Saturn's satellite theories and ring geometry

The question. Which theories and ephemerides give the positions of Saturn's satellites, in which versions and with what accuracy, and how does a program turn them, together with Saturn's figure and rings, into the apparent geometry that phenomena predictions need? The Galilean satellites share much of this geometry, and the Jupiter satellite topic treats them separately.

Theories and ephemerides, by generation

A satellite ephemerisephemerisA table or function giving the position and velocity of a body as a function of time. Modern planetary and lunar ephemerides such as JPL DE440 are numerically integrated orbits fitted to observations and distributed as Chebyshev polynomial coefficients. for phenomena work must give the satellite's position relative to Saturn's centre, not only its sky position. The relative position decides whether a satellite is in front of the globe, behind it or in its shadow. Two kinds of model exist. A semi-analytical theorysemi-analytical theoryA theory of motion that expresses a body's orbital elements as series of periodic terms in time, fitted to observations. TASS and Dourneau's theory are of this kind. Evaluating one needs only the series, not a numerical integration. writes the orbital elements as series in time and is cheap to evaluate. A numerically integrated ephemerisnumerically integrated ephemerisAn ephemeris made by integrating the equations of motion of all the bodies together and fitting the initial conditions and physical parameters to observations. JPL's SAT series and IMCCE's NOE ephemerides are of this kind. integrates all the bodies together and is delivered as Chebyshev polynomialsChebyshev polynomialsThe piecewise polynomial form in which SPK type 2 ephemerides store positions. Velocity is obtained by differentiating the polynomial, and evaluation needs no interpolation tables. coefficients in a file.

Dourneau (1987 thesis, 1993 paper)

Dourneau collected "nearly 22000 satellite observed coordinates" from H. Struve's visual work of 1886 to photographic plates of 1980 to 1985, and fitted theories of motion of all eight major satellites to them ¹. The fit corrected mean motions, node and pericentre rates, and the parameters of the Mimas-Tethys, Enceladus-Dione and Titan-Hyperion librations ¹. The paper estimates position uncertainties for recent years "between 350 and 500 km for the satellites II-VI, from 800 to 1000 km for Iapetus and Mimas, and about 2500km for Hyperion". It adds that comparison with the 1980 mutual events gives "lower uncertainties, evaluated between 50 and 100 km" for Tethys, Dione and Rhea ¹.

The Institut de mécanique céleste et de calcul des éphémérides (IMCCE) distributes the theory as Fortran subroutines that return "saturnicentric rectangular coordinates of the 8 major Saturnian satellites, in the FK4 system (B1950.0)" ¹¹. The Bureau des longitudes used it with the planetary theory VSOP82 for its printed 1996 predictions of Saturnian phenomena ¹². The 2009 predictions still listed it beside TASS as "Do93" ¹³.

Harper and Taylor (1993)

Harper and Taylor augmented the Strugnell and Taylor catalogue "by several thousand additional observations" and fitted theories of all the satellites except Hyperion ². They determined the masses of Mimas to Dione and Saturn's form factors $J_2 = 0.016478 \pm 0.000038$ and $J_4 = -0.00110 \pm 0.00028$. Those form factors were "calculated assuming the equatorial radius of Saturn to be 60000 kilometres" ². They place Saturn's equator plane at node $168.8387^{\circ} \pm 0.0035^{\circ}$ and inclination $28.0653^{\circ} \pm 0.0016^{\circ}$ on the ecliptic and equinox of B1950. Their Table 2b sets Dourneau's values $168.8112^{\circ}$ and $28.0817^{\circ}$ beside these ².

TASS 1.6 and 1.7 (Vienne and Duriez, 1995 and 1996)

TASSTASSThéorie Analytique des Satellites de Saturne, the semi-analytical theory of Mimas to Iapetus by Vienne and Duriez. Version 1.6 was published in 1995 and version 1.7, which adds Hyperion, in 1996. is "a new theory of high precision (about ten kilometers)" for Mimas, Enceladus, Tethys, Dione, Rhea, Titan and Iapetus ³. It is analytical in 55 dynamical parameters: the satellite masses, $J_2$, $J_4$ and $J_6$, six initial conditions per satellite, the inclination and node of Saturn's equator on the J2000 ecliptic, and Saturn's mass ³. TASS 1.6 was fitted to about 50,000 equations of condition from the Strugnell and Taylor catalogue ³. Each satellite is located by four complex osculating elements referred to Saturn's equator and its node on the ecliptic of 2000:

Here $N$ is the "mean mean motion", whose product with time is the linear part of the mean longitude, and $n$, $\lambda$, $e$, $i$, $\varpi$, $\Omega$ are the classical elliptic elements ³. The printed tables keep terms above 20 km. The full series include terms above 1 km ³. TASS 1.7, dated 1996 November 5, adds Hyperion and is distributed as the Fortran file tass17.f. Its positions are "referred to the center of Saturn and to the mean ecliptic and mean equinox for J2000.0 epoch" ¹⁴.

By 2008 TASS had "a drift reaching about 100 km", and Arlot and Thuillot judged JPL's Cassini-fitted SAT252 more accurate. They still used TASS for the 2009 predictions because the event times differed by only 10 to 30 seconds of time ¹³.

IMCCE numerical ephemerides (NOE, 2011 onward)

The IMCCE ephemeris NOE-6-2011-MAIN, a NOE ephemeridesNOE ephemeridesNumerical Orbit and Ephemerides: the numerically integrated satellite ephemerides of Lainey at IMCCE for the satellites of Mars, Jupiter, Saturn and Uranus, fitted to observations and published as SPICE kernels. A name such as NOE-5-2010-GAL identifies one fitted solution. solution, "was produced from the numerical integration of the orbital motion of the eight main Saturnian moons" with mutual perturbations, Saturn's gravity field to order six, solar and Jovian perturbations and tides ⁴. It was fitted to astrometry from 1886 to 2009, including the 2009 mutual events. Its internal accuracy is "a few hundreds of meters". Its expected accuracy over the whole period is "about 600 km for the six innermost moons, and 1,800 km for Hyperion and Iapetus" ⁴. It is published in SPICE format ⁴.

The same integration underlies Lainey's tidal studies. The 2012 fit found Saturn's tidal ratio $k_2/Q = (2.3 \pm 0.7)\times 10^{-4}$ ¹⁵. The 2017 fit added Cassini astrometry and gave $k_2 = 0.390 \pm 0.024$ ¹⁶. The 2020 fit found Titan receding at 11.3 ± 2.0 cm per year ¹⁷. These tidal terms matter for a developer only as a reason: an ephemeris without them drifts over centuries.

NOE stands for "Numerical Orbit and Ephemerides". It is an N-body code that was applied to the Martian moons before the Uranian ones ¹⁸. IMCCE's Miriade service labels its Saturnian satellite positions "INPOP19 + LTE-NOE" ¹⁹. LTE stands for Laboratoire Temps Espace, the name that IMCCE's campaign page carries ²⁰. The MULTI-SAT phenomena server offers five theory sets for satellites S1 to S8: Lainey et al. (2015), Lainey (2011), Vienne and Duriez, Dourneau, and Harper and Taylor ²¹.

JPL SAT ephemerides (SAT351 to SAT441)

JPL delivers numbered SATxxx ephemerisSATxxx ephemerisA numbered JPL ephemeris of Saturn's satellites, such as SAT415 or SAT441, distributed as SPICE SPK kernels. The number names a fit, not a version order of quality. files as SPICE SPK kernelsSPK kernelA SPICE binary file (.bsp) holding ephemeris segments, each giving the position of a target relative to a centre over a time span. The Moon is segment 3 to 301 and the Earth 3 to 399, both relative to the Earth-Moon barycentre.. The current main-satellite file is SAT441, and JPL's table cites Jacobson (2022), "The Orbits of the Main Saturnian Satellites, the Saturnian System Gravity Field, and the Orientation of Saturn's Pole" for it ^{22 23}. The NAIF copy sat441.bsp was created 2022-01-29 from SAT441.24. It covers Mimas to Phoebe, Helene, Telesto, Calypso, Methone, Polydeuces and Saturn itself from 1749 December 29 to 2250 January 5 ⁵. It is consistent with the planetary ephemeris DE440, uses the model SATORBINT, and states a Chebyshev interpolation error of 7.5 m ^{5 24}. The small inner satellites such as Janus and Epimetheus are in the separate file SAT415 ^{25 22}.

Each satellite SPK "provides the state ... of the named satellites relative to the barycenter of that planetary system", and the planet is carried "as a satellite of its own barycenter" ²⁶. A program must therefore subtract Saturn's own state, body 699, to get a saturnicentric vector. Using the barycentre, body 6, puts every satellite off by Saturn's reflex motion.

Accuracy, side by side

The 2009 mutual events give the only common yardstick in this topic. For the 23 best light curves, the root-mean-square O − C residualO − C residualObserved minus computed: the difference between an observed position or time and the value predicted by an ephemeris or theory. Its root mean square over a campaign measures how well the theory fits the data. residuals in right ascension and declination were ⁴:

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 Saturn is about 6 km, since Arlot and Thuillot equate 0.001 to 0.01 arcseconds with 6 to 60 km ¹³. Direct ground-based astrometry reaches "0.1 arcsec at best, which corresponds to 600 km" ¹³. The 1980 mutual events already reached post-fit residuals of 80 km, "approx. 0.01″ in astrometric accuracy" ²⁷.

Meeus chapter 46: satellite positions in the sky

Meeus's Astronomical Algorithms, second edition, chapter 46, computes apparent positions of Mimas to Iapetus ²⁸. The formulas below are quoted from the MIT-licensed Go transcription by Sonia Keys, not from the book. Its test reproduces the book's Example 46.a ⁶. A JavaScript port carries the same constants ²⁹.

The theory behind it. The chapter's constants are Dourneau's. Mimas's mean longitude $127.64^{\circ} + 381.994497^{\circ}\,t_1$ with $t_1$ counted from JD 2411093.0, its eccentricity 0.01905, and the Mimas-Tethys libration epoch 1866.39 all appear in Dourneau's element table ^{6 1}. The equator node $168.8112^{\circ}$ and inclination $28.0817^{\circ}$ are the values Harper and Taylor attribute to Dourneau ². The positions therefore carry Dourneau's accuracy of hundreds of kilometres, not TASS's or SAT441's.

Time arguments. With JDE the Julian Ephemeris Day at 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.-corrected instant:

Nine angles follow, in degrees ⁶:

One satellite, Mimas. Each satellite gets a mean longitude, an equation of the centre $E_c$, a radius in Saturn radii, an inclination $i_j$ and a node $\Omega$ on Saturn's equator ⁶. Meeus writes C for the equation of the centre and γ for the inclination. They are renamed here because $C$ and γ name other quantities in this corpus:

Enceladus, Tethys and Dione follow the same pattern with their own constants. Rhea, Titan, Hyperion and Iapetus use longer series and a common routine that solves the equation of the centre to fifth order in $e$ ⁶.

From orbit to equator to ecliptic. With $u = \lambda - \Omega$ and $w = \Omega - 168.8112^{\circ}$, the saturnicentric coordinates in Saturn radii are ⁶:

The vector is rotated by $28.0817^{\circ}$ about the X axis and by $168.8112^{\circ}$ about the Z axis into the ecliptic and equinox of B1950. It is then rotated by Saturn's geocentric longitude $\lambda_0$ and latitude $\beta_0$, which the code precesses to B1950 first ⁶. A fictitious ninth body at $(0, 0, 1)$, the pole of Saturn's equator, goes through the same rotations. The angle $D = \operatorname{atan2}(A_0, C_0)$ of its projection then turns the result so that Y lies along the projected pole ⁶:

Two small corrections. The code then applies, for satellite $j$ with orbital radius $r_j$ ⁶:

with $K_j$ = 20947, 23715, 26382, 29876, 35313, 53800, 59222, 91820 for Mimas to Iapetus and $\Delta$ the Earth-Saturn distance in au. The factor $W$ is perspective. Since $Z'$ is in Saturn radii and $\Delta$ in au, the constant 2475 implies a unit radius of 149,597,870.7 / 2475 ≈ 60,444 km. The $K_j$ term is a differential light-time correction. A satellite nearer or farther than Saturn's centre by $Z'$ radii is seen as it was $Z'R/c$ earlier or later, and in that time it moves $v Z'R/c$ along its orbit. This gives $K = c/v$. With the chapter's own orbital radii, the 60,444 km unit and periods from the mean motions, $c/v$ is 20,946 for Mimas and 53,794 for Titan, against the tabulated 20,947 and 53,800. This check is arithmetic on the transcribed constants. The book's own explanation of $K_j$ is not quoted here.

Light time. Saturn's geocentric distance is iterated twice with $\tau = 0.0057755183\,\Delta$ days. The satellite theory is evaluated at the retarded instant $\mathrm{JDE} - \tau$ ⁶. Meeus's planetary positions come from VSOP87 ⁶.

Test vector. For JDE 2451439.50074 the transcription's test reproduces Example 46.a, giving $X, Y$ in Saturn radii of (+3.102, −0.204) for Mimas, (+14.568, +4.738) for Titan and (−48.760, +4.137) for Iapetus ⁶.

Meeus chapter 45: the ring's aspect

Chapter 45 returns six quantities: $B$ and $B'$, the saturnicentric latitudes of the Earth and the Sun referred to the ring plane. These are the ring opening anglesB and B′ (ring opening angles)B is the saturnicentric latitude of the Earth referred to the ring plane and B′ is that of the Sun. They set how open the ring ellipse looks and how far the ring shadow falls on the globe. Both are zero at a ring-plane crossing.. It also returns $\Delta U$, the difference between the saturnicentric longitudes of the Sun and the Earth (see ΔUΔUThe difference between the saturnicentric longitudes of the Sun and the Earth, measured in the ring plane. With B and B′ it fixes on which side of the globe the shadows of the globe and rings appear.), the position angleposition angleAn angle measured on the sky at the centre of a disk, from celestial north through east, used to locate a point on the limb of the Sun or Moon. Contact points and limb heights are indexed by it. $P$ of the northern semi-minor axis, and the axes $a$ and $b$ of the outer edge of the outer ring ⁷. The same caveat applies: the formulas are quoted from the Go transcription, whose test reproduces the book's Example 45.a ⁷.

Ring plane. With $T$ in Julian centuries from J2000.0, the inclination and ascending node of the ring plane on the ecliptic are ⁷:

Geocentric Saturn. The Earth's heliocentric $l_0, b_0, R$ and Saturn's $l, b, r$ come from VSOP87, converted to FK5. Saturn's are taken at $t - \tau$ in a two-pass light-time loop ⁷:

Opening angle seen from the Earth and ring axes.

The inner edges are fixed fractions of $a$ and $b$ ⁷. The fraction is 0.8801 for the inner edge of the outer ring and 0.8599 for the outer edge of the inner ring. It is 0.6650 for the inner edge of the inner ring and 0.5486 for the inner edge of the "dusky" ring.

Opening angle seen from the Sun. Saturn's heliocentric position is corrected with $N = 113.6655^{\circ} + 0.8771^{\circ}\,T$ ⁷:

Longitude difference.

Position angle. The pole of the ring plane is at ecliptic $\lambda_0 = \Omega - 90^{\circ}$, $\beta_0 = 90^{\circ} - i$. Saturn's $\lambda, \beta$ get the annual aberration ⁷:

Nutation in longitude $\Delta\psi$ is added to $\lambda_0$ and $\lambda$. Both points are converted to $(\alpha_0, \delta_0)$ and $(\alpha, \delta)$ with the true obliquity, and ⁷:

Test vector. For JDE 2448972.50068 the test reproduces Example 45.a: $B = 16.442^{\circ}$, $B' = 14.679^{\circ}$, $\Delta U = 4.198^{\circ}$, $P = 6.741^{\circ}$, $a = 35.87''$, $b = 10.15''$ ⁷.

Saturn's figure and the rings in kilometres

The globe. NASA's fact sheet gives an equatorial radius of 60,268 km and a polar radius of 54,364 km at the 1 bar level, and a flattening of 0.09796 ⁸. The NAIF constants kernel carries the same triaxial radii, 60268, 60268 and 54364 km, and the IAU pole at right ascension $40.589^{\circ} - 0.036^{\circ}T$ and declination $83.537^{\circ} - 0.004^{\circ}T$ ⁹. Three other radii appear in the sources and must not be confused with the Saturn equatorial radiusSaturn equatorial radiusThe radius of Saturn's equator at the 1 bar pressure level, 60,268 km in the IAU and NSSDC values. It is not the 60,330 km or 60,000 km reference radii that some gravity models use.:

• 60,330 km is the reference radius of the gravity harmonics in SAT441 ⁵.
• 60,000 km is the reference radius of Harper and Taylor's $J_2$ and $J_4$ ².
• About 60,444 km is the unit implied by Meeus's perspective constant 2475, derived above from the transcribed code ⁶.

A predictor of transits and occultations by the globe must use the oblate outline. The cited equatorial and polar radii differ by 5,904 km, computed from the 60,268 km and 54,364 km values, so a spherical model misplaces a contact on the polar part of the limb by up to that distance. Some services ignore it, as the phenomena note records.

The rings. NASA's ring fact sheet gives the main edges in kilometres from Saturn's centre, with normal optical depth $\tau$ ¹⁰:

Meeus's $a$ and $b$ are full axes, so the radius of an edge is half the axis. The third column converts half of Meeus's $a = 375.35''$ at 1 au into a radius with 725.27 km per arcsecond. That scale follows from the au of 149,597,870.7 km used in SAT441's constants ^{7 5}. Meeus's outer edge is 0.5 per cent smaller than the fact sheet's. His inner edge of ring A is 2.1 per cent smaller, which narrows the Cassini Division on a drawn ring. These differences are below the resolution of a sketch. They matter for a prediction of when a satellite enters the Cassini Division's shadow. The B ring is thick enough, with $\tau$ up to 2.5, to hide a satellite almost completely. The C ring and Cassini Division pass much of the light ¹⁰. How these optical depths enter an eclipse calculation, and how a sky-plane position becomes a ring-plane radius, is in the phenomena note.

The ring plane is not fixed. Solar torques make it precess. The 1993 occultation results implied "an advance of ~2 hours in the time of ring plane crossing after a 15 yr interval" ²⁷. Meeus's polynomial for $i$ and $\Omega$ and the IAU pole in the PCK are both smooth fits. Neither carries that precession explicitly ^{7 9}.

Software and the theory each one uses

Stellarium's rearrangement changes the independent variable to $T = \mathrm{JD} - 2444240$ for all satellites. It also precomputes constant terms and merges terms of equal frequency ³⁰. Cartes du Ciel states that its satellite code is a Pascal rewrite of Xplanet's libmoons, so that Cartes du Ciel and Xplanet draw satellites in the same places ³². The Astronomical Almanac's satellite section has been printed from compact series. A 1995 Royal Greenwich Observatory abstract describes moving it to the Chapront and Vu mixed-function form ³⁴. No primary source in this corpus states which ephemeris feeds the current Almanac's Saturn pages.

Sources compared

What a developer should do

1. Use SAT441 through SPICE for anything that will be compared with observations. Load sat441.bsp with a DE440 kernel, and subtract body 699 to get saturnicentric vectors ^{5 26}.
2. Use TASS 1.7 when an analytic, file-free model is needed. Expect residuals of tens of milliarcseconds in right ascension, about three times SAT351's in 2009 ^{4 14}.
3. Treat Meeus chapter 46 as a drawing aid. It carries Dourneau's hundreds-of-kilometres accuracy and VSOP87 planets. Use its two test vectors to validate a port, and do not use it for event times ^{1 6}.
4. Use Meeus chapter 45 for B, B′, ΔU and P. These angles are smooth and the chapter is adequate for them. Take the ring radii in kilometres from the fact sheet rather than the fixed fractions if the product will predict ring-shadow events ^{7 10}.
5. Model the globe as an oblate spheroid of 60,268 by 54,364 km. Keep the gravity reference radii out of the geometry ^{9 5}.
6. Read first: Arlot et al. 2012 for the accuracy table, Vienne and Duriez 1995 for the theory, and the SAT441 comment file for the current JPL fit.

What this changes

The eclipse pipeline's ephemeris layer gains a second level. Planet positions come from DE440, and satellite positions relative to their planet come from a satellite file such as SAT441. The two must be consistent, which SAT441's own DE440 header guarantees ⁵. The body-shape layer gains a non-spherical primary with a ring system, which the solar-eclipse pipeline never needed.

Open questions

• Meeus's source statement. Obtain Astronomical Algorithms, second edition, chapters 45 and 46, and record the book's own attribution of the satellite theory and of the ring fractions. The attribution here rests on matched constants.
• The Astronomical Almanac's current Saturn source. Obtain the "Notes and References" section of a recent Astronomical Almanac and record which SAT or NOE file feeds Section F.
• TASS 1.7 drift today. Obtain or compute a TASS 1.7 minus SAT441 difference series for 2024 to 2026, the PHESAT season, to replace the 2008 "about 100 km" estimate.
• Jacobson 2022. Read AJ 164, 199 for SAT441's stated position uncertainties per satellite, which the comment file does not give.