Philosophy 167: Class 4 - Part 11 - Keplerian Orbital Theory: Defining the Keplerian Ellipse.
Smith, George E. (George Edwin), 1938-
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So, how do you define a Keplerian ellipse? Well, his way of defining it is, the first thing you do is locate aphelion and. You do that by where it's fastest in the sky versus where it's slowest in the sky. Of course relative to the sun. So that's not a trivial location.
Then you get the length of the major axis which is nowadays usually called A. Very often that's designates the semi-major axis, but I'm perfectly happy with it being A. So I'm defining a small a here, the semi-major axis half the total length of the line of absolutes and then eccentricity you get CS over CA by the ratio of the fastest speed to the slowest speed.
Again, relative to the sun. That's not the fastest speed and slowest speed from the earth. You've gotta go through the whole theory and relate everything to the sun. At that point you set up the center and you introduce this angle E, called the eccentric anomaly, with a radius from it.
And then you use, that's going to be the primary angle on the circle that you use to locate the planet at any time. Okay? And you'll see how in a moment, So then you take as given the angle ACQ and you ask, where is the planet? And the solution is, you draw from S the diametral distance.
You draw from E From C, which angle E, the line to the circle Q, you drop a perpendicular from Q and that's exactly where the diameter of distance is going to meet that perpendicular and give you the locust of point to the ellipse. Okay? So you end up, there are two ways once you have this.
The principle way is simply drop those perpendiculars from Q and then use the diametral distance rule. That's the way you wanna think of it. And notice the area rule's not coming in here yet. We're simply doing it in terms of E. Everybody see that? So this is just the diametral distance rule, all I've done is trigonometry to give you the angles that let you determine what the angle ASP has to be.
And if you know that angle, you just draw it out, and where it intersects the perpendicular, that's the same thing as the diametral distance. Just two ways of doing the same thing. Keplerian orbits, there are seven of them. Orientation of the line of apsides, that's a big deal, where it sits in relative to the Zodiac the length of the semi-major axis, that's the mean distance of the planet from the sun.
By the way, you should be able to see that immediately from the diametral distance rule. The mean distance is simply half the major axis, because the other term varies as a cosine. I'll back up and show that. Here's the rule. What's the mean of this? A. That happens to be a feature of ellipses that the semi-major access is always the meaning of the distances from the focus.
I had a big dispute with Steven Weinberg over that because he insisted it was wrong. I'll tell the story nicely. I then worked it up and I was afraid to send the email until I'd slept on it and I got up at six in the morning and there was an email from him saying you're right.
Gave me his argument and said now I can go back to sleep.
At that moment, I really respected him that he felt the need to inform me I was right about something when he told me I was wrong about something. That's where I'm going tomorrow is be seeing him.
But regardless, I'm struck that a physicist that good would insist that that rule was wrong. And the only argument I gave him initially was Newton would never have been wrong about that. I didn't even remember the diametral distance rule. I should have, but he didn't know, Weinberg didn't know the diametral distance rule.
But this is a way to get the ellipse. The key thing is those first steps. Once you get the first steps, other things fall into place. And it's that circle around that's so crucial. Okay, eccentricity, that comes off velocity ratios. A period, you get that from time back in the same position.
Orientation of the line of nodes, that's where the intersection between the plane of the orbit and the plane of the ecliptic intersect. That has to be oriented around the Zodiac. And it doesn't have to be at a nice angle, it can, that line of nodes and the line of apsides can be at screwy angles with respect to one another.
The inclination itself i and then the mean longitude at epoch, or the last time at aphelion. Mean time at epoch means you can go forward and backward indefinitely. If you have the last time at aphelion, then you can do the next orbit. Because you know the period and next two orbits, etc.
This is a comparison of Kepler's values for the elements and ones we can infer from Simon Newcomb's tables at the beginning of the 20th century. Which are extraordinarily accurate tables. And presumably project into the past. I'm only gonna dwell on one thing here. You can look at it for yourself.
The serious error here that makes a very large difference is the error in the eccentricity of the earth, where he has it as half of Tycho's 0.018. And a realistic number is 0.01688. Why is it wrong? Tycho had the sun way too near the earth and therefore had way too large a correction for parallax.
By the time the Rudolphine tables, Kepler actually mentions this. And he says almost certainly Tycho's three minutes of arc for the horizontal parallax. That's a thousand earth radii away. It's way too large. The earth is at least 3,000. Excuse me the sun is at least 3,000 earth radii away and this needs to be redone, but his comment was, until we get the proper number, it's not worth going back and doing it.
Okay but, that infects everything else. Almost every other error in here is arising from the fact that we're sitting on an orbit that's mis-described. Okay and mis-described not from bad observations, from bad corrections. You'll see that next week. It's gonna be a big deal because one of the things Horrocks decides very quickly is these numbers for parallax, they're way too large.
Kepler's is four times too large. Horrocks is the first one to start getting it right, and he's getting it right because he sees when he changes the number for the earth-sun orbit, he gets better results than the Rudolphine tables. You'll see that next week, but that's what I wanted to discuss.
This is a table of current values for the six planets we're concerned with. You'll notice that in each case they're given the value of the element. Followed in, for all but the mean distance, which is held constant, followed by a term times t, followed by a term times t squared.
T is measured in Julian centuries, not in years. Okay, so that's 100 year periods over which they change. I don't claim these are up to date, but they give you the general magnitude of how all of the elements are slowly mirroring. By the time of the Rudolphine Tables, Kepler had discovered that the elements of Jupiter and Saturn were changing significantly.
And he thought the ones from Mars were. And what he said, and he tells you that in the Rudolphine Tables, that, and raises the issue whether astronomy is perfectable at all. But in a letter, he makes the remark, it's gonna take centuries of observation to figure out what's going on.
It didn't take centuries of observation, it took Newton. But it took a hundred years after Newton to do it even then, but that was the turning point. But the point is, he's already picking up things are not exact here. And all of it was showing out of Danby, this is a very nice book.
The Danby book. To me it's the easiest to use instruction manual for celestial mechanics that I've ever seen. All that is it's right at my level. Okay. So it may not be at yours, as it may be too simple or too hard. But it's striking to see that there is this slow variation.
If you look through all those coefficients though, you're gonna see Jupiter and Saturn are immense compared to any of the others.