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.