
#19
Jan2114, 08:09 PM

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#20
Jan2614, 10:08 AM

P: 39

In absolutely every case (even with Ceres in the EarthMoon Lagrange Points 3, 4, and 5), the extra mass was sufficient to slightly slow down the Earth's velocity around the Sun (from 29.8 km/s to 29.3 km/s), causing Earth's orbit to increase slightly (1.02 AU SMA) and increasing the length of a year (adding about 8.4 hours per year). The amount of energy required to move an object the size of Ceres would be huge. Well beyond all the energy of our combined nuclear weapons. However, once in either the EarthMoon Lagrange Points 3, 4, or 5, no additional energy would be required to keep it in a stable orbit. 



#21
Jan2614, 11:00 AM

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Of course an object at the first and second Lagrange points is going to be booted out of the system. Those points aren't stable. So is the third. Only the two triangular Lagrange points are stable.
That you couldn't find any stable orbits suggests that you might just be seeing an artifact of your numerical integrator rather than something real. If you are using the classical fourth order RungeKutta integrator, you can almost certainly blame the booting on the integrator rather than reality. The same applies for low order symplectic integrators. The large truncation errors associated with these techniques means the results are pure fiction after a few dozen orbits. A good integrator for the four body problem (the Sun is going to be a big perturber) is hard to come by. There has been a lot of interest of late in selenocentric distant retrograde orbits (google that term). These orbits appear to be stable for hundreds of revolutions given the right evection angle (between 90 and +90 degrees, more or less). Some are very distant indeed. At a distance of 70,000 km from the Moon, these orbits are anything but elliptical when viewed from the perspective of a Mooncentered frame, and they are rather exotic when viewed from the perspective of an Earthcentered frame. 



#22
Jan2614, 12:44 PM

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In the same way, you can know that the Kuiper belt has many objects we did not discover so far, as our telescopes are not good enough and/or did not scan the whole sky with the same precision. 



#23
Jan2614, 05:40 PM

P: 28

It would screw up our lives. That's all I can say.




#24
Jan2614, 09:31 PM

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#25
Jan2614, 10:15 PM

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You can't use RK4 and expect anything remotely resembling reality after even one orbit. It is not a stable integrator.




#26
Jan2714, 06:23 PM

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The RK4 integrator I use in Gravity Simulator can hold a Ceresmass object in the Moon's L4 or L5 indefinitely. If I input solar system data for today, and run it backwards, I can accurately postdict solar eclipses from hundreds of years ago. My predictions for future eclipses are in excellent agreement with the predictions given in Nasa's eclipse website. This is after not just one orbit, but thousands of lunar orbits, hundreds of Earth orbits, and hundreds of orbits of the other planets that perturb the system.
My original EulerCromer integrator can do these things too provided the time step is low enough. Beyond hundreds of years, the postdictions start to fail, not because of integrator error, but because of forces not modeled (i.e. Earth is not a perfect sphere like I pretend it to be). Likewise, I can integrate forward and find the circumstances of Venus' next transit of the Sun, and be in excellent agreement with published values. But I can't predict the next time Io will occult Europa, as Jupiter's oblateness is significant but not modeled in the code. 



#27
Jan2714, 10:01 PM

P: 39

I also agree that the margin for error is directly proportional to the step increment. 



#28
Jan2714, 10:29 PM

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#29
Jan2714, 11:28 PM

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#30
Jan2714, 11:44 PM

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#31
Jan2814, 12:42 AM

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But with all due respect, you helped me get my degree in Physics and Astronomy. Google "tony87004 thanks DH". I used to be an electrical contractor. Now I'm a high school physics teacher :) 



#32
Jan2814, 05:58 AM

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#33
Jan2814, 06:47 AM

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If you are using native floating point (specifically, double precision; if you are using floats you are completely lost) and if want any hope of capturing the dynamics you need to set up a hierarchy of reference frames. If you want to do all of your computations in a common solar system barycenter frame, you *must* use some kind of extended precision arithmetic to avoid the huge truncation error problems in your integrators that would otherwise result with using double precision numbers. For example, it's best to represent an object orbiting the Earth using an "Earthcentered inertial" (aka ECI) frame. I put that in quotes because (a) it's not an inertial frame and (b) it is a very commonly used term. There's even a wikipedia page on Earthcentered inertial: http://en.wikipedia.org/wiki/Earthcentered_inertial. This ECI frame is not inertial. It is an accelerating frame. The astronomical / aerospace engineering term for the fictitious forces that results from using this accelerating frame is "third body force". (Physicists would call these apparent forces "tidal forces", but that term is used to denote something else in modeling solar system dynamics.) You need to model these apparent forces or you need to use extended precision arithmetic. If you use the latter, your integration will proceed extremely slowly. These very distant SDROs are of increasing interest to NASA, other space agencies, and also to the (not anywhere close to ready for prime time) asteroid mining community. So we've studied them. You need a very good integrator to model the behaviors. RK4 just doesn't cut it. Things get wonky after just an orbit or so when one uses RK4 against these exotic orbits. 


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