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Lecture 18 A105 Dark Matter

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The Milky Way galaxy's center hosts a supermassive black hole, and its rotational velocity profile deviates significantly from predictions based on visible matter, strongly suggesting the presence of a dominant, unseen component known as dark matter.

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All right, guys. Let's go ahead and

begin. Uh, so we'll pick up where we

left off last time. We were

analyzing the matter at the center of

the Milky Way uh, galaxy. So, I'm just

bullet here analyzing matter at center

Milky Way continued and we left off.

we're able to determine using Newton's

laws or if you could call Kepler's third

law if you want to the mass at the

center of the galaxy is 2.5 * 10 6* the

mass of our sun. So that's 2 and a half

million times the mass of our sun at the

center of the

uh Milky Way.

Now continuing on

uh the so to get that number remember we

had you know the formula that we used

just to rewrite that but we had it

yesterday. So it's r

v^2 over g. And we're again we're you

know you're analyzing the hydrogen

uh particles that are in a tight orbit

around that central mass and and so the

particles that's r is the distance out

and then the v is the uh the

velocity. And um if you then try to pin

down a size on the on the mass that's at

um 10 raised to the 10 m which is close

to 1 AU 1 AU being the distance from the

sun to the earth. Um, so just to put

this in perspective, this is we're able

to kind of, you know, pin down the size

of what that that mass that's at the

center of the uh, galaxy. And of course,

there's the actual value. So now we have

these two

numbers, the size and we have the

mass. And and after we got those then it

became apparent well we should check

something and so what we want to check

is we want to do the calculation on the

Schwarz child radius of of this uh mass

this. And so we'll go ahead and write that

down. We had that earlier.

So any black hole uh will satisfy this

equation. Uh that's the mass of the

black hole and then that's the

shortchild radius which is the basically

the size of it. You know the radius out

to the event horizon. So you know

because we have these two data points

here about the size of this object and

the mass of this object we could put

those in the Schwarz child radius and

and see what we get. If we get uh an

agreement there, then we can make a

statement about that object that's at

the center of the Milky Way galaxy. So,

let's just go ahead and put these

numbers in and see what we get. So based

upon that mass, let's see what the

Schwarz sheld radius is and then we'll

compare it to this number we got here 10

the 10th and just see what what we Okay,

so we've

got two the gravitate ah come on two the

gravitational constant the mass of the

object and then we divide by the speed

of light squared. So go ahead and put

those numbers in. So two is two

gravitational constant 6.67

67 10 - 11. Now I got to do the

mass. So

that's 2.5 * 10 6 * the mass of the sun.

The mass of the sun is 2 * 10 30. So

I've got 2.5

* 10 6 and then I multiply that by the

mass of the sun which is 2 * 10 30 and

we divide that by uh the speed of the

light squared. So 3 * 10 8 and we square

that. Okay. So when you do the

calculation what you see

is you get a Schwarz child radius that's

10 the 10 m and that's what we see when

we measure the size of that. So we can

make this statement now this is a very

big black hole okay and it's you know

two and a half billion times the mass of

our sun and because it's so large we

call this a super massive black hole. So there

Okay. And that just comes out of the the

two data points, you know, that we've

calculated. What's the mass there? Uh,

using Newton's laws. And then based upon

the visual evidence, what constraining

that that mass, what size do we get? And

And that tells us it's a black hole. So

super massive because very large mass.

Now there's additional evidence that

supports the existence of a very large black

hole. So let's do a new bullet here. Other

evidence. So if we look at uh the center

of the Milky Way galaxy, what we see is

we see three pieces of uh basically

radiation that uh would be produced as

production

Uh let me tell you about synretron

radiation. Of course X-rays are high

energy photons. Posetrons are like

electrons are the the antiparticle of

electron. Anytime matters violently

smashed together you can get posetrons.

Okay. Now this but let's talk about the

synretron I got too many

here. Spell this.

radiation. Anytime you have uh charged

particles in particular, they're usually

electrons and they're trapped due to a a

field either gravitational or magnetic

some sort of field where they travel in

a tight orbit. They give off what's

called synretron

radiation. So synretron radiation so

uh high-speed electrons trapped by fields

A field could be magnetic. It could be

gravitational. Whatever. High-speed

electrons trapped by a

um electromagnetic

radiation and this radiation is because

it's traveling in a circle. It's

accelerating due to its acceleration due

to its uh

uh well centrial acceleration but that's

due to the fact that the velocity vector

is changing direction. Let me let me

just say due to its acceleration. We'll

just leave it at

that. That's synretron radiation. high-speed

high-speed

electrons which are charged particles

and because they're traveling in a

circle by virtue of Maxwell's equations

anytime you have a charged particle

that's undergoing acceleration it'll

give off electromagnetic radiation the

EM stands for electromagnetic radiation

and that's that's synretron radiation

okay uh and the X-ray production is

anytime you take matter and smash it

together high speeds you can produce X-rays

X-rays

X-rays again are high energy

electromagnetic radiation. So what's

happening you see as matter falls into

the black hole the tidal forces slam it

together. You just think of a black hole

is kind of a funnel and matter gets

slammed together as it falls in and then

that v that violent collision is then

producing these three pieces of

uh radiation that we see. Okay. So we

see that coming from the center of the

Milky Way and that's what's giving us

additional evidence that we have a very

large black hole there that we call a

Um, okay. So, now I want to talk

about we get away from the galactic

nucleus and start talking about

the outer structure of the the Milky Way

get into kind of a problem. Okay. So

tangential

velocity of the Milky Way. And I know this

sounds

technical, but it's a very important

important

study. All right. So, just

picture, the Milky Way galaxy is the

following. This is the model you should

think about. There's a very large very

massive black hole at the center and of

course that exerts a tremendous

gravitational pole on all the stars and

the solar systems in the Milky Way. Now

just like if we think about our own

solar uh system, we have the sun in the

center and then orbiting the sun are the

planets and the sun exerts a

gravitational pull on those planets. But

the planets do not fall into the sun

because each planet has a tangential

velocity that gives rise to a

centrifugal force that counters the

gravitational pull of the sun. And so

that's why you know it same applies to

the moon in orbit around the earth. The

earth exerts gravitational pull on the

moon but the moon has a tangential

velocity that gives rise to a

centrifugal force that counterbalances

that gravitational pull that the earth

exerts on the moon. And the same with

the planets in orbit around the sun in

our solar system. Each planet has a

tangential velocity and that counters

the gravitational pole of the sun. And

in the same model, we we envision that

the the Milky Way galaxy is the same

way. I mean, you've got a a very large

black hole at the center and then you've

got all the stars and everything that

make up the the Milky Way and they're

that that black hole is exerting a

tremendous gravitational pull on all the

stars, all the solar systems. But

because that they're moving with a

tangential velocity, they're able to

counter that immense gravitational pull

uh that the black hole exerts on the

matter. So what we were wanting to do is

to make sure we completely

understood the tangential velocities,

these orbital speeds of the stars that

are in the Milky Way and measure those

speeds and then hopefully you know they

would agree with you know basic uh

Newton laws of

motion and uh that everything would be

would be fine And of course the story

ends in a different manner and that's

what we want to investigate. So what we

actually looked at is we would wanted

wanted to measure the the Doppler shift

on that uh

the spin flip radiation the 21 cm spin

flip radiation of hydrogen because the

the galaxy's full of hydrogen. So

hole. Okay.

And so we wanted to very accurately

measure all the velocities as if you

know for each star each location.

Actually we looked at each location

because we actually looked at the hydrogen

hydrogen gas.

observations of the

hydrogen. Uh

reveal the

these rotation velocities or the

tangential. I'll say

rotation, but then I'll put in here

tangential. Those two words are

interchangeable. I use tangential here

in the title of the bullet, but it's

they're both the

And of course what they do is they you

measure the Doppler shift on that spin

flip radiation and then get the velocity

and you do it for certain distances out

you know map out the entire Milky Way

galaxy. So you're going to be able to

then have a velocity profile versus

distance. Okay? And so that was the

goal. Now before I get you to the answer

of what we found, I I kind of got to

talk about uh velocity

profiles for objects that move that have

rotation. Okay, so we want to look at

uh velocity profiles for

for you know different situations,

different objects. So, the first one

we're going to look at is what's called

rotation. All

right. So, this is the situation where

imagine you have a platform

uh like a merrygoround with no no none

of the horses just just flat platform,

no horses, no nothing. And

uh I put you on the platform. I turn it

off and then I allow you to walk up,

step on the platform, and then I'm going

to turn on the platform.

Now, I'm going to first tell you, go to

the center of the platform. And so,

you're standing at the center. Now, I'm

going to turn it on. And now what you'll

notice is if if you stand there kind of

in the center, you you know

your your little circular speed isn't

very fast. But then if you start to walk

out towards the

edge, your tangential speed, your

rotational speed will be much quicker.

And that's what we mean by solid body rotation.

rotation.

There's a linear relationship between

the distance out from the center and the

the rotational speed or the tangential

speed and any body any object that

exhibits that then we say well that has

solid body rotation. Okay. So solid body

this. Okay. So the horizontal axis is

the distance out from the center.

Another word for that is

radius. But I just want you to think if

we go back to the merry-go round, this

is the center of the merrygoround. And

then out here's the edge of the

merrygoround. Okay? I allow you to step

on it. Then I turn the power on so it

starts rotating. Okay? And then the

vertical axis is your tangential speed,

your rotational speed. Okay, so I'm

gonna put that in

here. I know the word tangential, you

know, it just means tangent to the edge

of the circle. That's what tangential

And when you have solid body rotation,

any object that's undergoing solid body

rotation like that

merrygoround, then the plot looks like

this. It's just a straight

line. And so as you're if you're in

close to the center, you're down here

this distance, you go up to the line and

you have a very small tangential

speed. But then as you walk out to the

edge, you're way out here near the edge.

go up here and go over, you have a very

large tangential speed. So there's a a

linear relationship with the distance or

the radius out from the center and

versus the tangential speed. That's

solid body rotation. Any object that's a

solid that is

rotating, the farther out you are from

the axis of rotation, the center point,

the greater the tangential speed. Okay,

that's what we mean by solid body

rotation. Okay, that's one type of

rotation. Now, we're going to talk about

another that's based upon gravity and

that's called capillarian rotation. So,

we're going to do a new bullet here. So,

velocity is

proportional to the radius for solid

body. V, this is V and the radius is R.

Okay. And so as you increase r the

tangential speed the velocity increases.

So there that means proportional to the

velocity is proportional to the radius.

Okay. It's nice linear line there. The

slope is actually the angular rotational

speed of the merrygoround of the solid

body. But I don't want to get into all

the little

uh kinematics there. just velocities

proportional to R. Now we're going to

look at is we're talking about another

type of rotation. This is the rotation

that exists actually in our own solar

system or for that matter any uh bodies

that are orbiting around a central mass,

you know, like a plan a planet if it's

got moons or a sun if it's got planets

around it and so forth. And this is

called Keplerian rotation in honor of Kepler.

Now to explain Keplerian rotation, I'm

going to just kind of walk us through

just the data in our own solar system.

So, this is what I want to say. Let's

plot the orbital speed of each planet in

our solar system and we'll see what we

get. So, we're going to move this up.

We're going to do

again a two-dimensional

two-dimensional

plot. All right.

So this is the excuse me the distance

out from the sun. So you know the orbit

radius of the planet. So I'll just put

radius here. So orbit

radius how far away is that planet from

the sun. So right there's the sun and

then you know there's Mercury and you know

know

not the scale but just Mercury, Venus,

Earth, Mars, asteroid belt, Jupiter,

Saturn, Uranus and Neptune. Okay. So we

go out and then this will be the or orbital

orbital

orbital. All right. So if we do that

plot, the first one of course is we got

Mercury and it's up. So this is the sun

and then this is out at the edge here

out here where Neptune is. Okay? Because

you're far away. This is right up close.

So the closest planet is Mercury. So

Mercury has a very high orbital speed.

So here's

Mercury and then

Venus and

Earth and

Saturn, Uranus, and

Neptune. So if we go ahead

those. So let's just it this is

Mercury. Don't

um Venus, Earth. I mean, I'm not going

to label them all. These are just all the

the

planets, you know, Mars, Jupiter,

Saturn, Uranus. Here, let's put Neptune

down here just so you understand. Here's

system. Now, this is this line that fits through

through

there just comes from Kepler's

laws. Okay? So and if

so this is what we call capillarian

rotation where the plot looks like this.

rotation and the scaling of the velocity

the velocity is proportional to one over

the square root of the radius. And that

just comes out of Kepler's laws or

any system of where you have objects

that are gravitationally bound to a

central host object. Now those objects

are in in orbit around that host

object because of the laws of gravity

they should obey capillarian rotation.

rotation.

Okay. Now so we have the solid body

rotation. the velocity is proportional to

r and then we have kept rotation where

the velocity is proportional to one over

the square root of r. So what we see is

the velocity is very large and then it

rapidly drops off. So you get farther

and farther away.

Now, prior to looking at the data in the

Milky Way, we thought it had to look

like this because the Milky Way, you

have a central gravitating mass, i.e.

the super massive black hole, and then

you've got a bunch of matter, star in

solar systems, and they're

gravitationally bound. And so, it should

model exactly our own solar system. So

when we looked at the data from the Milky

Milky

Way, what we

got was not Kepler but goes by the name

of differential rotation. And

um I want

to draw

But before I do that, I'm also going to

just say we have the profile for the

Milky Way, but we've also done the same

thing for other

um galaxies. And then if you kind of all

average them out, they kind of always

look like this. So differential

So again this is radius distance out

from the center and then this is the

object. And so all the profiles if you

kind of average them out they kind of

look like this. They start here at the

center and they have a steep climb and

then they kind of flatten out and then

that looks nothing like Keplerian

rotation. Actually, if you look at it in

closer, it's solid body. So, in near the

super massive black hole, there's so

much matter that the way the force of

gravity is, it kind of holds that matter

together and it all rotates like a solid

body. But then out here, you don't

really see anything like

that. Okay? You don't see this Keplerian

drop off that we would expect. And so

this is the differential rotation of the

galaxies. Now I'm going

to draw more

accurately the one for the Milky Way and

then then let's see I'm on page five. So

the Milky Way actually looks like this.

of this is a big problem because you go

wait a minute why isn't it'll have that

Keplerian drop

off that we see now this decay here is

goes by the name Keplerian decay because

it drops

off. Go back to this other drawing. just

decay. So, Milky Way differential

rotation and so drawing the Milky Way

one and again this is kind

of excuse me you've smooth we've

smoothed it out but I just want

to again

speed. So there's a little kind of

undulation here. So you come

up and then there's a little curly cue

like that. Then it comes down and then a

slight rise and then this and then

there's a last kind of kick up like

that. And believe it or not, our our

solar system is right there. So our sun

is right

there. you know about 27,500

27,500

lys. Okay. And of course out here's 40

somewhere and then here's the center.

Okay. So this is a Milky Way again. It's

it's this differential where you have no

concrete capillarian drop off. So we

don't know

there's see if it was Kepler and we'd

expect this thing to start to really

drop off here and we don't see that.

This is the Keplerian drop off. We don't see

that. So we're

baffled what's going

on. Okay. Now in close we see

hints right here of solid body effects.

here.

Okay. But when you don't have solid body

and things are gravitationally bound by

the laws of Kepler and Newton you expect

the Keplerian drop off just like we see

in our

own solar system. We don't have that.

Now I want to give you some just some

important data related to the Milky Way

and related to our sun. Actually let

here. So you know this number. So this

about 2.3 * 10 5 m/ second and that's

about about 500 miles hour. So our solar

system is in orbit

around you know traveling in a giant circular

circular

orbit about

27,500 lys

um at a velocity of 2.3 times 10. That's

space now because you know how you know

you know let me go to another picture.

overhead.

Okay, there's our sun and the planet.

system 2.3

* 10 5 m/s. That's the speed. And

then there's the RRS, the

27,500 lys.

So because you know the speed and you

know the circumference of a circle, it's

2 pi r. You know the total distance

around here. So if you take the total

distance around here and you divide by the

the

speed 2.3 * 10

5 m/s, you could compute the time that

And we can we can find we can find the

orbit. All right. And that time is about

years. So that's how long it takes our

solar system to complete one orbit as it

orbits around that super massive black

hole that's, you know, right there,

right there in the center. 230 million

years. So the last time you're at this

location, the dinosaurs though, the

ancient dinosaurs, not the more modern

that really ancient

dinosaurs, you're talking 230 million,

you know, not 80 million, 230 million.

The ancient dinosaurs were ruling the

that's little bit of data we get just

because we know our

distance and we know our speed we're

able to determine our orbit time. So

this is called the orbit time tech

orbital period. I should write it orbital

period orbital time timing period mean

the same thing.

Okay, now I want to go

back because we need we really want got

to talk about something that's really

important. So, we want to go back. So,

I'm on to page

seven and we're going to do a bullet

here. And I let the cat out of the bag

by writing the the name of this bullet,

but I don't care. Here we go. Dark

matter. Okay, so now you've heard about

dark matter. Here's where dark matter

comes from. So, we're going to look at

again. Okay. For the Milky Way. So, here we

right. Big steep climb here. Little lat

uprise. Little kick and a kick up.

I'm just redrawing what we did right there.

there.

Okay. And here we are. Here's our sun

system. Okay. And as we said, we don't

non-existent. No Kepler decay. Oh, I

should say

This is the radius distance out from the

center. And then this is the orbital

speed. So here's

the here's what we think. There

There

is some mysterious matter in the

galaxy that is effectively gluing

together all the matter in the galaxy.

All the stars, the solar systems, gluing it

together so that it has this sort of

profile where it kind of it basically

flattens out. I mean if you go from here

to here this is basically yeah there's

these little undulations but basically

we get this flattening out. So I used to

call it flattening

out after you get beyond the the steep

climb. Then if you just if you go back

to this one where we you know the

average of all the the

galaxies and trauma and all the other

ones that we can get the data on you

average them all this what they look

like. You basically get yeah a little

kick here and there but basically

flat and that's the sometimes they say

the flat rotation curve. So instead of

rotation the technical is differential

differential

but the flat differential whatever.

So the idea is that on this dark matter

is that there must be a bunch of

matter in the Milky Way that acts

through gravity and gravity alone. So it

doesn't and doesn't give off any other

signature. No electromagnetic so we

can't see it. That's where the word dark

comes from. And there's got to be a lot

of it because it's got to basically act

like a gravitational kind of glue and

just glue everything together such that

all the matter as you get out away, you

know, not in close, but it

all just has this flat rotation curve.

So it acts like kind of a glue.

expected capillarian drop off or capillary

because we really thought we'd see, you

know, we should see a drop off just like

we see in our own solar

system. We don't see that. You get out

far away from the center of the Milky

Way and it just flattens out

basically. So, the idea is that okay,

there has to be some matter that we

can't see that's acting kind of like a

glue. It's gluing together together all the

the objects

objects

that make up the Milky Way

galaxy. So thus and I say we think

because you know we we are looking for

the dark matter and we

haven't we have no candidates and we say

oh we found dark

matter that

acts like a glue. [Music]

gravity. Now, because it's hidden, what

that means is we can't see it. Doesn't

give off any signature, any

electromagnetic radiation. So, if you

can't see it, it's like dark. So, that's

where we get the word

dark. And then because it's matter, it's

where we get the word dark [Music]

[Music]

matter. Okay, so we envision that the

the Milky Way galaxy is just full of

this stuff. This dark

matter gives off no electromagnetic

radiation, no signal that we can

pinpoint and but yet we believe it's

there because it's acting like kind of a

gravitational glue and gluing the stuff

that we can see the luminous stuff such

that we're getting this sort of profile.

Okay? And that's the basis of dark matter.

Now, even, you know, if you kind of

think about it, swallowing mad, I've

just thrown up this thing, dark matter.

I mean, just like

crazy. You can't we can't see it. We

don't know what it is. We don't have

any. We have some candidates, particle

candidates, but we'll get into those

later. But none of them are all really

that that good. They just would count

for a trivial amount. We need quite a

bit. I want to talk about quite a bit

here shortly. Um, but it's matter. So,

it's acting through the force of

gravity. That's it. It's kind of gluing

together everything in the in the galaxy

and but we can't see it. So, dark

matter. Now, we get to the

question swallowing this is hard enough,

but it even it gets a little more challenging.

challenging. is

um okay.

sure. You see that's kind of a problem

when you don't really now I can tell I

can tell you this you're not sure a

fraction very small fraction and the

reason it's fractional is based

upon big bang

uh nucleioynthesis and the amount of

matter that's produced at the big bang

we kind of know what the distribution of

these other things I'm going to talk

about are but they would not account

enough for what we need to glue together

the Milky Way such that we get that

profile that flattening out. So a

matter could

could

And we know nutrinos do not give off.

They they do not interact. They do have

a little mass. So they could they would

be a candidate for dark matter. Okay?

Because they have mass. And so there's a

small fraction of dark matter could be nutrinos.

nutrinos.

holes and we would think of like

primordial black holes. These very small

black holes that would have been

produced very early stages of the big

bang. those because they have matter,

they would act.

Okay. And you know, then you get in

possibly cold gases and some other

things, but

but that's really all that we have right

now. Okay. So, of of particles that we

know and uh it'd only be a fraction of

those because we know the amount of

these on average in the universe. And

you expect kind of the distribution in a

galaxy to be roughly the average what

you'd see in the universe. There's just

not enough. So let me get to the point

then is is you would ask to me, well,

how much dark matter do we need? So we

can run the computer models. If we start

out with the computer models and we put

in just the stars, the solar systems in

the Milky Way, then you see the

Keplerian drop off because the only you

have the central gravitating body, the

the super massive black hole and then

you've got stars in orbit around that.

So those are going to obey Kepler's law.

So you can do that Kepler and drop off.

So then if you start putting in your

computer simulation, just putting in

little particles, little they're

actually little strands of of of dark of

dark matter, and you start to put it in,

then you you see how much you need to

get that that profile.

So how much

much dark

dark

All right. Now, here's the this is this

will blow your mind. So, if we look in

the in the in our galaxy, so in the Milky

Way, all the normal matter, all the

stuff that we can

see is this. So

all the I don't even like to use the

word normal, ordinary, whatever. All the normal

matter would be

10%. Dark matter is

90%. That's

insane. That's insane. We have a galaxy

and we only know 10% of it. the ordinary

stuff, the stars, the planets,

everything that we can see, it's

luminous, you know, luminous matter,

they'll say use that instead of normal

or ordinary. And then all this the other

stuff that we need, the dark matter

stuff is 90%. That's like crazy. It's

like you don't really know what the

thing because 90% of it you don't know

yet. The 10 only 10% of it you don't know.

know.

So that's the one that's really hard to

glue together the Milky Way such that we

have the flattening out of the rotation

curve means

that 90% of the

galaxy has to be comprised of dark

matter. And so we're searching for dark

matter. And and that's that's the big

one of the big pushes that we're wanting

to uncover. What is the dark matter? All

right, guys. We'll end there and

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