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We have studied the solar system for a long time and have detailed 3 dimensional knowledge of all the various bodies in the solar system and how they move in space in relation to one another and also other stars. My question relates to the accuracy of this knowledge. Do we know the accuracy to +/- 1km, 10km, 100km or what? Is the accuracy different for different bodies ? I might expect we have more accurate data for the Earth and Moon than we do for Pluto for example? How can we be sure our error bars are correct?

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  • $\begingroup$ If you think about it, and (for simplicity) ignore relativity, if you have the precise position and velocity of all the planets (and the sun, and all the moons, and the larger asteroids) at some instant in time, you should be able to predict their positions at any later instant. It's "simply" a matter of calculating the gravitational forces between all the bodies. Except that the solar wind from the sun is variable and affects Mercury and Venus in unpredictable ways. Easy-peasy. $\endgroup$ – Hot Licks Mar 26 '17 at 22:35
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    $\begingroup$ @HotLicks the n-body problem is actually not very simple, as it turns out. $\endgroup$ – PyRulez Mar 27 '17 at 0:49
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    $\begingroup$ @HotLicks You should consider solving some of those Millennium Prize Problems then, I heard they pay well for these. Navier-Stokes specifically is simply a matter of calculating solutions to the equation which is already written down. $\endgroup$ – Dmitry Grigoryev Mar 27 '17 at 11:28
  • $\begingroup$ I wrote an answer that has reference to a paper (by Franz and Harper) that explains the details for much of what you ask at: http://physics.stackexchange.com/a/246552/59023. $\endgroup$ – honeste_vivere Mar 29 '17 at 13:28
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This one is tricky unless you know the magic term: ephemeris. An ephemeris gives the position of celestial bodies over time. Once you know that one, finding out information about their uncertainties is easier.

The uncertainties are actually rather interesting in that they are planet specific. For example, the dominating factor for Mercury's uncertainty is that its hard to calculate its position in orbit to better than about 1/1000th of an arcsecond (an arcsecond is 1/3600th of a degree). We update our understanding of its path using optical sensors, but its hard to beat down that angular uncertainty. On the other hand, Mars is very easy to predict. We can apparently predict where it will be 1 year later within 300m. Why? Well, we've got a pile of instrumentation that has landed on the planet and is orbiting the planet, so it's much easier to take good measurements!

The article linked above offers a quick snapshot of the known uncertainties on the ephemeris of the planets. They vary wildly. Neptune, for instance, is hard to predict within 1000km 30 years from now!

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  • $\begingroup$ Interesting stuff. So the historical ephemeris are really lists or tables of where various bodies are expected to be in the (usually night) sky as observed from Earth. Over time (2000y !) they have become more accurate as our instruments on Earth have become better at measuring things like very small angles and distances. Now we have instruments off Earth we can do even better. Clearly one way of testing how good we are is to compare a prediction (e.g 1 year ahead) with actual observation. $\endgroup$ – BetterBuildings Mar 26 '17 at 20:51
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    $\begingroup$ Note well: The linked paper by Folkner is now rather outdated. The ephemerides of Mercury, Jupiter, and Pluto are now known with considerably more accuracy than they were at the time that that paper was written thanks to the MESSENGER, New Horizons, and Juno spacecraft. Every flyby of a planet by a spacecraft improves our knowledge of that planet's orbit to some extent, and every vehicle sent into orbit about a planet improves our knowledge of that planet's orbit to a much greater extent. $\endgroup$ – David Hammen Mar 26 '17 at 21:15
  • $\begingroup$ @DavidHammen Have NASA or anyone else published anything showing how the ephemerides have been improved by the various probes you mention ? Presumably these days the best ephemerides exist in some data base we can access ? $\endgroup$ – BetterBuildings Mar 26 '17 at 21:37
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    $\begingroup$ @BetterBuildings: Interesting thing is, NASA provides a simple telnet interface to query their up-to-date ephemeris data. Presumably if you were to build a spaceship with internet access you can update your onboard database with NASA's data (or if you were to launch a rocket you can update just prior to launch) $\endgroup$ – slebetman Mar 27 '17 at 2:33
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    $\begingroup$ @slebetman Roaming charges may apply... $\endgroup$ – Cort Ammon Mar 27 '17 at 2:40
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NASA's Navigation and Ancillary Information Facility (NAIF) at JPL is responsible for knowing the exact positions of all the planets and their astroids, many asteroids, and every space mission bigger than a toy pop rocket. JPL NAIF site

NAIF provides data and software tools as SPICE. The data comes as "kernels" of various types covering natural bodies, spacecraft, instruments on spacecraft, leap seconds and so on. The SPK kernels describe the planets.

The data is all text files, so it can be read using Python, C, Matlab, whatever, without the trouble of fiddling with binary.

In one of the technical notes for the latest SPK, named DE431, published in 2013, comparing it to the previous one, it says: "The difference in the positions of the planets agree to better than 0.001 km over the time period covered by DE430, a difference which is not statistically distinguished by the currently available data."

By "available data" they mean: all the observations by telescope, from spacecraft far from Earth such as Cassini, Juno and New Horizons, Hipparcos, measurements from radio astronomy facilities such as EVLA, VLBA, and ALMA, occultations of stars by planets, and whatever other reliable sources I'm too lazy to look up.

If a one meter difference in data files can't be distinguished by observation, that's no surprise. But the fact that scientists as mission planners care about that level of accuracy, says something about the kind of accuracy we are able to deal with.

Apart from NASA, but thanks to those cubic mirrors left on the Moon by Apollo astronauts, astronomers can measure the distance between certain established reference points on Earth and the Moon to a few centimeters. Could be down to millimeters these days. Differences between these measurements and predictions of various models have helped us reach conclusions such as: the Moon is drifting away from Earth at 3.8 cm/year; the Moon has a liquid core; and once again, Einstein's General Relativity works out fine.

On the other hand, we haven't pinned down the outer planets so well. Pluto's exact position could be off by many kilometers, even after the New Horizons flyby. If you would enjoy reading a detailed analysis of the errors, read this note by W. M. Folkner (PDF)

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This question can't really be answered without a timeframe. As @PyRulez implied in the comments, the n-body problem is very complicated. In particular, when n > 2, the system is chaotic, meaning that your error margins will grow exponentially with time. This answer goes into some detail about how far ahead (in theory) orbits may be predicted. The other answer of course is far more practical.

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    $\begingroup$ While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review $\endgroup$ – Yashas Mar 27 '17 at 5:22
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    $\begingroup$ @YashasSamaga The linked page is another thread on Physics Stack Exchange, so it is likely that the link will stay "good" forever. Of course I agree the content in the other thread may change, and link-only information should be a comment only, not an answer. $\endgroup$ – Jeppe Stig Nielsen Mar 27 '17 at 7:50
  • $\begingroup$ In the case of any particular practical mission whether manned or unmanned the timeframe is going to be the expected duration of the trip (T) plus a bit extra to be on the safe side. However it might be prudent to have a look at say (2 x T) just in case things are on the edge of going "interesting" e.g due to 4 large asteroids happening to come into alignment and give a gravitational peak in one dimension or a 1 in 20,000 year wobble in a planet's orbit. $\endgroup$ – BetterBuildings Mar 28 '17 at 8:20
  • $\begingroup$ (continued) So lets look at a mission to Mars. How about a 4 year time frame. What is the known accuracy of the relative positions of Earth v Mars on a 4 year time frame ? This will be one factor to take into account in calculating reserve fuel for possible delta-v corrections. And required fuel mass is of course very high in importance when returning from a planet. Every 1 kg of extra reserve fuel may need several kg of fuel just for launch ! $\endgroup$ – BetterBuildings Mar 28 '17 at 8:26
  • $\begingroup$ @YashasSamaga the question, as stated, does not necessarily have a meaningful answer. On long timeframes, our accuracy is zero. If the poster is interested in a full analysis of a mission profile, that would presumably be a problem for NASA's GMAT software. $\endgroup$ – canhascodez Mar 28 '17 at 11:16

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