Measure angular motion

1. Measure change across the requested instant

Suppose a body’s longitude is 10.2° half a day before our requested time and 11.4° half a day after it. The change is 1.2° over one day, so the estimated speed is +1.2°/day. The plus sign means longitude increased. Sampling on both sides centers the measurement on the instant we want to describe, rather than putting the whole measurement before or after it.

speed ≈ signed_difference(longitude(t + h), longitude(t − h)) / (2h), with h = 0.5 day

Here h is half the sampling interval, measured in days. The denominator is 2 × 0.5 = 1 day. The implementation adds and subtracts time-library durations, converts the resulting instants to ephemeris seconds, and evaluates longitude twice. Because our speed unit is degrees per day, we divide by the interval in days, not by its equivalent 86,400 seconds.

This is a centered finite difference: “finite” means we use a definite separation in time, rather than a limiting separation of zero. The output estimates motion around the central instant; it is not the analytic derivative of the Chebyshev polynomial. A curved trajectory can change speed inside the interval, and the two samples may even fall in different stored segments. The engine keeps this half-day choice fixed.

The place accessor evaluates longitude and declination at the central instant, then computes speed from the two offset longitude samples. A successful central position does not guarantee a successful place: either offset sample can lie beyond fitted support. With support [0, 172800] s, the central instant 21600 s is readable, but its earlier sample would be −21600 s, so speed evaluation fails. The public domain’s preparation margins are intended to prevent this for a correctly prepared dataset.

2. Subtract directions without crossing the wrong way

Ordinary subtraction works for nearby directions away from the seam. It fails if our earlier longitude is 359.8° and the later one is 0.2°: subtraction gives −359.6°. On the circle these directions are only 0.4° apart in the forward direction. We need to choose the shortest signed change, adding or subtracting full turns as necessary.

Let a be the later angle and b the earlier one. First take a − b and its nonnegative remainder modulo 360: the value left after removing complete turns. Then shift and wrap once more to place the result between −180° inclusive and +180° exclusive. The source uses the following expression; rem means nonnegative remainder.

signed_delta(a, b) = rem(rem(a − b, 360) + 540, 360) − 180

For a = 0.2° and b = 359.8°, the raw subtraction is −359.6°. Its nonnegative remainder is 0.4°. Add 540° to get 540.4°; removing one full turn gives 180.4°. Subtracting 180° gives +0.4°. With the one-day sampling interval, that is +0.4°/day. Reversing the two inputs instead gives −0.4°, so the order matters.

At exactly a half-turn there are two equally short directions. The implementation selects −180°, consistent with its interval [−180, 180). Thus signed_delta(180, 0) is −180°, and signed_delta(0, 180) is also −180°. This is a deterministic tie convention, not evidence that either physical direction was observed.

Taking the shortest change also loses information about complete turns between samples. The two endpoint angles alone cannot reveal whether a body made extra revolutions, or moved more than half a turn during the interval. This calculation reports the chosen signed angular change, not an unwrapped travel history. The same helper later supports event searches.

const normalize360 = (value: number): number => { const remainder = value % 360; return remainder < 0 ? remainder + 360 : remainder; };
const signedDelta = (after: number, before: number): number => normalize360(normalize360(after - before) + 540) - 180;
const dailyMotion = (before: number, after: number): number => signedDelta(after, before) / (2 * 0.5);