Mutual events of the Galilean satellites

The question. When do the Galilean satellitesGalilean satellitesJupiter's four large moons, Io (I), Europa (II), Ganymede (III) and Callisto (IV), discovered by Galileo in 1610. They orbit close to Jupiter's equatorial plane at about 5.9, 9.4, 15.0 and 26.3 Jupiter radii. produce mutual eventsmutual eventAn occultation of one natural satellite by another as seen from the Earth, or an eclipse of one satellite by another's shadow as seen from the Sun. Such events happen only for months around a planet's equinox, when the Earth or the Sun lies close to the satellites' orbital plane., occulting and eclipsing one another? How are those events predicted, what photometric model turns a light curve into an astrometric position, and how accurate are the positions from each campaign?

When mutual events happen

Saquet et al. give the condition in one sentence. "Occultations can occur when the Joviocentric declination of the Earth becomes zero, whereas eclipses can occur when the Joviocentric declination of the Sun becomes zero" ¹. These are the jovicentric declinationsjovicentric declination of the Earth or SunThe latitude of the Earth (D_E) or the Sun (D_S) above Jupiter's equatorial plane as seen from Jupiter's centre. It stays within about 4 degrees, and it crosses zero at each Jovian equinox. $D_E$ and $D_S$. Meeus's low-accuracy method gives them ¹⁰:

with $\lambda = 34.35^{\circ} + 0.083091^{\circ}\,d + 0.329^{\circ}\sin V + B$, $d$ in days from J2000, and $\psi$ the phase angle, as in theories and ephemerides. $\lambda$ advances by one revolution in one Jovian year. $D_S$ crosses zero twice per revolution, so the seasons recur every 5.93 years. Jupiter's equator is inclined about 3.1 degrees to its orbit, so $D_S$ never exceeds that. The fastest rate of change of $D_S$ is $3.12^{\circ} \times 0.0831^{\circ} \times \pi/180 = 0.0045^{\circ}$ per day, or 1.6 degrees per year.

A rough window follows. Two satellites at radii $a_i < a_j$ in conjunction on the same side of Jupiter are separated in the sky-plane Y direction by about $(a_j - a_i)\sin D_E$. An occultation needs that to be smaller than the sum of their radii. For Io and Europa, $(671{,}100 - 421{,}800)\sin D_E < 1829 + 1563$ km gives $|D_E| < 0.78^{\circ}$, with the radii from the IAU values in NAIF's PCK ¹¹ and the orbital radii from Westfall ¹². At 1.6 degrees per year that window lasts about a year. This estimate neglects the satellites' own inclinations to Jupiter's equator. It is consistent with the 2014 to 2015 season, which ran from 2014 September 1 to 2015 July 20 ¹.

Saquet et al. expected the 2021 season to be less favourable, "since the maximum of events will occur at the conjunction of Jupiter with the Sun, and 192 events will be observable" ¹. The 1979 season had the same problem, so "very few phenomena were observed" ³. Arlot's 2014 slides give the recurrence for the other giant planets: Jupiter every 6 years, Saturn every 15, Uranus every 42 ¹³. Saturn's satellites are treated in their own topic.

Mutual events come in several shapes. The observing guide of IMCCE's Natural Satellites Data Center (NSDC) lists "occultation or eclipse, total, partial or annular", and notes that the light curve differs between these types ¹⁴. An eclipse and an occultation of the same pair can also coincide. So et al. call this a quasi-simultaneous mutual event and observed one between Europa and Ganymede on 22 August 2021 ¹⁵.

Campaigns from 1973 to 2021

The first predictions were made in 1956, but no observation of them is known. Brinkmann and Millis predicted the 1973 events and called for observations ³. Aksnes predicted them independently from Sampson's theory with Andoyer's method. He found discrepancies of "up to 30 minutes of time for J3 and J4" between the two prediction sets ³. The 1973 campaign of Aksnes and Franklin collected 91 light curves. It gave the radii of Europa, Ganymede and Callisto to 27, 32 and 75 km and corrections to Sampson's orbital elements ³. Lieske used 85 mutual events in his 1978 and 1980 fits, which led to E-2 ^{3 16}.

IMCCE, formerly part of the Bureau des longitudes, has organised the PHEMUPHEMU campaignAn international campaign, coordinated by IMCCE, to record light curves of mutual events of the Galilean satellites in one equinox season. Named by year, for example PHEMU15 for 2014 to 2015. campaigns since. Saquet et al. give the statistics in their Table 1 ¹.

The PHEMU97 catalogue paper itself reports "275 observations of 148 mutual events from 42 sites" ¹⁷. The PHEMU03 catalogue reports "377 observations of 118 mutual events from 42 sites" ¹⁸. For PHEMU15, IMCCE received 643 light curves and computed astrometric results for 609. One event was recorded 17 times ¹. For PHEMU21, MULTI-SAT predicted 242 events, of which 192 could be observed between 2021 March 3 and November 16 given Jupiter's angle from the Sun ¹⁵.

Predicting the events

IMCCE published predictions as plain tables before each campaign. The 1997 file lists "469 mutual eclipses and occultations", with local visibility files for 19 observatories and PC software to compute local conditions ¹⁹. The current tool is MULTI-SAT, run jointly by IMCCE and the Sternberg Astronomical Institute (SAI) in Moscow. Its paper describes it as a facility that produces ephemerides of all natural satellites with "special features ... such as predicting the phenomena and providing configurations useful for the observers" ²⁰.

The prediction tables list, per event, the start and end in UTC, the type in the form "nEm" or "nOm", the duration, the impact parameterimpact parameterThe smallest apparent distance during a mutual event between the centres of two satellites in an occultation, or between an eclipsed satellite and the axis of the other's shadow cone in an eclipse. Zero is a central event., the combined magnitude and its expected drop, and the distance from Jupiter's limb ². The impact factor "is equal to zero if the event is a central passage" and "equal to unit in the case of a grazing event" ². MULTI-SAT's PHEMU21 predictions came from the theory of Lainey et al. (2009) ¹⁵. PHEMU15 predictions used NOE-5-2010-GAL with INPOP13c ¹. Occult also predicts Galilean mutual events ¹⁵.

The geometry is the satellite analogue of a solar eclipse. For an occultation the shadow is replaced by the cone of visibility from the observer, and the reduction works in a topocentric frame. For an eclipse it works in a heliocentric frame ¹. The eclipsing satellite's penumbra at the eclipsed one has the width $w = 2\,\delta d\,R_\odot/r$ from the Jovian shadow, with $\delta d$ the distance between the two satellites along the shadow. For Io eclipsing Europa at $\delta d \approx 250{,}000$ km that is about 450 km, about 0.3 of Europa's radius. So et al. simplify this with one effective shadow radius "17% larger than the apparent radius of umbra" and "13% smaller than apparent radius of penumbra", from formulas of Emelyanov (2003) ¹⁵.

The photometric model

The observable is the combined flux of the satellites in one aperture against time. The NSDC guide explains why these events give better astrometry than eclipses by Jupiter. Most of the satellites "have no atmosphere disturbing the light curve because of the refraction of the light" ¹⁴. The photometric signal depends on how light is spread over each disc. The models have grown in steps:

The Hapke modelHapke modelA physical model of how a rough, porous surface scatters light as a function of incidence, emission and phase angles. Mutual-event fits use it, or the Oren-Nayar model, to set the brightness distribution across a satellite's disc. describes a rough particulate surface. VBO took the Hapke parameters of Domingue and Verbiscer (1997) for the leading and trailing sides of Europa, Ganymede and Callisto, and those of McEwen et al. (1988) for Io ⁶. They chose each satellite's brightness distribution so that it reproduces the ground-based rotational light curve. They found that albedo maps alone fit Io and Europa but not Ganymede or Callisto ⁶. 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. shift matters. For central events Vasundhara (2002) estimated shifts of "-50 to +90 km ( -14 to +25 mas) for Io and -30 to +50 km ( -8 to +14 mas) for Europa" ⁶. A uniform, limb-darkened model produced "a bias along the satellite track" consistent with that shift, and the albedo model removed it ⁶.

Catani et al. used "the complete version of the Oren-Nayer model" for the satellites. For a mutual eclipse they modelled the Sun's limb darkening with the law of Hestroffer and Magnan (1998) ⁵. This is the same physics as the solar limb-darkening functionlimb-darkening functionThe run of surface brightness with distance from the solar limb. Its steepness determines when a Baily's bead becomes visible, so bead timings depend on both the function and the detector's sensitivity. in solar eclipse bead work. In a mutual eclipse it shapes the penumbral part of the light curve. They measured the albedo ratio of the two satellites on the same night with the same instrument where possible ⁵. So et al. neglect light reflected from Jupiter in the flux model, citing Emelyanov et al. (2011) for the argument ¹⁵.

From light curve to position

Two reduction methods are in use. They give the same kind of result.

Emelyanov's method (IMCCE and SAI). Let $\Phi$ be the normalised flux, with $\Phi = 1$ outside the event. It is a function $\Phi(X(t), Y(t))$ of the projections of the satellites' angular separation on the celestial parallel and meridian. The ephemeris gives $X_{th}(t_i)$, $Y_{th}(t_i)$ at each observation time $t_i$. The method assumes constant corrections, $X(t) = X_{th}(t) + D_x$ and $Y(t) = Y_{th}(t) + D_y$. It then solves

for $D_x$, $D_y$ and the scale $K$ by linearised least squares, where $E_i$ is the observed flux ¹. The published result is $X = X_{th}(t) + D_x$, $Y = Y_{th}(t) + D_y$ at the instant of closest apparent approach. Here $X = \Delta\alpha\cos\delta$ and $Y = \Delta\delta$, "occulting minus occulted" or "eclipsing minus eclipsed" ¹. For a total event only the position angle $P$, with $\tan P = X/Y$, can be found, and only to within 180 degrees ¹.

The impact-parameter method (Brazil, VBO). This method fits the impact parameter $d_0$, the central instant $t_0$ and the relative velocity $v_0$, with the apparent distance $\rho(t) = \sqrt{d_0^2 + v_0^2 (t - t_0)^2}$ ⁵. It generates about nine hundred simulated light curves for each event around the ephemeris values and keeps the minimum chi-square ⁵. VBO fits a shift $s$ along the track and a change $\Delta IP$ in the impact parameter. With $P$ the position angle of Jupiter's pole, they convert these to

assuming the track is perpendicular to the pole ⁶. The observed instant of closest approach is $T_{obs} = T_{pred} - s/v$ ⁶.

Timing must be good to better than 0.1 s. "The satellite Io ... has a velocity of 17.2 km s⁻¹, so that an accuracy of 0.1 s of time corresponds to an accuracy of 1.7 km in space". The internal accuracy of the theory is "around 1 km" ¹. The NSDC asks for each photometric measure to be referred to UTC "within an accuracy at least of 0.1 second of time" ¹⁴.

Achieved accuracy

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. is about 3 km at Jupiter. Saquet et al. convert 24 milliarcseconds to 75 km and 50 milliarcseconds to 150 km ¹. Catani et al. use 5 milliarcseconds for 15 km ⁵. The campaign-wide O-CO − 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. figures mix professional and amateur light curves, so they measure the data as much as the ephemeris. The single-site results show what the method can do. Catani et al. found no mean offset larger than one sigma against DE440 with JUP365. They concluded that the ephemerides agree with the 2021 data "at the level of 5" milliarcseconds ⁵.

What the data have been used for

Mutual-event positions enter every modern ephemeris fit. NOE-5-2010-GAL and JUP310 both use the PHEMU85 to PHEMU03 data, with different astrometric data sets otherwise ¹. In Lainey et al. (2009), "the observations of mutual events, known to be among the most accurate observations, have a 1σ accuracy of about 0.025 arcsec and provide the best constraint of the satellite orbits for the past decades" ⁹. That fit gave $k_2/Q = 0.015 \pm 0.003$ for Io and $(1.102 \pm 0.203)\times 10^{-5}$ for Jupiter. It gave secular accelerations that shift Io, Europa and Ganymede by 55, 2125 and 2365 km over 116 years ⁹.

The events also carry physics. Infrared observations of mutual events measure the volcanic activity of Io ³. The 1973 light curves gave satellite radii and the first albedo maps of Europa ³. The method also reaches the inner moons. A 2021 eclipse of Thebe by Ganymede gave Thebe's position to 32 milliarcseconds, 96 km ⁵.

The 2026 to 2027 season

The next season is under way. A February 2026 observer presentation lists 305 predicted events between 2026 May 11 and 2027 August 10, generated with the SAI natural satellites service ². It quotes Saquet et al. on why the observations matter: "the precision of the mutual event observations is up to 20 mas for the Galilean moons (Arlot et al. 2014), or 60 km at the distance of Jupiter" ^{2 1}. Between seasons, mutual approximations give central instants at any time of year, with 11.3 milliarcseconds internal precision from 66 events in 2016 to 2018 ⁸.

Sources compared

What a developer should do

1. Predict with a modern ephemeris and both frames. Compute occultations topocentrically and eclipses heliocentrically, with JUP365 and DE440 or NOE-5-2021 ^{1 21 22}.
2. Report what MULTI-SAT reports. Give start and end in UTC, the "nEm"/"nOm" type, duration, impact factor, magnitude drop and distance from Jupiter's limb ².
3. Model the penumbra of the satellite's shadow. Its width is $2\,\delta d\,R_\odot/r$, a large fraction of the eclipsed satellite's radius. A single effective radius is a known simplification ¹⁵.
4. For astrometry, use a surface model. Use an albedo map or Hapke parameters with phase, and solar limb darkening for eclipses. A uniform disc biases positions by up to tens of milliarcseconds ^{6 5}.
5. Validate against the PHEMU catalogues in IMCCE's NSDB and against the published O-C distributions ¹.

Read first: Saquet et al. (2018), Catani et al. (2023), Vasundhara et al. (2017).

What this changes

Mutual events add a third use of the shadow-cone code, between two small spheres instead of between the Sun, a large body and a small one. The geometry is the Besselian construction with the eclipsing satellite as the Moon and the eclipsed satellite as the Earth. The new part is photometric. The pipeline would need a model of the target's surface brightness and of the Sun's limb darkening, integrated over the overlap of two discs at each time step. The solar-eclipse pipeline already computes the overlap of two discs for obscuration.

Open questions

• Arlot et al. (2014) and Emelyanov et al. (2022), full texts. Obtain both to quote their photometric models, the Hapke parameters used, and their per-satellite accuracy.
• Emelyanov (2003) penumbra formulas. Obtain the paper that So et al. cite for the effective umbra and penumbra radii, to quote the formulas.
• Saquet et al. (2018) model section. The paper gives the reduction method but cites Emelyanov and Gilbert (2006) for the photometric model. Obtain Emelyanov and Gilbert (2006), A&A 453, 1141, for the model equations.
• PHEMU27 predictions. Obtain IMCCE's or SAI's published prediction list for the 2026 to 2027 season, with the theory used, to replace the count taken from a presentation.
• Oren-Nayar against Hapke. Find a comparison of the two reflectance models on the same mutual-event light curves.