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Imagine a satellite of known mass, orbiting the sun outside the Earth-Moon system. Assuming that we made a serious attempt to measure both it's position and attempted to predict it's orbital path, which would be the most accurate?

Or rather, since "accuracy" is tricky, which would contribute most uncertainty?

On one hand you have the hard problem of measuring exact position and velocity at light minute distances, Vs the n-body problem for the entire solar system, outside the immediate area around Earth, where it swamps any other influences.

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  • $\begingroup$ By "satellite", did you mean a natural satellite (i.e., a moon of another planet) or did you mean an uncrewed human-made spacecraft? $\endgroup$ Feb 10 at 17:20
  • $\begingroup$ Human made satellite (or small rock). $\endgroup$ Feb 28 at 14:46

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Can we more accurately predict orbits or measure position?

As a general rule, orbits, by far.

The state of a planet, asteroid, comet, natural satellite, or human-made spacecraft has six degrees of freedom: Three for position, and three more for velocity. Orbital elements similarly have six degrees of freedom. Orbit determination mandates having a good fix on all six degrees of freedom. As determining position on the other hand requires only having a good fix on the three positional degrees of freedom, it seems to be contrary to say that orbits are more precisely measured than position.

Determining position requires accurate measures of distance (range) and bearing (azimuth and elevation). This typically is not possible as a one time measurement. Optical telescopes only determine bearing; range is difficult if not impossible with an optical telescope. The Keck Observatory claims an accuracy of 1 arcsecond, which at 1 astronomical unit (AU) distance corresponds to an uncertainty of 725 kilometers. The transmitter / receivers used by NASA's Deep Space Network have a half-beamwidth of about 0.017 degrees, which at 1 AU corresponds to an uncertainty of 44000 kilometers.

On one hand you have the hard problem of measuring exact position and velocity at light minute distances.

That it takes time for a signal to be transmitted from an antenna to an object in space and then be reflected back to Earth is a feature rather than a flaw. Time and frequency are the two things that humans measure most accurately. Modern measures used for orbit determination, particularly for human-made satellites cruising between planets, forego using bearing (azimuth and elevation) because those measures are lousy compared to the range and range rate measurements. Range (radial distance) measurements can be accurate to the sub-meter level by measuring the time delay. Range rate can similarly be measured to high precision via at the doppler shift in the transmitted versus received signals. Human-made spacecraft that go beyond the Earth-Moon system are outfitted with equipment that augment range and range rate measurements. However, even natural objects benefit greatly from range and range rate measurements made via radar astronomy.

Precise orbit determination relies on having captured dozens, if not hundreds or even several thousands of these imprecise and incomplete measures. Statistical techniques coupled with physical / mathematical models of orbits reduce the uncertainties and fill in the incompleteness. Orbit determination is much more precise than any one measurement, especially when used for interpolation.

Extrapolation (e.g., future predictions) remains a challenge. One of the biggest challenges for Near Earth Objects is the non-gravitational forces such as solar radiation pressure and the Yarkovsky effect that act on such objects. Is a NEO a potential hazard or not, and if a NEO is a potential hazard, when might it hit the Earth? Those non-gravitational forces make predictions a bit murky. Assessing these non-gravitational forces is one of the key science objectives of the OSIRIS-REx mission, which has already collect a sample from Bennu and is intended to rendezvous with Apophis in 2029.

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I am assuming here that "position" should be interpreted as angular position in the sky (R.A., Dec.) and not radial distance which is always hideously inaccurate using Earth-based means. Precision orbit determination often starts with precise data collection of the bearing and bearing rates. That is, the angular position and its rate of change with respect to some known point in the sky, usually a far more distant object regarded as "fixed" (e.g., quasars or pulsars). Observations have to be collected over sufficient spans of time to permit acceptable accuracy. Angular positions can be measured precisely and compared with precision clocks to derive precise angular rates.

A classic method is Gauss' method that only requires three position vectors (this was used with radar measurements in the early days of artificial satellites). Time of position fixes helps refine the computations for orbit determination but are not essential to the basic computations. Complicating this is Earth's movement, another reason for atomic time keeping and keeping track of advantageous angular positions with respect to the observer's instruments and location.

The short answer is that no one measures radial distance to a distant object orbiting the Sun. Instead precision angular and time measurements are the default approach used today. Doppler is sometimes used with space probes for deducing radial position and reducing the error of radial distance estimates but it is not a direct method.

BTW, unless the object is truly massive, mass is often ignored. This is always the case for distant artificial objects. And recall the Sun is far more massive than all the planets and asteroids put together. This means the "reduced mass" is by far dominated by the primary mass of the Sun.

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  • $\begingroup$ Position can be measured in any way you care for, to maximise accuracy. If it helps, assume it has retro reflectors and and can send back data from it's own star tracker $\endgroup$ May 4, 2021 at 5:22
  • $\begingroup$ @user2702772 As far as I know, a star tracker can only give you orientation information (not position) $\endgroup$ May 4, 2021 at 11:23
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    $\begingroup$ Yes, we do measure distance to the Moon quite accurately. (My gen rel professor at Univ. Md. was part of team who did this in 1970s using Apollo corner reflectors.) But they don't do it directly. Instead it's done by precisely measuring time. Likewise radar pulses, which are much mushier as pulses go. Celestial position determinations are all done using angles and clocks. (No long strings allowed! :) ) $\endgroup$
    – Chris Ison
    May 4, 2021 at 19:34
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    $\begingroup$ "no one measures radial distance to a distant object orbiting the Sun." This needs to be either reduced or removed. "Doppler is sometimes used with space probes for deducing radial position" is misleading. The technique is called "delay-Doppler" and is used for both spacecraft in deep space and for orbiters and landers on other solar-system bodies. Just for example How will InSight's RISE antennas end up pointed in the right direction? $\endgroup$
    – uhoh
    Feb 9 at 4:52
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    $\begingroup$ While I'm not going to downvote, I was strongly tempted to do so due to the highly inaccurate first sentence. Range (radial distance) and range rate are by far the most precise measures of satellites orbiting the Sun outside the Earth-Moon system. Bearing and, even worse, bearing rate measurements are so lousy in comparison to range and range rate that bearing and bearing rate typically are not used in modern orbit determination of human-made spacecraft. Presumably the OP meant human-made spacecraft when referring to satellites as natural satellites orbit a planet rather than the Sun. $\endgroup$ Feb 10 at 17:29
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I think the answer is easy, but it's conditional.

If:

  1. there are no unexpected or undetected events like a malfunction causing a change in orientation (e.g. substantial change in solar photon pressure or heat radiation direction) or outgassing or venting of gasses
  2. there have been regular measurements tracking the spacecraft
  3. it hasn't been a very long time since the last measurements were made (i.e. if you don't have to extrapolate for a long time into the future)

then the calculation will always be more accurate simply because the errors are reduced by fitting to so many measurements.

As far as we know, there's nothing mysterious or hocus-pocus about Newtonian gravity or necessary corrections due to General Relativity, and there has been such a tremendous effort made to build complete models based on the masses and motions of all the major players (the Sun, planets, their moons, many asteroids) that the major uncertainties for a spacecraft are basically coming from the fitting of a calculated trajectory to all of the previous measurements.

See for example:

especially the plots of the residuals.

However, if you've lost contact with the spacecraft for a very long time and then ask if a new measurement will be more accurate than extrapolation from old data, in that case the new measurement could certainly be considered more accurate.

That's where the n-body "gotcha" comes into play. Since trajectories are not conic sections but instead need to be calculated using the gravitational pull of n bodies, small differences can lead to bigger divergence over time. Over millions of years we call these divergences chaotic. But for only months or years, for heliocentric orbits, it's just a normal (but complicated) propagation of errors problem for extrapolation.

And if there's been an unexpected or undetected or otherwise difficult to understand event of change in the condition of the spacecraft, then the answer depends on the specific details.


"Figure 11. Residuals of the Cassini range data against DE440. The rms residual of the Cassini ranges is about 3 m." from "The JPL Planetary and Lunar Ephemerides DE440 and DE441" https://iopscience.iop.org/article/10.3847/1538-3881/abd414

Figure 11. Residuals of the Cassini range data against DE440. The rms residual of the Cassini ranges is about 3 m. Source: The JPL Planetary and Lunar Ephemerides DE440 and DE441)

note: the vertical axis is in meters, the horizontal axis spans about 13 years!

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