A star winks out behind a small body and a handful of telescopes each measure one chord across its silhouette. Every published
size is then an ellipse fitted to those chords. This asks what the chords force on their own — the silhouette is any convex
shape at all — and the whole question turns out to be one-dimensional and closed form.
168–268
km, certified from convexity
206 ± 15
km, the published ellipse
309
km of ceiling bought by the misses
0
optimisers; every area is exact
why this is one-dimensional
A convex silhouette has a concave chord.
Occultation chords are parallel: the shadow sweeps one way and every station cuts the silhouette along that direction. Put the
chord direction along x and let y be the cross-track coordinate. A convex silhouette has slices
[L(y), R(y)] with R concave and L convex, so the chord length w(y) = R(y) − L(y) is concave on
the body's support and zero outside it. Every measured chord is a value of w, every station that saw nothing is a y where w
vanishes, and the silhouette's area is ∫w. That is the whole problem, and it is the calibration envelope again with
concave where that one had monotone with a slope band.
So the two bounds are read off, not searched for. The floor is the concave hull of the measured lengths between the
outermost chords — the negatives play no part in it, and nothing forces the body to extend past its outermost chord. The
ceiling is concavity read backwards: beyond two samples, w is capped by their extrapolated secant, taken with an upper
value at the near sample and a lower one at the far, and that envelope is then clipped where the nearest station saw nothing.
The asymmetry is structural. The floor is bought by the chords; the ceiling is bought by the telescopes that recorded
nothing at all. Move this campaign's 23 negative stations out of reach and the ceiling goes from 267.5 km to
577.0 km — the misses are worth 309 km of upper bound, more than the object's own diameter, and
they are the part of a campaign least often published.
the chords, and every concave function that fits themsolid: the certified floor and ceiling · dashed: joining the dots, which is an assumption, not a boundtwo silhouettes the same five chords allow168.3 km and 267.5 km acrosswhat the error budget buysthe top row is the chords at face value
At face value the chords are not consistent with any convex silhouette. Taken with zero error, the concave hull of the
measured lengths runs 5.5 km above Javalambre's own chord — no convex body passes through all five.
About half of one stated error bar is enough to admit one, which is why the row above is drawn open. The published analysis meets the
same fact from the other side: its fit to the chords as timed scores χ² = 29.5, and only after shifting three chords in time does it
reach 4. This page does not take a side in that debate: a timing shift slides a chord along its own line and does not change how
long it is, so nothing here depends on which fit one prefers.
And a disagreement in the literature dissolves. The paper reports an occultation diameter of 206 ± 15 km and notes that its
mean 3-D estimate is not in agreement with the radiometric 237 ± 8 km from Herschel, Spitzer and ALMA. Both numbers sit
inside [168.3, 267.5] km. The tension is between two models, not between two measurements.
station
cross-track, km
chord, km
status
Latrape Observatory (0.35 m)
224.6
—
negative
Latrape Observatory (0.30 m)
224.6
—
negative
Sabadell Observatory
123.9
—
negative
Forcarei Observatory
123.6
—
negative
Sant Esteve Observatory
119.3
—
negative
Allariz Observatory
102.5
90.1 ± 13.8
positive
Javalambre Observatory
11.8
172.7 ± 3.8
positive
Aras de los Olmos Observatory
4.4
185.4 ± 39.0
positive
Nunki Observatory
-6.4
—
bad — bad data
La Hita Observatory (0.77 m)
-34.1
170.2 ± 40.9
positive
La Hita Observatory (0.40 m)
-34.1
179.3 ± 33.7
excluded — authors excluded it from the limb fit; kept as a consistency check
Kryoneri Observatory
-84.5
—
technical — technical failure
Univ. of Athens Observatory
-88.1
133.3 ± 104.3
positive
La Sagra Observatory
-136.7
—
negative
La Murta Observatory
-138.7
—
negative
Adiyaman University Observatory
-182.0
—
negative
Calar Alto Observatory
-188.8
—
negative
Sierra Nevada Observatory
-205.1
—
negative
13 further negative stations between 268 and 731 km from the centerline are omitted from this table; they are in the data file and change nothing.
Every number on this page is exact. The published values are decimals, so they are exact rationals, and the areas are
exact rationals too — there is no optimiser here and nothing converges. The envelope is discontinuous at every chord, because the
constraints anchored at a chord stop applying the moment you pass it; integrating it as a polyline understated the ceiling by 16%
and made it fall when chords were removed. Between breakpoints the envelope is affine, so the midpoint rule is exact and
blind to the jumps. Twenty cases and six reds, including a control that must fail: a dumbbell silhouette, which is not convex, has to
land outside the interval — and does.
Data transcribed from Santos-Sanz et al. 2021, MNRAS; arXiv:2012.06621, Tables 4 and 6. Companion to the transit instrument (frontier-apps; not yet ported here),
which finds the same asymmetry in exoplanet photometry, and to the calibration envelope,
whose closed form this is.