. We now have our designated population of
stars. We have a collection of them.
We're going to try to understand whether the way we understood the sun with
modeling, we can understand how these stars work.
We're not quite ready to start with the modeling because if you tell us stellar
modeler, I want you to model a ball of hydrogen.
I know its surface temperature and I know its luminosity.
He's going to ask you well, yeah, how much hydrogen is there?
We need a way to find the mass of a star. So how do you weight the star?
All we can do is look at it. How do you measure the mass of a star
just by looking at it? You recall that we found the mass of
Saturn just by looking not at it, but at a moon orbiting it and using Newton's
laws. Well,
wouldn't it be convenient if we found stars with things orbiting them.
Not moons, but something we could actually see.
So that we could use the same kind of calculation through Newton's law to
measure the mass of stars. Well luckily for us about a fifth of all
the stars in the sky, somewhere around that, are not single stars.
They are gravitationally bound to one or more partners.
And so, if we can see the partners, we can measure orbital parameters, and try
to make an estimate of the mass that way. these, if you recall, the picture on the
right, is our friend Albireo, the head of Cygnus the Swan, back from lecture one.
And you remember that we talked about the fact that these are two stars.
this is what astronomers, amateur astronomers like to call a double star.
but not all double stars are binaries. A double star is just two stars that are
very near each other in the sky. There are two ways this can happen.
The two stars could actually be near each other in the world or they could happen
to lie in the same direction in the Earth but it's vastly different distances, and
be nowhere near each other. As we saw for example, for the members of
the big dipper, which are nowhere near each other in the sky.
In that case, by the way, we call a false double star or a double star that is not
a binary is an optical double. A visual binary like Albireo is a binary
where the two members of the pair can actually be seen and separated.
So is Albireo a visual binary or an optical double?
The answer is we're not quite sure. How do we tell?
Well we can measure parallax angles to these stars.
And they are quite near each other in the sky.
Though it's not quite certain that they are bound.
For that we measure their motions and try to see whether their total energy is
negative or positive. it's not clear whether these stars are
bound or not. But they're close to that so that,
they're at about the same distance. And, with that in mind, let's stop for a
minute, and appreciate what we've learned, just looking at this picture.
What can we say that we couldn't say a few weeks ago, when we started out, and
looked at Albireo? Well, we see two stars, we clearly see
that one is orange and one is blue. So, we realize black body spectra tells
us that the blue star is hotter than the orange star.
We also see that one appears brighter than the other.
Since I told you their parallex angles are about the same, that tells you that
they're about the same distance. The one that appears brighter is more
luminous, so the orange star is more luminous than the blue star.
That tells us a third thing. These cannot both be main sequence stars.
Our long domain sequence, the harder the star is the more luminous since here we
have a cooler orange star more luminous than a harder blue star.
We have no doubt that one or more of these stars is not a main sequence star
just looking at the picture combining it with things we have learned systematized
we ca say some things about this. we'll see later that we have many other
techniques. Binary stars are so important that you've
develop many techniques for discovering them
binaries discovered through non visual techniques.
Where we can't resolve, even in a telescope.
The two members are going to be non visual binaries and we'll talk about how
to find them. But for now let's talk about a famous
visual binary to see what we can learn from it.
the star in question is called Alpha Centauri.
Alpha Centauri is famous because it's one of the nearest stars to Earth.
It's only, about a parsec. A little more than a parsec away from
Earth. it has, in fact two members.
It's a binary star. The two members are Alpha Centauri A and
B, A being the brighter, the primary, B being the secondary.
And in fact we think that the system is a triple star.
The third member would be Proxima Centauri, the actual closest star to
Earth very famous. we're not quite sure if Proxima Centauri
is bound to these two or not. If it is it's a very large, very slow
orbit. Let's ignore Proxima Centauri for now,
focus on Alpha Centauri. you see here a plot.
We can see both stars. So what we've done here is we've plotted,
we've fixed the brighter A component at one point.
And then plotted where in the sky the B component would appear.
These are right ascension and declination axis and the scale or the angles are
marked in arc seconds since we know the stars are about.
Both, are about, 1.3 parsecs from Earth. Well, at a distance of a parsec, an
angular separation of one arc second corresponds to an astronomical unit.
So the ticks on the scale, the arcsec ticks on the scale, correspond to
distances of about 1.3 astronomical units.
That means that these stars are orbiting within planetary distances they're
orbiting as close to each other, as our planets orbit the sun. This is a very
close binary system. and because we know the small angle
formula we can translate angles to distances, that's glorious in fact this
is not the actual orbit. the actual orbit, the apparent orbit is tilted
elipse here, the actual orbit if you flatten it out so that you see it face on
would be this more broader, that's here, how do we know
that? Well we can extract the radial velocity
of B Centauri, B Alpha Centauri, by measuring doppler shifts, and that tells
us how fast it's moving towards us, or away from us.
Combine that with what we see in the tangent plane and we get a full
description of the motion. Just as we did for other proper motions.
And this is, in fact the ellipse seen head-on and we can measure the
Eccentricity of this ellipse is about 5.. It's a rather eccentric motion.
The semi-major axis is 23 astronomical units approximately.
That's about the radius of Saturn at their closest approach these stars are
only eleven astronomical units apart and their period is about eighty years.
we can basically, you know we can plot an ellipse.
We measure the speed. We know exactly how
fast these stars are going to move and so we have an 80-year period at a semi-major
axis of 23 astronomical units. Now, before we go how to take advantage
of this, I should tell you I didn't pick this system, just because it is nearest
to, to earth, it is also, well of course as such it has been of interest to many
science fiction writers but it is also recently been of interests to scientists
because in October 2012 an earth size planet was detected orbiting Alpha
Centauri B remember we said that it was strange to find stable orbits in binary
systems. Even in this very close binary system
there is a stable orbit. Note that the planet orbits very, very
near to its star at a distance ten times closer than Mercury to the sun.
it's period is only three and a half days, or so.
This is a very close in much too hot for liquid water to exist, but the
discovery of an Earth sized planet, in a binary system is exciting nonetheless.
All the imaginations of what it would be like to live on a planet with two suns
get fired up and so on. So,
so much for Alpha Centauri. Why do we like binary stars?
We like binary stars, because when we have a visual binary we can measure the
positions. we can actually predict just as we did.
A projection of the orbit on the, tangent plane.
Remember the tangent plane is the plane, perpendicular to our line of sight.
If we can augment this with radial measurements of the radial velocity from
doppler shift, we can figure out both the motion in the tangent plane.
Remember, if this is us, and here is the star.
This is the tangent plane. This is the radial direction.
Doppler shifts give us the measurement of in this direction the astrometry where we
take the distance and the angle and convert it to distance over here with the
small angle formula, give us the projection of the orbit on the tangent
plane. We can actually produce the orbit of the
planet and in particular we can measure the orbit of the stars, and we can then
measure semi major axis and certainly periods.
And if we know semi major axis and period, we can use Kepler's formula,
scale it to the motion say, of the motion of the Earth around the sun.
And find that the total mass of the star divided by the total mass of the Earth
sun system, which to our approximation is the mass of the sun is the ratio of the
radii cubed divided by the ratio of the period squared.
So we have if since we measured the radius and we measured the period, we can
find the total mass of the system. We can do better than that because we
plot the motion of both partners in the sky, relative to the fixed stars far
behind them. We actually find our one and our two
separately. Remember, the two stars orbit each in a
circle, or an ellipse in this case of semi major axis or radius in our
approximation. R1 and R2 and since they're both orbiting
their center of mass these radial satisfy M1, R1 is M1, R2.
That's the distance from the center of mass.
We know the ratio of the two masses, because we measured the radii.
We know their sum, because we measured the period between these two.
We can write an easy algebraic equation, and find the individual masses of stars.
We have a way of weighting stars. This is why we like binaries.
not always are binaries so close that we can distinguish both partners.
It's worth figuring out other ways to study binary systems.
Let's try that.