B squared is now going to replace R squared in this term here,
to say how frequently the impacts occur.
And you can see, that it's going to depend not only on R squared,
as R gets bigger, it's going to depend on the escape velocity,
which of course depends on how much mass the object has, as the object gets bigger.
The escape velocity increases, but the most important term is this V infinity.
It will end up depending on how fast the other particles around are going.
If V infinity becomes very small,
if the objects are moving very slowly with respect to each other, you'd think gee,
they're never going to impact, no big deal.
But no, if they're moving very slowly with respect to each other,
the only influence they have is of gravity.
V infinity becomes zero, b squared becomes infinity.
And the fact that there's a V infinity squared on the bottom here and
a V infinity on the top here means that the overall impact
rate scales as 1 over V infinity.
So if the velocities get small the impacts go up dramatically.
Why would the velocities get small?
Let's do one other process, we had gravitational focus,
the other process that's important is called Dynamical friction.
It has nothing to do with friction, so it's kind of a weird term.
But what it really means is, if you have a bunch of objects around.
Let's say, you have some big ones, massive ones, and
then you have a bunch of small ones, little ones.
And they are moving around through space,
gravitationally interacting with each other.
An interesting thing is going to happen.
First, if we start them out all at the same velocity, going around the Sun,
they're all going at the same velocity.
What's going to happen is that the big ones are going to slow down.
They're not going to slow down going around the Sun.
They're all going around the Sun.
But they're going to slow down their relative motions to very small, relative
velocities, while the small ones will end up with very high relative motions.
The process is similar, though not exactly the same to what you could imagine.
What if you had a bunch of bouncy balls.
Let's say some big beach ball size ones like this.
And then a bunch of little super balls.
And you put them inside of a big box and
you initially start them all moving at the same velocity.
Well, the small ones are going to hit the big ones and start go faster and faster
every time whenever a big one hits a small one it comes off on very fast speed.
Every time a small one hits a big one it slows the big one down by just
a little bit eventually the big ones will have very similar speeds,
while the little ones will be moving around really quickly.
There are a couple of ways of thinking about this,
the other way you could think about it is called equipartition of energy,
the particles end up having similar amounts of kinetic energy.
Kinetic energy is one-half MV squared.
If M is really big, you better have a very small v.
If M is small, you have a very big V.
The other way of thinking about it is this.
If you had a single object sitting here.
And a sea of small bodies was coming around going this way.
Well, as they go by, the ones that don't hit are deflected like this.
And so, in front of the object, the object is moving this way.
In front of the object, there's a uniform sea of particles.
Behind the object, there's a little bit more density of objects right behind it
because they've been deflected this way.
A little bit less out here.
And that little bit extra density gives a little bit of a tug here and
slows this body down.
While these evolved and sped up a little bit.
Yeah, there are a lot of different ways of looking at it.
But the important point is, in a sea of particles, small ones will end up
going fast with respect to each other, and with respect to the large ones.
And the large ones will get increasingly slowed down.
The larger they are, the slower they will get with respect to each other,
their relative velocities will be very small.
What does that mean?
That means that in this initial protoplanetary disc,
in this initial disc of gas and dust.
Particle start to impact each other.
They start to stick.
They starts to be a little bit of gravitational focusing and
then there's feedback.
The particles starts to get bigger particles that
objects starts to get bigger.
Have more gravitational focusing.
They get even bigger still, then they start to slow down with respect to
all of the objects that are around there.
And maybe there's another one over here doing the exact same thing.
Suddenly, these objects have zero velocity with respect to each other.
Forget about all these little objects going around here, but
think about these big ones now.
These big ones now have almost no velocity with respect to each other and so
their gravitational cross section,
their gravitational focusing cross section becomes huge.
And they all merge.
This is a process that we call Runaway Growth, and it's a significantly faster
process than you would get by simply saying how many objects are going to
hit this or even how gravitationally focused are you going to be?
It's that combination of gravitational focusing and
Dynamical friction, which leads to this process of runaway growth.
This process of Runaway growth can continue until
everything in a region of the disk is combined into one single object.
We'll talk about what those single objects are in the next lecture.
