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4. Frontiers/Controversies in Astrophysics: Discovering Exoplanets - Hot Jupiters
Poziom:
Temat: Edukacja
Professor Charles
Bailyn: We were talking
last time about the objects in
the Solar System.
And we'd gone through kind of
two of the three stages of the
scientific method as it's
applied to observational science
rather than experimental
science.
And the first thing was just
making observations,
finding a bunch of things.
And so I gave you a little
slide show depicting some of the
objects in our Solar System.
And after observations the next
thing to do is classification,
and we did some of that too,
and I divided all these objects
that had been discovered into
six categories.
And then, once you've done
that, once you have some
categories that kind of make
sense,
then the next thing is to
interpret these results and to
try and explain where these
categories come from,
how they arise,
and actually figure something
out.
And I've described that as
interpretation,
and I want to offer you a
little bit of interpretation
about what we've found about the
Solar System.
Now, I only want to explain
some of the classifications.
This is actually a common thing
to do.
When you find a whole bunch of
different things and you've got
twelve classes and three
sub-classes and two exceptions,
you kind of want to explain the
big features first and then
worry about the little things
later.
This is commonly done.
So, what I'm going to do is I'm
going to talk about only the
inner terrestrial planets.
You'll recall that these are
small, rocky things in
relatively short orbit,
and contrast them with the
outer planets,
the Jovian, the Jupiter-like
planets, which are large and
have not only rocks,
but also lots of ice and gas.
And these things are in wider
orbit, but the orbits of both of
these are, basically,
more or less circular.
Not precisely circular,
they're actually elliptical,
but quite close,
and they're all in the same
plane.
That is to say,
they're all going around the
same way.
There's nothing that's going
this way instead of this way.
So, they're circular and
co-planar.
And let's try for an
explanation of those particular
features.
Okay, so this would be
something like a Theory of
Planetary Formation:
how the planets formed,
why they get that way.
This word, "theory," is a
serious problem.
This is one of the foremost
examples of a word that means
something different when
scientists use it than when
normal people use it.
In scientific parlance,
a theory is something which has
a lot of support,
which explains a lot of
observed or experimental fact.
In everyday life,
of course, a theory means a
wild guess.
So, there's a pretty stark
difference between those two
definitions.
This gets our friends the
evolutionary biologists,
in all kinds of trouble because
they keep talking about the
theory of evolution.
And a certain segment of
society interprets that as the
wild guess of evolution and this
creates various kinds of
difficulties.
The problem is,
there isn't another word that
one can readily use for what the
science definition of this--you
could use "paradigm,"
you could use "scenario."
These are kind of ugly sorts of
words, so I think we're stuck
with "theory."
But I mean this in the sense of
something that explains a lot of
facts, rather than in the sense
of wild guess.
Okay, so here we go:
Theory of Planetary Formation.
So, the idea is that the
planets form from a disk of
material around the Sun.
So, the Sun and the planets are
created out of a collapsing
cloud.
The cloud collapses,
some parts of the cloud are
rotating, that prevents them
from collapsing.
And so, you have a situation,
after a while,
where you have a sort of
star-like thing in the middle,
and then a kind of disk of
stuff around it.
So, this is a side view,
and in the top view,
all these things are in orbit
here.
And this disk consists of
basically the same material as
the Sun, which is to say,
lots of gas,
by which, I mean hydrogen and
helium in particular.
Some ice, of course,
when it's in a star it's all
melted.
But these are elements like
carbon, nitrogen,
oxygen, that go into making
ices,
and a little bit of heavier
elements that could,
if you put them all together,
form dust and rocks,
and things like that.
A little heavier element:
things like silicon and iron.
So, that's what the Sun
consists of, and so does this
disk orbiting around it.
Then what happens?
In the disk,
things gradually stick
together.
So rather than having
individual atoms or molecules,
the molecules and atoms sort of
run into each other,
form dust grains,
the dust grains run--or ice
crystals or whatever,
those run into each other,
form bigger things.
And you gradually make bigger
things as the little things
collide with each other and
stick.
So in the disk,
things gradually stick together
and become tiny little objects,
which are sometimes given the
name "planetesimals."
And then the "planetesimals"
bump into each other and stick.
These stick together until
you've got a situation where in
each region, each orbit,
each distance from the Sun,
you've coalesced everything
into one large object.
And so you end up with one big
object in each region,
by which I mean distance from
the Sun, distance from the star.
And this explains the orbits,
because since these large
objects have been created by
running a lot of small objects
together,
the ellipticity of any of the
elliptical part of the orbit of
any of these things tend to
cancel out.
Because one of these objects
will be elliptical in one
direction;
another will be elliptical in
another direction.
If you put them all together
they'll--that orbit will tend to
be circular.
Similarly, some of the objects
will be going up out of the
plane of the Solar System,
some will be going down,
but you'll run them together
and they'll all end up with kind
of similar circular co-planar
orbits.
So makes approximately--that's
the approximate sign
[~]--circular,
co-planar orbit.
So, that's good because that's
one of the things we're trying
to explain.
And there is an expected
difference between how this
works out in the inner parts of
the Solar System and how it
works out in the outer parts of
the Solar System.
In the Inner Solar System the
ice and gas in the
"planetesimals" evaporates and
does not become part of the
planet that gets formed.
And so the planets are only the
rocky parts.
They actually hold onto a
little bit of the ice and gas,
but not very much.
Whereas, in the Outer Solar
System, the ice is frozen,
and so it behaves just like
rocks.
And so planets have rocks and
ice.
This means that they're
substantially more massive.
And if they're sufficiently
massive, then they have enough
gravity to hold onto the gas
also--onto gas as well.
It's also true that in the
Outer Solar System there's more
volume, there's more stuff,
and so there's more stuff to
build the planets out of in the
first place.
And so, this nicely explains
the difference between the Inner
Solar System and the Outer Solar
System.
You build these things up out
of little chunks.
But in the Inner Solar System
the temperature is high enough
that the chunks of ice
evaporate, and so you can't
build them up out of that.
And so you get much,
much smaller things made almost
entirely out of rocks,
in contrast to the Outer Solar
System where you have enough
ice,
you build much bigger planets,
and in some cases you hang on
to a lot of the gas as well.
So, this is interesting because
it makes a prediction about how
other Solar Systems ought to
look.
Namely, that this difference
between inner planets and outer
planets ought to exist
everywhere.
That planetary systems--there
should be a general feature,
because there's nothing of what
I've said so far that's unique
to the Sun.
So planetary systems should
have inner rocky planets and
outer Jupiter-like planets,
sort of gas-plus-ice sorts of
planets, much bigger.
And the dividing line between
these two kinds of planets is
determined by temperature.
Because there will be some
temperature where those ice
things melt and therefore you
don't expect it to be at the
same distance away from a star.
You expect it to be at the
place where the temperature is
the same.
Therefore, if you have a very
bright star, hot star,
you'll have inner planets out
further from that star,
inner-type terrestrial planets
will go out further.
And if you have a very dim
faint star, which doesn't
generate as much heat,
the gas planets will--the
dividing line between the gas
planets and the terrestrial
planets will be much closer in.
So the dividing line is
determined by temperature,
hence, by the luminosity--the
amount of energy given off of
the star.
So now, this tells us what
we're supposed to do next.
Namely, go out and find a whole
bunch of other Solar Systems and
verify this prediction.
Namely, that if you've got a
really bright,
hot star, you ought to have
rocks out fairly far in the
Solar System,
whereas, if you have a dim
star, you'll have Jupiter-like
things coming in much closer.
And so that's what I want to
talk about now,
is how people went about doing
this and what the results were.
And I'll tell you the punch
line in advance,
which is that this totally
doesn't work.
But it's an obvious prediction
from our theory of planet
formation that came about by an
examination of what was going on
in our own Solar System.
Okay.
So, observing exoplanets.
How do you find these things?
And you'll recall we started
down this track in the last
lecture.
The key point here is that
stars move too.
It's not just the planet going
around a star,
it's the planet and the star
both go around the center of
mass of the system and the stars
move, too.
You can't see the planets
independently.
And there was a little
equation--the velocity of the
star times the mass of the star
is equal to the velocity of the
planet times the mass of the
planet.
This is basically an equation
of momentum.
The distance of the star to the
center of mass,
times its mass,
is equal to the distance of the
planet to the center of mass,
times the mass of the planet.
This is kind of a definition of
the center of mass.
And sometimes,
you don't want to deal with the
individual velocities or
distances, you want to deal with
the total velocity or distances.
And so that's just defined
where if you want to talk about
the total that's obviously just
D_star plus
D_planet.
Similarly, for the total
velocity and the total mass,
and that's just defining terms.
And then, in order to relate
these distances to things that
have shown up in our equations
and Kepler's Third Law,
it's true that the maximum
value of the total
distance--that is to say,
the distance between these
things can vary,
because they can be in
elliptical orbits.
Sometimes they're closer than
others.
And if you take the maximum
distance between them--that's
the maximum of D_tot
al--that's a,
that's the semi-major axis
of the orbit.
And now, for nearly circular
orbits, as the planets turn out
to be, then
D_total is
always the same,
because if it's circular the
distance between them doesn't
vary.
So then,
D_total is more
or less equal to the semi-major
axis because it's always the
same,
therefore it's always near its
own maximum.
And then, you can also say
something interesting about the
velocity.
The velocity--what's the
definition of velocity?
Velocity is miles per hour,
or something like that.
So, it's distance per time.
And let's take a time period of
one orbital period and ask the
question, how far does something
go in one orbital period?
Well, it goes all the way
around its orbit.
And you may recall from high
school geometry that if you know
the radius of a circle,
you also know its
circumference.
The circumference is the
distance it would have to
travel, that's 2π times the
radius.
This is the basic fact from
geometry and so that's 2π times
the distance,
in this case,
the semi-major axis.
But this is only true for
nearly circular orbits and the
reason is that in highly
elliptical orbits,
the velocity changes by a
substantial amount.
It moves much faster when it's
closer.
And so you can only really
define what the overall velocity
of the thing is,
if you've got a nearly circular
orbit.
But in that case,
as is true for planets,
this 2πa over P
gives you a value for velocity.
Now, so we have another little
equation here,
V equals 2πa /
P.
This is an important one,
so you'll want to remember
that.
And I should say,
which kind of Vs and
as, remember up here
there is
V_star,
V_planet,
V_total,
all these different kinds of
things.
What do I actually mean by that?
And it can mean any of them,
but it has to be consistent.
So if you're dealing with the
velocity of the star then
a is equal to
D_star.
Remember these are all nearly
circular orbits so D
isn't going to change.
And if you've--if you're
dealing with V of the
planet, then a is equal
to D of the planet,
and that is approximately equal
to D_total.
Because the mass of the planet
is so low that almost all the
motion in the system comes from
the planet.
So, you can also deal with
V_total
_, which is
equal to
V_planet.
And so, all these things go
together.
But if you're worried about the
velocity of the star,
you have to be careful,
because it's not a of
the orbit as a whole,
it's just a tiny piece of the
orbit that involves the motion
of the star.
Okay?
Let's do an example.
How fast does the Earth move?
V = (2πa) /
P.
Well, we know a is equal
to one Astronomical Unit.
P is equal to one year
for the Earth's orbit,
and so the velocity of the
Earth is 2π Astronomical Units
per year.
Pretty straightforward but not
very informative,
because we don't have a feeling
for measuring velocities in
Astronomical Units per year.
There's a joke in--a kind of
physics joke that you convert
all velocities into furlongs per
fortnight just to be annoying.
But let's not do that;
let's convert it instead into
meters per second,
because then we have the hope
of understanding what's going
on.
V is equal to 2π.
One Astronomical Unit is 1.5
times 10 to the 11 meters.
And a year is 3 x 10^(7)
seconds.
π / 3 = 1.
2 x 1.5 = 3.
10^(11) / 10^(7).
11 - 7 = 4.
So, this is 10^(4) meters per
second, that's 30 kilometers per
second.
So, we move right along as we
go around the Sun.
If you did this for Jupiter,
plugged in the various values
for Jupiter--I won't actually do
that calculation,
you can do it on your own--you
discover that Jupiter moves
about half as fast as the Earth.
So, it goes around 15
kilometers per second,
that's 1.5 x 10^(4) meters per
second.
So now, we can ask the
question, "How fast does the Sun
move in response to the orbit of
these planets?"
So how fast is the solar motion
induced by Jupiter?
Okay, and now we go back to
this momentum equation.
Velocity of Jupiter times the
mass of Jupiter is equal to the
velocity of the Sun times the
mass of the Sun.
What we want to know is the
velocity of the Sun.
And so that is equal to the
velocity of Jupiter,
which we just calculated,
times the mass of Jupiter,
divided by the mass of the Sun.
1.5 x 10^(4),
that's the velocity of Jupiter.
Mass of Jupiter,
as it happens--I think I wrote
this down last time,
2 x 10^(27) kilograms.
Mass of the Sun, 2 x 10^(30).
Twos cancel, obviously.
We get 1.5 x 10^(4).
Times 10^(27), that's 10^(31).
10^(31) / 10^(30)=10^(1).
Is equal to 15 meters per
second, not kilometers now,
meters.
And so the Sun--Jupiter moves
15 kilometers per second,
the Sun moves 15 meters per
second.
That makes perfect sense,
because the Sun is 1,000 times
more massive than Jupiter,
so it has to be going 1,000
times more slowly.
So instead of moving at some
number of kilometers per second,
it's moving at some number of
meters per second.
This, it turns out,
can be detected with modern
equipment.
"Detectable," let's say,
in distant stars.
You can see things that move by
15 meters a second.
We'll come back to how that's
done in a minute.
How about the Earth?
Solar motion due to Earth.
Now, you might think is going
to be bigger,
because the velocity of Earth
is bigger than the velocity of
Jupiter.
But, of course,
the mass is much,
much smaller.
So we have M_earth
V_earth
is equal to
M_sun
V_sun,
where V_sun
now means the motion induced by
Earth.
This number is bigger than it
was for Jupiter,
but this number's a whole lot
smaller.
And so the overall effect is
that the V_sun
is going to be smaller,
V_earth
M_earth /
M_sun.
That's 3 x 10^(4),
that's the velocity.
Mass of the Earth,
as it turns out,
is 6 x 10^(24) meters per
second.
The Sun, down here at 2 x
10^(30), same Sun.
6 / 2 = 3.
3 times 3 is 10,
so we get (10^(1) x 10^(4) x
10^(24)) / 10^(30).
One--five--29 over 30.
That's 10^(-1) = 1 / 10 of a
meter per second,
or 10 centimeters per second.
So that's much,
much slower than the Sun moves
in response to Jupiter.
Why?
Because the Earth is so much
less massive.
So, 15 meters a second for--as
the result of Jupiter,
only 10 centimeters a second,
a tenth of a meter per second,
as a result of Earth.
And with current technology,
things that slow are not
detectable, yet in
other--around--in other stars.
So, we have a situation,
and this is what was happening
about ten years ago,
where instruments had been
developed that could,
in principle,
see the reflex motion of stars
due to planets like Jupiter,
but weren't yet capable of
seeing the motion of stars due
to planets like Earth.
But, we expect that Solar
Systems ought to have planets
like Jupiter,
and so people went out to try
and look.
All right, how do you find--How
do you observe these things?
And now, if you've taken high
school physics,
you will recall,
perhaps, something called the
Doppler Shift.
This is the key.
And this is a way of measuring
velocity and it turns out oddly
enough that velocities are some
of the most easy and
straightforward things to
measure in astronomical objects
because you can determine them
by the Doppler Shift.
And so just to remind you or to
inform you, if you haven't seen
this before,
and there is some help sheets
and things that you can look at
about this too.
Light is characterized by its
wavelength, which is usually
given the Greek letter Lambda
[λ].
And light that is something
like 4 x 10^(-7) meters.
A wavelength has units of
length.
This looks blue to us.
Light that is--let's color code
this for your convenience,
5 x 10^(-7) meters looks green,
7 x 10^(-7) meters kind of
looks red.
Longer wavelengths are what we
call "infrared."
And shorter wavelengths that we
can't see are called
"ultraviolet."
And so, ultraviolet up here.
And if you get really,
really long--if you have,
like, meter wavelengths--that's
radio waves, out here.
And if you have really short
wavelengths, those are x-rays
and gamma rays.
So, all of these kinds of
radiation are basically the same
thing, called "electro-magnetic
radiation";
again, there's a help sheet.
These are all electro-magnetic
radiation, and what kind of
radiation it is depends on the
wavelength.
And the key to the Doppler
Shift is that the observed
wavelength changes,
depending on the relative
motion of the thing emitting the
light in the observer.
Motion of source and observer.
In particular,
if they're moving toward each
other, then the wavelength gets
shorter,
and if they're moving away from
each other, the wavelength gets
longer.
Do you feel an equation coming
on?
Because obviously this is going
to need to be quantified,
right?
How much shorter?
How much longer?
But before we do that,
let me just point out that this
motion towards is sometimes
called a "blueshift" because it
makes--it pushes the light from
the red end of the visual
spectrum towards the blue and
this kind of thing here,
the motion away from each other
is called the "redshift."
And let me show you why this is
true before I write down the
equation.
Let's see, it's just--let me
get out of that and try this one
instead.
It's just a property of how
waves look, so look what
happens.
If the thing is stationary
there, in the middle,
that's emitting the waves,
then the waves propagate
equally in all directions,
and both observers see the same
distance between successive
waves.
That is to say,
the same wavelength.
And you can see that there.
Then, when the thing is moving
in some direction,
each successive wave is emitted
a little bit closer to one
observer,
and a little further away from
the other observer.
And so, because the waves are
emitted at different places,
the wave fronts here--I'll wait
until this cycle goes through
again.
The wave fronts for this
observer are closer to each
other and the wavelengths looks
shorter.
So, when the thing is coming
towards you that's emitting the
wave, it looks shorter to the
observer it's going towards.
Whereas, for this guy,
the waves--each successive wave
is emitted a little bit further
away.
And so the wave fronts are
further away from each other
when they pass,
and then the wavelength becomes
longer.
So, that's the kind of
conceptual thing that's going
on.
And the key thing is that the
velocity that's relevant here is
velocity toward and away from
you.
If the thing is going sideways,
it doesn't make any difference.
And so, it's not actually
velocity that you observe by
looking at the Doppler Shift.
It's radial velocity,
which is the technical term
for--is the thing coming towards
you or moving away from you.
And how fast is it coming
towards you and how fast is it
moving away from you?
Okay, let me turn this off here.
So here's the equation for
that--let's see here.
Lambda is the wavelength--I'll
explain all these terms in a
minute.
And this is important.
And the terms mean the
following things:
this is radial velocity
[V_r],
and it's positive when it's
going away from you.
It's negative when it's going
towards you.
And it's zero when it's going
sideways.
And it's a velocity,
it's in meters per second or
whatever the appropriate units
of--furlongs per fortnight,
or whatever the appropriate
units are.
The only restriction on the
units is, it has to be in the
same units as the thing in the
denominator, here.
That's C,
that's the speed of light.
That's 3 x 10^(8) meters per
second.
Maybe we should have you work
it out in furlongs per
fortnight.
No, no, no, we won't do that.
But as long as the velocity
here is expressed in the same
terms as you express the speed
of light, then the units will
work out.
This λ with the little zero at
the bottom is the rest
wavelength, so that's the
wavelength you would observe
from whatever light source,
electro-magnetic radiation
source you have,
if nothing was moving.
And Delta Lambda [∆λ],
this is not ∆ times λ.
That's one symbol,
confusingly enough.
Delta always means change;
you may remember this from
calculus if you've taken
calculus.
Delta always means change,
so this is a change in
velocity--sorry,
change in the wavelength.
And it is defined such that the
observed wavelength is equal to
the rest wavelength,
plus the change in the
wavelength induced by the radial
velocity.
So now, look how this works.
If this side of the equation is
negative--if it's coming towards
you--then this quantity is
negative.
That means this quantity is
negative.
That means this quantity is
negative.
That means the observed
wavelength is shorter than the
rest wavelength,
which is exactly how it's
supposed to be.
When we come towards you,
it's blueshifted,
the wavelengths get shorter.
Similarly, if this is going--if
something's going away from you
then V_r is
positive,
this is positive,
and you end up with a longer
wavelength.
Okay?
All right, example:
how fast do you have to go to
turn a red light green?
This is potentially useful
should you ever be pulled over
for running a red light.
You can just say,
"it looked green."
How fast to make a red light
green?
And let's call green light 5 x
10^(-7) meters.
Red light is 7 x 10^(-7) meters.
So ∆λ had better be equal to
(7 - 5) x 10^(-7).
That's 2 x10^(-7) again in
meters.
And we want this to be
negative, because we want
λ_0 to be the red.
That's what it would be like if
nothing was moving and we want
λ_observed to be
green.
And λ_observed =
λ_0 + ∆λ.
And this had better be,
- 2 x 10^(-7) meters.
And so ∆λ / λ _0,
that's (- 2 x 10^(-7)) over (5
x 10^(-7)).So that's - 2/5 is
equal to the radial velocity
over the speed of light.
So, if you're going at - 2/5
the speed of light,
then the red light looks green.
Now, the minus just means you
have to be moving toward that
light.
Now, two things about this,
first of all,
don't use this as an excuse,
because it'll cost you much
more in the ticket for going
over the speed limit if you're
going at 2/5 of the speed of
light.
Second of all,
be wary of this a little bit
because there is,
in fact, a change to the
equation that happens when
you're going close to the speed
of light,
and we'll talk about that when
we get to relativity.
And so this is just an example
of how this works out.
But when you're dealing with
the motions of stars – 15
meters per second,
10 centimeters a second--you're
nowhere near the speed of light,
and so the equation that I
wrote down is actually fine.
So, what do you expect to see
when you're looking at a star,
which has a planet in orbit
around it?
Looking at a star in some kind
of orbit--so here's the radial
velocity as a function of time.
And the key thing about orbits
is that the radial velocity
changes, because first the
thing's coming towards you then
later in its orbit it turns
around goes the other way.
Then it comes back and it comes
towards you, turns around and
goes the other way.
So, the radial velocity will
change from positive to negative
and back as the object first
comes towards you,
then away from you.
And so, it'll look like this if
you make a whole bunch of
observations of this.
It turns out that for circular
orbits, this is a sine wave.
And if you were to observe
this, you could observe directly
from such a plot,
if you made repeated
observations of the radial
velocity of a star,
or some other thing in orbit,
you could observe two things.
First of all,
you would immediately be able
to tell what the orbital period
is.
That's the amount of time--this
is a time axis--it takes for the
object to come back to the same
place in the orbit for a second
time.
Second of all,
you could tell something about
what the velocity is.
The amplitude of this sine wave
is something to do with the
overall velocity,
because that's the maximum
velocity it has coming towards
you.
But you have to be a little
careful here,
because that's only true if the
object--if the orbit is edge-on.
Let me explain what I mean by
that, we'll come back to this
later.
If the orbit's going this way,
then it never comes towards you
or goes away from you--it's
always going sideways.
If, on the other hand,
the orbit's going this way,
then first it comes towards
you, then it goes away from you.
And if it's somewhere in
between, you only see part of
the motion of the orbit in terms
of radial velocity.
So, this amplitude is V
if the orbit is edge-on.
If not, V is going to be
more than that,
because you're only seeing part
of the motion.
That's a detail we'll come back
to later.
So, this is what you expect to
see if there's a planet going
around the star,
and if you have enough
sensitivity in your measurements
of the Doppler Shift to be able
to actually see that motion.
Okay, so here's what they saw.
This is a star called 51
Pegasus, in the Constellation of
Pegasus.
It is a "solar analog,"
so-called, by which they mean,
it's about as much like the Sun
as they can--as you can find if
you go out and look at other
stars.
So it's very,
very much like the Sun.
This is a radial velocity curve.
I have taken the axis off for a
reason I'll point out later.
This is radial velocity versus
time, and this is exactly what
you expect to see.
It goes up;
it comes down--very good news.
This is exactly what you expect
to see for a planet.
There are two problems here,
as it turns out.
Problem number one is the
X-axis,
because it turns out,
the amount of time it takes you
to go one orbit around for this
object is a little more than
four days.
The shortest orbital period of
a planet in our own Solar System
is that of Mercury,
which is eighty-eight days,
so this thing is way closer to
the star than anything in our
solar system.
X-axis problem is that
P is equal to around four
days.
Problem number two is the
Y-axis.
And now let me put some units
on to this thing.
I've done a very bad thing of
showing you a graph with no
units, but it was just to
prolong the suspense here.
V_R.
Zero.
So, that's going sideways.
This is 50 meters a second.
This is negative 50 meters a
second.
And so first it--here it's
coming towards you,
there it's going away from you.
And so, the amplitude of this
thing, and therefore the
velocity of the star,
is something like 50 meters a
second.
Now, it's less obvious why
that's a problem,
but it is.
And let me show you why.
Let me do the equation--do the
equations a little bit.
Okay, so this is a solar analog.
What is the semi-major axis of
the planets orbit?
Axis of planet.
We know P is equal
to--let's see 4 / 365.24 is
equal to 1 / 100 is equal to
10^(-2) in years.
That's four days over a year.
And M is equal to 1
solar mass because it's a solar
analog.
So, a^(3) = P^(2).
M = (10^(-2))^(2)
M is equal--one.
10^(-4)-- So a is equal
to 10 to the--well let's--we got
to do this right.
(100 x 10^(-6))^(1/3).
5^(3) = 125.
So the cube root of 100 is 5.
5 x 10^(-2) Astronomical Units.
Or let's put it in real units
here.
5 x 10^(-2).
An Astronomical Unit is 1.5 x
10^(11) meters.
5 x 1.5 is like 7 x 10^(9)
meters.
So that's how--that's the
semi-major axis--and if you
compare it, if you go look at
the lists of planets in our own
Solar System,
what you'll discover is that's
way closer than any of the
planets in our own Solar
System--way closer to the star
than Mercury is.
I haven't used the
Y-axis yet.
Now I'm about to,
because what I'm going to do
is, I'm going to take--I'm going
to figure out the velocity that
this planet is moving at.
That's 2 π a / P.
You wrote that down a little
while ago;
a, I just figured out.
So, 2 times π times 7 times
10^(9).
P is 1/100 of a year.
So, that's 10^(-2),
times 3 x 10^(7),
which is the number of seconds
in a year.
π / 3 = 1.
2 x 7 = 15.
10 ^(9) – 10^(-2) x 10^(7).
10^(5) -- So,
this is 15 x 10^(4),
or 1.5 x 10^(5).
So this is thing is going--the
planet now, this is
V_total,
which is approximately equal to
V_planet,
is going at 150 kilometers a
second.
Much faster than the Earth is
going around;
well, that makes perfect sense.
It's in closer--got to move
faster to stay in its orbit.
And so, this is 150 kilometers
a second if you prefer those
units.
Now, we know the velocity of
the planet, we know the mass of
the star, we know the velocity
of the star and so we can figure
out the mass of the planet.
Here's how it works.
M_p
V_p =
M_star
V_star.
V_planet,
we've just figured out,
is 1.5 x 10^(5).
Actually, let's leave it as 15
x 10^(4), that'll turn out to be
easier for the arithmetic.
The mass of the stars,
the same as the mass of the
Sun, 2 x 10^(30).
The velocity of the star we
just observed.
We saw it moving back and
forth, it's 50 kilometers a
second, 5 x 10^(1).
Okay, so now--and this is
multiplied by the mass of the
planet.
So, the mass of the planet is
equal to 2 x 10^(30),
times (5 x 10^(1)) / (15 x
10^(4)).
5 / 15 is a third,
so this is a third times
10^(-3), times the mass of the
Sun.
10^(-3) times the mass of the
Sun, that's the mass of Jupiter.
So, this is equal to 1/3 of the
mass of Jupiter--which is,
by the way, bigger than the
mass of Saturn or any other
object in our own Solar System.
Okay, so that's catastrophic,
right?
That's a hopeless disaster,
because the first part of the
lecture and the second part of
the lecture have entirely
contradicted each other.
Because in the first part of
the lecture I gave you a whole
song and dance,
and you all wrote it down,
and it sounded believable at
the time.
How inner planets were going to
be these small rocky things.
And now the very first planet
we go out and find turns out to
be a very close planet that's
quite massive.
And this is impossible
according to this nice little
theory of planetary formation
that I promulgated to you
earlier in the lecture.
And so, the very first planet
that was observed turned out to
be a screw-up.
Worse than that,
there were soon dozens more
like it discovered.
These are given the name "Hot
Jupiters," and you can see what
the problem is.
If you figure out what the
surface temperature of these
things ought to be,
it's over 1,000 degrees.
There's no way you could have
gas or ice on such a thing,
it would totally melt.
And there aren't enough rocks
in the whole rocky
elements--silicon and iron,
and so forth--in the whole of
the planetary system of our own
Solar System to add up to 1/3 of
a Jupiter,
even if you put them all
together.
So how is this done?
What is going on here?
It seems clear that one of the
two things must be true.
Either this thing isn't a
planet and there's some other
explanation for this attractive
bunch of data here,
or something has gone seriously
wrong in our understanding of
how planets form.
And stay tuned,
we'll talk about that next
time.
