In an earlier post, I asked 'martins' a question:
"On another issue: in an earlier comment relating to making public integrators you mentioned that "this would come in handy in various places, e.g. for mission planning tools, or for creating our own power series ephemeris solutions for celestial bodies (I'm still looking for a solution for Hyperion)." What by way of ephemeris solutions are you looking for?"
To this, 'martins' replied:
"Essentially I was thinking about a machinery that can generate a series expansion of a perturbed orbit from a numerical solution over a reasonably long period centered at the present - essentially what VSOP does for the major planets. In the case of Hyperion, this would be referenced to the true position of Saturn, or barycenter position of Saturn and its moons (and ideally in the same frame of reference as VSOP, to avoid the frame transformations you mentioned earlier).
As far as I know, Hyperion's may be a particularly difficult orbit to model, because of resonances with Titan and high eccentricity, so a series expansion diverges either very quickly or requires a large number of terms. But since you have already created a precise numerical gravity simulator, it might be interesting to use it to see just how well it can be approximated with a series solution."
This exchange got me thinking about how one would go about building a VSOP-like ephemeris of, say, an 'irregular' body such as Hyperion. After some reflection, and a little bit of a literature search, I decided that the mathematical machinery to do this exists and so I thought I would 'give it a go'. So, this post is a 'place-holder' for a number of (yet to be produced) results arising from this exercise. Results will be posted here as I work through this exercise.
For those that might not be familiar with the vocabulary, an ephemeris is essentially a mathematical 'look-up' table that encodes the position of the Sun, the planets and moons. It serves two purposes, the first of which is entirely practical: although a lot of effort may go into putting the ephemeris together in the first place, ephemeris users need only employ a simple algorithm to extract the encoded information. An ephemeris, VSOP87, lies at the heart of Orbiter. It is used by the numerical integrator to work out the location of the principle gravitation bodies in the Solar System. The second is more theoretical. Because of the theoretical underpinnings of the ephemeris construction, the ephemeris solution may yield useful information about the nature of perturbations - e.g., the rate of perihelion advance of an orbit - that would otherwise not be self-evident. For Orbiter, this is less relevant - but in many respects the more interesting aspect.
There are two kinds of ephemerides in common usage. The first is the VSOP series, principally VSOP87 and ELP series, produced by the Bureau des Longitudes. These encode the positions of planets (and the Moon) as a series of cosine terms which, when summed, yield the planetary positions with high accuracy. The full VSOP87 solution contains many thousands of such terms for the major planets and although a little daunting when first encountered are actually quite simple (and fast) to use in practice.
The other kind of ephemeris is the 'DE' series produced by Nasa's Jet Propulsion laboratory - e.g., DE405/406. These ephemeris series are more recent and are used for mission planning purposes. However, the encoded solution of planetary positions consists of an unwieldy set of Chebyshev interpolation schemes. Data files for the full ephemeris solution are extensive to such an extent that they make the extensive VSOP cosine tables look positively succinct. Although, it is older, the VSOP87 is still widely used since for, most purposes, it is sufficiently accurate - and far more compact.
In this thread, then, I'm going to focus on the construction of a VSOP-like ephemeris solution - that is, a cosine table for the saturn-centric position of Hyperion over, say, one or two centuries from the present.
And why is this such a hard task? Surely, fitting a few cosine terms to an integrated solution of Hyperion can't be that hard. Basically, the issue is largely one of accuracy: to produce an ephemeris capable of accurately determining the position of Hyperion one hundred years from now requires a great deal of precision in determining the values of the ephemeris cosine terms: being off by 1 part in 10^10 in the frequency of a cosine term can lead to inaccuracies of thousands of kilometres. And then there is the sheer number of terms that may need to be included. Because the forces perturbing Hyperion's motion are considerable, and because it is a high eccentricity orbit, and because it has a resonance with Titan, the number of cosine is terms needed to accurately determine Hyperion's position may be very large. Or, indeed, Hyperion's orbit may be chaotic and predicting its position one hundred years from now is impossible. At any roads, having thought about it, the construction of a realistic and accurate ephemeris for Hyperion is not a trivial thing to do. (And if you don't believe me, give it a go yourself.)
In fact, the mathematical and computational challenges that this task imposes are such that it doesn't make much sense to attack the problem with a direct frontal assault. Better, methinks, to learn how to walk first by practicing on simpler, less demanding problems.
So, this is my general plan of attack: to work up to the task of an accurate ephemeris for Hyperion, by working progressively through a series of progressively more challenging problems. The specific sequence that I propose to work through is as follows:
1. Determine a VSOP-like ephemeris for a high-eccentricity two-body elliptical orbit around a single stationary gravitational source, e.g., the Sun. Although there is no need for an ephemeris solution here, it contains many of the same challenges of the Hyperion problem without making the problem overly complicated.
2. Determine a VSOP-like ephemeris for the Sun/Earth/Moon system. This is now a three-body problem and contains a number of significant perturbations that lead to phenomena such as 'orbital precession'. Although quite simplistic by modern standards, until well after the Lunar landings, ephemeris based on this three body problem were central to lunar mission planning.
3. Determine a VSOP-like ephemeris for the Sun/Saturn/Titan/Hyperion system. This is now a four body-problem. It captures the essence of resonance between Titan and Hyperion as well as Hyperion's high eccentricity orbit. It ignores significant perturbations due to the gravitational influence of the other planets - principally, Jupiter.
4. Determine a VSOP-like ephemeris for Sun/Jupiter/Saturn/Titan+other moons/Hyperion. This will be a semi-realistic ephemeris for Hyperion. It will include perturbations from Jupiter as well as those from the rest of Saturn's moons. It will not, however, be consistent with the VSOP87 ephemeris solution used by Orbiter
5. Determine a VSOP87-like ephemeris for Hyperion based on the full VSOP87 solution. The end result.
I've already completed Step 1 - and I'll report on this next. Step 2 is underway.
"On another issue: in an earlier comment relating to making public integrators you mentioned that "this would come in handy in various places, e.g. for mission planning tools, or for creating our own power series ephemeris solutions for celestial bodies (I'm still looking for a solution for Hyperion)." What by way of ephemeris solutions are you looking for?"
To this, 'martins' replied:
"Essentially I was thinking about a machinery that can generate a series expansion of a perturbed orbit from a numerical solution over a reasonably long period centered at the present - essentially what VSOP does for the major planets. In the case of Hyperion, this would be referenced to the true position of Saturn, or barycenter position of Saturn and its moons (and ideally in the same frame of reference as VSOP, to avoid the frame transformations you mentioned earlier).
As far as I know, Hyperion's may be a particularly difficult orbit to model, because of resonances with Titan and high eccentricity, so a series expansion diverges either very quickly or requires a large number of terms. But since you have already created a precise numerical gravity simulator, it might be interesting to use it to see just how well it can be approximated with a series solution."
This exchange got me thinking about how one would go about building a VSOP-like ephemeris of, say, an 'irregular' body such as Hyperion. After some reflection, and a little bit of a literature search, I decided that the mathematical machinery to do this exists and so I thought I would 'give it a go'. So, this post is a 'place-holder' for a number of (yet to be produced) results arising from this exercise. Results will be posted here as I work through this exercise.
For those that might not be familiar with the vocabulary, an ephemeris is essentially a mathematical 'look-up' table that encodes the position of the Sun, the planets and moons. It serves two purposes, the first of which is entirely practical: although a lot of effort may go into putting the ephemeris together in the first place, ephemeris users need only employ a simple algorithm to extract the encoded information. An ephemeris, VSOP87, lies at the heart of Orbiter. It is used by the numerical integrator to work out the location of the principle gravitation bodies in the Solar System. The second is more theoretical. Because of the theoretical underpinnings of the ephemeris construction, the ephemeris solution may yield useful information about the nature of perturbations - e.g., the rate of perihelion advance of an orbit - that would otherwise not be self-evident. For Orbiter, this is less relevant - but in many respects the more interesting aspect.
There are two kinds of ephemerides in common usage. The first is the VSOP series, principally VSOP87 and ELP series, produced by the Bureau des Longitudes. These encode the positions of planets (and the Moon) as a series of cosine terms which, when summed, yield the planetary positions with high accuracy. The full VSOP87 solution contains many thousands of such terms for the major planets and although a little daunting when first encountered are actually quite simple (and fast) to use in practice.
The other kind of ephemeris is the 'DE' series produced by Nasa's Jet Propulsion laboratory - e.g., DE405/406. These ephemeris series are more recent and are used for mission planning purposes. However, the encoded solution of planetary positions consists of an unwieldy set of Chebyshev interpolation schemes. Data files for the full ephemeris solution are extensive to such an extent that they make the extensive VSOP cosine tables look positively succinct. Although, it is older, the VSOP87 is still widely used since for, most purposes, it is sufficiently accurate - and far more compact.
In this thread, then, I'm going to focus on the construction of a VSOP-like ephemeris solution - that is, a cosine table for the saturn-centric position of Hyperion over, say, one or two centuries from the present.
And why is this such a hard task? Surely, fitting a few cosine terms to an integrated solution of Hyperion can't be that hard. Basically, the issue is largely one of accuracy: to produce an ephemeris capable of accurately determining the position of Hyperion one hundred years from now requires a great deal of precision in determining the values of the ephemeris cosine terms: being off by 1 part in 10^10 in the frequency of a cosine term can lead to inaccuracies of thousands of kilometres. And then there is the sheer number of terms that may need to be included. Because the forces perturbing Hyperion's motion are considerable, and because it is a high eccentricity orbit, and because it has a resonance with Titan, the number of cosine is terms needed to accurately determine Hyperion's position may be very large. Or, indeed, Hyperion's orbit may be chaotic and predicting its position one hundred years from now is impossible. At any roads, having thought about it, the construction of a realistic and accurate ephemeris for Hyperion is not a trivial thing to do. (And if you don't believe me, give it a go yourself.)
In fact, the mathematical and computational challenges that this task imposes are such that it doesn't make much sense to attack the problem with a direct frontal assault. Better, methinks, to learn how to walk first by practicing on simpler, less demanding problems.
So, this is my general plan of attack: to work up to the task of an accurate ephemeris for Hyperion, by working progressively through a series of progressively more challenging problems. The specific sequence that I propose to work through is as follows:
1. Determine a VSOP-like ephemeris for a high-eccentricity two-body elliptical orbit around a single stationary gravitational source, e.g., the Sun. Although there is no need for an ephemeris solution here, it contains many of the same challenges of the Hyperion problem without making the problem overly complicated.
2. Determine a VSOP-like ephemeris for the Sun/Earth/Moon system. This is now a three-body problem and contains a number of significant perturbations that lead to phenomena such as 'orbital precession'. Although quite simplistic by modern standards, until well after the Lunar landings, ephemeris based on this three body problem were central to lunar mission planning.
3. Determine a VSOP-like ephemeris for the Sun/Saturn/Titan/Hyperion system. This is now a four body-problem. It captures the essence of resonance between Titan and Hyperion as well as Hyperion's high eccentricity orbit. It ignores significant perturbations due to the gravitational influence of the other planets - principally, Jupiter.
4. Determine a VSOP-like ephemeris for Sun/Jupiter/Saturn/Titan+other moons/Hyperion. This will be a semi-realistic ephemeris for Hyperion. It will include perturbations from Jupiter as well as those from the rest of Saturn's moons. It will not, however, be consistent with the VSOP87 ephemeris solution used by Orbiter
5. Determine a VSOP87-like ephemeris for Hyperion based on the full VSOP87 solution. The end result.
I've already completed Step 1 - and I'll report on this next. Step 2 is underway.
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