We did pretty well understanding terrestrial worlds which are relatively
similar to Earth. Now we can get ambitious, we can see what
we can say about the Jovian planets and see how much of what we've understood
translates to these more interesting systems.
looking at their motion in the solar system the things I'd like to point out
is the inclinations of the orbits are very small.
the periods of the orbits are interesting.
Jupiter and Saturn are locked, as you can see if you do the calculation, into a two
to five resonance. So, the one to two resonance that
destabilize things has locked into a stable two to five resonance which is
anchoring if, in some sense, the stability of the rest of the solar
system. in terms of the rotation periods, the
spin periods, notice how fast all of these objects spin.
The slowest is Uranus with a seventeen hour, spin.
Jupiter and its entire mass rotates about its axis every ten hours.
If we think about these giant planets forming out of a gravitational collapse,
a gravitational instability then we remember that this will acquire angular
momentum from more distant parts of the cloud.
This explains the rather rapid spinning motion and will have consequences as we
go along. the other thing to note is that the tilt
angles are moderate except for Uranus, which has a axial tilt of 97 degrees,
which means its axis essentially is not perpendicular to the ecliptic but is
essentially in the plane of the ecliptic. this gives extraordinarily
marked seasonal effects on Uranus and even from observations from Earth will be
able to remark those. Notice that those seasons are rather long
since the year is 84 years long. And so in some sense, Neptune with the
longest year was discovered about a year ago.
This is what we see in terms of their motions.
We talked about how they got there. looking at the surface or the atmosphere,
which in these cases is the same thing since this atmosphere, primarily
hydrogen, molecular hydrogen and helium, is all we're going to see because there's
a sufficiently deep atmosphere in all of these that we cannot see through them.
the atmospheres are always hydrogen and helium.
Uranus and Neptune acquire their blue tint from traces of methane.
What do we see looking at the surface? Well, the surface is the same as the
atmosphere because all of these planets have sufficiently thick
high atmospheres of hydrogen and helium that we cannot see through them with any
instrument. And so, the atmosphere is what we see the
atmospheres are hydrogen and helium. In the case of Uranus and Neptune a trace
amount of methane tinges them blue. And what we see when we look at this
wonderful image of Jupiter is the lines of latitude tracing convection cells the
atmosphere is heated by the core from below, just as on Earth.
This in conjunction with the more rapid rotation leads to these bands and zones
which are tracing the patterns of the convection cells. together with the rapid
rotation, you get very massive global winds.
And at the interfaces between direct, differing directions of global winds, you
get intense storms most famously the Giant Red Spot on Jupiter. A storm the
size of the Earth which we conjecture has been ongoing for 100,000 years.
but there are many other persistent storms on Jupiter.
comparing this to an image of Saturn, we see similar things.
We see the same band structure, although less pronounced.
The dark coloring of some of the regions on Jupiter
are is attributed to the rising of some form
of complex. Perhaps, hydrocarbon molecules.
But nobody's precisely sure what causes the coloring.
And in particular, nobody can explain why once in a while, one of those dark belts
disappears for a few months, as happened about six months ago.
And then, reacquires its colors. It's not a change in the global
circulation patterns. But something else, and nobody completely
understands it. we see the same kind of storm pattern on
Saturn. We see simliar effects on Uranus and indeed similar effects on Neptune. On
Neptune, what we notice is the hemispherical variations with season.
this happens on Uranus, it also happens on Neptune despite the
tilt not being as extreme. Simply because the seasons are so long
that global hemispheric effects take place in the course of the 164 year
orbit. Looking inside the planets, well we don't
do seismology on these giant planets. we conjecture the structure of their
inside from modelling as we do for the terrestrial planets combined with
measurements of the obliqueness, the rapid spin causes all of these planets
like as indeed, the Earth does, to be slightly thicker around the equator than
around the poles. This is because of the effect of spin.
By measuring how ablate they are compared to how they spin, we can estimate produce
some estimate of the internal structure. And the internal structure that seems to
obtain for Jupiter and Saturn is that inside the external atmosphere of
hydrogen and helium, is a mantle made of metallic hydrogen.
Hydrogen, if you look at its place in the periodic table, the, should be a metal.
Metallic hydrogen is certainly not found naturally on Earth.
It occurs only under very, very high pressures it was only produced on Earth
in I think the 80s.' and but it exists naturally under the
extreme compression of the immense gravity and the immense weight of the
heavy atmospheres of these planets. We have the usual hydro-dynamic
equilibrium where pressures in the cores of these planets are very, very extreme
and decrease as you move out. But since the atmosphere is thousands of
kilometers thick and very heavy, even at the bottom of the atmosphere, pressures
are intense. metallic hydrogen is a conductor.
The rapid rotation leads to a strong dipolar magnetic field for all of these
planets. in the case of the gas giants Jupiter and
Saturn this magnet is aligned very nicely with the planet's rotation axis.
in the case of Uranus and Neptune, the mantle is actually compressed ices. So
water, ammonia, methane, etc., at high density.
these too can conduct, but the mechanism is a little bit different.
the understanding is incomplete but the one of the manifestations of the
difference in the mantle is that the magnetic fields of Uranus and Neptune are
60 degrees tilted relative to their direction of rotation.
inside each of these is a core, the, presumably the seed that started the
planet's growth. as you can see in this image, the core is
roughly Earth-like in size probably more massive because compressed. And has an
iron-rich inside, and perhaps a silicate outside if it chemically differentiated.
so set, the core of each of these planets is about the size of Earth itself.
Again, the interiors of these planets are both extremely compressed and very, very
hot. in the case of these planets, much of the
internal heat, remember the only the core is rich in heavy elements.
Here this is a tiny fraction of the actual huge planet.
and so, radioactivity does not play a large role in maintaining and producing
internal heat. Internal heart is produced by Kelvin
Helmholtz processes by precipitation and by increased continuing compression of
the interiors of the planet. On the case, in the case of Saturn,
precipitation of liquid helium. So, liquid helium rain at high pressure
is the mechanism by which Saturn is chemically differentiating.
And this is significant. If you do the
calculation of surface temperature based on our black body calculation, compared
to the measured temperature of the surfaces of these planets, you'll
discover that both Jupiter and Saturn radiate about twice as much energy as
they absorb from the sun. This is the difference is being made by
continual heating from the core through mostly Calvin Helmholtz processes.
And we have the strong magnetic field. This is an aurora on Jupiter's north
pole. These are aurorae on Saturn's poles.
And if we look for aurorae on Uranus, you can see that if you look at the ring
structure that indicates the equator of the planet, this is not the pole.
The pole is way over here to the left. And the aurorae are showing up at
intermediate latitudes. This is indicating to you the offset of
the magnetic field. indeed, this reminds us that the most
impressive structures carried by these giant planets are their rings.
And we understand, in some sense, where these might come from.
These planets are formed by a gravitational instability in the nebula
collapse. And the collapse will be accompanied by a
flattening into an accretion disk from which the planet acquires gas and dust.
near, some of this leftovers of the nebula will form moons.
So, most of these planets have moons that orbit them in their equatorial plane.
these moons formed within the accretion disk, just as the planets formed in the
sun's accretion disk. but,
close to the planet, tidal forces can prevent gravitational accretion.
This would have been true in the case of the Sun as well, except that close to the
Sun whatever didn't accrete into Mercury was blown away by T Tauri winds.
Planets do not generate T Tauri winds. So whatever accreted near the planet, too
close to form a star a moon will produce ring like structures.
And they have ring structures around Jupiter,
and ring structures around Uranus, and ring Neptune,
and ring structures around Uranus. And the beautiful magnificent rings
around Saturn. for reasons nobody completely understands
the rings around all of the three giants other than Saturn are dusty and dim.
Whereas, with Saturns rings are almost 99.9% water ice.
Water ice, as we know, is brilliant and shiny which is why the rings appear so
bright. we see here that the rings are discrete,
the large gap here, the Cassini gap, is visible in a moderate telescope from
Earth. The slightly smaller gap farther out, the
Enke gap, is visible with a more advanced telescope.
But upon closer inspection, it turns out that these rings have a very, very fine
internal structure. We'll get into that in a second.
Let's understand what it is that I'm saying when I say that too close to a
planet a moon cannot form. the limiting distance, the closest
distance at which it can form is called the Roche Limit.
it was computed in 1848 by Roche. And the idea is this, we have a planet
over here, we're trying to form a moon over there.
We will say a moon can accrete and form provided that the tidal forces of the
planet are not enough to blow it apart. So we've done enough to do this
calculation that Roche did. We give this a mass and planet.
We give this a mass and moon. Set them a distance D apart.
And the question is, basically, this moon is held together by gravity.
Is the moons gravity enough to hold it together against the tidal forces? If we
drop a pebble on the surface of moon, will it be lifted by tidal forces or not?
And so, we know how to do this. we need to compare the tidal acceleration due to
the planet's tidal forces on the moon. We computed this in the past.
It's 2 * G times the mass of the planet, times the radius of the moon divided by
the distance cubed. This is the tidal acceleration with which
the planet tries to stretch the moon as we have it here in the horizontal
direction. And we need to compare this to the
gravitational acceleration due to the moon's own mass at its surface.
And this, of course is G times the mass of the moon divided by the radius of the
moon squared. As D gets smaller, the tidal acceleration
increases. When it overwhelms the moon's own
gravity, the moon will simply fall apart as long as it's gravitationally bound. Or
if we're trying to form a moon by gravitationally binding moonitesimals,
the moon will never form. And we'll end up as we'll see with ring
structures. So, let's write these two and set them
together. This is setting the gravitational
acceleration on the moon's surface equal to the tidal force generated by the
nearby planet. dividing through by R and canceling the
Gs as usual I'm going to put this down over here.
And I find that I have this interesting expression that says, the mass of the
moon divided by its radius cubed is equal to twice the mass of the planet divided
by the distance cubed. Which it's interesting to write in a
different way. You write the mass of the planet divided
by the distance cubed. But then, you divide and multiply by the
radius of the planet cubed. Now, why is this useful way to write it?
Because the cube of the radius is a measure of volume.
Remember that the volume of a ball is four pi over three R cubed.
And that if I know the average density of an object, then its mass is the density
times the volume. So, dividing the mass by R cubed up to
irrelevant cancelling factors of four pi over three gives me a measure of the
density. In other words what this equation tells
me is that the density of the moon is twice the density of the planet times the
ratio between the cube of the ratio between the radius of the planet and the
distance between them, when that distance is at the Roche limit. When tidal forces
are just enough to rip the planet apart. In other words, I can now solve for D in
terms of the densities and the solution that I get.
So, here's the expression written that way.
Again, I I rewrite things in terms of the densities, and this says that the density
of the moon is twice the density of the planet, times the radius of the planet
divided by D cubed. Now, multiplying through by D cubed
moving dividing by row M, I finally find and then taking a cubed route to get my
answer, I find the fearsome equation on our splash page that says that the Roche
limit, the dist, closest distance at which a moon can survive, that's D, the
Roche distance, is given by the radius of the planet times the, the 1/3rd power of
the ratio of twice the planet's density to the moon's density.
to make a reasonable calculation let's assume for example, that we're describing
Earth and Earth's moon. What is the nearest distance to Earth
that our moon could have possibly have, have formed?
Remember, it formed much nearer. Well, let's give the moon half of Earth's
density. So twice the Earth density divided by the
moon's density would be about four. four to the one third is about one sixth.
So in the case of a planet twice as dense as its moon,
you find that the minimum distance is about 1.6 times the radius of the planet
if a rogue planet is twice real moon, which is the case for our Earth, our
moon. And so, our moon could not have formed
less than about two Earth radii from the center.
In other words, about 7,000 kilometers away from the Earth's surface.
nearer than that, our moon would have been ripped apart by Earth's tidal
forces. And
this is a very useful expression. Objects inside the Roche radius cannot
possibly be gravitationally bound. Now, that does not mean that they are
completely ground to dust at some point. In the solar system, the, the rough scale
is about a kilometer. We will do a homework problem that will
help us understand how that happens. At some small scale, remember the
planetesimals, were held together chemically.
Smaller things are held together by chemical forces.
They at small distances are able to withstand gravitational
tidal forces. But gravitationally bound objects will be dispersed over time.
And so, the way that this will work is if you bring a moon to within the Roche
radius which is indicated by this white ring, tidal forces will elongate it.
And when it crosses the Roche radius it will start to fall apart.
remember it's being extended so parts near to the planet will fall towards the
planet, parts far from the planet will fall away
from the planet. And because they're all moving, imagine
this thing was in orbit, all of the planet was, the moon was
moving with the same orbital velocity. the bits that are falling too close to
the planet are now moving at a speed too small for their now lower orbit so they
will fall into eliptical orbits with this as [UNKNOWN].
And the bits that are falling away from the planet will be moving so slow, so
they will fall into elliptical orbits. And when everything settles down into
circular orbits the guys that have, that fell towards the planet will have
advanced ahead. The guys that fell into the planet, away
from the planet will have been left behind.
And basically, the whole structure is smeared out along a ring.
We'll see this happening again and again when tidal forces rip something apart.
They generate that is in orbit, they generate a ring structure.
So this is where the rings could have come from or why the rings formed but no
moons. But stability over the long term of such a ring structure, you'd expect
collisions between ring particles, you'd expect that mixture.
looking at Saturn's rings, we see that there are changes in composition between
thousands and thousands of ringlets. it's not completely understood what
maintains the stability of this amazing structure.
for the record, Saturn's rings are about 150,000 kilometers in width.
but by some measures, only 400 metres in thickness.
This is an extremely thin object, precisely on the equatorial plain and
made up of essentially of chunks of water ice. But, with various additives or
various trace amounts of other materials so that they have slightly different
properties in the various ringlets. And
one of the things that keep maintains this, helps maintain this structure is
the existence of moons that orbit both within the ring system and outside it
they produce these gaps, these gaps that are marked here, are gaps at which
objects orbiting in the ring would be in resident orbits with the moon.
Just as in the asteroid belt, this would cause them to be moved into eccentric
objects and essentially ejected. So, nothing orbits in the gaps and this
separates ringlets from each other. Moreover, there are moons that orbit
inside the rings, inside the Roche limit where a moon cannot exist.
Indeed, those moons are slowly being ripped apart and perhaps resupplying
material to the rings as the material is being lost to
collisions. And, these, these objects are both very beautiful and very intriguing,
and I don't think we completely understand what maintains their
structure, but they're certainly a generic property of accretion systems.