So, that's what went down in the inner solar system.
with 90% of the non-solar mass being in Venus and Saturn,
you wouldn't be surprised to learn that the real action Went wn in the outer
solar system beyond what we call the snow line.
The snow line is that line that separates the inner solar system where water was a
vapor from the outer solar system where water was a solid.
Why is this so important? Because those initial dust, solid dust
fragments that coalesced chemically to form planetesimals slowly and
painstakingly in the inner solar system were very rare.
They were made of these trace elements, iron, nickel, alumina, they are very rare
in the solar system. In the outer solar system, water was a
solid. Little grains of ice could stick to each
other chemically. Water is much more prevalent in the solar
nebula and in the solar system today, than iron or nickel or aluminum or all of
them put together. Despite the smaller overall density at
larger radius, the fact that water was available to coalesce as solid particles
meant that the whole process of planetesimal and protoplanet formation
goes much, much faster. This is especially true right at the snow
line, about five astronomical units from the sun.
And so, Jupiter, which lives, right at that radius,
grows very, very fast. it goes through the initial accretion of
planetesimals to protoplanets much, much faster than the inner planet and reaches
very rapidly a massive ten to fifteen earth masses.
And, this is a crucial borderline
what is this object made of? Well, by this time, it's certainly
spherical, it has melted. It has chemically differentiated.
Iron will have dropped into the central core.
There will be an outer core of silicates and rocky materials and then the outer
mantle will be water. It was made, much, much of the planet
would have been made of ice. The ice would have melted, so we now have
this sort of layered structure with an inner iron core, an outer silicate core,
and a watery mantle. And this object, when it reaches ten to
fifteen Earth masses, achieves an important milestone besides melting and
chemical differentiation, one that the Earth, which never got this big, never
had a chance to achieve. At this point,
the gravitational attraction of this large object, which is become quite
compact under the force of its gravity, because it's held up by pressure is large
enough to bind an atmosphere of hydrogen and helium.
Now, this is a breakthrough because, of course,
while water is more prevalent than the solar nebula then, nickel water and
nickel together are both trace elements. What the solar nebula is really full of
is hydrogen and helium gas, that's 98% of the mass.
Once you have an object massive enough and to bind hydrogen and helium,
remember, we saw that hydrogen and helium are not bound to earth, but at the lower
temperatures in the outer nebula, they will bind to Jupiter.
Jupiter is rapidly able to gravitationally attract all of the gas
essentially in its vicinity. in fact, one can imagine the formation of
Jupiter as a gas giant essentially as a gravitational collapse, a gravitational
instability, similar in smaller scale to what happened at the center that produced
the protosun. In other words, the dense core that is
the core of Jupiter forms an instability, causes a chunk of the gaseous disk to
collapse onto that core. What does this produce?
Well, it produces the usual phenomena, it produces increased angular velocity as
the gas collapses, we find that the rotation speeds up.
Indeed, Jupiter rotates about its axis once every nine hours.
And a giant object like Jupiter, indeed, spins much faster than Earth does.
In addition this causes the flattening into a disk, just as the solar system
flattened into a disk. So, we get the phenomenon around Jupiter
of an accretion disk which is the place from which gas is falling onto the
nascent planet. And in that disk,
some of the material, because of the centrifugal barrier, will not fall on to
Jupiter. Some of it will be icy.
And we have as with the inner solar system, we have the formation of planets.
Here, the leftovers will form moons, and as we'll see closer to the planet rings,
and so these structures in the equatorial plane of Jupiter are just mimicking on a
smaller scale, the entire planetary system that lives on the equatorial plane
of the sun, it's the same issue of a cloud
collapsing, spinning up, and flattening. In addition, of course, the core is
heated by Kelvin-Helmholtz heating, but we have a, a rapid runaway growth,
especially towards the end. I takes about ten million years,
not a, coincidentally, a very good time scale because that's when the gas gets
blown away by the sun for Jupiter to accrete roughly its current mass and over
the same time it essentially exhaust all the
available gas near its orbit. For whereas, Earth was able to trap all
the planetisimal matter, but the gas was essentially unbound and the the inner
nebula was still full of hydrogen and helium gas until the T Tauri wind blew it
away. And the outer nebula before the T Tauri
went, cleans up Jupiter and Saturn have essentially sucked up all the gas in
their orbits. Now, Saturn is not as massive as Jupiter.
Well, Saturn is a little farther out. The density is smaller there.
There is less water Saturn accretes to critical helium and hydrogen binding mass
a little bit later. The density is also smaller but mostly it
starts later. It has less time, has less gas left, and
so Saturn never quite attains the dimensions of Jupiter, but it repeats the
same process, gravitational instability, collapse of a cloud, speeding up of the
rotation, flattening out of an of accretion disk and moves in the
equatorial plane. So, Jupiter and Saturn lives the rich
life outside the snow line and so become fat and big.
What about Uranus and Neptune? Well, those are qualitatively different
object. They're giants on terrestrial scale.
They're not as big as Jupiter and Saturn and they're not predominantly hydrogen,
despite what we might have heard. They're predominantly ices.
each of them has only about a sole Earth mass, of hydrogen bound to it.
presumably, they started, yet later there was not much gas left.
But even so, we have a problem. In their current orbits, there is no way
that the modeling tells us that these planets would have grown in time to bind
any hydrogen at all in the time before the sun blew the hydrogen away.
What must have happened is that they must have formed closer to the sun where the
nebula was denser and then migrated out, migrated out.
I thought that we solved Newton's equations and they told us that an object
orbiting the sun will move in a closed orbit, which is an ellipse.
Newton solved these equations, he solved the correctly, what is this, migrating,
why would something move in an orbit that is not an ellipse?
This is an important topic so let's pay attention to it.
So indeed, the motion of an object around the sun or, in fact, the slightly more
complicated problem of two objects orbiting each other is a problem that
physics students solve in their first or second year of doing physics.
And it's completely solved when the solutions are ellipses or hyperbolas if
the motion is not bound, but we know how to solve that problem.
Things become complicated when we have not one planet orbiting the sun but, say,
two planets orbiting the sun and you want to take into account not only the
gravitational attraction of the sun on each of these objects but also the
gravitational attraction of one upon the other.
This brings you into the three or four or whatever body problem and the three body
problem in Newtonian physics is not a freshman exercise.
In fact, it's not solved and in some technical sense, not analytically
solvable because the problem has a property called chaos.
it is the fact that small, infinitesimal changes in these initial conditions,
where and with what velocity you start the objects out, can lead to
qualitatively different results. And so, a direct solution is difficult.
How do we actually understand what's going to happen in something complicated
like the solar system, which is at the very least a nine body problem?
Well, one possibility is we numerically
simulate the problem. In other words, we run through the
exercise that I described at the end of our Newton's Laws discussion.
We start with the initial positions and velocities, we compute an acceleration.
Figure out from that, where things will be a few years hence.
Repeat the, recalculate the forces, recalculate the accelerations, and
basically go through that process on a computer.
It's difficult but it's doable. That's how we know what the solar system
will do billions of years hence and what it did, billions of years ago.
but under some circumstances, it's useful to think the, under most circumstances,
the gravitational force of the sun dominates gravity in the solar system
simply because the sun is so much massive than everybody else.
This might not be true when two objects come very near to each other.
So certainly, the dominant gravitational force acting on me right now is that of
the Earth, not of the sun. Despite the sun being more massive, the
Earth is much closer. So, if two objects are moving in solar
orbits but then, approach each other very close, what I can do is sort of stitch
together approximations as long as they're far enough apart, I'm going to
ignore their gravitational attraction on each other.
When they get close, then for an instant, I'm going to forget about the sun and let
them collide with each other and only treat the gravitational attraction
between them. And then, once they've moved far apart
again, they will have new, changed velocities.
I will plug those in to the equations for solar orbit And essentially, the objects
will have transitioned through the collision from one solar orbit to a new
solar orbit. And this process goes under many names,
gravitational slingshot, gravity assist. It's often used by NASA to propel its
spacecraft to higher orbits. This is how Voyager got to leave the
solar system, it's how Galileo got to travel as far as Jupiter.
And the way it works is that if you take into account that the, the scattering
process, the collision is with the gravitational collision, is with a moving
planet and you adjust the initial conditions right,
you can set it up. So, that in the process of collision, the
light or object, in this case, say, the spacecraft picks up some extra orbital
velocity from the moving planet and here is, for example, a demonstration.
This is the trajectory of the Dawn Craft, whose images of Vesta, the asteroid, we
were looking at before. This was launched from Earth on one of
those Hohmann Transfer, those elliptical orbits we talked about, such that almost
at the antipodal point it was meeting up with Mars.
And then it wasn't quite antipodal, it didn't go into Mars orbit.
But as it zoomed past Mars, Mars accelerated it, and then, of course, as
it passed Mars, Mars slowed it down. Then in effect though, because Mars is
moving, is that it acquired some extra energy from the collision with Mars went
from one solar orbit into a new solar orbit and the design of the orbit was
such, this was precisely a minimum energy trajectory to the orbit with which we
wanted, in, to which we wanted it to arrive, which is Vesta's orbit out here,
at a larger distance from the sun, and it arrived and then slowed down and orbited
along with Vesta, fell into orbit around Vesta.
This is object of careful design and there, it took those beautiful images.
And when it leaves, Vesta will hop up to an, the orbit of Ceres, another asteroid
and will take images of Ceres. And so this gravity assist is nothing
new. This is a, this perturbative
approximation works well when the objects encounter each other once.
And then, are not, not any very close in the future because Dawn is now in Vesta
orbit and will not get close to Mars the future.
It looks different when this situation is that two objects are in
closed orbits around the sun and so will be at a nearest approach every time their
orbital periods coincide. We'll see that in a moment in this nice
demonstration and maybe that will clarify the situation.
So this is our old planetary configurations simulator which I'm going
to re-purpose to a different goal. So, imagine that we have two planets,
both orbiting the sun, initially circular orbits.
I'm going to start, as always, when the two planets are in opposition or
conjunction, or whatever you want to call it, but they're lined up with the sun,
this is the moment of their closest approach.
At this point, what is going on? Well there's an attraction and as the
inner planet has been catching up with the the outer planet the
gravitational attraction between them has been attempting to distort the inner
orbit this way, because the inner planet is attracted to the outer planet, so it,
it, it acquires a slightly higher velocity.
At the same time, it's slowing down the orbit of the outer planet, and so the
outer planet's orbit is being distorted slightly in this direction and, of
course, I grossly exaggerate what is going on.
Now what I'm going to do now is allow the animation to run for a bit and we'll see
that after a while as usual, the inner planet outruns the outer planet and if we
wait long enough they come into conjunction again about here.
Well, this time, again, in conjunction, there's a distortion of the orbit.
The inner orbit is being distorted in this direction while the outer orbit is
being distorted in that direction. and you can see that we can go do one
more round of this crazy race. And what is going to happen is that they
will come into conjunction again at some other
position along their orbits. And I've got it timed so it's kind of
close to where it was before, but not too close.
This time, the inner orbit is being distorted into this shape.
And of, again, I am grossly exaggerating, and the outer orbit into the
perpendicular shape, something like this.
You see that what's going on is that the net result of all these deformations is
not going to be very much because at each conjunction, the orbit is deformed in a
different direction. Aha.
Now, I'm going to try the same thing, except, I'm going to set up the ratio of
the radii of the two planets to be one to 1.59.
If you check with Kepler's laws, this will make the ratio of their periods one
to two. and we'll see that this changes things
considerably. what's going on here, again, during this
conjunction, the inner orbit is being stretched out in this direction while the
outer orbit is being stretched out that way.
and now, if we let the animation run we'll see that because the ratio of the
orbits is two [COUGH] the next conjunction will happen precisely after
the great, slower outer planet has completed one rotation and the inner
planet has completed two and guess what, the deformation of the orbits this time,
is in the same direction. What happens this time is that at each
encounter, the orbit is further deformed in the same direction.
When you have orbital periods, this was the situation when P2 was 2P1.
When the ratio of the periods is a fraction with small numerator and
denominator, we'll get once every denominator number
of periods a conjunction with extends the orbit in the same direction.
And the orbit over time will become more and more deformed.
This is called orbital resonance. What we saw then is this phenomenon
called orbital resonance. from a distance, the gravitational
interaction between two planets might perturb their orbits slightly.
If the periods of the solar orbits are resonant, what does resonant mean?
That means that nP1 is nP2 where n and m are sufficiently small integers because
this means that if P2 is the larger orbit so that n is the larger number, every n
orbits of this one conjunction happens at exactly the same place.
And so, every n orbit, the effect is enhanced.
If n is not too large, then we get this resonant perturbation.
And the net result is that the orbit will be elongated in a particular direction.
And notice that as it's elongated in that direction, the distance of closest
approach only gets closer and so this again is a nonlinear process that can
continue and so success of perturbations add.
Now, there's two things that can happen in the details of that dynamic, is beyond
the kind of math that we're want to, going to do.
one possibility is you get meta-stable resonances, not quite stable orbits that
could go on forever, but orbits in which two things can
continually orbit for a long time and a great example of that is Earths mosrt
famous other moon, it's name is Cruithne or whatever the pronunciation is and it
actually orbits the sun in a solar orbit that is one to one resonance with that of
the Earth. The objects will never collide, the
orbits are tilted and the timing is off. But what happens is that Cruithne orbits
in a more elliptical orbit than the Earth does but its period is precisely one to
one. You will see that every time the two
objects or arrive at this sort of intersection, from our point of view in,
in right ascension of the orbits. they arrive there in precisely the same
configuration. This is orbital locking or resonance.
And what this looks like from Earth is if you imagine fixing the earth in one
place, so this is the sun,
the slight wiggle in the Earth's orbit about the sun is due to the ellipticity
of the Earth's orbit. And here's Cruithne orbiting the sun but
as we see it from Earth, Cruithne goes around the sort of bean-shaped orbit.
This is often very confusing. People describe this bean-shaped orbit
and try to figure out why anything would orbit the sun in a bean-shaped orbit.
It looks to us as though Cruithne is orbiting the sun in a bean-shaped orbit.
That is really just the point of view from Earth because we're moving along our
orbit. And indeed, there are three or four other
objects that are locked into resonant orbits with Earth.
And so, we have more than one sort of moon, but only one that is both naked eye
visible and actually in Earth's orbit. But this is sort of a bound orbit between
the Earth and the sun, where
together, controlling the motion of this piece of rock, Cruithne which is also
quite small. This is not the standard outcome of
orbital resonance. Typically, what happens in orbital
resonance is that because the orbit is deformed in the coherent way in a
particular direction, the entire situation become, because of this
nonlinearity, the entire situation becomes unstable and the orbit is
destabilized. This is probably best characterized in
the asteroid belt. The asteroid belt, remember, lies between
two to four astronomical units from the sun.
and in this region fall all of the resonances with, with small numbers of
periods with Jupiter and with Saturn. And what we observe indeed is the
distribution of asteroids as a function of their distance from the sun.
And so this is the position of Jupiter.
the fact that there are asteroids in Jupiter's orbit that's a resonance
situation. There's other Trojans.
In fact, how those Trojans fall into a stable resonance is again, something
that's a little bit beyond us. But there are these Lagrangian points
around which there's an unstable resonance and these asteroids orbit that
unstable position.
And so, they end up being locked for long terms into these positions, 60 degrees
away from the planet along its orbit. But more importantly, in the more inner
reaches, this is the main asteroid belt, over here
is Earth orbit. And what we see, is that there's a
distribution of asteroids and then there are gaps.
There are distances from the sun at which nothing orbits.
These are called Kirkwood Gaps. And what they correspond to is distances
from the sun at which the period of an asteroid, were it to orbit there, would
be a third or a half or 2/5 and so on of the period of Jupiter. These are objects
which would be coming into orbital resonance with Jupiter that would distort
their orbits into eccentric orbits and if there were a little bit of a tilt to
their orbit, that too would get accentuated, and eventually they get
completely ejected from the asteroid belt.
So, nothing orbits at these orbits and in particular, not only are these guys,
or are, is there nothing orbiting there now, but in the original asteroid belt as
Jupiter and Saturn were forming, they started to distort the motion of things
in the asteroid belt. This lead to these objects not moving in
the nice circular orbits into which everybody was trying to settle.
But into these eccentric orbits, this leads to violent, destructive collisions
between the asteroids or the planetesimals and protoplanets that are
forming in the asteroid belt. This is one of the ingredients into why a
planet never managed to form the larger objects were colliding destructively at
high speed rather than sort of overtaking each other very slowly with slow relative
velocities as was the case in the inner solar system farther from Jupiter.
And this process of resonant emptying of the asteroid belt eliminates most of the
stuff that was there. Astronomers estimate there were between
one and three Earth masses of material in the region from the sun between two and
four astronomical units and most of that has gotten kicked out less than a tenth
of a percent is now present, and we attribute this to the, the gravitational
effects of Jupiter and partially, Saturn. Now, that we understand that orbits can
change we can understand something about the currently accepted, and I should say,
it's relatively recent and might still change understanding of the early
evolution of the relevant part of the solar system, mainly the outer solar
system. This is called the Nice model, it was
formulated in a sequence of papers that came out of the University of Nice.
These are where the simulations, the simulations were done.
And the idea is this, remember out at 30 astronomical units where
Neptune orbits today, there is no way to form a giant.
In this model, the four giants form at distances between five, were slightly
away from where Jupiter is today, 5.5 astronomical units out to seventeen
astronomical units so closer in than they are today.
And here, four giant planets form and they completely exhaust the gas in the
disk beyond this orbit all the way out to 35 astronomical units, planetesimals are
formed. There are ices, methane, water farther
out even more and they do form planetesimals.
And so, there's about 35 Earth masses worth of planetesimals in a disk outside
the orbits of these giants. Now over,
time the outer giants, it seems that probably Uranus was formed farther from
the sun than Neptune and they cross. So, Uranus will encounter some of the
inner part of this disk. So, when Uranus which is in an inner
orbit meets these fragments in the outside of the disk, then it slows them
down. And Uranus slows down the fragments,
shifting them inwards and acquiring a little bit of their energy, it shifts
slowly outwards. Now, this is a very slow process because
planetesimals relative to Uranus, here in this orbit doesn't change much,
but there's this constant rain of planetesimals moving inward propelled by
collisions with the other giants, where they, they, they encounter Uranus.
And the next, they encounter Neptune, and next, they encounter Saturn.
And eventually, they encounter Jupiter. And yeah, this slowly shifts the giants,
other, over millions of years, all of these giant orbits are slightly growing.
Now, this is very slow, until after about 600 million years, something happens.
And what happens is that this motion brings Jupiter and Saturn, who are not
too far from that right now, into a two to one orbital resonance where, the
period of Saturn is precisely twice the period of Jupiter.
At this point, we have problem the problem that we discussed.
There's this instability. The orbits are both elongated, one in, in
orthogonal directions, in a coherent way but these are two massive objects.
The gravitational impact on each other is nontrivial.
They both get deformed into eccentric orbits.
They, that in turn, influences the rest of the
giants. And what one sees in the simulations, and
you can I'll post a link and you can see these simulations yourself, is that once
this point is reached, then very rapidly, the planet planetary orbits change, and
Uranus and Neptune swap places and move, get pushed farther out.
Jupiter gets moved a little bit inwards as a result of all of this and in the
process the entire outer asteroid belt is depleted.
all of those planetesimals beyond the disk are scattered either, either into
higher orbit trans-Neptunian orbits beyond 30 astronomical units where
Neptune today resides, and they form what we think of now as the Kuiper belt.
Or some of them are rejected into very eccentric off-access, off-plane
trajectories. And we think of that as the origin of
what we now see as the Oort cloud. And notice that as Jupiter and Saturn are
moving resonantly, they apply a resonant and shifting a influence on the asteroid
belt, and this really is what completely depletes it and prevents the formation of
anything there. Also, the nearest planet to this whole
mass is Mars and the perturbations applied by Jupiter and Saturn, to the
orbit of Mars are what prevent, we think, Mars from becoming a planet, the size or
mass of Venus or the Earth the main difference is that it was subject to more
perturbations as it was trying to grow. Moreover,
following this resonant, the resonant period, Saturn moves out.
it, it, it collides in the sense of orbital encounter with Uranus and
Neptune, pushing them out into eccentric orbits.
They encounter the planetesimals, as I said, destroying the disk.
Scattering things into either trans-Neptunian orbits or
higher orbits. Some of the planetecimals that they
encountered, in fact, slowed. And if they're slowed down sufficiently,
they rain down into the interior solar system.
this means that there will be a time period in the history of the inner solar
system, a time, and the outer solar system actually, where there will be this
barrage of things coming from this outer disk of planitesimals, and raining down,
all the way down into the inner solar system this period, and we see its traces
in the history of cratering in the inner solar system.
It's called the time of heavy bombardment.
And when we talk about cratering in the inner solar system, in our moons, we will
be able to trace a timeline that matches up with this, and
then, the remnants of the disk create enough friction to settle everything down
into the circular orbits which are essentially stable, which we find them
today. So, here is a graphic representation. We
start out with the four giants note that light blue Uranus is farther out than
dark blue Neptune, and then Jupiter and Saturn are inside.
Then after a, a few hundred million years,
just before the resonance the orbits have shifted.
They're a little bit more eccentric. Neptune and Uranus have exchanged places.
And Neptune is starting to plow through this disk of planetesimals.
And indeed, the disk is thinning out on the inside.
And some of these planetesimals are being blown down into the inner solar system.
Some are being scattered out into the outskirts of the disk.
And then, very rapidly, once resonance is achieved
the disk is completely emptied, we see a scattering of population beyond Neptune's
orbit. Some of them have been scattered entirely
out of the plane to form the Oort cloud and the giant has settled into their
accepted orbits that we've seen today. And this is what we our models predict,
was the history of the solar system. This is how we got to where we are.
We've answered, at some level, the question, the last question we asked,
where it all came from and when? And in the process, we've learned
something about how orbits change, though we haven't addressed the question of will
planetary orbits change in the future, I can tell you that the solar system is
stable on time scales of hundreds of millions of years at least as far as the
planets are concerned. what happens after a few hundred million
years, it is hard to predict. But as we'll see, there are other issues
that are going on in the solar system on those time scales.
We know why there's no planets where the asteroids are.
we understand why there are two kinds of planets.
We know why the asteroids didn't form a planet.
We know why things are round and why thing, some things aren't.
And we understand why all planetary orbits are circular in a plane,
friction slowed them down to circles. The plane is the original plane of the
rotation of the solar nebula, and comets move on whatever orbits they happen to
have been thrown into. Because they are the result of some near
collision with a heavy object of some old planetesimal.
So, wWe've made some progress though there
are still questions we need to ask. It might be a good time to summarize what
we've learned about the history of the solar system with a timeline.
So, we started our clocks at what I call time zero, 4.56 billion years ago.
Something triggers the collapse of the molecular cloud that will form the solar
nebula within a 100,000 years in the inner solar system and less in the outer
solar system, we have planetesimals. within 10 million years the inner system
has formed protoplanets and is beginning to start the accression of planets.
The outer planets by this time are pretty much formed.
The T Tauri winds start to sweep away the gas and the dust,
leaving planetesimals that are more massive and dense objects.
And so, the inner planets have take about till a hundred million years from this
event to complete their formation and settled into their orbits.
This includes the massive impacts that formed the moon and stripped Mercury of
its outer layers and so on. After about 600 million years
we hit the Jupiter Saturn resonance. And this is the time that the asteroid
plates, belt is depleted, the outer planets migrate out pretty much out to
their current orbits. And in the process
some of the material from the planetesimals beyond Neptune's orbit gets
ejected out into the Kuiper belt. Some into off-access orbit, off-plane
orbit, producing the Oort cloud farther out, and a lot gets thrust into the inner
solar system and we find the period of heavy bombardment.
So, we should expect lots of 3.8 billion year old cratering, right, because that
is the time of the heavy bombardment and by about 700 million years, the stable
configuration has basically been achieved, planets exist and of roughly
their current mass, in pretty much their current orbits and not coincidentally,
the first signs of life appear on the planet we call Earth.