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.