Orbital Simulator with general relativity

[darkdave3000]

Is it possible to calculate with precision the orbits of the Earth, Moon and Sun with general relativity instead of Newtonian Physics?

How would this work? Would the software have to record past positions of each body and have the Earth be attracted to the old sun's position 8 minutes ago? Would the Moon be attracted to a lagged position of the Earth a tiny fraction of a second old?

Keep in mind the Sun wobbles around the barycentre of the solar system and the Earth and Moon likewise also orbit around an Earth Moon barycentre.

David

 

[PeterDonis]

Is it possible to calculate with precision the orbits of the Earth, Moon and Sun with general relativity instead of Newtonian Physics?

Sure, it's just more tedious since the equations are more complex--but you also get more accurate predictions.

How would this work?

You adopt an appropriate specialization of the Einstein Field Equation for the system you are studying, and then use standard techniques for evolving systems described by partial differential equations. The following paper is fairly advanced but should give you a general idea:

https://arxiv.org/abs/1607.06298

 

[PeterDonis]

Would the software have to record past positions of each body and have the Earth be attracted to the old sun's position 8 minutes ago? Would the Moon be attracted to a lagged position of the Earth a tiny fraction of a second old?

In GR, generally speaking, gravity is not a force, it's spacetime curvature. The Sun and planets follow geodesics in a particular curved spacetime geometry.

For the weak field, slow motion approximation, which works well for the solar system, you can view gravity as a force, but it's not a purely Newtonian force; there are also velocity-dependent components to the force (somewhat like magnetism) that can be thought of as compensating for the fact that gravity "propagates" at a finite speed (the speed of light). This classic paper by Carlip discusses some of the issues involved:

https://arxiv.org/abs/gr-qc/9909087

 

[darkdave3000]

So having each body being attracted to a previous position of each other is not going to work as a modified Newtonian engine? I was hoping not to have to do all that field coding!

Also where would I get the exact position of Earth, moon and sun along with their exact velocities at that time? I need a snapshot of these 3 bodies as starting positions in my software. Any time after 2000AD should be fine.

And what frame of reference should I use for position? Solar System center of gravity?

David

 

[PeterDonis]

having each body being attracted to a previous position of each other is not going to work as a modified Newtonian engine?

No. Newtonian gravity with a time delay put in by hand will give you very wrong results.

where would I get the exact position of Earth, moon and sun?

And what frame of reference should I use for position? Solar System center of gravity?

These questions are way beyond the scope of a PF discussion. People spend years learning how to do these simulations accurately. The papers I linked to before will give you a start.

 

[darkdave3000]

I remember I bought a booklet that had all the positions of the planets in the solar system. I forget the name of it, I remember it started with "a" Almanac or something like that. This kind of book, is it available online to help me with initial positions of the planets in my software for any instant in time?

David

 

[darkdave3000]

http://asa.usno.navy.mil/static/files/2018/lunarpoly_2018.txt

I found this with what looks like 3D coordinates (X,Y,Z) for the moon, but what are they referencing? What is at 0,0,0?

 

[Ibix]

RA is right ascencion. Dec is declination. I'm not sure what HP would be. The description suggests that they're polynomial coefficients, presumably for a polynomial in terms of time after the listed time. It will give you position on the sky - the origin is the Earth's centre. No idea if it will give you distance. You'd be better to ask in the astronomy and astrophysics forum, I would think.

You are aware, I presume, that relativity is negligible to a ridiculous degree of precision in the solar system. Rounding in those polynomial coefficients and the whole concept of using polynomial approximation to the trajectories are almost certainly larger sources of error.

 

[Bandersnatch]

I'm not sure what HP would be.

That's likely horizontal parallax:
https://en.wikipedia.org/wiki/Parallax#Lunar_parallax

 

[PeterDonis]

You are aware, I presume, that relativity is negligible to a ridiculous degree of precision in the solar system.

No, it isn't. We have accurate enough measurements now to see the perihelion shift of all of the planets.

 

[Ibix]

No, it isn't. We have accurate enough measurements now to see the perihelion shift of all of the planets.

I'm aware of that - you mentioned it to Dale in jeremyfiennes' recent thread. How much precision will his simulation have to carry to detect it? And are the USNO data he linked good enough?

I'm all for doing this kind of thing for the fun of it, but it might be easier to start with a more extreme case where the effect he's looking for isn't tiny, is all.

 

[PeterDonis]

How much precision will his simulation have to carry to detect it?

Hm, fair question. The relativistic perihelion shift of Mercury is about 7 percent of the total shift; but the total shift is roughly one arc-second per orbit, and I would have to work out what accuracy of observation of Earth that translates to.

 

[Ibix]

Depends what he's simulating and why, of course. A naive estimate would be to note that the anomalous precession is 43 arcseconds per century, in which time Mercury completes 414 orbits. So precision would have to be at least 43 parts in 414×360×3600, or about one part in 13 million, just for Mercury.

Another naive estimate is that relativistic effects are likely to be of scale $r_{orbit}/R_{Schwarzschild}$ (give or take a factor of two), where the Schwarzschild radius for the Sun is about 3km. That comes out to about one part in 16 million for Mercury's orbit, one part in 50 million for Earth, and about one in 1.5 billion for Neptune.

 

[Klystron]

I remember I bought a booklet that had all the positions of the planets in the solar system. I forget the name of it, I remember it started with "a" Almanac or something like that. This kind of book, is it available online to help me with initial positions of the planets in my software for any instant in time?

David

The general term for a 'celestial almanac': ephemeris https://en.wikipedia.org/wiki/Ephemeris

 

[pervect]

You might want to look at Folkner, et al theory paper, "The Planetary and Lunar Ephemerides DE430 and DE431"

And also the JPL horizons ephermeides referenced by this paper, which is already implemented, does what you describe, and has a public interface so I believe you could personally access it. But learning to operate it appears to be quite technical - I know of the existence of the program but not how to operate it. Documentation is available on the JPL website, but appears quite technical.

Here is a list of what is considered taken from the above paper.

The translational equations of motion include contributions from: (a) the point mass inter-
actions among the Sun, Moon, planets, and asteroids; (b) the effects of the figure of the Sun
on the Moon and planets; (c) the effects of the figures of the Earth and Moon on each other
and on the Sun and planets from Mercury through Jupiter; (d) the effects upon the Moon’s
motion caused by tides raised upon the Earth by the Moon and Sun; and (e) the effects on
the Moon’s orbit of tides raised on the Moon by the Earth.

Thus we have are relativistic equations for point masses (based on linearized approximation of General relativity), a Newtonian analysis of what happens because of the "figure" (i.e. the non-spherical shape) of the various solar system bodies, and some tidal effects (the figure of the Earth's oceans doesn't stay constant, it changes due to the tides).

Goldstein's book, "Classical mechanics", has some discussion of the effect of figure on the Earth-moon system if you want more details of how the whole idea of "figure" is handled. But note it's a graduate level text (though it's introductory at the graduate level). Basically one expands the (Newtonian) gravitational potential in terms of series expansion via spherical harmonics.

Folkner points out that the moon needs a particularly complex model, due to lunar mass concentrations. So one needs a lot of the spherical and zonal harmonics, more so than for other bodies.

It's possible to do the analysis of the effects of "figure" in the context of full GR, but it's simpler to analyze it with Newtonian methods.

There are effects not covered in the JPL program that don't matter for the solar system models, but could matter for more exotic problems, effects such as the emission of gravitational waves due to the orbital motion of the bodies. Simulations that take into account more than linearized gravity are needed if one wants to include these sort of effects. Such simulations have been done for black hole mergers (binary inspirals), but I'm afraid I don't know the details, other than getting them working was a project for the entire scientific community.

JPL's program is rather old, so the way it is set up may not follow the most modern conventions. Probably with the right options, it can be configured to output the numbers according to modern conventions, but that's really just a guess.

An overview of the modern conventions would probably be the wiki article on the ICRS, the Inernational Celestial Reference System, see for instance <<wiki link>>.

The technology behind the ICRS is very long baseline interferometry, which gives extremely precise measurements of the position of extragalactic radio sources, typically quasars.

A general observation - getting into the full level of detail of how we handle the solar system simulations is interesting, but to fully appreciate it one probably needs to understand General relativity first. It's not necessarily a good project to take on to learn about how GR works. The usual approach would be to learn the basics of GR, learn about approximations to GR such as PPN, then learn about the refinements to PPN needed to allow conversion between geocentric (earth-centered) and barycentric (solar system centered) coordinates.

 

[darkdave3000]

What is PPN?

You might want to look at Folkner, et al theory paper, "The Planetary and Lunar Ephemerides DE430 and DE431"

And also the JPL horizons ephermeides referenced by this paper, which is already implemented, does what you describe, and has a public interface so I believe you could personally access it. But learning to operate it appears to be quite technical - I know of the existence of the program but not how to operate it. Documentation is available on the JPL website, but appears quite technical.

Here is a list of what is considered taken from the above paper.
Thus we have are relativistic equations for point masses (based on linearized approximation of General relativity), a Newtonian analysis of what happens because of the "figure" (i.e. the non-spherical shape) of the various solar system bodies, and some tidal effects (the figure of the Earth's oceans doesn't stay constant, it changes due to the tides).

Goldstein's book, "Classical mechanics", has some discussion of the effect of figure on the Earth-moon system if you want more details of how the whole idea of "figure" is handled. But note it's a graduate level text (though it's introductory at the graduate level). Basically one expands the (Newtonian) gravitational potential in terms of series expansion via spherical harmonics.

Folkner points out that the moon needs a particularly complex model, due to lunar mass concentrations. So one needs a lot of the spherical and zonal harmonics, more so than for other bodies.

It's possible to do the analysis of the effects of "figure" in the context of full GR, but it's simpler to analyze it with Newtonian methods.

There are effects not covered in the JPL program that don't matter for the solar system models, but could matter for more exotic problems, effects such as the emission of gravitational waves due to the orbital motion of the bodies. Simulations that take into account more than linearized gravity are needed if one wants to include these sort of effects. Such simulations have been done for black hole mergers (binary inspirals), but I'm afraid I don't know the details, other than getting them working was a project for the entire scientific community.

JPL's program is rather old, so the way it is set up may not follow the most modern conventions. Probably with the right options, it can be configured to output the numbers according to modern conventions, but that's really just a guess.

An overview of the modern conventions would probably be the wiki article on the ICRS, the Inernational Celestial Reference System, see for instance <<wiki link>>.

The technology behind the ICRS is very long baseline interferometry, which gives extremely precise measurements of the position of extragalactic radio sources, typically quasars.

A general observation - getting into the full level of detail of how we handle the solar system simulations is interesting, but to fully appreciate it one probably needs to understand General relativity first. It's not necessarily a good project to take on to learn about how GR works. The usual approach would be to learn the basics of GR, learn about approximations to GR such as PPN, then learn about the refinements to PPN needed to allow conversion between geocentric (earth-centered) and barycentric (solar system centered) coordinates.

 

[jbriggs444]

What is PPN?

[Note that you do not need to quote an entire post to ask a simple question]

The term was new to me as well, but @PeterDonis described it in these forums recently here. Given that hint, one can quickly google one's way to this.

Parameterized Post Newton -- a way to systematically characterize members of a class of gravity models using a particular parameterization.

 

[Yukterez]

Would the software have to record past positions of each body and have the Earth be attracted to the old sun's position 8 minutes ago? Would the Moon be attracted to a lagged position of the Earth a tiny fraction of a second old?

Not really:

The observed absence of gravitational aberration requires that "Newtonian'' gravity propagates at a speed ς>2×10¹⁰c. By evaluating the gravitational effect of an accelerating mass, I show that aberration in general relativity is almost exactly canceled by velocity-dependent interactions, permitting ς=c. This cancellation is dictated by conservation laws and the quadrupole nature of gravitational radiation.

 

[pervect]

What is PPN?

Parameterizied Post-Newtonian Formailsm. See <<link>>

Post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields.

...

The parameterized post-Newtonian formalism or PPN formalism, is a version of this formulation that explicitly details the parameters in which a general theory of gravity can differ from Newtonian gravity. It is used as a tool to compare Newtonian and Einsteinian gravity in the limit in which the gravitational field is weak and generated by objects moving slowly compared to the speed of light.

PPN is general enough that with different values for the parameters, it can be used for metric theories of gravity other than general relativity. In this application, though, we are really only interested in the version where the parameters are set to the values appropriate for General Relativity.