate how many times it passed around the galactic center, so that

simple calculation. In 4.6 billion — in scientific notation, billion is 10^9 — it’s 4.6*10^9

years since it was formed the Sun completed. I need to calculate. I need to see how many

intervals of time equal to one orbital period, which is 230 million years or 2.3*10^8 years

fits into 4.6*10^9 years. That will give me the number of revolutions. We have 4.6*10^9

years. I should divide that with 2.3*10^8 years per revolution. Years drop out. The revolution is going to come on top, so

4.6/2.3=2. 10^9/10^8=10, so 2*10=20, so 20 revolutions. Since it was formed 4.6 billion

years ago, the Sun made 20 complete revolutions around the galactic center. What about the halo stars? It turns out that,

while the disk stars move along circular paths, slightly undulated, the halo stars move in

highly elliptical orbits at any angle with respect to the disk and in random directions,

like these two move this way, this one moves the other way. In a sense, they resemble bees

in the beehive. The halo stars move in random directions along

highly elliptical orbits. We see that the two groups of stars, those in the disk and

those in the halo, also differ in the way in which they move around the galactic center. Finally, the third part, the bulge, basically,

the motion is similar to the motion of stars in the halo. They move along highly elliptical

orbits every which way. There is no definite trend, like in the disk, that all move coherently

in the same direction. All these things — the structure, the composition

of the three major parts, the way the stars in those three parts move — would give us

hints as to how, perhaps, a galaxy like our own, the Milky Way, was formed. One can deduce the mass of a galaxy from the

orbital data of its stars by applying, essentially, the third Kepler’s law as formulated by Newton

or a period, because if you know the orbital period and you know the radius of the orbit,

you can figure out the speed. Or the other way around. If you know the radius

of the orbit and the speed, you can figure out the orbital period, how long it takes

a star to revolve around the Sun once. Let me remind you of what is involved. It’s really

simple. You combine the second Newton’s law of motion with his law of gravity, and you

can actually see what is the correlation between the mass that is forcing a body to revolve

around it and their distance, and the orbital speed. Let’s review this quickly. Here, we have a

central body, say in the case of the solar system it could be sun, then another body

revolving around it — let me just draw one more; that should do it — at a distance d

with the orbital speed v. Of course, there is a force of gravitational

attraction that is responsible for this. Basically, what the Newton’s second law tells us is that

the mass of this body here, times its acceleration and it’s accelerating because it’s moving

along a circular orbit. Its velocity is changing the direction all the time — the velocity

is not constant. That acceleration, the rate of change, and the velocity direction is caused

by the force of gravity. The Newton’s second law tells us that the

mass of this body times its acceleration which for uniform circular motion with the speed

v and radius of the orbit d happens to be like this. That must be equal to the force

causing that acceleration, force of gravity, which according the Newton’s law of universal

gravitation, is some gravitational constant times the product of the masses between two

objects over their mutual distance squared. What happens here is that the mass of this

body that is revolving actually can be canceled out. I can multiply both sides with one power

of d, which will eventually remove d from the left-hand side and reduce this power of

two to one. Then I can solve this to get the mass of the central object. I get that the mass of the central object

is given by a product of the distance between the two objects, the orbital speed squared

over gravitational constant. That’s how the mass of, say, the Earth is deduced from the

knowledge of the orbital data of the Moon. We know the distance to the Moon. We can measure

its speed from orbital period. That’s how we determine the mass of the Earth. Or this

could be a planet revolving around the Sun. From the orbital data of the planet, from

its orbital speed and distance from the Sun — it could be measured, in principle — we

can determine the mass of the Sun. Or, if this here represents most of the mass

in a galaxy, so we are looking here at a star moving at the edge of the galactic disk, this

would effectively give us the mass contained within the orbit of that particular star. Or another way to phrase this is to say that

I can solve this for v. Then I get that the orbital speed is related to the distance to

the center of the force of gravity. It’s basically the root of the gravitational constant times

the mass of the central object. Then I have to divide with the root of distance. This

is so-called Keplerian

rotation curve. We can apply this to the planets in the solar

system, including dwarf planet Pluto. You can see that indeed all the planets fall on

this curve that represents their orbital speed as a function of the distance from the Sun.

This was a great triumph of Newton’s mechanics and his theory of gravitation that in fact

one could explain in great detail how the planets move around the Sun. Why don’t we try to do the same thing for

the galaxy, to look at the dependence of orbital velocity of a star in a galaxy on its distance

from the galactic center and use this relation here to figure out how much mass is contained

in the orbit of the star? When that was done, there was a big surprise

that changed our understanding of the matter in the universe. They were first measured

by American astronomer Vera Rubin in 1970s. I like to show the pictures of people. After finishing her undergraduate studies

in Physics, she wanted to study Astrophysics Astronomy at Princeton. She was not allowed.

In fact, women were not allowed to take up Astronomy at Princeton until late ’70s, until

1977. Instead, she went to Cornell and did very well for herself. Why are they so different? Let me just show.

Here is a particular galaxy. Catalog number is M33. You would say — one would argue like

this — “If we look at the distribution of light, the most of the light is coming from

galactic center because we have the highest density of stars there in the bulge surrounding

region. As you move away from the galactic center, there’s less and less light intensity,

because we have fewer and fewer stars. You would expect that this distance, anywhere

between 5 and 10 distance in kilo-parsecs. One parsec is 3.26 light years, and kilo-parsecs

is 3.26 thousand light years. This is the unit here is kilo-parsec. You would expect that at seven or so, that

most of the mass in the galaxies exhausted. It’s contained within that distance of seven.

You would expect that beyond seven or so you would start seeing a Keplerian rotation curve.

Where the orbital speed decreases with distance from the galactic center as a square root

of distance, but that is not what is seen. It’s essentially slightly rising, and in many

cases it’s simply flat. It turns out that in our own galaxy beyond

the star, beyond the Sun, the orbital speed is flat. It doesn’t depend on distance, it

doesn’t decrease with distance. It’s at about 220 kilometers per second just like the orbital

speed of the Sun. That’s very puzzling. What’s wrong? Are the

Newton’s Laws of Motion wrong? If you believe that they are correct, what this observation

implies, is that as you move beyond the visible edge of the galaxy, that there is more mass

there that you can see. Look at the relation that we obtain here.

If this stays constant beyond certain distance, in spite of increasing distance, that means

that the mass has to increase with distance. As we go farther away from the galactic center,

if this is constant, the mass enclosed within that radius has to go up, but it’s not producing

any light. That mass that was initially call missing mass is now called Dark Matter. We

don’t know what it is to this day. If you believe that the Law of Gravity is

correct at these large distances, then there is a mass that is not luminous, that does

not produce light, that does not interact with light, but it is a source of gravitational

force just like any other mass. Then the observed curve is something like

this. We have most of the intensity close to the center and then it decays, because

we have most of the stars closer to the center of the galaxy. If we assume that

the mass, at least the one that produces light,

is distributed in the same way…Here I have a distance from galactic center. If I assume

that just like light, the mass is distributed in the same way, I would expect to get the

same bell-shaped curve. Most of the mass is close to the center of

the galaxy. Then I would expect that by this distance here most of the mass is exhausted.

Based on this distribution of mass, my expectation for what I should see as a rotation curve

would be something like this. Beyond this distance I would expect to get

a Keplerian dependence. That’s not what is observed. What is observed is that essentially

the rotation speeds remain constant. As I said, if one accepts the Newtonian gravity,

or the Theory of Gravity in general, and Newton’s Laws of Motion, then that would imply that

there is more mass that we can see. There is also non-luminous mass or so-called Dark

Matter. When you do the calculation as to how much

mass in addition you have to have to account for these observed rotation curves, it turns

out that typically this Dark Matter is five to six times more mass. It contains five to

six times more mass than the luminous mass, the mass contained in the stars.