Is gravitational force represented among the orbital elements?

[Meithan]

Start with a guess. In this case, E₀ = M is a good starting guess.

Unless the eccentricity is high (e > 0.8), in which case it seems M = À (pi) is a better choice.

Edit: after checking I think this applies to Newton's method, not to this fixed-point method. For high eccentricities, Newton's method will converge much faster, it seems. Still, a few hundred iterations is peanuts to any modern computer.

Edited June 18, 2015 by Meithan

[KSK]

It does, but part of it is calculating the eccentric anomaly given the mean anomaly. There is no closed-form solution for this:

So it says "follow these steps," but it doesn't give all the steps.

Ahh - missed that. Sorry, my bad - glad more knowledgeable folks were here to help out.

[YNM]

Start with a guess. In this case, E₀ = M is a good starting guess.

Unless the eccentricity is high (e > 0.8), in which case it seems M = À (pi) is a better choice.

Hmm, thought that any arbitrary number could work ? (granted the equation contains sine which means possible dellusional results, but I just want to know whether it's truly the case for iteration bar diverging results which stems from wrong placement, been wondering since a bit long)

[ZetaX]

Hmm, thought that any arbitrary number could work ? (granted the equation contains sine which means possible dellusional results, but I just want to know whether it's truly the case for iteration bar diverging results which stems from wrong placement, been wondering since a bit long)

The comment was not about any choice being wrong, but what choice converges faster. But in general, a bad choice could lead to divergence or convergence to a second fixed point.

[Meithan]

Hmm, thought that any arbitrary number could work ? (granted the equation contains sine which means possible dellusional results, but I just want to know whether it's truly the case for iteration bar diverging results which stems from wrong placement, been wondering since a bit long)

The fixed point method will work alright for large eccentricities, it just will converge more slowly.

Here, I ran a test for M = 3.0 (it seems values near pi are bad for this method): with e = 0.05, it takes 6 iterations to find E to 7 decimals. With e = 0.95, it takes 264 iterations. If we push it really far, say e = 0.999, it now takes 3997 iterations to find E.

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But let's not go off-topic. I'm interested in knowing what T.C. has to say about what's been discussed so far.

Edited June 18, 2015 by Meithan

[Brotoro]

The orbital elements simply describe the path of the satellite. You could get the elements by just observing the motion of the object long enough without knowing anything about the forces involved (but if you want to calculate an orbit quickly from fewer observations, it helps to know things about the force of gravity operating in the system).

Similarly, I could describe the motion of a rock circling around on the end of a string without having to go into the forces causing it to do so. It would also just be parameters that describe the motion (and from which you could derive the Physics involved, if you wanted).

[T.C.]

But let's not go off-topic. I'm interested in knowing what T.C. has to say about what's been discussed so far.

You pretty much settled everything when you said "the mass (or gravitational parameter) of the central body... is a quantity so important in orbital dynamics that I guess it's generally assumed to be known"

When you're learning something new, the hard part isn't getting the solid facts down, it is getting your mind around the squishy context. Conceptually grouping mass among the physical constants, and not among the orbital parameters, is the kind of thing that a novice like me wouldn't necessarily do instinctively. It takes a discussion like this to get a feel for that kind of thing. And this discussion has done its job well; I consider my question resolved. You've all helped me a lot. Thanks to everyone who posted. Feel free to go off-topic as much as you'd like now.

-TC

[Meithan]

You pretty much settled everything when you said "the mass (or gravitational parameter) of the central body... is a quantity so important in orbital dynamics that I guess it's generally assumed to be known"

When you're learning something new, the hard part isn't getting the solid facts down, it is getting your mind around the squishy context. Conceptually grouping mass among the physical constants, and not among the orbital parameters, is the kind of thing that a novice like me wouldn't necessarily do instinctively. It takes a discussion like this to get a feel for that kind of thing. And this discussion has done its job well; I consider my question resolved. You've all helped me a lot. Thanks to everyone who posted. Feel free to go off-topic as much as you'd like now.

-TC

I'm glad the discussion was of help to you . It's quite common to forget the "obvious" assumptions people make when they're talking within a very specific and agreed-upon context.

Now, regarding the Wikipedia article, I think the question of whether we should introduce a clarification still stands. Your points in the talk page are valid, and while they may be clearer now to you, the fact that initially they weren't even after reading the article suggests it is perhaps not as clear as it ought to be.

Would you like to discuss a modification here? PM? Directly on the talk page?

[T.C.]

Now, regarding the Wikipedia article, I think the question of whether we should introduce a clarification still stands. Your points in the talk page are valid, and while they may be clearer now to you, the fact that initially they weren't even after reading the article suggests it is perhaps not as clear as it ought to be.

Would you like to discuss a modification here? PM? Directly on the talk page?

The talk page is best, I think. I will not undertake to change the Wikipedia article(s) myself until I am more confident of my knowledge, but I welcome any changes you choose to make as a result of this discussion.

-TC

[Meithan]

The talk page is best, I think. I will not undertake to change the Wikipedia article(s) myself until I am more confident of my knowledge, but I welcome any changes you choose to make as a result of this discussion.

-TC

Alright then, I'll post some comments in the Talk page and think about what modifications we could do to make the article clearer. Your pointing out that μ is not defined, for instance, is cause enough for an edit.

[K^2]

I knew of calculating a fixed point of f by trying \lim_{n\to \infty} f^n(x_0), but I never heard that name. Is that what physicists call it or where did you get it from¿

Linear algebra and associated numerical methods is the most common place where you find it called that. Specifically, finding highest eigen value and associated vector can be done by taking the matrix to a high power. But any time you might be thinking of function as an operator, the name comes up.

Oh, and yes, convergence is only fast for small eccentricities. I should have clarified that. That is the case for most objects in stable orbits, so in practice, you rarely need to worry about it. But if you happen to have a comet on near-escape trajectory, you might want into other methods. Newton's works, of course.