Two, 
what? If we can see both members of a 
binary pair, and if we can resolve them, 
and if they are near enough, that we can 
actually distinguish their motion, then 
we can use astrometry to measure the 
period and the size of the orbit and 
learn a lot. 
We're interested in binary stars. 
Most binary stars, like most everything, 
is not going to be near enough for us to 
measure their motion, and often, we won't 
be able to even resolve the two stars. 
We'll see the light of the two as one 
star just as our eye unaided, does not 
distinguish the two members of Albireo. 
We see one star, even with a big 
telescope, the stars are far enough or 
one of them is dim enough, we'll only see 
one star. 
How do we know by looking at a star that 
it might actually be a binary? 
Well, one way is to measure its spectrum, 
that's what we do and what will we 
recognize when we're looking at the 
spectrum? 
What about the spectrum will tell us that 
we're looking at a binary? 
Well, if we have a binary system that is 
not oriented such that its orbital motion 
of the two members is completely 
tangential. If there's some radial 
component, then as we saw, that means 
that some of the time the stars will be 
moving towards us and sometimes away from 
us. And furthermore, when one is moving 
towards us, the other will be moving 
away. 
What this will give us is a periodic 
Doppler shift. 
We'll see that if you look at the 
spectrum, there will be some of the lines 
that will be blue shifted and some others 
will be red shifted. 
And then, the rows will exchange, 
so we'll find a periodically shifting 
Doppler shift, a really 
characterically look simple version of 
this is two stars with one spectral line. 
And what we see here is two copies of 
this spectral line, moving greatly 
exaggerated amount to the right and to 
the left, to the red and to the blue. 
And so, when one star is moving towards 
us the other is moving away from us, 
we can use the maximal amount of the 
Doppler shift to estimate the speed with 
which those two stars are moving. 
the example below shows the same thing, 
again, caricatured one spectra line. 
But here we see that one of the stars is 
moving faster than the other. 
One of those lines shifts more than the 
other line does that tells us that the 
two stars are moving, orbiting each 
other, but one moves more. 
that means that, that star is lighter 
than the other stars. 
And in many cases, one of those stars is 
so dim that we only see one, then this is 
very much like discovering a planet 
through Doppler shifts. 
You see a star orbiting something, you 
can try to conclude what it is that, that 
star is orbiting if you know something 
about the star. 
Of, obviously, the more you observe, the 
more information you can get. 
So we learn by measuring the spectrum. 
We look at a star, we notice that it has 
sets of spectral lines that move are 
Doppler shifted in oposite directions and 
they move back and forth. 
Well, what can we measure? 
By measuring how long it takes for the 
lines to move back and forth, we 
certainly measure the period with which 
the two stars that we're actually seeing 
as one are orbiting each other. 
We can use the Doppler formula, 
we measure the maximal shift of some 
spectral line and we measure if it's a 
double line binary star, 
we can measure the speeds of each of the 
individual stars. 
Now, you may remember that we're only 
measuring the radial component of the 
speed, not the tangential component. 
So I'm going to assume that the system is 
aligned if you sort of imagine 
[SOUND] a set of axises like these. I'm 
going to put the, the orbit in the 
horizontal plane in this picture, so that 
the maximum value of radial velocity 
towards us or away from us is equal to 
the actual speed with which the stars are 
orbiting each other if the orbit is 
tilted like this, 
then this will be false. 
We'll talk about how we deal with that 
later. 
For now, let us assume for simplicity, we 
have circular orbits and no tilt and 
we'll have to do some algebra, 
so bear with me for a bit. 
So what do we do? 
We've measured the two speeds. 
If I know how fast something is moving 
and I know the period, 
how long it takes to go around the 
circle, 
I certainly can figure it out, the radii 
of the two circles in which the two stars 
move, 
this already tells me something. 
Remember that the radii of the two 
circles about which each of the stars 
move are, in fact, since these are 
circles about their center, common center 
of mass are related by this equation. 
so if I measure the two velocities, I 
already have the ratio of the two masses. 
I can write that M2 = * V1 / V2 as we 
predict it from guessing, by just by 
looking at the spectrum, the lighter star 
move faster than the heavier star. 
So far, so good. 
What else can we do? 
Lets collect some more information. 
In terms of that. 
we can write the total mass of the 
system, because remember that's what 
enters into Newton's expression for 
Keplers law M1 + M2. 
I write the expression I had for M2. I'll 
make a common denominator over here and 
add this relation between M1 and M. 
So if you tell me M, I can figure out the 
mass of the individual star. 
That's what I've done so far. 
Furthermore, the other parameter that 
enters into Kepler's formula is the total 
distance between the two stars. 
Remember, that's just the sum of the 
radii of the two circle,s because of the 
way they orbit, they're always on the 
opposite sides of their respective 
circles. 
I plug in the expression that I had for 
the radii of the two circles and I find 
that the radii, total distance between 
the stars is related to the period and 
the velocities or the speeds that I 
measure. 
In this way, remember, periods and V, 
period and V's is what we measure, 
this allows me to compute r. 
So, 
now, we're going to take all this 
information that we've accumulated here 
and plug it into Newton's form of 
Kepler's Law. 
Remember that P, R and the total mass are 
related by this expression. 
I remind you, on the previous slide, we 
found this relation between R and the 
observed quantities, and this relation 
between M, the observed quantities, and 
M1. 
Remember, the mass of one of the stars is 
what I am after. 
So, what I do, first, is I plug this 
expression for R into that expression. 
I solve this equation for M, multiplying 
by M dividing P^2. 
And then, I, for this R^3, I plug the 
cube of this expression and then I see 
some handy dandy cancellations going on 
here. 
pi is cancel all over the place and I get 
left with one power of P up top. And I 
can write an equation that says, that M 
is P / 2pi G * (V1 + V2)^3. 
But I also have this other expression for 
M in terms of my friend M1 and that's (V1 
+ V2) M1 / V2. 
And so, I see that one power of V1 + V2 
cancels and multiplying both sides by V2, 
I finally have an equation for M1 in 
terms of P, 
the speeds, get a measure and Newton's 
constant. 
I am done by measuring this Doppler 
shift, which gave me the speeds, and the 
period, I can actually measure the mass 
of of a star. 
Of course, it's easiest to assess this by 
scaling it relative to we know that for 
the earth and the sun, I have that the 
mass of the sun is equal to the speed of 
earth times one year times nah, 
the speed of the sun is negligible. 
Let's write the speed of the earth 
squared divided by 2pi G. 
And then dividing one expression by the 
other. I get my happy scaling formula 
that says, since I know the speed of 
earth and it's orbit is 29.78 kilometers 
per second, 
I can get the mass of each star in units 
of the solar mass this way and so, we can 
find masses. 
spectroscopic binaries or other binaries 
stars are a way to measure masses, 
this is why it's so useful. 
The two caveats, of course, if the orbits 
tilt, we're only measuring the radial 
component of the speed. 
How would we know if the orbit is tilted? 
Well, if we're lucky, we would have 
astrometric measurements and we can make 
both as in the alpha 
Centauri version. 
We can make measurements both in the 
tangential direction and in the radial 
direction. 
And otherwise, we have to infer what 
really we're measuring is one component 
of the velocity, 
so the inclination shows up as an overall 
factor here. 
and often, we only see one of the stars 
and then we're only measuring V1. 
We don't know v2, 
we can't get complete information, 
but we can learn a lot of things from 
this. 
Binary stars are cool and useful objects.