00:01
Here i'm going to be looking at some parts of an ellipse, and we are going to tie these into orbits of planets in our solar system.
00:11
So the first thing to note is i'm going to line up the ellipse such that it has its center at the point zero zero, so the origin, and it is lined up horizontally with its major axis, which we're going to call a is the semi -major axis.
00:35
And b that we'll call the semi -minor axis.
00:43
So b.
00:59
And we'll note that the equation of an ellipse is x squared over a squared plus y squared over b squared is equal to one.
01:15
So in order to write that equation down, we need to know what both a and b are.
01:21
And there are some subtleties.
01:24
So remember that an ellipse is a curve that remains the same distance between two foci.
01:33
And we're going to go ahead and put the sun at one of those foci.
01:38
And we'll put a planet in orbit.
01:43
Maybe it's the earth, but it doesn't have to be.
01:45
But we'll have a planet somewhere in the orbit.
01:50
So the first thing to note is that there are two special.
01:54
Distances along the axis.
01:56
They are known as parahelian.
02:00
So we'll denote that by dp is the closest distance to the sun.
02:08
Parahelian means near the sun, and that's a distance dp.
02:19
And there's also an appelian distance.
02:22
We'll kind of show that on the opposite side.
02:36
And that means furthest from the sun or far from the sun.
02:43
But the way an ellipse is defined is that as the planet goes in its orbit, it remains a fixed distance total between those two foci.
02:56
And what that means is you could draw the ellipse by taking a string of certain length, taking two thumbtacks, separating them by some amount, and using that string, to trace out the ellipse as long as the string both ends are squeezed onto those thumbtacks, tied onto those thumbtacks.
03:22
Now there's a special distance, a couple special distances that will denote.
03:29
Let's go ahead, give ourselves a little bit more space.
03:37
So the mean distance, i will call that d is equal to the mean distance of planet from the sun.
03:54
Sun, and that is basically the same as the semi -major axis.
04:08
Furthermore, okay, there is another parameter for an ellipse called its eccentricity.
04:14
I'll call that little e, and i'll show where it comes about.
04:21
But the eccentricity is a measure, a relative measure, of how far off -center those two foci are.
04:30
And i'll just kind of denote it.
04:32
It's a symmetric quantity, but each of those thumbtacks, if you will, has been displaced from the center by the amount a .e.
04:44
So i'll write down a .e is equal to amount foci are displaced from center.
04:55
So eccentrics means off center.
05:03
That's a very good word to used to describe this ellipse.
05:08
So how are all these distances related? here's where things get tricky, but we can see that the sum, so here we'll come up with some relationships, the sum of the perihelian and the appelian distances is equal to two times a, the length of the entire major axis.
05:40
Furthermore, we have a relationship between the perihelian distance and the eccentricity and a.
05:50
So just notice that if we take ae, we can do some things with it.
05:59
We have it on both sides.
06:02
So we have two relationships with a and e in them.
06:16
Ae plus the perihelian distance is equal to a is the same thing as the appelian distance minus ae is equal to a.
06:36
Yeah, if you add those two together, you'd get twice a, as you should.
06:42
Now, the hardest one to determine is what about the semi -minor axis? and here i'll draw a little sketch of the ellipse to kind of show how to figure that out.
06:55
Okay, so redraw the ellipse.
07:05
I'll just kind of schematically show it with the major axis in there.
07:18
And i'll draw just a semi -minor axis.
07:29
Okay, but if the planet is sitting at the point right along the semi -minor axis, it is an equal distance from each of the two thumbtacks.
07:43
So there's this nice symmetry.
07:51
And so we have kind of a right triangle.
08:02
Okay, so the length of the l that's shown squared is equal to b squared plus a, e squared.
08:14
So if we knew what the length was, we could find b in terms of a and the eccentricity.
08:21
And in order to figure out that length, recall that the length of the string must remain fixed as you're drawing in ellipse.
08:34
So twice l equals length of string to draw an ellipse.
08:44
So we're kind of using a geometry thing here and not worrying about planets so much.
08:53
But if we were looking at the point along the perihelian, let me try to make that look more earth -like, the length we could write it down as let me just kind of show a string drawn in there, maybe with brown.
09:18
So it's going to go all the way from the leftmost thumbtack all the way over.
09:27
And that length is ae plus a.
09:34
And then it's got to go back, which is a minus ae.
09:46
So it turns out that l is the same as a.
09:50
So what's true is that if the planet is sitting in its orbit right along the semi -minor axis, make sure i use the right word there, it is exactly at the average distance from the sun.
10:19
So we'll say that on the minor axis planet is distance a from the sun.
10:26
So we'll forget about our string and talk in terms of planets.
10:33
Okay, and using this, we can then figure out the semi -minor based on all sorts of things.
10:44
So what we basically have is a squared minus a.
10:49
A squared, e squared is equal to b squared.
11:10
Okay, so that's basically the relationships that we have, a relationship for the semi -minor axis, a relationship for the semi -major axis in terms of the perihelian and appheelian, as well as a couple relationships that can give us the eccentricity.
11:39
So those are all fairly valuable.
11:43
So here we're going to take a look at some examples actually using some planets.
11:49
So i'll set up a table.
11:54
We'll look at three planets, earth, mars, at jupiter.
12:04
And we want to fill out all the information about them.
12:09
Earth, mars, jupiter.
12:15
What do we want to know? so we want to know, perihelian distance, the apphulian distance, the eccentricity, and a and b.
12:37
With those latter two, you can then put into the equation x squared over a squared plus y squared over b squared is equal to one.
12:50
Now, a little bit of a caveat here.
12:53
And that is that we've really simplified the orbit down to a horizontal ellipse.
13:02
In reality, the ellipse can be rotated at an angle, which is different for each planets.
13:10
And the center of the ellipse can be forced to be equal to the origin, that's for sure.
13:19
But each planet should have its own angle of rotation.
13:22
And without further information, we can't really figure that out.
13:29
So we'll go ahead and assume the simple picture here.
13:35
Okay, and we'll put some data in here.
13:40
The data is in units of 93 million miles.
13:51
It could also be put in astronomical units, but we'll go ahead and use a million miles.
14:02
And we'll call one, two, and three, because we have different data.
14:06
For the earth, we have, let's see, we have the mean distance, i believe.
14:16
Yes.
14:17
So for the earth, we have a, and we have the apeelian.
14:33
For mars, let me just get all this date in here.
14:39
We have the perihelian and a.
15:03
Let's see is that what we have for mars? let me check...