Properties of the Irregular Satellite System around Uranus Inferred from K2, Herschel, and Spitzer Observations

Continuous photometry from the Kepler space telescope may provide accurate and unbiased rotation rates and amplitudes for solar system targets. The telescope gathers light in a wide visual band spanning from 420 to 900 nm, and follows a step-and-stare scheme in the K2 mission, observing different fields for up to 80 days along the Ecliptic plane (Howell et al. 2014). Kepler observed Uranus and its vicinity during Campaign 8 of the mission for 78.73 days, between 2016 January 4.55 and March 23.28. Apart from the planet, four irregular satellites Caliban, Setebos, Sycorax, and Prospero, were also proposed and selected for observations (GO8039, PI: A. Pál).⁷

We applied the same pipeline for the reduction of the Kepler observations that we used in previous works, e.g., to determine the light variations and rotation rates of various Trans-Neptunian Objects, main-belt and Trojan asteroids, or of the moon Nereid (Pál et al. 2015, 2016; Kiss et al. 2016; Szabó et al. 2016, 2017). The method is based on the FITSH⁸ software package (Pál 2012): the processing steps are detailed in the previous papers and in the companion paper that describes the light curves of main-belt asteroids drifting though the same image mosaic (Molnár et al. 2017), also providing information on the limitations and capabilities. In short, we created mosaic images from the individual Target Pixel Files (TPFs), which contain the time-series photometric information (time, flux, flux error, and background) for each downloaded pixel around a given target, as opposed to light-curve files that contain a pipeline-extracted brightness summed in an aperture as a function of time. For more details, the reader is directed to Kepler Archive Manual.⁹ We then derived the astrometric solutions for the mosaic images, using the USNO-B1.0 catalog (Monet et al. 2003), where the K2 full-frame images from the campaign were exploited as initial hints for the source cross-matching. Then, we registered the images into the same reference system, and subtracted a median image from each image. This median image was created from a selection of individual images that did not include the obvious diffraction pattern contamination from Uranus. We then applied aperture photometry at the positions of the satellites. The sharp images of the stars that were shifted to compensate the attitude changes of the telescope create characteristic residuals in the differential images that may contaminate our photometry. Therefore we filtered out the epochs when the scatter of the background pixels in the photometric annulus was high. The per-cadence photometric uncertainty values were derived from the shot noise of Kepler and from the estimated background noise.

All observations were collected in long cadence mode, with a sampling of 29.4 minutes. Three of the proposed satellites fell into, or near, the large mosaic that also covered the apparent motion of Uranus (Figure 1). Other satellites were also present in the same mosaic, and we successfully detected the light variations of a fourth one, Ferdinand.

Zoom In Zoom Out Reset image size

Prospero fell onto an adjacent CCD module, and its motion was covered with a narrow band of pixels. In that case, we collected the TPFs of nearby, unrelated targets into a small mosaic around the track of Prospero in order to have a good astrometric solution. The log of observations is summarized in Table 1. The duty cycle there shows the ratio of accepted photometric data points to the number of cadences when the satellites were present on the images. We note that Ferdinand was observed twice, at the beginning and at the end of the campaign: the two rows in the table refer to the entire length of the observation and the sections when Ferdinand was on the Kepler CCD array, respectively. The temporal distribution of data points we used for photometry is presented in Figure 2. We also checked other satellites that fell on the CCD array, e.g., Stephano (24.5 mag) or Trinculo (25.5 mag), but found no meaningful signals in their photometry.

Zoom In Zoom Out Reset image size

2.2.1. Sycorax

Sycorax was observed with the MIPS camera of the Spitzer Space Telescope (Rieke et al. 2004) at two epochs, on 2008 December 27 and 29, both at 24 and 70 μm. The summary of these observations is given in Table 2 below. Sycorax was successfully detected at both epochs and at both wavelengths (see Figure 3). We used the same data reduction and photometry pipeline as in Mueller et al. (2012) and Stansberry et al. (2008, 2012). The MIPS instrument team data analysis tools (Gordon et al. 2005) were used to produce flux-calibrated images for each band, and the contribution of background objects were subtracted (see Stansberry et al. 2008). Aperture photometry was performed both on the original and final images and the final flux values were obtained using the aperture corrections by Gordon et al. (2007) and Engelbracht et al. (2007). Color correction of the in-band fluxes were done following Stansberry et al. (2007). The flux densities obtained are presented in Table 3.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 2.  Summary of Thermal Infrared Observations

Instrument	AORKEY/	Target	Band	Date	r	Δ	α
	OBSID		(μm)	(JD)	(au)	(au)	(deg)
Spitzer/MIPS	28832512	Sycorax	24	2454827.702	20.081	19.672	2.68
	28832512		70	2454827.718	20.081	19.672	2.68
	28832768		24	2454829.672	20.081	19.704	2.71
	28832768		70	2454829.688	20.081	19.704	2.71
Herschel/PACS	1342221837–38	Sycorax	70	2455710.589	20.084	20.519	2.62
	1342221875–76		70	2455710.939	20.084	20.514	2.62
	1342221839–40		100	2455710.617	20.084	20.519	2.62
	1342221877–78		100	2455710.966	20.084	20.513	2.63
	1342221837–40		160	2455710.589	20.084	20.519	2.62
	1342221875–78		160	2455710.939	20.084	20.514	2.62
Herschel/PACS	1342236891–92	Caliban	70	2455933.979	20.119	20.359	2.73
	1342237436–37		70	2455940.001	20.117	20.457	2.63
	1342236891–92		160	2455933.979	20.119	20.359	2.73
	1342237436–37		160	2455940.001	20.117	20.457	2.63
Herschel/PACS	1342236891–92	Trinculo	70	2455933.979	20.051	20.293	2.73
	1342237436–37		70	2455940.001	20.048	20.390	2.64
	1342236891–92		160	2455933.979	20.051	20.293	2.73
	1342237436–37		160	2455940.001	20.048	20.390	2.64
Herschel/PACS	1342236891–92	Ferdinand	70	2455933.979	19.930	20.174	2.75
	1342237436–37		70	2455940.001	19.929	20.173	2.65
	1342236891–92		160	2455933.979	19.930	20.174	2.75
	1342237436–37		160	2455940.001	19.929	20.173	2.65

Note. The columns are (1) telescope/instrument; (2) AORKEY (Spitzer/MIPS) or OBSID (Herschel/PACS); (3) target—in the case of Caliban the target was caught in a larger, off-Uranus field; (4) band (nominal wavelength in μm); (5) date of the observation (julian date); (6) heliocentric distance; (7) distance from observer; (8) phase angle.

Download table as:  ASCIITypeset image

Table 3.  Summary of Thermal Infrared Observations

Target	Detector/	${\lambda }_{\mathrm{eff}}$	Fi	${C}_{\lambda }$	Fm	${H}_{V/R}$
	Filter	(μm)	(mJy)		(mJy)	(mag)
Sycorax	MIPS 24	23.68	3.017 ± 0.045	0.96 ± 0.01	3.14 ± 0.16	7.50 ± 0.04
	MIPS 70	71.42	14.68 ± 2.78	0.92 ± 0.01	16.07 ± 2.89	(Grav et al. 2004)
	MIPS 24	23.68	3.109 ± 0.046	0.96 ± 0.01	3.17 ± 0.17	⋯
	MIPS 70	71.42	19.12 ± 2.70	0.92 ± 0.01	20.78 ± 3.11	⋯
	PACS 70	70.0	16.7 ± 0.6	0.98 ± 0.01	17.0 ± 1.0	⋯
	PACS 100	100.0	15.3 ± 1.6	1.00 ± 0.01	15.3 ± 1.8	⋯
	PACS 160	160.0	5.3 ± 3.1	1.04 ± 0.01	5.5 ± 3.2	⋯
Caliban	PACS 70	70.0	1.4 ± 0.8	0.98 ± 0.01	1.4 ± 0.8	9.16 ± 0.016
	PACS 160	160.0	<3 mJy	1.04 ± 0.01	<3 mJy	(Grav et al. 2004)
Trinculo	PACS 70	70.0	<0.8 mJy	0.98 ± 0.01	<0.8 mJy	11.92 ± 0.18
	PACS 160	160.0	<3 mJy	1.04 ± 0.01	<3 mJy	(Grav et al. 2004)
Ferdinand	PACS 70	70.0	<0.8 mJy	0.98 ± 0.01	<0.8 mJy	12.5 ± 0.1(R)
	PACS 160	160.0	<3 mJy	1.04 ± 0.01	<3 mJy	(this work)

Note. The columns are (1) target—in the case of Caliban the target was caught in a larger, off-Uranus field; (2) detector and filter combination; (3) effective wavelength of the band (${\lambda }_{\mathrm{eff}}$); (4) in-band flux density (F_i); (5) color correction factor (${{\rm{C}}}_{\lambda }$); (6) monochromatic flux (${F}_{m}={F}_{i}/{{\rm{C}}}_{\lambda }$); (7) absolute magnitude in V or R Band; Herschel/PACS flux densities of Sycorax are taken from Lellouch et al. (2013).

Download table as:  ASCIITypeset image

Sycorax was also observed in dedicated observations with the PACS camera of the Herschel Space Observatory, in the framework of the "TNOs are Cool!" Herschel Open Time Key Program (Müller et al. 2009). The flux densities derived from these observations have already been presented in Lellouch et al. (2013).

We used both the Spitzer/MIPS and Herschel/PACS data for modeling of the thermal emission of the satellite (see Section 3.1).

2.2.2. Serendipitous Herschel/PACS Observations of Irregular Satellites

Caliban was identified as potentially present on some far-infrared maps taken with the PACS camera of the Herschel Space Observatory. Herschel/PACS observed the environment of Uranus at two epochs: on 2012 January 7 (OBSIDs: 1342236891/92/scan and cross-scan) and on 2012 January 13 (OBSIDs: 1342237436/37), under the proposal ID OT1_ddan01_1 in both cases. All measurements used the 70/160 μm filter combination in all four cases. The data reduction pipeline we used is the same as the one used in the 'TNOs are Cool! Herschel Open Time Key Programme (Müller et al. 2009), described in detail in Kiss et al. (2014), and identical to the one we used to reduce the Herschel/PACS maps of Nereid (Kiss et al. 2016). As our aim was to obtain photometry of a point source, we used the photProject() task with high-pass filtering to create maps from the time domain detector data. The photProject() task performs a simple coaddition of the frames using the drizzle method (Fruchter & Hook 2002), and the high-pass filtering applies a sliding median-filter on individual pixel timelines. More details on the procedure can be found in The PACS Data Reduction Guide.¹⁰

Maps were created from the detector scans in the co-moving frame of Uranus (see Figure 4), which was practically identical to that of the satellites due to the small relative velocities of Uranus and the satellites (<05 hr⁻¹).

We identified a faint source at both epochs in the 70 μm band at the expected location of Caliban, obtained from the NASA Horizons System considering the Herschel-centric observing geometry (see Figure 5), and derived a combined flux of F₇₀ = 1.4 ± 0.8 mJy. No obvious source could be identified on the 160 μm maps, and the general photometric accuracy obtained using the implanted source method (see Kiss et al. 2014) defined a 1σ upper limit of F₁₆₀ < 3 mJy for the 160 μm brightness of Caliban.

Zoom In Zoom Out Reset image size

While Trinculo and Ferdinand could potentially be present on the same set of Herschel/PACS images as Caliban, these were not detected either at 70 or at 160 μm, therefore we consider that their flux densities are below 0.8 mJy and 3 mJy at 70 and 160 μm, respectively.

Sycorax was also observed on the night of 2015 November 05/06 with the 1 m Ritchey-Cretien Coude (RCC) telescope of the Konkoly Observatory, located at Piszkéstető Mountain Station. In total, 70 frames were acquired with an exposure time of 300 s each, using an Andor iXon-888 electron-multiplying CCD camera. Although it is not so relevant for such long exposures, we note here that the camera was operated in frame transfer readout mode in order to have an effectively zero dead time between the subsequent frames. The frames were taken in Sloan r' filters and used the SDSS-III DR9 catalog (Ahn et al. 2012) for reference magnitudes of the comparison stars. Standard calibration procedures and aperture photometry were performed using the various tasks of the FITSH package (Pál 2012). The resulted light curve is plotted in Figure 6.

Zoom In Zoom Out Reset image size

The thermal emission of Sycorax was modeled using the flux densities listed in Table 3, applying a NEATM model. Due to the notable difference in observing geometry at the PACS and MIPS epochs, for a specific model (given size and beaming parameter) the corresponding observation geometries were considered at the PACS and MIPS epochs separately, and the ${\chi }^{2}$ values were derived accordingly. The best-fit model is presented in Figure 7, where the flux densities of the MIPS and PACS epochs are shown individually. The NEATM model provided a best-fit effective diameter and albedo estimate of D = 165 ± 13 km, ${p}_{V}={0.065}_{-0.011}^{+0.015}$, with a beaming parameter of $\eta ={1.20}_{-0.20}^{+0.25}$.

Zoom In Zoom Out Reset image size

In addition to the NEATM model, we also applied thermophysical modeling (TPM, see Müller & Lagerros 1998, 2002, and references therein) considering an absolute magnitude of H_V = 750 ± 004 (Grav et al. 2004), a default slope parameter of G = 0.15 and a wavelength-dependent emissivity (Müller & Lagerros 2002). We used P = 6.9162 hr for the rotation period, as determined from the combination of K2 and Konkoly 1m-RCC measurements (see Section 4.1); possible thermal inertia values were considered in the Γ = 0.1–50 ${\rm{J}}\,{{\rm{m}}}^{-2}\,{{\rm{s}}}^{-1/2}\,{{\rm{K}}}^{-1}$ range and surface roughness values were allowed between ρ = 0.1 and 0.9. Because the spin-axis orientation of Sycorax is not known, we considered three possible scenarios, a pole-on (spin-axis orientation of λ = 356°, β ≈ 0° in ecliptic coordinates), an equator-on (λ = 0°, β = 90°), and an intermediate one (λ = 356°, β = 45°). The equator-on and intermediate solutions provide a low best-fit reduced ${\chi }^{2}$ of ${\chi }_{{\rm{r}}}^{2}$ < 1, and give very similar best-fit values for the size. However, no acceptable solution with sufficiently low ${\chi }_{{\rm{r}}}^{2}$ could be obtained for the pole-on case (${\chi }_{{\rm{r}}}^{2}\,\gg 1$ in all cases) and a non-pole-on configuration is also supported by the presence of a definite visible-range light curve. Our best estimates for the size and albedo are $D={157}_{-15}^{+23}$ km and ${p}_{V}={0.07}_{-0.01}^{+0.02}$, associated with a likely intermediate surface roughness (ρ ≈ 0.5) and a close-to-equator-on spin-axis configuration. Based on this analysis, very low levels of thermal inertia (${\rm{\Gamma }}\lt 1\,{\rm{J}}\,{{\rm{m}}}^{-2}\,{{\rm{s}}}^{-1/2}\,{{\rm{K}}}^{-1}$) can be excluded for Sycorax.

Zoom In Zoom Out Reset image size

The diameter and geometric albedo obtained by our analysis is close to the best-fit values obtained by Lellouch et al. (2013) using Herschel/PACS data alone ($D={165}_{-42}^{+36}$, ${p}_{V}\,={0.049}_{-0.017}^{+0.038}$, see also Figure 8), but in our case with smaller error bars. However, the beaming parameter is much better constrained with the consideration of the Spitzer/MIPS fluxes, as the Herschel/PACS data could not restrict the models further due to the lack of short wavelength (≲40 μm) data ($\eta ={1.26}_{-0.78}^{+0.92}$ in Lellouch et al. 2013). Our best-fit η = 1.20 is very close to the median values obtained for Centaurs and trans-Neptunian objects based on a large sample of Spitzer/MIPS and Herschel/PACS measurements (η ≈ 1.2, Stansberry et al. 2008; Lellouch et al. 2013). The thermal inertia value of Γ = 3–4 ${\rm{J}}\,{{\rm{m}}}^{-2}\,{{\rm{s}}}^{-1/2}\,{{\rm{K}}}^{-1}$ we obtained for Sycorax is also in agreement with the Γ = 5 ± 1 ${\rm{J}}\,{{\rm{m}}}^{-2}\,{{\rm{s}}}^{-1/2}\,{{\rm{K}}}^{-1}$ value found by Lellouch et al. (2013) for Centaurs at heliocentric distances <25 au.

For all potentially detectable satellites, we used a NEATM model with a constant emissivity of  = 0.9 to estimate the expected flux densities in the 70 μm Herschel/PACS band for a range of diameters/geometric albedos and beaming parameters. The beaming parameter η was allowed to vary between 0.6 and 1.6, while the diameters were chosen to match a V-band geometric albedo range of $0.01\leqslant {p}_{V}\,\leqslant 0.3$. Phase integrals were calculated applying both the geometric albedo-dependent phase integral developed for the outer Solar system by Brucker et al. (2009) and the "standard" value using the canonical slope parameter of G = 0.15 (see, e.g., Muinonen et al. 2010). The difference between the 70 μm thermal emission flux densities obtained by the two methods were ≤4 μJy in all cases, and are negligible for our purposes.

As described above, Caliban was tentatively detected on Herschel/PACS 70 μm maps with a combined monochromatic flux density of F₇₀ = 1.4 ± 0.8 mJy. A generally assumed dark surface of p_V ≈ 0.04 and the corresponding diameter of ∼70 km would produce a 70 μm flux density of F₇₀ > 2.4 mJy, easily detectable on our Herschel/PACS maps, over 3σ of the 0.8 mJy 70 μm flux uncertainty of the Herschel/PACS maps. The fact that the detected flux density of Caliban is smaller than that already indicates a brighter surface. We obtained $D={42}_{-12}^{+20}$ km and ${p}_{V}={0.22}_{-0.12}^{+0.20}$ as the best-fit NEATM solution using the single available 70 μm flux density, with no real constraint on the beaming parameter in our originally chosen 0.6 ≤ η ≤ 1.6 range, with our best-fit model having a corresponding η of 0.8. The 70 μm flux of Caliban can, however, be equally well fitted with D ≈ 50 km and ${p}_{V}$ ≈ 0.15, using a higher η value of ∼1.4. The uncertainties in the size and albedo due to the unconstrained beaming parameter are reflected in the errors of the best-fit values quoted above. The geometric albedo we obtained for Caliban may seem to be surprisingly high in the Uranian irregular satellite system, taking into account the low albedo of Sycorax. On the other hand, relatively bright surfaces exist among other irregular satellites, e.g., Nereid has a surface with ${p}_{V}$ > 20% (Kiss et al. 2016), but it is notably larger than Caliban, ∼350 km in effective diameter.

Trinculo and Ferdinand were not detected on the Herschel/PACS images. However, considering the 0.8 mJy 70 μm flux uncertainties as an upper limit for both targets, we can put some constraints on their geometric albedos and diameters. We note that for Ferdinand we calculated the R-band absolute magnitude using data in the Minor Planet Circular MPEC-2003-S105, assuming an R-band specific linear phase correction of ${\beta }_{{\rm{R}}}=0.119$ (Belskaya et al. 2008) and obtained H_R = 12.06 ± 0.15 mag. For Trinculo, we used the value provided by Grav et al. (2004; see also Table 3). The 0.8 mJy 1σ upper limit indicates a geometric albedo of ${p}_{V}$ > 0.03 for both satellites, and correspondingly their diameters are D < 50 km.

Maris et al. (2001) performed the first detailed study of the light curve in the R band using the 3.6 m ESO NTT telescope at La Silla, on 1999 October 8 and 9. The amplitude of light variation they found was A = 0032 ± 0008 with a P = 4.12 ± 0.04 hr period. Measurements taken with the VLT in 2005 (Maris et al. 2007) provided the most likely light-curve period and amplitude of P = 3.6 ± 0.02 hr and A = 0067 ± 0004.

Our K2 measurements revealed a well-defined rotational period with a frequency of f = 6.9374 ± 0.0083 cycle day⁻¹ (P = 3.458 ± 0.001 hr, see Figure 10). We assume that the light curve of Sycorax is double-peaked, which is supported by the slight asymmetry of the light curve when it is folded with the half frequency/double rotation period (Figure 10). This gives a double-peaked rotation period of P = 6.9190 ± 0.0082 hr. The single-peaked period is very close to that obtained by Maris et al. (2007) using VLT measurements. The peak-to-peak light-curve amplitude obtained from K2 data is A = 012 ± 002, consistent with that obtained by Maris et al. (2007) as sinusoidal amplitude.

There is a second significant peak on the frequency diagram at 0.35 cycle day⁻¹ (Figure 9, upmost panel). Such secondary periods are often explained by tumbling rotation or a companion. Although a companion around Sycorax (a moon of a moon) would be an intriguing possibility, we suggest that this second peak is a sampling artifact. The interaction of a strictly periodic sampling (such as that of Kepler) and a strictly periodic process (such as a rotation of the asteroid) results in periodic phase shifts in the sampling, and hence, an emergence of a stroboscopic period that can modulate timing, brightness, shape modulations, etc. In Szabó et al. (2013), we calculated the stroboscopic period as ${P}_{{\rm{s}}}/P=1/\min (\,]P/C[,]1-P/C[\,)$, where P_s is the stroboscopic period, P is the observed actual period, C is the cadence, and $]\,[$ denotes the fractional part. Substituting the rotation period (6.9162 ± 0.0013 hr) and the cadence (1765.5 s) into the formula, the stroboscopic period is calculated to be 9.5–10.0-times the rotation period. Since the ratio of the two detected periods is 9.97, this is perfectly compatible with the stroboscopic origin of the long-period brightness variation. Stroboscopic periods were calculated from the single-peak periods of the other targets too (see below), as listed in Table 4. We note that a stroboscopic period is not accepted simply by an amplitude criterion as a possible rotation period, but should be rejected due to its stroboscopic nature.

Zoom In Zoom Out Reset image size

Since the series of ground-based observations with the 1 m telescope of the Konkoly Observatory was roughly three months before the K2 measurements and the precision of K2 measurements are rather fine, one can extrapolate the rotation cycle numbers in an unambiguous manner for such a relatively short period. This allows us to combine the two series of measurements (K2 and 1 m RCC) to get an accurate rotational frequency, for which we obtained n = 3.47012 ± 0.00067 cycle day⁻¹, i.e., P = 6.9162 ± 0.0013 hr.

Caliban: Maris et al. (2001) determined a light-curve period of 2.66 hr that is not confirmed by our data (see Figure 10). Instead, we obtained a most likely frequency of f = 4.8249 ± 0.0092 cycle day⁻¹ and the asymmetry of the folded light curve indicates that the real rotation period corresponds to the half frequency, i.e., P = 4.9742 ± 0.0095 hr. The single-peak period of Caliban is the only one apart from that of Sycorax for which the corresponding stroboscopic frequency (P_s = 7.0026 hr or f_s = 3.4256 cycle day⁻¹) is close to a significant peak in Figure 10.

Zoom In Zoom Out Reset image size

Prospero: In our analysis, the least unambiguous light-curve period was obtained for Prospero (see Figure 11). We identified the most likely period of 3.359 ± 0.044 cycle day⁻¹ (P = 7.145 ± 0.092 hr), but a strong, secondary peak is also visible at f = 4.415 ± 0.045 cycle day⁻¹ that corresponds to a single-peak rotation period of P = 5.346 ± 0.055 hr, very close to the light-curve period obtained by Maris et al. (2007).

Zoom In Zoom Out Reset image size

Setebos: For this satellite, we confirm the light-curve period of 4.38 ± 0.05 hr obtained by Maris et al. (2007) as we derived a most likely rotation period of P = 4.255 ± 0.017 hr, very close to the previously mentioned value, without indication of a double-peak light curve.

Ferdinand: The light curve period of Ferdinand was not determined earlier. The most likely frequency of 2.027 ± 0.039 cycle day⁻¹ corresponds to a rather long rotation period of 11.84 ± 0.22 hr, the longest one in our sample (assuming a single-peak light curve, see Figure 11). Such long rotation periods are, however, not rare and present e.g., in the Saturnian system where the rotation periods of 10 from 16 irregular satellites in a recently studied sample show rotation periods longer than that of Ferdinand (Denk & Mottola 2013a).