Two-body orbits: example from initial scalars

Introduction

I want to work out a problem similar to the previous one — but different.

Instead of being given the position and velocity vectors, we will be given their magnitudes, i.e. the distance (from the center of mass of the primary) and the speed — and the flight path angle.

These are enough to let us determine the scalar orbit; and that is what I want to show you how to do.
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Happenings – 2010 May 29

(Yes, the title used to say April 29. My bad.)

Okay, it’s later than usual for my weekly math diary post. I’ve been distracted so far today — by nothing bad, but distracted nonetheless by life.

Let’s see.

My newsgroup browsing this morning turned up an interesting read: “What Is Mathematics For?

And since Lockhart’s Lament (at the bottom of this link) discusses the same issue, it seems appropriate to have provided its link again.

Oh, and I was searching my own blog for something — and shocked to discover that command-F isn’t very useful anymore, because I have split the posts into a small portion which is displayed, and another portion which is not… command-F only works on what is displayed.

Well, I always wondered why WordPress offered me a “search” widget; I thought command-F was all we needed. Now I know: the widget will select every post — heck, it will select everything! — containing a search string.
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Two-body orbits: example from initial vectors

Edit: 2011 Apr 29, mu = GM not Gm!

Introduction & Review

I am going to work a numerical example of the equations I worked out in the previous post. To be specific, I am going to assume we are given the position and velocity vectors for an object in orbit, at one instant of time.

Let me begin by doing what I should have done at the end of the previous post: here’s a summary of the previous post.

We took the vector differential equation

\ddot{\vec{r}} + \frac{\mu}{r^3}\ \vec{r} = 0
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Happenings – 2010 May 22

Hello, again.

If you’re a regular reader, then you noticed that no posts went out last weekend — neither a happenings post nor a technical post.

Well, an old friend was visiting for a few days. If you can believe it, I spent time with my friend instead of doing mathematics. (Gee, sometimes I have a life!)

Now I’m back to mathematics — and I have been since he left, when I’ve had the time.
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Orbits: Vector derivations for 2-body orbits

Introduction

First, let’s take it for granted that the differential equation for the two-body problem (e.g. the earth orbiting the sun in an otherwise empty universe) is

\ddot{\vec{r}} + \frac{\mu}{r^3}\ \vec{r} = 0

where \vec{r}\ is the vector from the primary (e.g. the sun with mass M) to the secondary (e.g. the earth, with mass m) and \mu = G(M+m)\ ; and a simple r\ is the magnitude of the vector \vec{r}\ .

I want to derive

  • conservation of energy
  • conservation of angular momentum
  • a neat equation for r and v
  • the scalar equation for the orbit.

Furthermore, I want to use vector operations wherever possible. (I like vectors!)
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Happenings – 2010 May 8

Last weekend and this week have been interesting but not too interesting. (Too interesting can be bad.)

Logic.
As I moved on to writing up “universal generalization”, “universal instantiation”, and such things, I seem to have discovered that none of my books has a really good explanation.

Maybe that’s okay. I wasn’t planning on doing them rigorously — merely identifying them and showing a couple of the dangerous curves while using them… working at the level of Hummel and Exner. We’ll see whether I can sort it all out to my satisfaction and have a post ready for final edit Monday evening.

Still, I’m shocked that my library let me down.

Color.
I made some progress in Fairchild’s “color appearance models”. In addition, my freshly ordered copy of Kang’s “computational color technology” arrived… and I found time to read through it. I bought it because it uses Cohen’s approach, and it discusses the reconstruction of spectra from tristimulus values. It is very likely that I will be talking about Fairchild, and likely that I will be talking about Kang.

I also found time to work out explicitly x,y as functions of Y assuming that X and Z are held constant while Y varies. You can find it in this comment, but there aren’t any pictures.

If I decide that I probably can’t get a logic post ready this weekend, I might be able to do a color post about my monitor; that will illustrate properties of other monitors.

Modular functions.
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Logic: finding valid conclusions

Introduction

This is not a general post about all methods of extracting valid conclusions, but I will be looking at more than just syllogistic reasoning.

I want to talk about the four logic questions I asked in my most recent happenings post.

You may, but don’t need to, go back to that post… here are four pairs of premises. (And because we have two premises for each question, we may consider syllogistic reasoning, as well as other methods.)

some A are B
no B are C

no B are A
some C are B

no A are B
some B are C

some B are A
no C are B

What valid conclusion or conclusions, if any, may we draw?
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