Approximating the Position of Space Probes With Minimal Code | Iris Webb

Information about the Parker Solar Probe or the Voyagers can be as interesting as info about the Moon or Saturn. However, generating that info is significantly harder, the libraries that can<br>generate data about them are few and far between. And even when generating that data is possible, it&rsquo;s difficult and cumbersome to actually do in a program. This is enough that Stellarium, XEphem, and KStars don&rsquo;t generate data on interplanetary probes.

This is because generating this data involves either:

Running an n-body gravitational simulation.

Looking up the data in an extremely large interpolated table, generated by the previous method.

Neither of these are desirable for code that needs a small footprint and quick runtime. So tools that aren&rsquo;t an n-body simulator like<br>GMAT, ephemeris reader like<br>CSPICE, both like JPL Horizons,<br>or don&rsquo;t want to prepackage megabytes of data for each probe, usually choose to omit space probes entirely

The Difference Between Planets and Probes

There are fast, low footprint, accurate ways to generate the positions of planets.<br>Extremely Large Interpolated Tables still exist<br>and are used for exact accuracy. But for smaller utilities, Medium Sized Tables or Smaller can do the same nearly as well.

The accuracy error in VSOP97 nano, over the years 2000 BC to 2500 AD (source: celestialprogramming)

The orbits of planets are consistent and periodic, and since the main influence in their orbits is the Sun, they closely follow the laws of Keplerian Motion.

Keplerian motion, in this context, models a 2-body orbit of something around a fixed point, since<br>the path the orbit traces is unchanging and inertial. It can be calculated to whatever precision is needed, at any epoch, with basic trig. Only 6 parameters are needed to describe an Keplerian orbit, although these 6 change based on what type of orbit is specified.

JPL Horizons, can be used to return 12 elements from any object (including space probes!) at any time, although in practice half of these are needed, and some can be derived from others:

Parameter<br>Symbol<br>Describes

Eccentricity<br>How non-circular an orbit is

Periapsis dist.<br>The closest approach of an object

Inclination<br>Deviation of the orbit from a 2d plane of reference, into 3d space

Longitude of the Ascending Node<br>Ω or ☊<br>Where the orbit deviates from the 2d plane of reference

Argument of Periapsis<br>ω or w<br>Where the closest approach is

Time of Periapsis<br>Tp<br>When the object has, or will, come to its closest approach

Mean motion<br>The approximate angular speed of the object

Mean anomaly<br>The amount of the objects orbit it has completed, at epoch

True anomaly<br>The location of the object along its orbit at epoch

Semi-major axis<br>The average between the Periapsis and Apoapsis

Apoapsis distance<br>rap<br>The farthest approach of an orbit

Orbital Period<br>The time it takes for the object to complete an orbit

Planets don&rsquo;t follow Keplerian orbits exactly though, other objects influence their trajectories slightly. Jupiter and Saturn are large enough and exert enough force on each other for their positions to be several degrees off from what Keplerian orbits predict. This has to be accounted for explicitly.

As a rule:

The more external influence there is on an object, the more that objects trajectory will differ from a Keplerian orbit

For most planets, asteroids, comets, and even interstellar objects like Oumuamua, this is a small problem that can be corrected for. But a space probes trajectory is defined completely by other objects gravity.<br>Through gravity assists, capturing into orbit around objects, correction maneuvers, etc. Which means the entire trajectory of a space probe can&rsquo;t be modeled by one set of Keplerian elements.

The Life of a Space Probe

Any object in interplanetary transfer (not on the surface, or in orbit of anything, except the sun) will spend almost all its time waiting.<br>The closest approach to a planet could last days, part of a larger observation period, where data is collected and its future trajectory is fine-tuned.<br>But these periods are separated from each other by years, where the probe is too far away from anything to take pictures or collect useful data,<br>it hibernates and nothing happens.

The force gravity imparts on an object grows quadratically, inverse to distance. A space probe passing 1,000km from a planet will be affected by it 4 times more than it would at 2,000 km.<br>This also means that a space probe 0.1 AU away from an object will be affected ~22,000x less than it would at 100,000km. Outside of gravity assists, a space probes trajectory is affected<br>by other objects, but this affect is far, far smaller compared to being slingshot around a planet.

JPL Horizons confirms this:

Generating an &ldquo;Osculating Elements&rdquo; table for the Parker Solar Probe and graphing any orbital element that doesn&rsquo;t normally change, Jolts in...