The Cosmic Classroom - Kepler's Laws

Uploaded by vmargoniner on 16.09.2009

Welcome back to the Cosmic Classroom.
So this week I heard a lot of questions
about Kepler's Laws. Mostly, many of you had
questions about the understanding of what
Kepler's Law means. And also how can you memorize
the the different laws. There're also some questions
regarding Newton's version of Kepler's Law, which we'll
treat somewhere else. So lets review Kepler's
Law, so they are three laws, and I think that much
you probably remember. So there're three laws.
The first law, first law,is the one that says that
planets orbit in elliptical orbit around the sun,
with the sun in one of the focus of the ellipse.
So let me try to draw an ellipse here,there,let's
put the sun in one of the focus, and of course I'm
exaggerating the ellipse. Right?
The real, if you were to really draw this to scale,
it would look like a circle. Right?
You wouldn't be able to see it, how elliptical it is.
That's just to make a point here. So there's the sun and
there is a planet here and the first law says
that the planet orbits in this elliptical orbit, that's it.
Now the Second Law draws on it.
The Second Law, adds something else to it.
So the Second Law, Second Law, I'll use the
same drawing here. What the Second Law
is, says is that the time that it takes the
planet to move from this position here.
Let's say it's here, August first, and it takes
two months to travel here, and it would a day
in September first, yeah. So it says that while
traveling the planet covers an area here and that,
that area is constant. So if it takes two months
to go from here to here, there is another place in
this orbit that it will also take two months to go
to, from there to there. Because the area
here is the same. So it says, the Third Law says
that planets sweep equal areas and equal amount
of time in the orbit. Now it's often not clear how
is it possible that this area here is equal to this one.
How do I know that? So of course in this case
I just drew it approximately, but you could convince
yourself that this a very similar by getting something
for which you know the area and see how many
you can fit in there. So for example you get
some pennies see how many you can fit in there.
And you'll see that you'll be able to fit about
as many in this side. The pennies are a little bit
too big to demonstrate here, but that's the idea of area.
Right? So it's the same real state in
both, in both places the same real state is covered, but the
shape is different that's Newton's that's Kepler's Second Law,
excuse me, Kepler's second law. Alright?
So Second Law builds on the third one.
The third one gets more confusing. The third one is the one that
adds an, another planet to it. Ok?
So the Third Law now deals with comparison between
two planets orbiting the sun. The first and second law
are talking about one planet and thus,and the
third law lets add another planet here, in another orbit.
Alright? Around the sun. There.
So the Third Law states that the period of, that's
period the amount of time that it takes for a planet
to go around the sun square is proportional to
the average distance from the planet to the sun cube.
And this is an empirical relationship and Kepler didn't
know why it was that way. He just notice that, that
was a relationship... Notice that
it makes sense. If 8 increases lets, lets
imagine that this is Earth at one astronomical unit.
If you're now at two astronomical units of
course it's going to take longer to go around
because the path is longer. But not only it takes
longer because the path is longer but actually
the planets are moving slower further from the sun.
You can think about it as they attracted less to the
sun, so they slowed down. So a planet an outer planet
here is moving up to certain velocity while the inner planet
is moving at a faster speed. A faster velocity.
And it's got really busy. But I hope you followed.
Just to give you a little bit more on, on Kepler's
third law, which is the more complicated one.
It dares to have an equation in there,which you
really don't like I know. Kepler's third law, so I
would like to give you one more specific example.
So lets imagine, so we have the sun here,
right there, and then there is the Earth going
around the sun in something that should look more
or less like a circle,you shouldn't be able to say
that the sun ellipse. So here is the Earth,
here is the sun and you now that the
Earth is at the distance of one astronomical
unit from the sun. That's the definition of an
astronomically unit and it takes one year for the sun
to go around, for the Earth to go around the sun, one year.
Now at about three astronomical units from
the sun we have something called the Asteroid Belt.
So I'm going to try to draw something that's
three times further away. Alright?
Here, its called the Asteroid Belt it doesn't have any
planets on it, but it has a lot of little rocks there.
No planet was formed in there because Jupiter is nearby and
keeps, you know, messing them up. But they are a bunch of,
you know, little rocks in here, and those little rocks
are also orbiting the sun. The rocks are at three
times the distance,three astronomical units from the sun.
So let's see how long it takes for the Asteroids to
complete one full orbit. Right?
According to Kepler's third law we know the relationship
between that period and the distance from the sun.
We know that "P" squared equals "A" cubed. So if the distance is three,
so I have that "P" squared equals 3 cubed meaning "P" squared equals about 27.
And if you take the square root of this, P is
square root of 27, P is approximately, this design
for approximately, 5 years. So it takes 5 years for the
Asteroids to complete one orbit, but notice that they are only
traveling a distance that it's three times greater than Earth.
Right? The radius is three times
the distance is just three times greater, but it's taking 5
times as long to go around. Which leads us back to
the conclusion that the Asteroids are traveling at
the slowest speed than earth. And that's it for Kepler's Law.
I hope it helped. See you next time.