Prediction geometry and astrometry

The question. How is the ground track of a stellar occultationstellar occultationThe passage of a solar-system body in front of a star as seen by an observer. Because the star is effectively a point at infinity, the body's shadow is a cylinder of the body's own cross-section, and the event lasts from a fraction of a second to a few minutes. computed when the occulter is an asteroid, a trans-Neptunian objecttrans-Neptunian object (TNO)A small body whose orbit lies mostly beyond Neptune, at roughly 30 to 50 au or more. At such distances a body a few hundred kilometres across subtends tens of milliarcseconds., a CentaurCentaurA small body on an unstable orbit between Jupiter and Neptune, thought to come from the trans-Neptunian region. Chariklo and Chiron are examples. or a planetary satellite? Which quantities set the width of its uncertainty band, and how did the Gaia catalogues and occultation astrometry change that uncertainty? Lunar occultations use the same plane but a nearby, large occulter with a known limb, and are the subject of Lunar occultations.

The geometry: a cylinder on the fundamental plane

An eclipse shadow is a cone because the Sun has a finite size at a finite distance. A star is effectively at infinity and its angular size is far below the body's. SORA's authors state that "unlike an eclipse, the star is farther and its angular size is much smaller than the body's, which results in almost the same size for the body and its umbra" ¹. The shadow is then a cylinder parallel to the star direction, with the cross-section of the body's projected outline. Its track on the ground is the shadow pathshadow pathThe track on the Earth's surface swept by the shadow of an occulting body during a stellar occultation. On the fundamental plane its width equals the body's diameter. On the ground it widens as the star sinks towards the horizon..

The reference plane is the fundamental planefundamental planeThe plane through the Earth's centre perpendicular to the axis of the Moon's shadow. Its x axis lies in the equator pointing east, its y axis points north, and the shadow's cross-section on it is an exact circle. of Besselian elements, with the star's direction $(\alpha_\star, \delta_\star)$ taking the place of the point Z. Herald's reduction of asteroid occultations works on this plane "using Besselian elements", with the apparent positions of both the star and the asteroid defining its orientation ⁵. In the plane, the shadow radius is the body's radius $R_\circ$ at every distance along the axis. There are no cone angles and no $l_1$, $l_2$ variation. The path widthpath widthThe width of the umbral or antumbral track measured on the ground perpendicular to the direction of motion, given by Mikhailov's 1931 formula from the shadow radius, the height above the fundamental plane and the direction of motion. on the plane is therefore $2R_\circ$ for a spherical body.

SORA writes the plane coordinates as orthographic projections onto the plane perpendicular to the geocentre-star direction. For the observer and the body ¹:

and the observer's position relative to the body's shadow centre is $(\xi_\lambda - \xi_\circ,\ \eta_\lambda - \eta_\circ)$, which the paper calls $(x, y)$. Here $D_\lambda$, $\alpha_\lambda$ and $\delta_\lambda$ are the geocentric distance, right ascension and declination of the observer's position vector in the ICRS, and $D_\circ$, $\alpha_\circ$, $\delta_\circ$ those of the body. $\xi$ is positive towards increasing right ascension and $\eta$ towards the ICRS north pole ¹. The observer terms are the eclipse coordinates $\xi = \rho\cos\phi'\sin\theta$ and $\eta = \rho\sin\phi'\cos d - \rho\cos\phi'\sin d\cos\theta$ of Contact times and magnitude, with $d = \delta_\star$ and $\theta$ the star's local hour angle. SORA's paper writes the plane offset as $x$ and $y$ in kilometres. In the eclipse notes $x$ and $y$ are the Moon's coordinates in Earth radii, a different quantity.

The paper's exact form, used for the astrometric offset itself, is $x = D_\circ \sin[(\alpha_\star - \alpha_\circ)\cos\delta_\circ]$ and $y = D_\circ\sin(\delta_\star - \delta_\circ)$, with both positions astrometric ICRS coordinates centred at the observer ¹. Because the body's position is less certain than the star's in the Gaia era, the body coordinates in this equation are the ephemeris values. The fitted displacement of the observed centre is then an ephemeris offset ¹.

Two approximations hide in the geocentric form. First, the star is taken to be at infinity, so all observers share one line of sight. SORA's authors note that for Proxima Centauri the topocentric and geocentric star directions differ by up to 2.3 km projected at 100 au ¹. Second, all observers are referred to one instant at the geocentre. The light-timelight-timeThe travel time of light to the observer, about 499 seconds from the Sun and 1.3 seconds from the Moon. A reception correction evaluates the target at emission time and the observer at reception time. Observer aberration is separate. across one Earth radius is 0.021 s, which at 30 km per second is 600 m on the plane and matters for near-Earth asteroids ¹.

Finding the events

A prediction search crosses an ephemeris with a star catalogue. In SORA's prediction function the ephemeris is sampled at a fixed step, 60 s by default, and Gaia stars are downloaded for the box the path covers ³. The star positions are propagated to the middle of the interval with apply_space_motion, matched to the nearest ephemeris point, and kept when the separation is below

where $R_\oplus$ is added only for a geocentric search, the multiplier $n_\sigma$ takes the value of the program argument sigma, default 1, $\sigma_\alpha$ and $\sigma_\delta$ are the ephemeris errors in arcseconds, $\mu_\star$ is the star's total proper motion and $\Delta t$ the search interval ³. The paper describes the same search radius as "the body's radius increased by the ephemeris 1-sigma uncertainty", with the Earth's radius added for a geocentric search ¹.

Each candidate is then refined. The paper gives the closest approachclosest approachThe minimum angular or projected distance between the star and the centre of the occulting body as seen from a reference point, usually the geocentre, and the instant at which it occurs. An occultation is possible somewhere on Earth if this distance is smaller than the body's radius plus the Earth's radius on the fundamental plane. from two ephemeris positions bracketing the star, after Assafin et al. 2010 ¹:

with $\Delta_1$ and $\Delta_2$ the star's separations from the ephemeris positions at $t_1$ and $t_2$ and $\Delta_B$ the separation between those positions. The shadow velocityshadow velocityThe speed of the occulting body relative to the star on the fundamental plane, in km per second. It converts a timing error into a distance error and an exposure time into a spatial resolution. is $V_S = D_\circ \sin(\Delta_B)/(t_2 - t_1)$ ¹. The released code does something different. Function occ_params evaluates the projected separation on a grid of ±900 s at 20 s steps, then ±20 s at 0.5 s, then ±1 s at 0.02 s. It takes the minimum each time and recurses when the minimum falls at a grid edge ³. It computes the position angle as arctan2(-f, -g) of the projected offset (f, g), and the velocity as the change in that offset over one second, signed by the east-west direction of motion ³. The paper's two-point formula is the algorithm description. The grid search is the implementation.

Both assume straight-line motion of the shadow during the event. The paper justifies it by the short duration: at the Earth's mean orbital speed of 29.78 km per second a small body's shadow crosses the Earth in about 7 minutes. It warns that the linear path fails near quadrature, when the shadow is slow, and suggests a polynomial instead ¹. Herald's reduction uses "a cubic expression representing the motion of the asteroid's shadow on the fundamental plane" ⁵.

The amateur prediction services split the search from the update. In Preston's 2007 description, several searchers each scanned for events with their own orbits and catalogues. Goffin covered asteroids of 40 km and larger with a compiled catalogue to about magnitude 12. Donnell and Breit used astorb.dat and Occult's Tycho-2 catalogue for asteroids of 10 km and larger ¹⁰. Preston then refitted the orbit with recent astrometry and recomputed the path with Occult a month and a week before each event ¹⁰. Since 2024 Preston's site states that its posted predictions are not the most current, and that current ones are distributed through OccultWatcher ¹¹. The 2024 supplement of the Royal Astronomical Society of Canada (RASC) confirms that updates "with improved orbits during the year" are "now done only with Occult Watcher" ⁷.

From closest approach to a path on the ground

SORA's map draws the path from the prediction-table values alone ³. For each time offset $\tau$ from closest approach, the shadow centre on the plane is

where $CA_{\mathrm{km}} = D_\circ \times CA$ in radians, $PA$ is the position angle and $PA'$ is $PA$ folded into ±90 degrees ³. The two path edges are the centre moved by $\pm R_\circ$ along $(\sin PA', \cos PA')$. The 1-sigma lines are moved by $\pm(R_\circ + D_\circ\,\sigma)$ with $\sigma$ the ephemeris error converted from milliarcseconds to radians ³. Function xy2latlon converts plane points to longitude and latitude. It takes the depth along the star direction on a sphere of radius R_earth, transforms GCRS to ITRS, reprojects the resulting site on the ellipsoid at zero height, and iterates four times ³. The map is therefore a path at sea level on the ellipsoid. It ignores observer height.

Occult's map, as Preston describes it, draws solid edges assuming a circular body ². Preston states that the ground width exceeds the body's width when the star is not overhead: "The width of the shadow = (diameter of asteroid) / SIN (maximum altitude of star along the path)" ². That expression is the width measured along the ground in the direction of the star's azimuth. On the fundamental plane the width remains the diameter.

The 1-sigma lines1-sigma lineA line drawn parallel to a predicted shadow-path edge, offset outward by the 1-standard-deviation cross-track uncertainty. Between the two 1-sigma lines the true path is expected with about 68 per cent probability if the errors are Gaussian. are, in Preston's words, "the effect of a 1-sigma shift in the edge of the path", with "a 68% probability the actual path ... will fall somewhere between the 1-sigma lines" and the 2-sigma lines at twice the distance from each edge with 95 per cent ². Occult's plot header also gives the full uncertainty ellipse as major axis, minor axis and position angle ². Preston's event Rank is "the probability of at least one successful observation by a team of two observers where the two observers are positioned 3/4 path width apart symmetrically about the center of the path" ².

The uncertainty budget

Tanga and Delbo define the path uncertainty as the quadrature sum of the ephemeris and star errors ⁴:

where $\sigma_{\mathrm{CEU}}$ is the current ephemeris uncertainty of the asteroid and $\sigma_\star$ the star position error at the event epoch. They define a prediction efficiency $p = 2\theta/\sigma_{\mathrm{path}}$, with $2\theta$ the apparent diameter, and read $p = 1$ as a 68 per cent chance of seeing the event ⁴. The angular uncertainty becomes a cross-trackcross-track and along-trackDirections on the fundamental plane perpendicular to and along the shadow's motion. Cross-track error moves the predicted path sideways on the ground. Along-track error changes the predicted time. distance on the plane through the body's distance. One milliarcsecondmilliarcsecondOne thousandth of an arcsecond. At a distance of $D$ astronomical units one milliarcsecond spans $0.7253,D$ km, so about 3 km at Jupiter in opposition and about 6 km at Saturn. at 1 au is $1.495979\times10^{8}\ \mathrm{km}\times 4.8481\times10^{-9} = 0.7253$ km. Herald puts it as "an angular displacement of 1 mas is crudely equivalent to a displacement of the asteroid's shadow of about 1 km" for the main belt ⁵. Buie quotes 31.4 km per milliarcsecond for Arrokoth at 43.3 au ¹².

SORA's paper turns the budget into a target. The combined error must be of the order of the body's apparent radius, about 230 milliarcseconds for Ceres, 40 milliarcseconds for Pluto, and as little as 1 milliarcsecond for small distant bodies ¹.

The star

Before Gaia, the star term was large and catalogue-dependent. Tanga and Delbo give about 1 milliarcsecond for FK6, 10 milliarcseconds for Hipparcos, 70 for Tycho-2 and 20 for UCAC2 in the magnitude range 10 to 14 ⁴. Preston's 2007 process preferred UCAC2 "whenever available", and treated double stars as a standing problem ¹⁰.

The Gaia releasesGaia catalogue releasesThe star catalogues from ESA's Gaia astrometric mission: DR1 (2016), DR2 (2018), EDR3 (2020) and DR3 (2022). From DR2 onward they give positions, proper motions and parallaxes at the sub-milliarcsecond level for faint stars, which removed the star position as the main error in occultation predictions. made the star term small. SORA's paper cites Gaia EDR3 positions below 1 milliarcsecond to magnitude 21, about 10 microarcseconds for stars brighter than magnitude 15 at the catalogue epoch, and proper motions near 20 microarcseconds per year for those stars ¹. French and Souami state that DR3 proper motions are "a factor of 2 more accurate than in the DR2 catalog" ¹³. Three residual issues remain, and each source names them:

• Unresolved companions. A RUWERUWEThe renormalised unit weight error of a Gaia astrometric solution. Values above about 1.4 indicate that the star's motion does not fit a single-star model, often because of an unresolved companion, and occultation predictions for such stars have been observed to shift. above 1.4 flags a poor single-star fit, and positive occultations of such stars "showed a significant shift from the predictions" ¹. Preston's FAQ gives the same warning for DR2 and its duplicated-source flag ².
• Bright-star frame rotation in DR2. Herald lists that DR2 stars brighter than about magnitude 12.5 "are affected by a frame rotation error", and that this uncertainty was not yet in the data set ⁵.
• Proper-motion systematics in EDR3. SORA's paper cites a systematic error of up to 80 microarcseconds per year for stars brighter than magnitude 13. For the 1988 Pluto occultation star it amounts to 1.10 milliarcseconds in declination against a formal error of 0.35 milliarcseconds ¹. SORA can apply the Cantat-Gaudin and Brandt correction, and French and Souami applied it to DR3 ^{1 13}.

SORA propagates the star from the catalogue epoch with the rigorous model of Butkevich and Lindegren 2014, including the propagation of its uncertainty, and then applies parallax for the chosen observer ¹. Herald's astrometry corrects for proper motion "including foreshortening" and for parallax ⁵.

The ephemeris

The ephemeris term is now the dominant one. JPL Horizons outputs, for bodies with a covariance, the right ascension and declination uncertainty and a plane-of-sky error ellipseplane-of-sky error ellipseThe projection of a body's three-dimensional position uncertainty onto the plane perpendicular to the observer's line of sight, given by a semi-major axis, a semi-minor axis and an orientation angle. JPL Horizons quotes it at 3 standard deviations.. Both are "formal +/- 3-standard-deviation" values, and Horizons warns they "can be optimistic" when there are systematic biases or long extrapolations ¹⁴. A 1-sigma path therefore needs the Horizons value divided by 3. The SBDB API returns the orbital covariance itself, at the JPL orbit-solution epoch, as a full matrix or an upper-triangular vector ¹⁵. SORA does not read either. Its ephemeris errors error_ra and error_dec are user inputs in arcseconds, and both default to zero ³. A default SORA run therefore searches with no ephemeris margin. Its map draws dashed error lines only when the separate error argument, in milliarcseconds, is given ³.

For trans-Neptunian objects the Lucky Star group fits its own orbits with NIMANIMANumerical Integration of the Motion of an Asteroid, the orbit-fitting procedure of Desmars and co-authors used by the Lucky Star project to produce ephemerides of trans-Neptunian objects and Centaurs for occultation prediction, with observations weighted by bias and precision.. The least-squares correction and covariance propagation are ⁹:

where $\mathbf{q}$ is the six-element state vector, which the paper writes $C$, $A$ the matrix of partial derivatives, $\Delta Y$ the observed-minus-computed residuals, $\Lambda_0 = (A^T V_{\mathrm{obs}}^{-1} A)^{-1}$ the covariance at the epoch and $\Lambda(t)$ its linear propagation to date $t$ ⁹. The weighting prevents one well-observed night from dominating ⁹:

with $N_i$ the number of observations from the same observatory on the same night, $b_i$ a bias for that observatory and catalogue, and $\sigma_i$ the individual precision. The adopted biases are 300 milliarcseconds for Minor Planet Center positions, 75 to 300 milliarcseconds for the group's own positions depending on telescope and catalogue, and zero for occultation positions ⁹. The NIMA README records the occultation precision as 40 or 75 milliarcseconds. Since 2016 December 7, version 4 has also weighted offsets reduced with Gaia DR1 by $b = 40 + 20\,|\mathrm{yr} - 2015|$ mas and $\sigma = 20 + 20\,|\mathrm{yr} - 2015|$ mas ¹⁶.

The alternative is the ephemeris offset methodephemeris offset methodA prediction technique that measures the difference between new astrometry of a body and a reference ephemeris a few days before an event and applies that difference as a constant correction. It fails when the offset drifts over a few weeks.: measure the body against a reference ephemeris a few days before the event and apply that offset as a constant ¹. Desmars et al. showed that the offset between NIMA and JPL ephemerides drifts "even for short time scales of a few weeks". The constant-offset method "could not accurately predict 6 out of the 13 observed positive occultations successfully predicted by NIMA" ⁹.

For main-belt asteroids in 2007, Preston fitted orbits with OrbFit using JPL DE405, all planets and the Moon, and 13 large asteroids as perturbers. He gave a default a priori RMS of about 1 arcsecond to ordinary astrometry and 0.12 to 0.35 arcseconds to "high accuracy" data ¹⁰. For 2024, the RASC supplement takes orbital elements from JPL Horizons and star data from Gaia DR3 in Occult's own catalogue ⁷.

Orbit improvement: Gaia astrometry and occultation astrometry

An occultation is itself an astrometric measurement. Herald's procedure inverts the Besselian computation ⁵:

1. Find the plane coordinates of the shadow centre at the reference time.
2. Run an inverse Besselian calculation for the offset from the star's apparent position.
3. Convert that offset to the J2000 ICRS frame by removing precession, nutation and aberration.
4. Add it to the star's ICRS position corrected for parallax and proper motion.
5. Correct for the relativistic deflection of the asteroid relative to the star. This "can be significant for solar elongations as large as 145° or more".

Because the result is tied to one star, it "can be updated whenever the star position is updated" ⁵. Because it is measured from the full outline, it refers to the centre of figurecentre of figureThe centre of the Moon's mean limb figure. The ephemeris tabulates the centre of mass, which lies about 1.9 km away, mostly along the Earth-Moon line, so the sky-plane part is about 0.5 arcseconds, historically applied as +0.50 arcseconds in longitude and −0.25 arcseconds in latitude. A limb profile referenced to the centre of mass, as the LOLA and Kaguya grids are, supersedes the correction., or to the centre of masscentre of massThe point whose motion the lunar ephemeris describes. Modern DEMs express heights relative to a sphere centred here, which is why they need no centre-of-figure correction. when a shape model is fitted. Image astrometry, including Gaia's, measures the photocentrephotocentreThe brightness-weighted centre of a body's image, which astrometry measures. Uneven albedo and phase move it away from the centre of figure. ⁵. For Eros at a 51 degree phase angle the photocentre is displaced by about 5 per cent of the mean diameter ⁵.

Ferreira et al. fitted orbits to the whole occultation record. With pre-Gaia catalogues the residuals were 30 to 50 milliarcseconds, and 80 to 90 milliarcseconds for some catalogues. With Gaia EDR3 they fell to about the 10 milliarcsecond level, an average factor of about 5 ⁶. The core of the residual distribution has a full width at half maximum of about 10 milliarcseconds along-track and cross-track ⁶. The authors conclude that "occultation astrometry can reach the performance of Gaia on small asteroids (5-8 km in the main belt)" ⁶. The Gaia Focused Product Release fitted orbits of nearly 157 000 asteroids from Gaia data alone. It tested them against 5 774 occultation positions of 978 asteroids ¹⁷. For events within the Gaia time span the observed-minus-computed distance "peaks at about 10 mas", and away from it the ephemeris degrades by about 10 milliarcseconds per year ¹⁷.

For distant bodies one chord is enough. Rommel et al. derived 37 positions for 19 TNOs and 4 Centaurs, 68 per cent of them single-chord, "most have intrinsic precision at the submilliarcsecond level" on the Gaia DR2 frame ⁸. Camargo et al. state that occultation positions receive the largest weights in NIMA because they are accurate at the milliarcsecond level ¹⁸. SORA's paper adds that such positions "can highly improve the ephemeris ... even if it is from a single-chord event" ¹.

The Arrokoth campaigns show the loop working on a 30 km body at 43 au. The final prediction for 2017 June 3 had a 1-sigma cross-track uncertainty of 44 km ¹². After new Hubble Space Telescope (HST) astrometry reduced against pre-release Gaia DR2, the July 2017 predictions reached 14 km cross-track, 0.46 milliarcseconds. With the 2017 July 17 occultation position included, the 2018 August 4 prediction was 13 km cross-track and 1.8 s, or 36 km, along-track ¹². Buie et al. planned station spacing with a Monte Carlo of the central line drawn from the predicted distribution. They used 15.5 km spacing for the 44 km case and 4 km spacing once the uncertainty was 14 km ¹².

Typical numbers before and after Gaia

The count of events followed. Herald reports about 10 events a year in 1990 and 586 in 2019 ⁵. Ferreira et al. identify a sharp rise with Hipparcos in the 1990s, a plateau from 2005 to 2015, and a new rise with Gaia DR1 in 2016 and DR2 in 2018 ⁶.

Timing requirements

An occultation turns time into distance at the shadow velocity, typically several to tens of kilometres per second. Herald's list of improvements names accurate time as "the most critical item", seeking ways "to link accurate time (to within 10msec or less)" ⁵. The common time source in that data set is a GPS receiver stamping each field of an analogue video stream. The recent anomalies were "associated with the use of NTP as a time source, personal equation for visual observations, and observers attempting an event at the limit of their system's capabilities" ⁵. NTP is the Network Time Protocol. Observations are weighted by method: 5 for analogue video, 4 for digital video or photometer, 3 for sequential images or drift scans, and 1 for visual ⁵. Past reports without stated time uncertainties are to receive 0.5, 1.0 or 1.5 s ⁵.

The Arrokoth flight of SOFIA, the Stratospheric Observatory for Infrared Astronomy, found three aircraft clocks offset from each other by 1 s steps. The likely cause was out-of-date leap-second almanac data, so the team trusted only the GPS-slaved clock recorded with the images ¹². Herald's time scale for reported events is UTC ⁵. SORA expects UTC by default but accepts any Astropy time scale ¹.

The light-curve side of timing, including the sub-exposure estimate $\sigma_t = \sigma_{LC}\,t_{\mathrm{exp}}/\Delta f$, is in Light curves, chords and results.

Software, services and the data they use

The JPL planetary ephemerisJPL DE seriesJPL's Development Ephemerides integrate and fit the Sun, planets and Moon to tracking and ranging observations. Accuracy statements and inter-release differences have date and component limits. A fitted lunar range residual does not bound every three-dimensional orbit component. enters all of these through the asteroid orbit fit and the Earth's position. French and Souami's planetary predictions use DE440 ¹³. Preston's 2007 orbit fits used DE405 ¹⁰. None of the sources cited here states which DE version Occult itself uses for asteroid work.

Occult's catalogue is described both as Gaia EDR3 and as Gaia DR3 ⁷. For astrometry the two are the same: Gaia DR3 "neither contains new astrometry nor new photometric calibrations", and its astrometry is the one published as part of EDR3 ²⁴.

Sources compared

What a developer should do

1. Put the fundamental plane on the star direction and compute the body's plane coordinates with SORA's equations 7 to 9, in kilometres, from apparent or astrometric positions used consistently for star and body ^{1 5}.
2. Propagate the Gaia DR3 star with full space motion and its covariance, apply the Cantat-Gaudin and Brandt proper-motion correction for bright stars, and flag RUWE above 1.4 ¹.
3. Take the ephemeris from SPICE kernels or Horizons, and its covariance from the SBDB API. Map the covariance to the plane-of-sky ellipse at the event and divide Horizons 3-sigma values by 3 ^{15 14}. In SORA, set error_ra and error_dec explicitly, because they default to zero ³.
4. Form $\sigma_{\mathrm{path}}$ from the cross-track components of both errors, not from $\max(\sigma_\alpha, \sigma_\delta)$ as SORA's search does. Convert with $0.7253\,D_\circ$ km per milliarcsecond, with $D_\circ$ in astronomical units ^{4 3}.
5. Use a cubic in time for the shadow motion, as Herald does, so that slow events near quadrature stay correct ^{5 1}.
6. Project to the ground with the observer's height on the ellipsoid. Report the path edges, the 1-sigma lines and the along-track time uncertainty in seconds ².
7. Close the loop. Ingest observed occultation positions with zero bias and a precision of a few milliarcseconds into the orbit fit, as NIMA does ¹⁶.

Read SORA's paper and prediction/core.py first, then Herald et al. 2020 sections 4 and 5, then Desmars et al. 2015 section 3.

What this changes

The eclipse pipeline's Besselian layer carries over with three changes. The shadow axis is the star direction and is fixed for the event. The shadow radius is the body's radius with no cone angle. The ephemeris uncertainty becomes a first-class output, because for occultations the 1-sigma band is part of the product, not a footnote. The pipeline needs a star-catalogue stage with space-motion propagation and an orbit-covariance stage, neither of which a solar-eclipse pipeline has.

Open questions

• Obtain Occult's help file Occult.chm and read the asteroid-prediction topics, to record the Besselian element definitions, the DE version and how the uncertainty ellipse is built from orbit and star errors ¹⁹.
• Obtain Assafin et al. 2010 (A&A 515, A32) for the original closest-approach and offset method that SORA's paper cites.
• Obtain Herald's Journal for Occultation Astronomy articles on the Gaia-era prediction process to replace the 2007 Preston slides as the description of current IOTA updates ¹⁰.
• Obtain the orbit files of the Gaia Focused Product Release to measure the path uncertainty for main-belt events from those orbits ¹⁷.
• Run SORA's prediction and Occult on the same event and star, and compare the closest-approach time, the central line and the 1-sigma lines.