How accurate are simple models of the solar system?

[Jack21222]

How accurate are "simple" models of the solar system?

Exactly how much effect does the perturbation of other planets have on planetary orbits? That is, if we were to take a naive model of the solar system, just applying Kepler's laws to each planet individually, with no planet-planet interactions, how accurate would it be in 10 years? 100? 1000?

And since the laws are time-reversible, could we then run these models backwards, starting with the current positions of the planets, using just Kepler's laws, to get a position of all of the planets dating back to Kepler? Would the error in the orbits due to the system being an n-body problem be big enough going back 400 years such that the calculated position from this model would diverge from the positions Tycho Brahe measured?

In other words, if in 1620, Kepler had calculated where the planets would be in the sky in 2020 using only his laws, would there be a measurable difference?

I hope my question is making sense. I really want to know what time scale is required to really notice the difference between the 2-body solution to planetary orbits as compared to real life.

 

[Nabeshin]

To give you an idea of some of the corrections that need to be applied, check out this table about the precession of Mercury's perihelion:
http://en.wikipedia.org/wiki/Tests_of_general_relativity#Perihelion_precession_of_Mercury

 

[HallsofIvy]

That varys depending on the planet. The outer planets come closest to matching Kepler's laws. As you get closer to the sun, the "non-linearities" in general relativity become more important with Mercury varying most from Kepler's laws.

 

[Jack21222]

Even apart from general relativity, I'm also interested in the effects of perturbations from other planets. How much does Jupiter affect the position of mars, for example.

 

[D H]

That varys depending on the planet. The outer planets come closest to matching Kepler's laws. As you get closer to the sun, the "non-linearities" in general relativity become more important with Mercury varying most from Kepler's laws.

General relativity explains but a small part of the precession of Mercury, 43 arc seconds per century. Look at the wikipedia article referenced in Nabeshin's post.

Choose a substandard reference frame and you'll get a precession rate that is more than two orders of magnitude larger than that relativistic precession. That said, very few people use that frame nowadays. If you want to model the solar system you had dang well be using ICRF as the reference frame. Even with a reasonable frame such as ICRF, Mercury's orbit is still going to precess, and general relativity still plays a very minor role in that precession. The precession of Mercury due to Jupiter (and to a lesser extent, the other planets) is more than an order of magnitude larger than the precession due to general relativity.

It is this non-relativistic precession and related non-relativistic effects that the OP is asking about. Kepler's laws: Crawling. Newtonian mechanics: Walking. General relativity: Running. You need to learn to walk before you can learn to run.

Jupiter is the 600 pound gorilla of our solar system. To get an accurate model of the solar system, even a non-relativistic one, you need to account for Jupiter.

• For the inner planets, modeling Jupiter as a circular ring of mass (as opposed to a point mass) gives a fairly nice first order improvement to Kepler's laws. The net result is that the orbits of the inner planets precess.
• For the outer planets, modeling the Sun+Jupiter as a single point mass gives a fairly nice zeroth order improvement to Kepler's laws. The net result is that the orbital rate is a bit higher for the outer planets than Kepler's laws suggest.
• Jupiter itself needs to be treated specially, but Newtonian mechanics has a simple solution for two masses, neither of them negligible, orbiting one another.

If you want something better than these simple corrections you will need to resort to numerical integration. And if you want very high accuracy, you need to use something beyond Newtonian gravity. A linearized post-Newtonian formalism gives very high accuracy, and it still allows you to look at the solar system as being basically Newtonian plus some additional perturbative forces tossed into the mix.

 

[Jack21222]

Thanks, D H.

Do you happen to have any order of magnitude estimate of how long the divergence between the "basic" model and the improved models becomes noticeable? Or between the improved models and reality?

The reason I ask is I wanted to use the solar system to test a claim of "beyond the standard model" physics, possibly as an undergraduate capstone project. My thought was to backtest a current solar system model with standard physics to make sure it agrees with observations from 100 or more years ago, maybe even as far back as Brahe. Then, once the model can be backtested and agrees with observations that far back, use that to constrain the size of this "beyond the standard model" effect proposed.

I'm alternatively thinking "If there was even a tiny variation in the standard Newtonian clockwork model, we'd have noticed it," and "this n-body problem is too complicated to be able to model accurately even 50 years back or forward in time."

Thanks for the responses. I'm still waiting on my research advisor to see if my idea has any merit, or if it's a complete waste of time. It's fun to think about, anyway.

 

[Nabeshin]

Do you happen to have any order of magnitude estimate of how long the divergence between the "basic" model and the improved models becomes noticeable? Or between the improved models and reality?

Noticeable will always be dependent on the precision of your measuring apparatus! So one cannot give a general answer to this question.

The reason I ask is I wanted to use the solar system to test a claim of "beyond the standard model" physics, possibly as an undergraduate capstone project. My thought was to backtest a current solar system model with standard physics to make sure it agrees with observations from 100 or more years ago, maybe even as far back as Brahe. Then, once the model can be backtested and agrees with observations that far back, use that to constrain the size of this "beyond the standard model" effect proposed.

I don't know if this is such a good idea. Even if you backwards integrate the planets with pristine accuracy and find a discrepancy to observations a few centuries old, what have you shown? Not much, except that there is a discrepancy which could be explained in dozens of different ways (I imagine investigating all of these would be quite a big project!). Furthermore, even under the umbrella of beyond the SM, you really need to pick a theory or class of theories to investigate specifically. For instance, you could ask the question "If MOND is true, how does that affect the positions of the planets over the course of hundreds of years?". Now, given that simple MOND models have already been ruled out (although it might be a fun exercise to use them anyways), the more complicated ones would provide a significantly more difficult calculation.

I'm alternatively thinking "If there was even a tiny variation in the standard Newtonian clockwork model, we'd have noticed it," and "this n-body problem is too complicated to be able to model accurately even 50 years back or forward in time."

Well, just by the nature of the beast, numerical errors in the position and velocity of the planets will compound over time and eventually become important. So you're fighting a losing battle here -- the longer you integrate, the more likely you are to notice the effects of any kind of physical variation in parameters, but the more likely it is that numerical errors will have swamped your results. There's probably a middle ground where you might notice a change in physics before the errors become too significant, but without solving the equations beforehand, I don't see a clear way to find out where this might be.

Thanks for the responses. I'm still waiting on my research advisor to see if my idea has any merit, or if it's a complete waste of time. It's fun to think about, anyway.[/QUOTE]

 

[Jack21222]

I don't know if this is such a good idea. Even if you backwards integrate the planets with pristine accuracy and find a discrepancy to observations a few centuries old, what have you shown? Not much, except that there is a discrepancy which could be explained in dozens of different ways

Well, my hope was to not see any discrepancy, which would allow me to constrain this parameter. Let's say in a perfect world, a model using normal Newtonian physics run backwards matched Brahe's observations. I could then run the model again with the modified parameter at different values, and see at what value the discrepancy becomes so large that Brahe would have recorded a different value for the position of a planet on a certain day.

(I imagine investigating all of these would be quite a big project!). Furthermore, even under the umbrella of beyond the SM, you really need to pick a theory or class of theories to investigate specifically. For instance, you could ask the question "If MOND is true, how does that affect the positions of the planets over the course of hundreds of years?". Now, given that simple MOND models have already been ruled out (although it might be a fun exercise to use them anyways), the more complicated ones would provide a significantly more difficult calculation.

I already had one particular item in mind. Several "beyond the standard model" models predict a small violation of the equivalence principle. My research advisor has already looked at 3 solar system tests of the equivalence principle and had it published in Phys Review D back in 2000, but then didn't take it any further. Here's the paper: http://arxiv.org/abs/gr-qc/0007047

He dug it back out and wanted me to see if I could come up with any other solar system tests of the equivalence principle. This is just something I've brainstormed.

I know it isn't really a hot topic of research, and I realize that EP violation, if there is any, is very well constrained already, but I think it's a decent enough undergrad research topic. In his paper, he only looked at Jupiter when considering Kepler's laws. I was curious how a full simulation of the solar system would look with different values of EP violation. But, if even models with no equivalence principle violation noticeably diverge based on small perturbations that may be skipped over in more basic models, then I'm not sure what good the divergence in the EP violation models would do.

Well, just by the nature of the beast, numerical errors in the position and velocity of the planets will compound over time and eventually become important. So you're fighting a losing battle here -- the longer you integrate, the more likely you are to notice the effects of any kind of physical variation in parameters, but the more likely it is that numerical errors will have swamped your results. There's probably a middle ground where you might notice a change in physics before the errors become too significant, but without solving the equations beforehand, I don't see a clear way to find out where this might be.

Yeah, that's the wall I was running up against in my mind. Trick is, even if I'm looking at a non-reduced 3 body problem, the equations cannot be solved beforehand.

Thanks for the replies, even if nothing comes of this, this has been a good thread to organize my thoughts in.

 

[tony873004]

I just ran a simulation in Gravity Simulator to test this. I started with our solar system present day, with data from JPL Horizons. I then ran the solar system back to 1631. I then tried it again, but this time I made all the planets except for Earth massless. I needed Earth to keep its mass or the Moon went flying off. On December 9, 1631 00:00:00, I compared the locations of Mars between the two simulations. I transferred the massless Mars into the other simulation so I could plot them together. The Mars from the massless sim is called marsII.

http://orbitsimulator.com/BA/marsII.GIF

As you can see, they're separated by 15% of an AU. In angular distance as viewed from Earth, that's 7.4 degrees, a little more than the width of your fist held at arm's length. I didn't monitor the simulations during the 400 years, so massless Mars may have made a few laps around the Sun relative to the real Mars, but I doubt it.

Although you mentioned 1620, I choose 1631 because there was a transit of Venus that year, so I could see how accurate my simulation with mass is. It was run with a time step of 2048 seconds using RK4.
My simulation had Venus passing 937 arcseconds from the center of the Sun on Dec 7, 1631 at 05:30
Nasa's eclipse page says it passed 940 arcseconds from the center of the Sun on Dec 7, 1631 at 05:19
The difference isn't huge. The numbers from my sim assume the observer is at the center of the Earth. My sim also doesn't take into account light-travel time, or relativity.

 

[Nabeshin]

I already had one particular item in mind. Several "beyond the standard model" models predict a small violation of the equivalence principle. My research advisor has already looked at 3 solar system tests of the equivalence principle and had it published in Phys Review D back in 2000, but then didn't take it any further. Here's the paper: http://arxiv.org/abs/gr-qc/0007047

He dug it back out and wanted me to see if I could come up with any other solar system tests of the equivalence principle. This is just something I've brainstormed.

I know it isn't really a hot topic of research, and I realize that EP violation, if there is any, is very well constrained already, but I think it's a decent enough undergrad research topic. In his paper, he only looked at Jupiter when considering Kepler's laws. I was curious how a full simulation of the solar system would look with different values of EP violation. But, if even models with no equivalence principle violation noticeably diverge based on small perturbations that may be skipped over in more basic models, then I'm not sure what good the divergence in the EP violation models would do.

Ah, cool! That's actually somewhat along the same vein as the project I'm working on right now (particularly, I've become quite familiar with Cliff Will's book on this stuff). I agree, it's actually a nice little project now that you've given it some context! The unfortunate part is that the farther back in time you go the less and less reliable any astronomical data will be.

It seems from the paper you link that your advisor has already done most of the theoretical heavy lifting for you -- all that remains is to numerically solve the equations, no? A naive way to test for how much numerical errors are affecting your simulation might be to do something like this: Start with some initial data and integrate backwards in time some set amount of time. Then, slightly perturb the initial data, and see if your solution remains in a small neighborhood around the old solution. If it doesn't, you're experiencing chaos. Particularly, if you can't integrate back however far you want using the quoted uncertainties in position/velocity for the planets, you're simply going to have to integrate less.

(Someone who knows more about these kind of phenomena can likely give you a much more robust test of what's going on in your equations -- that's why I call this method naive )

At any rate, you can probably do a fairly quick numerical study to see how changing various parameters (in the Newtonian case!) affect the final result. That is, if you just include the earth, sun, and Jupiter, how do things change when you add Saturn, or any of the other planets. How many do you need to include to get results which are accurate to within the precision you're looking for?

After that, it's just a matter of performing a similar analysis with the modified equations?

 

[lpetrich]

One can estimate how far off by estimating the amount of perturbation each planet receives from the others.

Consider a planet orbiting at distance r from the Sun, which has mass M. It experiences gravitational acceleration G*M/r²

Inside planet

An inside planet with mass m orbiting at distance a will make gravitational acceleration G*m/(r+a)² ~ G*m/r² + G*m*a/r³ + G*m*a²/r⁴ + ...

I'm omitting numerical factors like factors of 2 and 3.

The Sun gets pulled by that planet: G*M/(r - a*(m/M))² ~ G*M/r² - G*m*a/r³ + G*m²/M*a²/r⁴ + ...

The first terms combine to make Sun + planet: G*(M+m)/r²
The second terms cancel
The third terms are about G*m*a²/r⁴

The perturbation: (m/M)*(a/r)²

Outside

An inside planet with mass m orbiting at distance a will make gravitational acceleration G*m/(a+r)² ~ G*m/a² + G*m*r/a³ + ...

It also pulls on the Sun with acceleration G*m/a², which cancels out the first term
The second term is about G*m*r/a³

The perturbation: (m/M)*(r/a)³

Summary:

a < r: (m/M)*(a/r)²
a > r: (m/M)*(r/a)³

It should be easy to estimate these quantities for the Solar System's planets.

 

[Nik_2213]

Uh, if you back-calculate two millennia to compare predicted eclipses with local observations, and take into account orbital precessions, slowing of Earth's rotation due tidal drag, changes in calendar etc etc, IIRC, they come out correct.

The flip-side, IIRC, is that a lot of effort goes into observing occultations, where a planet, moon or asteroid blots out a star for a while, the precise timing observed from different ground stations helping to refine orbits.

Then you have the high-tech tactics of bouncing laser pulses off the Moon's American & Russian retro-reflectors, plus radar pulses off passing asteroids. Studying the doppler shift from probes on or orbiting other planets provides lots of data, too...

 

[Chronos]

The mass of the planets is negligible campared to the sun. They are also very difficult to model [a three plus body calculation]. For all intents and purposes, you can ignore planetary perturbations and achieve spectactualarly accurate results..

 

[D H]

The mass of the planets is negligible campared to the sun. They are also very difficult to model [a three plus body calculation]. For all intents and purposes, you can ignore planetary perturbations and achieve spectactualarly accurate results..

Over what time scale, and what scale of distance? Jupiter is about 1/1000 of the mass of the Sun, so ignoring Jupiter's mass means the period is only good to 4 places for the outer planets. That error is going to lead to spectacularly inaccurate results over any extended period of time. The other planets cause Mercury's orbit to precess by ~530 arc seconds/century (c.f. general relativity, which results in a precession of ~43 arc seconds/century). Finally, if you want any kind of precision in modeling the Moon's orbit about the Earth you will need to do something very special.