The Orbits of Jupiter's Irregular Satellites

Orbits of the irregular Jovian satellites are strongly perturbed by the Sun and cannot be entirely described by an analytical theory. In order to obtain high-precision satellite ephemerides, we numerically integrated equations of motion as described in Peters (1981). The equations are defined in Cartesian coordinates, with Jovian system barycenter as the origin. The axes are referenced to the International Celestial Reference frame (ICRF). We used barycentric dynamical time (TDB) to remain consistent with Jet Propulsion Laboratory (JPL) planetary ephemerides. The gravitational field of Jupiter includes the zonal harmonic coefficients J₂, J₃, J₄, and J₆. The model includes the perturbations from the Galilean satellites, Saturn, Uranus, Neptune, and the Sun. The mass of the Sun was modified to include the terrestrial planets and the Moon. We used JPL's satellite ephemeris JUP310 (Jacobson, unpublished) for the positions of the Galilean satellites. JPL's planetary ephemeris DE435 (Folkner 2014) was used for the positions of the Sun and planets. The complete list of dynamical constants used in our integration is shown in Table 1.

The integration used a variable size time step and the variable-order Gauss–Jackson method. We imposed a maximum velocity error of 10⁻¹⁰ km s⁻¹ in order to control the integration step size. As a result, the average integration step size was ∼4800 s. We used a weighted least-squares procedure based on Householder transformations (Lawson & Hanson 1974) to adjust the epoch state vectors for the satellites. The GMs, the products of the Newtonian constant of gravitation G and the satellites masses M, were initialized at 10⁻²⁰ km³ s⁻² but were otherwise allowed to adjust.

The long data arc and good quality of astrometric measurements are the prerequisites for high-precision orbital determination. Most of the irregular Jovian satellites have periods of about 2 yr, which makes it relatively easy to acquire good orbital coverage. However, many of these objects are faint, H > 23 mag, which puts a requirement on large telescopes as the observing instruments.

Astrometric observations for the first-discovered Jovian irregulars, Himalia, Elara, Pasiphae, and Sinope, date back to the late nineteenth and early twentieth centuries, and long data arcs also exist for Lysithea, Carme, and Ananke. The rest of the satellites were discovered relatively recently, in the early 2000s, and some of them have not been reobserved since their discovery. Jacobson et al. (2012) described the efforts to recover some of the satellites that were considered to be nearly lost, and in the process, two more Jovian irregulars were discovered (Alexandersen et al. 2012).

The large majority of astrometric observations originate from Earth-based telescopes, although there are a handful of observations of Himalia and Callirrhoe from the New Horizons spacecraft flyby of Jupiter. The modern Hipparcos Catalog (Perryman et al. 1997) based astrometry is reported as positions in the ICRF. We convert the older measurements to the ICRF positions. The references to optical observations up to the year 2000 are documented in Jacobson (2000). We continued to use the Jacobson (2000) observational biases for the early measurements. We have since extended the data set with observations published in the Minor Planet Electronic Circulars (MPEC), the International Astronomical Union Circulars (IAUC), the Natural Satellites Data Center (NSDC) database (Arlot & Emelyanov 2009), the United States Naval Observatory Flagstaff Station catalog (www.nofs.navy.mil/data), and the Pulkovo Observatory database (www.puldb.ru/db). An extensive data set spanning 23 yr of observation of 18 irregular satellites (12 Jovian satellites) has been published in 2015 by Gomes-Júnior et al. (2015).

Our resulting data set contains more than 30,000 astrometric points from over 100 observatories. The measurements were weighted by either observer-assigned uncertainties or revised uncertainties that were scaled by an inverse of the rms of the residuals for a particular observer.

Table 2 shows the time span and number of points used in orbital fits, as well as the post-fit residual statistics. We list the rms of the post-fit residuals in right ascension (R.A.) and declination (decl.) in units of arcseconds and in units of standard deviation, σ. The residuals of 05 are expected for orbital fits to faint, distant objects such as the irregular Jovian satellites. Our weighting scheme intended to bring the rms of the residuals converted to the units of standard deviation close to 1σ. Some of the reduced values in Table 2 are smaller because we did not weight the data below the star catalog errors, which are estimated to be around 030.

Table 2.  Post-fit Residual Statistics

Satellite	Time Span	No.	rms		Reduced rms
			R.A.	decl.	R.A.	decl.
Himalia	1894–2016	3257/3251	0410	0373	0.632σ	0.619σ
Elara	1905–2016	1877/1876	0372	0365	0.692σ	0.710σ
Lysithea	1938–2016	772	0357	0317	0.727σ	0.690σ
Leda	1974–2015	308/309	0432	0396	0.755σ	0.738σ
Dia	2000–2011	64	0368	0395	0.789σ	0.848σ
Themisto	1975–2015	104	0551	0412	0.827σ	0.751σ
Carpo	2003–2009	39	0313	0294	0.832σ	0.960σ
Ananke	1951–2015	1042/1043	0399	0456	0.795σ	0.802σ
Harpalyke	2000–2011	84	0409	0343	0.916σ	0.842σ
Iocaste	2000–2015	78	0348	0408	0.931σ	1.008σ
Praxidike	2000–2016	70	0216	0330	0.719σ	0.952σ
Thyone	2001–2003	30	0204	0319	0.680σ	0.886σ
Hermippe	2001–2011	49	0329	0330	0.840σ	0.820σ
Euanthe	2001–2009	23	0258	0138	0.822σ	0.459σ
Orthosie	2001–2010	26	0233	0259	0.776σ	0.756σ
Mneme	2002–2011	51	0279	0284	0.846σ	0.811σ
Thelxinoe	2002–2011	32	0656	0679	1.041σ	1.078σ
Helike	2003–2010	44	0580	0334	0.959σ	0.841σ
2003 J3	2003	15	0235	0202	0.784σ	0.672σ
2003 J12	2003	11	0214	0189	0.713σ	0.630σ
2003 J15	2003	12	0220	0248	0.734σ	0.826σ
2003 J16	2003–2011	98	0222	0134	0.679σ	0.447σ
2011 J1	2011	17	0502	0350	1.005σ	1.000σ
2003 J18	2003–2011	40	0400	0339	0.678σ	0.447σ
Euporie	2001–2011	27	0528	0569	1.056σ	1.138σ
Carme	1938–2016	1483	0443	0447	0.787σ	0.783σ
Taygete	2000–2003	53	0221	0402	0.737σ	0.608σ
Chaldene	2000–2012	59	0346	0393	0.772σ	0.891σ
Kalyke	2000–2010	78	0336	0506	0.809σ	0.985σ
Erinome	2000–2010	62	0276	0409	0.735σ	0.871σ
Isonoe	2000–2011	88	0403	0392	0.839σ	0.899σ
Aitne	2001–2011	37	0302	0288	0.709σ	0.771σ
Kale	2001–2010	31	0401	0378	0.940σ	0.849σ
Pasithee	2001–2011	24	0431	0327	0.826σ	0.578σ
Arche	2002–2011	39	0253	0317	0.652σ	0.793σ
Kallichore	2003–2010	29	0457	0401	0.920σ	0.858σ
Eukelade	2003–2011	38	0370	0284	0.903σ	0.725σ
Herse	2003–2011	48	0275	0289	0.705σ	0.818σ
2003 J5	2003	22	0346	0302	0.831σ	0.969σ
2003 J9	2003	17	0256	0370	0.853σ	1.057σ
2003 J10	2003	11	0294	0556	0.839σ	1.112σ
2003 J19	2003	10	0118	0253	0.395σ	0.760σ
2010 J1	2003–2011	166	0396	0308	0.827σ	0.755σ
2010 J2	2010–2011	116	0343	0298	0.873σ	0.805σ
Pasiphae	1908–2016	2615	0447	0449	0.759σ	0.759σ
Sinope	1914–2016	1352	0490	0455	0.874σ	0.860σ
Callirrhoe	1999–2016	175	0338	0371	0.734σ	0.840σ
Magaclite	2000–2016	75	0403	0461	0.788σ	0.691σ
Autonoe	2001–2011	46	0327	0481	0.970σ	0.867σ
Eurydome	2001–2011	35	0206	0371	0.687σ	0.813σ
Sponde	2001–2011	30	0212	0233	0.667σ	0.638σ
Hegemone	2002–2011	39	0484	0297	0.881σ	0.661σ
Aoede	2002–2015	38	0320	0394	0.819σ	0.943σ
Cyllene	2002–2009	24	0179	0499	0.566σ	0.952σ
Kore	2003–2011	35	0198	0198	0.647σ	0.636σ
2003 J2	2003	8	0163	0109	0.542σ	0.365σ
2003 J4	2003	11	0137	0191	0.458σ	0.636σ
2003 J23	2003	16	0392	0300	0.989σ	0.966σ
2011 J2	2011–2012	25	0484	0481	0.850σ	0.808σ

Note. All R.A. residuals in this manuscript have been scaled with the cosine of declination.

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For satellites with long data arcs such as Himalia, Elara, Pasiphae, Sinope, Lysithea, Carme, and Ananke, our goal was to balance the χ² contributions from old and new astrometry. Figures 1 and 2 show residuals of the individual measurements of Himalia, Carme, Ananke, and Pasiphae. These plots are useful to get a quick assessment of the timeline of observations and to check whether any of the individual measurements have residuals that are too high. The flat scatter of the residuals demonstrates that both old and new data were fit at their assigned accuracy levels.

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Finally, we need to be careful when interpreting the residuals of fits to sparse data sets; such is the case for objects discovered in 2003. A number of these (2003 J2, 2003 J3, 2003 J10, 2003 J12, 2003 J15, 2003 J19) have less than two dozen measurements. Here, the rms of the residuals can be artificially small because the data points originate from few observing days and a single observer.

We list the epoch state vectors for Himalia, Carme, Ananke, and Pasiphae in Table 3, while the rest are shown in the supplementary material. These state vectors contain a large number of decimal places and can be used as the starting points in any future numerical integrations.

The Sun is the dominant perturber of the irregular Jovian satellites, but we also investigated the strength of the perturbations from Saturn, Uranus, and Neptune. In this part of the analysis, orbital fits were reconverged without a particular planet in the perturbers list and compared to the solution that had the full set of perturbers. The comparison of in-orbit, radial, and out-of-plane differences was done for 1900–2016. This is the length of time for which we have observations (at least for Himalia, Elara, and Pasiphae) and the influence of the perturbers could be detected.

The rms of the differences between the models that fit the data with and without Saturn is less than 1000 km for most satellites in the prograde group. The exception is Carpo, with rms ∼ 5000 km. Uranus has a very small influence on the progrades, a few tens of kilometers at most. The influence of Saturn is more significant for the retrograde satellites. The rms of the perturbations due to Saturn is tens of thousands of kilometers during the 1900–2016 period. The perturbations from Uranus are much less pronounced, up to a few hundred kilometers. These results are consistent with the preliminary conclusions in Jacobson (2000). For completeness, we also tested for the influence of Neptune, but the perturbations turned out to be on the order of several tens of kilometers, which is negligible when compared to the current data accuracy.

Emelyanov (2005b) noted that Himalia and Elara had an ∼65 K km close encounter on 1949 July 15 and that Elara's orbital fit depends on the mass of Himalia. Himalia's mass determination was based on a subset of 280–326 measurements of Elara before and after the 1949 encounter. The obtained GM for Himalia was 0.28 ± 0.04 km³ s⁻². This study also showed that observations of Lysithea did not improve Himalia's mass result despite one relatively close approach at 169 K km in 1954.

Our analysis had more measurements of Elara and Lysithea than Emelyanov (2005b). We used 129 R.A. and decl. measurements of Elara prior to 1949 and 1877 R.A. and 1876 decl. points between 1905 and 2016. There were 44 R.A. and decl. measurements of Lysithea prior to Himalia's encounter in 1954 and 772 astrometry points between 1938 and 2016. We obtained GM = 0.13 ± 0.02 km³ s⁻², a formal 1σ error, which is about a factor of two smaller than the Emelyanov (2005b) mass.

We tested the robustness of our result by hard-wiring Himalia's mass to 0.28 km³ s⁻² and 0.56 km³ s⁻², or twice the Emelyanov (2005b) mass. For each case, we reconverged the orbital fit and calculated the rms of the residuals for Elara and, for completeness, Lysithea. We estimated a statistically significant change in the rms of the residuals as rms $\sqrt{(}2N)/N$ (Hogg & Tanis 1993), where N is the number of degrees of freedom. This expression stems from the properties of χ² distribution and the fact that it has a mean and variance of N and 2N, respectively. From here, the standard deviation of the reduced χ² distribution is $\sqrt{(}2N)/N$ or $\sqrt{(}2/N)$. This metric is acceptable assuming that the data have normally distributed errors with standard deviation of one and no cross-correlation.

Table 4 lists Himalia's masses and the corresponding reduced rms of the residuals for Elara and Lysithea. We found that the mass of Himalia needs to be almost twice as large as Emelyanov (2005b) before a statistically significant change occurs in the rms of the residuals for Elara. Despite using more measurements for Lysithea than Emelyanov (2005b), we also concluded that its orbit shows no significant sensitivity to Himalia's mass.

Table 4.  Reduced rms of the Residuals of Elara and Lysithea as a Function of Himalia's Mass

Himalia Mass	Density	Reduced rms Elara	Reduced rms Lysithea
(km3 s−2)	(g/cm3)	($\sqrt{R.A{.}^{2}+{decl}{.}^{2}}$)	($\sqrt{R.A{.}^{2}+{decl}{.}^{2}}$)
0.13	1.55	0.9914σ	1.0027σ
0.28	2.26	1.0023σ	1.0057σ
0.56	6.52	1.0700σ	1.0192σ

Note. The number of points used in orbital fits is listed in Table 2.

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Figure 3 shows in-orbit, radial, and out-of-plane differences for Elara's orbit when we use our nominal mass for Himalia and the Emelyanov (2005b) mass for Himalia. The comparison is shown for the entire duration of Elara's data, 1935–2016. The largest difference of ∼2000 km is along the orbital track, and it is for the data taken just after the Himalia–Elara encounter in 1949. At Jupiter distance, 2000 km is ∼067, which is smaller than the observational errors for the data taken in the 1950s. This plot justifies our claim that the two mass values are statistically indistinguishable, but it also shows that the mass of Himalia cannot be that much larger than that given by Emelyanov (2005b) because the orbital differences would grow to a level that can be detected.

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Table 4 also lists densities of Himalia. The density was calculated based on Cassini estimates (Porco et al. 2003) of the satellite's radius: 75 ± 10 km and 60 ± 10 km. We adopted an average radius of 67.5 km. Our nominal mass result suggests that Himalia is a low-density object. A factor of two increase in the Emelyanov (2005b) GM has an unlikely density of 6.5 g cm⁻³, and the residuals appear significantly worse. We thus conclude that Himalia's mass is in the 0.13–0.28 km³ s⁻² range. We attempted to include other satellite masses in the orbital fits, but the data showed no sensitivity.

The dominant sources of errors in orbital fits are star catalog errors and measurement errors. These errors are particularly relevant for the data taken before the CCD technology and modern data reduction techniques. All other components of the orbital fit have errors that are significantly smaller. The orbital model and the dynamical constants used in the model are well established. The uncertainties in the DE435 ephemerides of Jupiter are on the order of few tens of kilometers. The GMs of Jupiter and perturbing bodies are well known from the spacecraft data, so they are also not a significant factor in the fit accuracy.

We used a linear covariance mapping as described in Jacobson et al. (2012) to asses the orbital fit quality. Jacobson et al. (2012) evaluated the projection of the uncertainties on the plane of sky in R.A. and decl. on a particular date that was either three or 10 orbital periods beyond 2012 January. This method produced a result that is dependent on the orbital geometry at a particular date. The R.A. and decl. uncertainties on the plane of sky can be small for a short period of time and grow orders of magnitude for other dates.

In order to account for the uncertainties changing with time, we mapped the covariance over 2 yr, 2009–2011 January, 2019–2021 January, and 2029–2031 January. Two years is an approximate duration of orbital periods for most Jovian irregulars. We examine the rms of the uncertainties in the in-orbit, radial, and out-of-plane directions during these times. For completeness, we also project the uncertainties on the plane of sky and evaluate the maximum plane-of-sky (combined R.A. and decl.) uncertainty. These maximum values should provide good constraints on the search region if further observations are planned. Furthermore, we get a quick assessment of the orbital status, i.e., the satellite position is known, recoverable, or lost. We consider anything with a plane-of-sky uncertainty larger than 1000'' to be lost. For comparison, Jupiter's average Hill sphere is 14,000''.

Table 5 list results for Himalia, Pasiphae, Ananke, and Carme, while other satellites are listed in the supplementary material. We note that in-orbit (or transverse) uncertainties grow as a result of uncertainties in the mean motion. We plot the growth of the plane-of-sky uncertainties from 2010 to 2030 for Himalia, Pasiphae, Ananke, and Carme in Figure 4. The general secular growth for the four satellites in Figure 4 is below 70 mas until at least 2030, which means that these satellites are in no danger of being lost.

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There are currently 11 irregular satellites, all with provisional IAU designations, that have observations spanning less than 1 yr. Ten of these satellites have not been observed since 2003, which means that they are likely lost. Likewise, one of the satellites discovered in 2011, 2011 J1, has the plane-of-sky uncertainty of hundreds of arcseconds and is on the verge of being lost. Figure 2 in Jacobson et al. (2012) shows that even a satellite with few hundred arcsecond plane-of-sky uncertainty could be recovered with a telescope that has large enough field of view. The observations also need to be carefully planned so that the satellite is observed when the uncertainties have favorable orbital geometry projection.

Results of orbital integration can be expressed in terms of planetocentric mean orbital elements (Brozović & Jacobson 2009; Brozović et al. 2011; Jacobson et al. 2012). We take the state vectors listed in Table 3 and propagate them backward and forward in time from 1600 February to 2599 September using the same set of perturbers as described in Section 2. This produces positions of the satellites over a period of 1000 yr of orbital integration. These "measurements" are compared to the positions from an analytical model, a precessing ellipse. The initial elements and rates of the model are refined via the least-squares procedure until the fit shows no further improvement. The ellipse captures the constant and secular properties of the integrated orbit, and the differences that remain are due to periodic perturbations. The mean elements listed in Table 6 are a descriptive representation of the integrated orbits; hence, there are no uncertainties associated with the elements. We have already discussed the orbital uncertainties for the integrated orbits in Section 3.5.

We use a ratio of the apsidal and nodal precession rates, ν, to classify the orbits similarly to Ćuk & Burns (2004). We define the longitude of periapsis as ϖ = Ω + ω for prograde orbits and ϖ = Ω − ω for retrograde orbits. Here, Ω is longitude of the ascending node and ω is argument of periapsis. If the argument of periapsis is circulating slower than the node, the object is marked as a reverse circulator (RC). If the argument of periapsis is circulating faster than the node, the object is marked as a direct circulator (DC). The satellites that are the Lidov–Kozai librators (Lidov 1962; Kozai 1962) have ν = −1. The objects that are close to the secular resonance with Jupiter have their rates of apsidal precession matched to that of Jupiter, $d\varpi /{dt}-d{\varpi }_{\mathrm{Jupiter}}/{dt}=0$, and consequently, they have ν = 0. Ćuk & Burns (2004) also classified objects as main-sequence circulators (MSC) or non-main-sequence (non-MS) residents with respect to how close they are to the secular resonance with Jupiter. The MSC objects are direct circulators with ν < 0.2, or reverse circulators with orbital parameters between the Lidov–Kozai and secular resonances. The non-MS objects have ν ≥ 0.2.

Table 6 shows that all objects considered to be part of Himalia's prograde group, that is, Himalia, Elara, Lysithea, Leda, and Dia, are direct circulators. Themisto and Carpo clearly do not belong to this group as one is a reverse circulator and the other is in the Lidov–Kozai resonance. Themisto orbits Jupiter at 75 M km, and its orbital period is only 130 days.

The retrograde group is about equally split into reverse and direct circulators, except Euporie, which is another Lidov–Kozai resonator, and Pasiphae, which is in secular resonance with Jupiter (Whipple & Shelus 1993). According to the present orbital solution, 2003 J18 appears to reside close to the Lidov–Kozai region. In addition, there are some interesting outliers, such as 2003 J2 and 2003 J12, which have orbital periods of 981 and 490 days, respectively, while all other retrogrades have periods from 550 to 760 days.

We used JPL planetary ephemeris DE431 (Folkner et al. 2014), which spans 17,000 yr (−8002 December to 9001 April), to calculate the heliocentric mean orbital elements in ecliptic coordinates for Jupiter and to obtain the nodal and apsidal precession rates (also listed in Table 6). A direct comparison of the apsidal precession rates of Jupiter with those of Pasiphae, Cylene, 2011 J2, Hegemone, and Sinope shows that Pasiphae is in secular resonance, but that the others are not too far from the resonance themselves.

Figure 5 is a summary of the spatial distribution of the Jovian irregulars in terms of $a\times \cos (i)$ and $a\times \sin (i)$. The radial distance from the origin is expressed in the units of Hill radii (0.36 au for Jupiter). Almost all satellites show large pericenter-to-apocenter differences in their orbits. Eighteen of the retrograde satellites form a tight group, with Carme as the largest member (panel (a) in Figure 5). Another nine retrogrades (Iocaste included) form another cluster, with Ananke as the largest member (panel (b) in Figure 5). The third obvious cluster is Himalia's prograde group, with Himalia, Leda, Elara, Lysithea, and Dia as the members (panel (c) in Figure 5). The Pasiphae cluster (Pasiphae, Megaclite, 2003 J4, and Cyllene) uncovered by Beaugé & Nesvorný (2007) does not stand out as much as the other ones, and a more sophisticated analysis, beyond the scope of this paper, would be required to attempt a formal clustering.

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