Our solar system's an exciting place. one of the most exciting developments in
the past few decades has been the realization that our solar system is not
unique. There are solar systems all over the
place and we're starting to learn something about them.
And it seems very fitting to end this week of solar system science with some
discussion of what it is that we found out there, the topic that goes under the
name extrasolar planets, or exoplanets for short.
And so, the fact that planets should exist around
other stars and that the sun is not unique in having planets is an old idea.
the fact that we saw protoplanetary disks around stars suggests that many stars
will have planets. and people have been looking for planets
around other stars for a long time. The first successful discovery was in
1988, so only 27 years ago. by now at last count, there are 853
confirmed planets orbiting 672 different stars, so we have whole solar systems
that we have found and the field is growing very fast.
I think we're just starting to learn what these extra solar systems have to teach
us. finding planets is a tough business.
what you're looking for basically is a chunk of rock or gas that is not hot and
is not glowing and is very near to something, a star, that is hot and is
glowing, so it is a difficult problem trying to actually see a planet at great
distance. we see the light of the star, but seeing
the reflected light off a planet is a difficult proposition.
despite this, it has been applied and all 32 planets and 38 systems have been
actually imaged. In other words were found in actual
images of a star, the way this is done is the same way we will learn next week.
technology developed to image the outer regions of the star, of the Sun, the
corona when basically superimposes a disc that blocks out the star light in the
one's telescope and then looks for the dim lights around it.
there are future missions that are designed to specifically image extra
solar planets and we might be able to increase the rate of discovery.
Anyone can learn a lot about a planet if one can actually image it.
But most of the 900 planets we've seen have not been discovered that way.
In fact, most of the time, all you can see is the light of a star.
So, how does the light of a star tell you that there is a planet orbiting it?
The following few simulations might help understand that.
One key to understanding how a star can tell us that a planet is orbiting it is
to remember that if a planet is orbiting the star, then the star is also orbiting
the planet. In fact, they are both orbiting their
common center of gravity as this animation here will show us.
We have here a planet and a star. I have just, for the purposes of
discussion, given them a mass ratio of one to four.
And, what we see is that they each orbit I've assumed a circular orbit and they
each orbit in a circle about the relative common center of mass, which is here this
green X. And what we see is that, the radii of the
two circles are of course distances from the center of mass.
In other words if the larger star here is four times as massive as the smaller as
the smaller planet, then the radius of the orbit in which the star orbits is a
fourth the radius of the orbit in which the
planet orbits. In other words, if this, orbital radius
is our and this orbital radius is our planet,
then, since they're orbiting their center of mass, these satisfy mass of the planet
times the radius of the planet's orbit is equal to the mass of the star times the
radius of the star's orbit. And of course, the distance between the
planet and the star, since they're on opposite sides, are the distance is
simply rs plus rp, the sum of the two radii.
And this distance is constant at all times,
of course, most planets don't orbit in circular orbits but as usual we will make
that simplifying assumption, it'll help us make calculations more
straightforward. Of course, since typically, a star is far
more massive than the planets that orbit it, the star will move a far small, in a
far smaller circle than well the planets and detecting the motion of a star might
be difficult because the motion is very small.
To give us a sense for, of how this works we'll try to detect, imagine detecting
from a distance, a solar, the planets of a solar system we understand, namely our
Solar System. And indeed, because earth orbits the sun,
the sun is also orbiting the earth, the center of mass of earth and sun lies
well deep within the sun. And it would be very difficult to
actually note the motion of the sun. In fact the only planet whose common
center of mass with the sun lies outside just outside the surface of the sun is
Jupiter. And in this animation, we see that if we
take into account only the gravitational effects of Jupiter, then we observe the
sun moving, around in a circle around a point just outside the surface of the
sun. Saturn's influence is the next most
remarkable and because we now have two periodic motions, the sun's motion is a
little bit more complicated and no longer moves in a circle. So by doing some
interesting analysis, we could, in fact, disentangle perhaps, the two motions due
to the sun, to Saturn and Jupiter if we can observe this motion.
So, we need to understand how far the sun is moving and try to see what
limits this places on our ability to detect planets.
So now, we, we, we have an idea. Maybe we can observe that a planet is
orbiting a star by watching the star wobble.
We need to understand how much the star will wobble.
We expect it not to be very much, but because it is a periodic motion, we
might be able to actually distinguish it and realize that what is going on is that
it, there's a planet orbiting it. And we said, that if we assume circular
orbits, there's a distance r between the star and the planet, then we can write r
as R is the radius of the circle in which the star moves plus the radius of the
circle in which the planet moves and the two radii are determined by the center of
mass relation. And plugging one relation into the other,
I see that the total distance is Rs + Rp, but Rp is Rs * Ms / Mp.
We write it this way and now, I can solve this equation for Rs,
Rs = to R / 1 + Ms / Mp. And then, I can pick a common denominator
on the bottom to write Mp + Ms and Mp comes here.
Mp + Ms is nothing but the total mass, which I'll call M.
And this is not surprising, their amount by which the star moves, is related to
the distance between the two by the ratio between the planet's mass and the total
mass, the larger the planet the more the star will move.
Not too surprising, lets see what that tells us for example about an idea,
let's try to detect Jupiter. How far would the sun move?
Well, plugging in this result we just figured out.
I find that the radius of the circle in which the sun moves due to the
gravitational effect of Jupiter is the ratio of the mass of Jupiter to the mass
of the sun times the radius of Jupiter's orbit, which I've put in here and that's
indeed right outside the sun. This is 740,000 kilometers.
The sun itself is about 696,000 kilometers in radius.
This point lies right outside the sun. Detecting the wobble of a star due to the
effects of a planet that orbits it is going to be difficult.
In fact, of all of the planets that we know, one was detected by this method,
this is called astrometry. So,
finding the motion discerning the motion of the star is not going to be our most
successful strategy. What else can we do?
Let's go back to the simulation and see a different approach.
The secret, it turns out, is that while it is very difficult to
measure the position of a star with the precision necessary to detect a planet
surrounding it, we may be able to detect its wobble by
measuring its velocity. How can you measure velocity without
measuring position? That's an interesting question.
Well, there is a sense in which we can measure velocity directly without
measuring position and this is the Doppler effect.
Remember that the spectrum of a star will contain absorption lines for whatever
gases are in the star's atmosphere, as we saw for the spectrum of the sun,
we can identify a particular absorption line and find that it is shifted in
frequency or in wavelength from its natural wavelength.
We can use the Doppler formula. I remind you that lambda is lambda 0 * 1
- v / c, where c is the speed of light and v is the speed with which something
is moving. Since spectral lines are very, very
narrowly known we can measure rather small
velocities compared to the speed of light.
In fact, down to a few meters per a second.
So one part in 10^8 we can make pretty accurate velocity measurements provided
there is a Doppler effect. Remember, the Doppler effect only obtains
if the object to the source of light is moving towards us or away from us.
So in the position in, that, this animation currently depicts, of course,
we have no such luck, because as this planet orbits the star and the star
orbits the planet, there is no motion at all
no change in the star's distance from us so there, that there is absolutely no
radial velocity component. The star or the planet neither approach us nor
recede. On the other hand, if I tilt and I
imagine that the system happens to be oriented in this direction,
then what I observe is that now the planet, and therefore the star,
periodically approaches earth and recedes from earth and maybe we can detect the
variations in wavelength of some absorption line by measuring the velocity
with which the star is moving. And, the way we do this is precisely by
doing spectroscopy. Let's take a look.
This animation sort of shows you the set up.
We have here a planet orbiting the star, therefore the star orbiting the planet,
albeit in a much smaller circle. We cannot see the planet, but we can see
the star and things here are perfectly aligned.
The motion of the system is in such a plain that the entire velocity of this
vector of the star will be visible to us. There is no motion out of the plain of
this image, which would be invisible to us and we observe some absorption lines
in the spectrum of the star. And when the star is receding from us,
the absorption lines are shifted to the red, when the star is approaching us,
they are shifted to the blue. And in, we see this periodic shift in the
Doppler shift in the absorption lines and by
detecting the periodicity, we can measure the period with which the planets orbits
the star and perhaps learn some more. Of course, the Doppler shift here is
greatly accelerated. Let's do a little calculation to see what
kind of velocities we'd be trying to detect.
So perhaps, the way that we can detect the fact that a planet is orbiting a
star, hence the star orbiting a planet, is to hope that the orientation of the
orbit is such that at least some fraction of the star's motion is radial towards us
or away from us and we can use Doppler shifts to detect that velocity.
Let's figure out how fast the star would be moving.
so we have the system. how fast is the star moving?
Well, the star moves around the circle of radius Rs, as we said,
and it moves with a velocity speed Vs, whose velocity is continually changing.
So, as we know, the acceleration, from our
second week of class, the acceleration with which the star accelerates towards
the center is Vs^2 / Rs and if I multiply this by the mass of the star this is the
force acting on the planet, on the star to accelerate it and of course this is
Newton's gravitational force. In other words, this is G * Mp * Ms / by the
square of the distance between them. So let's put that in.
Here's our equation. what do we do with this equation?
Well, of course. we first realize that the Ms completely
cancels from here, this is characteristic of gravity.
now what I want to do is figure out v. I want to know how fast the star moved.
so from here, I have that vs^2 is given by G times the mass of the planet times
the radius of the circle in which the star moves divided by R^2.
But it would help to remember that we computed that the radius of the circle in
which the star moves is related to R by the mass of the planbet divided by the
total mass. And so, using that in here, I find that
this is G * Mp / R^2 * R * Mp / total M. And what I find is that this is given by
well, I can cancel one of the Rs here and I find that vs^2 is g *
Mp^2 / M * R, which is almost nice, except I will find
it more convenient to write G * M / R, this is a quantity with dimensions of
velocity squares, * ratio of Mp to the total M^2.
All I've done here is multiply and divide by the mass the total mass.
And so, this is my expression for vs and when I plug it in, I see that the
following things are true for M is usually typically just the mass of the
star, the mass of the planet is small. Anyway what we see is that massive
planets will cause more motion. This is certainly true,
but, unlike the case in the when we were trying to do astrometry, in the case of
radial velocity measurements, the closer a planet is to its star,
the smaller R, the more radial velocity the star will acquire.
So this method will be useful in discovering planets that are massive and
orbit close in. We can solve this equation now for vs by
taking the square root and this is the expression we get.
Again, the largest values of velocity will arise from heavy planets that orbit
near their stars. We can plug in the numbers for Jupiter.
If you look at this quantity in the square root, it's simply the orbital
velocity of Jupiter to good approximation.
And so, plugging in the numbers we see, that due to Jupiter orbiting it, the sun
moves at 12.5 meters per second. This is a small velocity relative to the
speed of light, but wavelengths can be measured with extreme precision.
We would have very little trouble if orbits were aligned correctly,
identifying the motion of the sun by ten meters per second, and in fact, this is
one of the most productive ways of detecting extrasolar planets.
a total of almost 500 planets in almost 400 solar systems have been detected by
what we call radial velocity measurements.
And remember, this is only possible for a systems so
aligned that part at least of the motion is radial.
If the system happens to lie so that the motion of the planet in the star is
perpendicular to our line of site, there just is no radial component,
there is no Doppler shift. We're out of luck trying to find it.
despite this, this has been one of the most productive ways of measuring it.
But if the system is aligned properly, we can do even better.
Let's check this out. One great advantage of the radial
velocity measurement detection is that if we have some way of estimating mass of
the star, independently, we'll see that we have such methods, then, it gives us
information about the mass of the orbiting planet,
ass well as the distance at which it orbits.
And so, we can extract the beginnings of more data about what kind of planet we
see, and what kind of orbit. And there some circumstances, if the
alignment is just right, we can do even better.
if the alignment is such that the planet in fact transits in front of the star,
then the planet will partially eclipse its star.
And the way we will observe this is that the star light appears to dim when the
planet is transiting and we call this measurement measuring the light curve.
The light curve is the dependence of the brightness of the star on time.
And, what we will see is a periodic change as the brightness of the star dips
periodically, when the planet transits it, we can use
this to compute the period. We can use this to study from the details
of the shape of the light curve. You can learn many things about the
planet and such as its size, its mass. If we're lucky, we might even see the
planet's atmosphere beginning the eclipse process, and see the star through the
planet's atmosphere, and attempt to do spectroscopy, and understand the
chemistry of this star's atmosphere. So in principle, the transit method is a
very constructive method, which gives us a lot of information.
Some 300 planets and over 200 systems have been detected.
There's a dedicated space telescope, the Kepler Space Telescope, which is doing a
survey of the sky and already has over 2,000 candidate planets and over a 1,000
candidate systems. But the transit system the transit method
is not considered a complete detection until verified by some other means, so
these are undergoing verification. The sample of planetary systems that we
have outside the solar system is certainly growing very quickly and we
should have a lot of information. Ideally, when detects the same planet
both using radio velocity and transit method and then one can extract lots and
lots of information. We will talk more about transits when we
talk about binary stars and light curves. There are a few other methods that have
been used. The lights of a star can be intensified
by the place, presence of the planet, and some sixteen planets have been
observed this ti-, this way. If we have one transiting planet, we can
study its, orbit and find protobations to its orbit, and describe them to another
planet, and learn about the existence of the other planet in the same system.
Some planets have been detected this way. In all, we're compiling the collection of
solar systems and it helps us to see, well,
how do they stand to our understanding? How does this match our understanding of
our solar system? We developed sort of a good theory that
describes the one solar system in the universe,
but now that we suddenly have thousands of them, you can ask does this theory
work? And so, what have we found?
well this, as this image clearly shows, we haven't looked very far.
We've looked to about 300 light years around the solar system,
that's a tiny fraction of the galaxy. We have all these thousands of planets,
depending on how exactly you make the estimate, and as I said, this is a young
science between one and 40% of sunlight stars have planets.
Either number means that in a galaxy with a few hundred billion stars, there are
lots and lots and lots of planets and planetary systems all over the place.
no, the solar system is not unique. Now, the methods that we've used, both
the, the most constructive ones, both the transit method and the radial velocity
measurement method are most sensitive to what we call hot Jupiters planets that
are very massive and orbit very near their star.
Remember, we saw that with the radial velocity measurement.
It's also true that it's more easy to arrange an alignment if the planet is
close or methods are sensitive to hot Jupiters.
Maybe that's not, it's not surprising that mostly we find, at least initially,
lots of planets with masses similar to or larger than Jupiter's orbiting at
Mercury-sized orbits. Now,
well yes, there is a sample bias going on here, clearly a selection bias.
if one had a detector that was only sensitive to unicorns and one found no
coyotes, this would not be a surprise. But if one had a, a detector sensitive
only to unicorns then found unicorns, that would still be a discovery.
Discovering all these gas giant planets orbiting within a half an astronomical
unit of a sunlight star should still be jarring, because, remember we explained
that great length, Y, it is impossible for such planets to form so close to the
star, because the solar nebula is too warm and
volatile materials are not solid. What we learn from the fact that they
exist is presumably they did not form, did not position,
they formed farther out where the nebula was colder, and migrated in.
So the kind of migration that the Nice model proposes for the solar system
apparently is occurring all over the place.
And this indeed, seems to be a ubiquitous feature of the solar systems we're
finding the kind of sedate stability that we see in our solar system is not
actually the standard. There's lots of strongly interacting,
very resonant orbits, instabilities in that sense, at least, the orbits we
are find, we find in our solar system are a little bit uncommon.
What are the planets like? Well again most of the planets we find
are very massive and lots of them are very closed in.
If you look at this logarithmic plot right in the center of the plot over down
here is our Earth with and then the horizontal axis is the
distance from the star, the vertical axis is the mass of the
planet. Both of these are logarithmic.
So most of the planets that have been found are much more massive than earth.
In fact, much more massive than Jupiter up here.
And many of them orbit very close in, these are these hot Jupiters that, we
spoke of, but we're also finding planets that orbit
out at far larger distances from their stars and we've even found some planets
that are on the order of magnitude of earth's mass or less.
And so if you take the selection bias into account, the prediction is that the
fact this data suggests that there are earth-like planets in much larger numbers
than there are Jupiters or gas giants. Because, the fact that we're detecting
them at all means they are far more common.
we also find some super duper Jupiters, some gas giants with masses,
15, 20 times the mass of Jupiter. Much larger than one thought planetary
systems could create. And some interesting, strange cases like
a planet called Kepler-16b that orbits a binary system.
So two stars orbiting each other with a planet in an almost perfectly circular
orbit around the center of mass of the system.
this had been thought to be impossible to arrange in a stable orbit and Kepler-16
shows us that nature manages to find a way.
So, the field of extrasolar planets is despite the fact that it's been a few
decades since we first discovered the first ones.
This is really a field in its infancy, we're just starting to learn what we can
learn from these exciting systems. And well, I anticipate a lot of news,
both in terms of discovery and in terms of insight into how solar systems work in
the coming decades. Let's sum up a busy week.
Well, we found a lot to learn, right, in our own solar neighborhood.
A lot of it, both in our solar system and certainly beyond is just being learned.
I hope you're starting to develop an appreciation for the fact that we tried
to understand some of the fundamental physics, because whatever we can, we can
apply, we can actually understand what is going on,
we can make calculations, that tell us what to expect and match them to
experiment. I think understanding the physics,
roughly of the greenhouse effect might be a good example of this.
And, as I said exoplanets are probably going to eventually completely change the
way we think of solar systems and so stay tuned.
we've not covered many of the interesting topics in the understanding of our solar
system. I want you to think of this week more as
a hunting license. You now have
the tools to go and read about the moons of Jupiter and why it is bizarre that
Neptune has a moon that orbits completely outside the ecliptic, what that probably
tells you about Triton and so on and so forth.
And so I think you are equipped with at least the tools to go and acquire
information. next week, we aim for the stars.