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For fun I'd like to use Hubble Astrometric data for example the observations listed at the bottom here to numerically estimate the orbit of 2014 MU69 as well as its uncertainties.

My plan is to use some combination of JPL Horizons, Hubble's TLEs, and Skyfield to get the J2000.0 position of the HST at the time of each exposure, and to get the position of the sun and major planets to generate the gravity field in which to integrate MU69's motion.

I understand I will have to retard the gravity from each source by its particular light-time, as well as correct for the light time for the HST images.

This will not be fast or efficient, it's strictly an exercise. At each time step I'll have to iterate and interpolate to find out, just for example "where would Jupiter have been in it's orbit such that it's gravity would be arriving right now".

I'd do that and calculate an initial orbit for MU69, then use that to calculate apparent positions for the HST data, calculate an error, then try a different starting state vector for MU69 and see if that's better or worse, and just use steepest descent to find a nominal orbit. From that, I can see how sensitive the fit is to various combinations of deviations from nominal.

I'm aware there may be smarter ways to do this, but to appreciate them it's better to do it brute force at least once. In the age of giga-flop laptops it's a viable option.

My question: Are there other things I need to consider?

Just for example, do I need to worry about time moving at different speeds at different distances from the sun (general relativity) or forces on MU69 besides gravity from the Sun and outer planets in order to get the level of accuracy relevant to comparison with the HST astrometry?

Again: ...in order to get the level of accuracy relevant to comparison with the HST astrometry — so I'm not looking for a list of arbitrarily small effects.

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    $\begingroup$ You might want to look over naif.jpl.nasa.gov/pub/naif/toolkit_docs/C/index.html in particular how they note that computing light travel time doesn't make much of a difference. Also, you might want to use a numerical DFQ solver instead of the iteration you propose. I've done similar work for planets, nearly replicating Horizons' answers. Let me know if you want more details. $\endgroup$ – barrycarter May 11 '17 at 17:19
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    $\begingroup$ @barrycarter Thank you for those - I'll definitely take a look! I want to start by doing it in the lowest-tech way first, so that I can better appreciate more advanced algorithms in the future, but it's great to know they exist. I only let myself start using the scipy ODE methods after I wrote and tested an RK45 with automatic variable step size script myself first. It's just my way of learning how stuff works. $\endgroup$ – uhoh May 11 '17 at 17:34
  • $\begingroup$ This object's orbit has a semimajor axis of 44 AU. Stated at the crudest level, we need general relativity rather than newtonian gravity when gravitational fields are strong. Therefore the outer solar system is the worst possible place to look if you're hoping to get fun, relativistic effects that need to be taken into account. This is why Mercury's orbit was a classic test of GR, and is also why the Shapiro time delay was measured for sun-grazing rays: adsabs.harvard.edu/cgi-bin/… $\endgroup$ – Ben Crowell May 11 '17 at 19:49
  • $\begingroup$ @BenCrowell the calculation involves observation of MU69 at 44AU from the HST which is 1 AU from the Sun. So I'm asking about the difference between 1 AU location and the 44 AU location. $\endgroup$ – uhoh May 12 '17 at 3:26
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This is not a complete answer. It is instead an extended comment to the following:

I understand I will have to retard the gravity from each source by its particular light-time, as well as correct for the light time for the HST images.

While you do want to correct for light time travel with regard to seeing a moving remote object, you definitely do not want to do the first part (retarded gravity). That is not how Newtonian mechanics works, nor is it not how general relativity works.

There is no lag in Newtonian mechanics; gravitation is instantaneous in Newtonian mechanics. In general relativity, there are some terms that act like lags, but there are other terms that act like leads. Those lag-like and lead-like terms in general relativity nearly cancel for small gravitational sources such as our Sun. That near cancelation is what makes Newtonian mechanics very close to correct. Keep in mind that the even for Mercury, the relativistic effect is very, very small: Only 43 arc seconds per century of precession that is not explainable by Newtonian mechanics.

A couple centuries ago, Laplace investigated whether gravitation is instantaneous. He found that adding any significant lag to Newtonian gravity makes the solar system become unstable in short order and this concluded that the speed of gravity had to be very high, at least $7\times10^{6}$ times the speed of light. A couple of decades ago, another highly respected astronomer published a paper in Physics Letters A (a highly respectable physics journal) that came to the conclusion that the speed of gravity is at least 20 billion times the speed of light.

Both Laplace and this more recent author were wrong. Laplace can be forgiven for not having possession of a time machine that would carry him a century into his future. The latter author cannot. His article has been cited 175 times (per google scholar), but almost all of the citations are essentially "You're wrong. So very, very wrong, and here's why ..." The "here's why" is that that is not the way general relativity works.

The easy way to solve what you are trying to do is to ignore relativistic effects. Just assume Newtonian physics, in which gravitation is instantaneous, but the speed of light is not.

The hard way is to incorporate general relativity, to some extent. You'll need a relativistically-correct time scale (e.g., JPL's Teph) and some kind of post-Newtonian model of gravitation that is, at least to first order, consistent with general relativity. For example, see The Planetary and Lunar Ephemerides DE430 and DE431. Do that and you'll be on par with the groups that develop extremely accurate solar system ephemerides.

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  • $\begingroup$ To double check, are you saying that if I calculate the force based on where the Sun and outer planets would have been, I'd get a worse result than if I treat gravity as instantaneous, or just that that's not the best, most correct way to do it? $\endgroup$ – uhoh May 11 '17 at 17:43
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    $\begingroup$ @uhoh - Adding a lag (and nothing else) will yield worse results (far worse results) than if you simply treat gravity as instantaneous. If you want to do GR right, you have to go whole hog. Well, almost whole hog. Except for very simple systems (e.g., a pair of neutron stars orbiting one another), nobody goes whole hog. They instead use some sort of post-Newtonian approximation or parameterized post-Newtonian formalism. $\endgroup$ – David Hammen May 11 '17 at 18:19
  • $\begingroup$ OK it's morning now for me and I've given your answer a fresh read, and I understand just what you are saying, though I don't mean to suggest I understand much about GR. This reminds me of a read this some day paper I've been holding on to. After learning about modeling non-gravitational forces on comets for this answer I've been meaning to try out Equation 1 in lpi.usra.edu/books/CometsII/7009.pdf (linked in comments there). It seems the second line is such a correction. In the mean time, I wont use a lag. $\endgroup$ – uhoh May 12 '17 at 3:57
  • $\begingroup$ Thanks again for the comprehensive answer and history lesson to boot. As an aside, somewhat related though it's a different question. $\endgroup$ – uhoh May 12 '17 at 3:58
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    $\begingroup$ Your previous reference used the Gaussian gravitational constant $k$ rather than $GM_E$ in the above. Otherwise, they're identical. $\endgroup$ – David Hammen May 22 '17 at 11:35

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