Uncertainty: how sure is a position?

Every asteroid position ARCKEEP reports could be some distance from the truth. This page explains what that distance is made of, how ARCKEEP will measure it, and why it reports none until it can.

Status: planned. ARCKEEP leaves the uncertainty fields out of its ADES files today (how it works, section 8). The notes marked "for researchers" open to give method and assumptions.

1 The unit

Sky positions are angles. One arcsecond (1″) is 1/3600 of a degree. A one-euro coin, 23.25 mm across, seen from about 4.8 km away spans about 1″. The statistical weights the professional community gives individual observing stations range from 0.2″ to 1.5″ (Vereš et al. 2017, tables 2 to 4).

2 Two different uncertainties

	The question it answers	Symbol
Measurement uncertainty	How well did we measure where the asteroid appeared?	M
Prediction uncertainty	How well do we know where the asteroid should be?	P

They are easy to confuse, and both are needed.

3 Measurement uncertainty, M

Three independent sources contribute:

The plate solution. The pixel-to-sky mapping is fitted to catalogue stars, so it has its own error. The error is smallest near the image centre and grows towards the edges.
The centroid. Where exactly is the centre of the blurry blob? The fainter the blob, the harder it is to say. A common rule of thumb for a round blob is: error ≈ blob width ÷ (2.35 × signal-to-noise ratio). A trailed blob is less certain along the trail than across it.
The timing. A position is only as good as the time it belongs to. If the clock is off by 0.5 seconds and the asteroid moves 30″ per minute, its position is off by 0.25″, along the direction of motion.

Independent errors combine in quadrature:

σ_total = √(σ_plate² + σ_centroid² + σ_timing²)

They are not simply added, because independent random errors partly cancel. It is the same error-propagation rule used in laboratory work.

The two sky directions are not independent either. A trail running diagonally makes the RA and the Dec errors large together. So the full description is an error ellipse, not a circle: the variance in each direction plus how strongly the two move together. Those three numbers are the covariance, and ADES has a field for each.

for researchers

The solver already computes the parameter covariance of its fit but keeps only diagonal summaries (centre error, scale, rotation). The planned change propagates the full WCS covariance to the target's pixel, giving σ_RA, σ_Dec and their correlation. The solver is vendored from AstroEquip, so the change is made there and arrives by refresh. Gaia position and proper-motion errors at the observation epoch belong in the plate term. No centroid or timing term is measured yet; both require calibration (section 6).

4 Prediction uncertainty, P

For a well-known asteroid, P is very small. Hundreds of observations over years pin the orbit down, so its predicted position is usually known far more precisely than any single measurement.

For a newly found asteroid seen on one night, P is very large. One night tells you the direction to the object and how fast that direction is changing. It does not tell you how far away the object is. A close, slow object and a distant, fast one can look the same tonight. It is like a light moving across the night sky: a small drone nearby, or an aircraft far away? You cannot tell until you have watched for a while.

night 1 a week later: one possible future per distance you ---> + ----------> . . . . . . . night 2 picks the distance that fits: you ---> + ---> + ----> .

Each possible distance implies a different orbit, and those orbits separate over the following days, so the predictions fan out across a large region of sky. A second night's position rules out most distances, and the fan collapses to a small region. This is why follow-up observations matter, and why ARCKEEP exists: an object nobody re-observes in time can be lost.

for researchers

One night constrains an attributable: two angles and two rates. Range and range rate are unconstrained within the admissible region (Milani et al. 2004). Planned methods by arc length: systematic ranging (Farnocchia et al. 2015) for a single night; a preliminary orbit refined by weighted least squares for a few nights; and propagation of the MPC state and covariance for catalogued objects. Covariance is carried forward by linearisation, computing numerically how the output moves with each of the six inputs, with sampling where the region is too large for a linear treatment. The size of the one-night fan and its collapse after a second night were measured in simulation outside the repository and are not quoted, because ARCKEEP quotes only numbers its own code reproduces.

5 Where M and P meet: the gate

Identification today uses fixed review limits (how it works, section 7). The planned statistical gate replaces them. In one dimension it asks:

d = (measured − predicted) / √(σ_M² + σ_P²)

This is the difference counted in units of the combined uncertainty. A value of 1 or 2 is normal: a match. A value of 10 means a different object or a mistake. The full gate does the same in four dimensions, with ellipses instead of single numbers.

6 Calibrating ourselves

For a well-known asteroid, P is much smaller than M. So when ARCKEEP measures a known asteroid, the O-C against its orbit is almost entirely ARCKEEP's own measurement error. Repeated over many archive images of known objects, that tells us:

whether the claimed error bars are the right size: the errors divided by the claimed σ should scatter with a spread of about 1;
how the error grows with fainter targets or longer trails;
whether the clock is off: if measurements sit consistently ahead of or behind the prediction along the direction of motion, that offset divided by the speed is the clock error.

The clock check already works: comparing positions of known asteroids with their orbits showed that Tautenburg's DATE-OBS marks the middle of the exposure (how it works, section 3). Only when the full calibration exists will ARCKEEP fill the ADES uncertainty fields.

for researchers

P must be checked per object, since it is not small for short arcs or very fast close approaches. The MPC orbit was fitted using the archive's original measurements of the same frames, so a remeasurement is not fully independent of it: exclude sparsely observed objects, or refit without the matching observation. The Vereš et al. weights are statistical weights for orbit fitting, not per-measurement uncertainties, and must not be written into rmsRA or rmsDec.