So, I hope you realize not only the 
binary stars are cool but that our 
understanding of the physics allows us to 
understand how you learned so much about 
them. 
as this is the case with the planets, if 
the inclination of the orbit is indeed 
zero, in the sense that the planets orbit 
in a plane that includes our line of 
sight, then something even more exciting 
is going to happen. 
not only will it measure the full radio 
velocities but halfway between those 
points where you measured the full radial 
velocities just as with planets, one star 
will transit the other. 
And so, there will be times during the 
evolution, the, the history of transiting 
binary like this where we see both stars. 
We can't resolve them, but we see and we 
see both sets of spectroscopic absorption 
lines. 
And then, there will be times when we 
only see one of the two stars or we see 
parts of one and all of the other. 
We see a non-trivial behavior of the 
light curve. 
And the light curve, as I promised with 
planets, will tell us things about the 
periods, the sizes, the temperatures of 
the two stars. 
And if we can combine this with Doppler 
measurements, we can learn a lot about 
the system. 
I want to spend some time looking at what 
we learn from a light curve because I 
think it will interest you and I promised 
that when we get to planets, 
let's take a look at an eclipsing binary 
simulator. 
This is our eclipsing binary simulator. 
It has many features, I hope you'll play 
with it, let's see what we can learn from 
an eclipsing binary, so here is our 
system over here on the left. 
we can change its parameters if we want, 
but I've had set it up so we have star 
number one, this is a hot blue star. Its 
temperature is 7850 K, its radius is five 
solar radii. 
Star number two is a smaller reddish 
star, it's only got 2.8 so the radii and 
the temperature of 6,200 Kelvin. And 
[COUGH] I've chosen to view the system 
[UNKNOWN] let me try to show you what it 
looks like if ypu turn it around. 
this is our a system. 
Notice that astronomers call inclination 
the angle that is zero when you're seeing 
the system face on and 90, when you're 
seeing it edge on. 
We're using a complementary notation for 
silly historical reasons, but I wanted 
you to know that other conventions exist. 
And we can't, we're imagining that we 
cannot resolve these two stars. 
So, what we see is the combined light of 
both in our telescope. 
And we can make photometric measurements. 
We can measure the intensity. 
And over here on the right, we see the 
light curve. 
In other words, the time dependence of 
the combined intensity of the two stars. 
This will be a periodic periodically 
changing. 
And we folded it back, so we see a 
complete period. 
And this red cursor shows us the current 
situation at any given instant. 
While on the left, we can see the 
configuration of the system. 
And so, as I start the animation, the 
stars are orbiting each other. 
And if I've stopped it at an interesting 
point, this is the point at which the red 
dim star is moving away from us, the blue 
more luminous star is moving towards us. 
So, if we measure the Doppler shifts, 
this is the moment when you'd find 
maximal red shift in the spectral lines 
of the red star and maximal blue shift in 
the spectral lines of the blue star. 
And we could use that, for example, to 
measure their speeds. 
And that will be useful, of course, 
because we'll have some information from 
the Doppler shift, from the radial 
velocity curves. 
And as time goes by, 
well, the Doppler shifts will re, will 
decrease. 
And if I stop the animation at this point 
we're seeing that as the red star begins 
to be eclipsed by the blue star the total 
combined intensity begins to dip, of 
course, because I'm losing the light from 
the red star. 
And so, just looking at the depth of the 
dip that is being created, 
when the difference between full 
intensity and intensity during the full 
eclipse of the red star by the blue star, 
this difference is precisely a 
measurement of b2. 
And, of course, if I know the brightness 
of star number two, then by subtraction, 
I know the brightness of star number one. 
So, I have a measurement independently of 
the brightnesses of the two stars just by 
looking at the light curve. 
Step number one, 
I'm going to have to erase this because 
I'm going to annotate this diagram very 
heavily. 
another thing we note is that, at this 
point, 
when they're about to eclipse each other 
the red star, star number 2, is moving 
precisely to our left with a speed, v2, 
which presumably may be measured by a 
Doppler shift. 
And at the same time star number one is 
moving to the right at its speed, v1. 
And so, the speed of relative motion is 
just v1 + v2. 
And this tells us something because what 
we are seeing is, over here at this time, 
we are seeing the beginning of the 
eclipse right over here. 
And, of course, if we let time pass, 
we'll see that as star number two is 
increasingly eclipsed by start number 
one, the light curve yet deepens, and it 
reaches it's maximum depth at this time 
right over here, and if we call this time 
interval, t2, 
then we can describe what happened during 
this time interval. 
What happened during this time interval 
is that the combined motion of the two 
planets carried them through the diameter 
of star two from the point when star two 
was just touching the outside of star one 
with its left edge until it just 
disappeared around, behind it. 
So, v1v2. 
+ v2 * t2 is twice R2. If we measure the 
speeds using Doppler shift we, now have 
an independent measurement of the radius 
of star number two and, of course, if I 
let the animation run a little bit more 
then we can see that 
when I get, 
I when the red star reemerges, 
I now have a measurement of the combined 
intensity again, and about this point, I 
get maximal blue shift on the red star's 
spectral lines, maximum red shift and the 
blue star's spectral lines. 
And when the next eclipse starts, first 
of all, the directions of the speeds are 
of course reversed. 
But I'll leave them there. 
Please do not get confused by this. 
But now I can measure a different time 
interval. 
Of course, the length of this time 
interval during which the dip develops, 
is again the same length. 
It's the time that it takes the red star 
to move relative to the blue star twice 
the red diameter. 
But if instead, I start my clock now and 
measure the time it takes until this, 
until the following happens, 
until the eclipse begins to, to be 
decreasing until this point. 
This time interval over here is the time 
that it took the relative motion of the 
two stars to carry them through the 
diameter of the blue star. 
So, v1 + v2 times this time interval, t1, 
is just twice R1. 
So, we have a measurement of the radii of 
the two stars and at least their ratio, 
if we don't have Doppler shift 
measurements, 
if we do have Doppler shift measurements, 
we can get the exact radii. 
But certainly, we can get a ratio of the 
radii. 
We found the brightnesses of the two 
stars independently. 
We can also get an independent measure of 
the temperature. 
Notice that the dip created when star 
number one, 
when the blue star is occluded, is deeper 
than the dip created when the red star is 
occluded. 
This has to do simply with the fact that 
the blue star is harder. 
Notice we don't get a total eclipse of 
the blue star. 
It's not measuring its total luminosity. 
What I'm missing here is a disk out of 
the surface of the blue star equal to the 
size of the red star. 
what this means of course, is that the 
reason this deep is, this dip is deeper 
is because this corresponds to loosing a 
disk with a temperature t1 to the fourth 
and this corresponds to loosing the disk 
with the temperature t2 to the fourth, 
the ratios of the depths of the two dips 
tell me about the ratio of the 
temperature of the two stars to the 
fourth. 
And you can see that as I adjust the 
relative temperatures, 
I will find that this will the relative 
depth of the two dips will change when I 
set the temperatures equal the dips are 
equal in 
depth as long as the smaller star is 
cooler, the dip went, it is obscured as 
less deep. 
And if the, I make the smaller star 
hotter, then the arrangement is reversed. 
So, we can read lots of things off of 
this figure. 
I hope you'll play with the simulator. 
I should show you two other things if I 
can erase all my notations. 
One is that if the orbit happens to be 
elliptical, I can give it some 
eccentricity. 
And what happens then is that depending 
on the orientation of the ellipse, the 
two dips might not be symmetrically 
positioned within the 
orbit, within the period, because it 
might take the two stars. 
what we see is that this 
in between this eclipse and this eclipse 
falls perihelion. 
When the two stars are close together, 
they go fast. 
In between this eclipse and the following 
eclipse this way, lies aphelion that 
takes longer so we can get the a measure, 
but the asymmetry gives us a measure of 
the electricity, the eccentricity of the 
orbit. 
And furthermore if we have stars whose 
radii are comparable then, of course, 
when the radii become close we never get 
this flat region on the bottom because 
total eclipse occurs only for an instant. 
But even more fun 
if, as the stars 
combined radii become closer and closer 
to the radius, we get what we call a 
contact binary. 
A contact binary, 
such that the stars are essentially 
touching each other. 
That's the situation I have here. 
What a contact binary does is it causes a 
flat region where we get constant full 
luminosity to disappear because 
essentially the stars will eclipse each 
other and then immediately start the 
other eclipse and we never get this 
constant period of a full illumination. 
Lots and lots of information to be read 
off a light curve, 
the same applied to planets. 
I promised that, that I would show it to 
you at some point. 
I hope this has been instructive. 
So, eclipsing binaries are a rich source 
of information. 
In general, we can learn a lot from 
binaries. The most famous example of 
binaries, eclipsing binaries, of course, 
as we call it Algol, or perhaps Alghoul 
which is the ghoulish looking star in 
Perseus that dims for a few hours every 
few days because it turns out, it's an 
eclipsing binary. 
As we'll see later, there is yet more 
information that we can extract from 
binaries. 
Binary stars as I like to call them are a 
star with a built-in probe, 
and you can learn a lot about both 
members of the pair because there's 
something for them to interact with. 
So, we'll come to binary stars again and 
again as the class, class goes on. 
For now, let's observe that our friend, 
Alphecca. 
Alpha Coronae Borealis happens to be a 
double line eclipsing binary with a 
period of about 17.4 days and we can do 
the Doppler measurements. 
again, I quote in the credits section, 
the paper from which I lifted these data, 
and it would be fun to go and read the 
paper. 
And you can see you can learn a lot from 
that. 
Doppler measurements give me the radial 
velocities of the two stars. 
And I give here the speeds as we measure 
them. 
I plug into our formula for the masses of 
the two members of the binary. 
Of course, this was our formula for M1 
replacing everyone here with a two, I get 
the formula for M2. 
And I predict that 
Alphecca is a pair of stars, one with a 
mass three times the solar mass, one with 
a mass approximately a solar mass. 
I can compare this with my spectro 
measurements because it's a double line 
binary, I see the spectra of both stars. 
Indeed 
we talked about Alpha Coronae Borealis A 
last time. 
We said, there was an A-type star. 
that mass of three solar masses and the 
radius, we said then, about three solar 
radii, is about right for A star. Alpha 
Coronae Borealis B, turns out to be G5V, 
which makes it a little bit cooler than 
the star. 
It's radius should, than the sun I mean, 
it's radius should perhaps be a little 
smaller, but another main sequence star. 
you can read about the life curve fits in 
the paper that I quote. 
You can also see how life is never quite 
as simple as what I presented in this 
idealized scenario in particular, for 
example, the eccentricity of the orbit is 
very high and so our circular orbit 
calculation is bound to be a little bit 
inaccurate. In fact, from the paper 
including both the eclipse data and the 
radial velocity data, the best estimate 
they find is about 2.6 solar masses for A 
and 0.9 solar masses for Alpha Coronae 
Borealis B. 
A radius of about three hm, 
we got that right because temperature and 
radius don't care for radial 
measurements. 
so indeed, the radius of, of a Alpha 
Coronae Borealis A is three solar radii 
and of secondary 0.9. 
The temperature we had right for the 
primary the secondary is about the solar 
temperature and the [UNKNOWN] ratio of 
luminosities. 
the luminosity in units of the solar 
luminosity is what you would expect from 
their spectral type. 
The beautiful thing is, yeah, these 
people did some very careful numerics, 
but we understand and can actually 
emulate at least the physical principles, 
if not all the details, that went into 
this calculation. 
I hope you appreciate how much you've 
learned.