Herschel-PACS photometry of Uranus' five major moons

Aims. We aim to determine far-infrared fluxes at 70, 100, and 160$\mu$m of the five major Uranus satellites Titania, Oberon, Umbriel, Ariel and Miranda, based on observations with the photometer PACS-P aboard the Herschel Space Observatory. Methods. The bright image of Uranus is subtracted using a scaled Uranus point spread function (PSF) reference established from all maps of each wavelength in an iterative process removing the superimposed moons. Photometry of the satellites is performed by PSF photometry. Thermophysical models of the icy moons are fitted to the photometry of each measurement epoch and auxilliary data at shorter wavelengths. Results. The best fitting thermophysical models provide constraints for important thermal properties of the moons like surface roughness and thermal inertia. We present the first thermal infrared radiometry longward of 50$\mu$m of the four largest Uranian moons, Titania, Oberon, Umbriel and Ariel, at epochs with equator-on illumination. Due to this inclination geometry there was heat transport to the night side so that thermal inertia played a role, allowing us to constrain that parameter. Also some indication for differences in the thermal properties of leading and trailing hemispheres is found. We specify precisely the systematic error of the Uranus flux by its moons, when using Uranus as a far-infrared prime flux calibrator. Conclusions. We have successfully demonstrated an image processing technique for PACS photometer data allowing to remove a bright central source. We have established improved thermophysical models of the five major Uranus satellites. Derived thermal inertia values resemble more those of TNO dwarf planets Pluto and Haumea than those of smaller TNOs and Centaurs.


Introduction
The planet Uranus is a well suited primary flux standard at the upper end of the accessible flux range for a number of contemporary far-infrared space and airborne photometers, like ISOPHOT (Lemke et al. 1996), Herschel-PACS (Poglitsch et al. 2010) and HAWC+ (Harper et al. 2018). Uranus is also an important flux/amplitude calibrator for submm/mm/cm groundbased observatories, e.g. IRAM (Kramer et al. 2008) or JCMT (SCUBA-2) (Chapin et al. 2013).
Uranus was routinely observed during the Herschel mission (Pilbratt et al. 2010) as part of the PACS photometer 70, 100, and 160 µm filter flux calibration program, in particular for a quantitative verification of the flux non-linearity correction for PACS (Müller et al. 2016).
Due to its flux density of > 500 Jy, Uranus exhibits an extended intensity profile in the PACS maps which reaches out to radii > 1 , and overwhelms the emission from its moons. An ex-Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. ample is the Uranus image shown in the left panel of Fig. 1. Nevertheless, by detailed comparison of the Uranus image with a PACS reference PSF ( Fig. 1 middle), it is possible to trace extra features on top of the Uranus PSF. That is how we recognized the two largest and most distant of the five major Uranian moons, Titania and Oberon, in the PACS maps. (Titania and Oberon were discovered by the name patron of the Herschel Space Observatory, William (Wilhelm) Herschel, himself). In the following sections we will describe the method used to generate the Uranus reference PSF and subtract it from the maps in order to extract FIR fluxes for all five major moons of Uranus. This photometry will be compared with thermophysical modelling of the moons. and Vesta (Lutz 2015) 1 did not provide adequate PSF subtraction results. We therefore decided to construct a Uranus reference PSF (Ref PSF from now on) out of the individual Uranus maps in each PACS filter.
The Herschel Science Archive contains twenty individual scan map measurements of Uranus, taken over the entire course of the mission at five distinct epochs (cf. Table A.1). Within each of those five epochs, four scan map observations were taken approximately 6 min apart from each other. The PACS photometer could take data simultaneously in the 160 µm filter and either the 70 µm or 100 µm filter. The starting point of our PSF analysis were the ten 70 and 100 µm and their twenty 160 µm, highpass filtered and flux calibrated level 2 scan maps produced for the Uranus photometry as published in Müller et al. (2016). The data reduction and calibration performed in HIPE 2 (Ott 2010) up to this level is described in Balog et al. (2014). A general description of PACS high-pass filter processing is given in the PACS Handbook (Exter et al. 2018). In order to determine any dependence of our PSF photometry on the data reduction, we re-processed the maps with a variety of map parameter combinations for HPF radius and pixfrac, as listed in Table 1. The variation of the results among the nine different created maps of the same observation identifier (OBSID) is one component in our photometric uncertainty assessment. The related uncertainty is listed under σ red in Tables A.1 -A.6.

Establishment of Uranus reference PSF
As a first step the WCS (world coordinate system) astrometries of the images were corrected by finding the centre of Uranus. This was crucial to correct the majority of astrometric uncertainties of the images. In addition to the standard flux calibration in HIPE a final flux calibration step was done by removing the dependence of the detector response on the telescope background, a calibration feature which is described in Balog et al. (2014). Table 1. Used scan map parameters for the input maps of the PSF fitting step. FWHM PS F is the average full width-half maximum of the point spread function for a point-like source in the corresponding filter. "outpix" is the output pixel size in the final map. This was kept constant, which means a sampling of the PSF FWHM by 5 pixels in each filter. "HPF" is the abbreviation for high pass filter, "pixfrac" is the ratio of drop size to input pixel size used for the drizzling algorithm (Fruchter & Hook 2002)  Notes. (a) This parameter determines the elementary section of a scan over which the high-pass filter algorithm computes a running median value. Its unit is "number of read-outs". The spatial interval between two readouts is α ro = vscan νro . For the standard ν ro = 10 Hz read-out scheme in PACS prime mode, and a scan speed v scan = 20"/s, the spatial interval α ro between two read-outs corresponds to 2". The entire width of the HPF window (") = [(2 × HPF radius) + 1] × α ro .
The relation of detector responsivity with telescope background could be established from the Uranus observations themselves with a very high S/N. All images were then flux normalized to a mean Uranus-to-Herschel distance and rotated to the same reference angle. The distance correction was in the order of 6%, while the detector response correction with telescope background was in the order of 1%. Details of these flux corrections are detailed in Appendix B. After these corrections the uncertainty of Uranus flux was within a remarkable 0.19% -0.27% depending on the filter, proving the outstanding flux stability of the PACS instrument. This was important, because flux variation could have a negative effect on the creation of the median image for the Ref PSF in the next steps. On the other hand, an arbitrary normalization compensating for the flux differences would render any photometry afterwards unreliable.
Four times oversampling was used for the Ref PSF (FWHM was sampled by 20 pixels) to mitigate the information loss by the re-sampling of the data back and forth. A separate Ref PSF was generated for each of the two scan directions, due to small differences between them. The very first Ref PSF was generated by a simple median over the individual images on each pixel. The median removed the orbiting moons for most of the pixels around the PSF centre. However, for some areas of the Ref PSF the moons were overlapping multiple times. To remove the remnants of the moons at these spots the generation of the Ref PSF was done in an iteration loop. The iteration loop also corrected small distortions and flux differences between the images (called PSF matching, see Section 2.3) and further enhanced the astrometry of the images. The iteration loop is shown on Fig. 2 Similarly, 5 flux parameters were used for the Moon component, fitted for each moon to take into account the relative flux difference of Uranus and its moons. The Uranus PSF shape was changing slightly between images. To adjust these individual differences we convolved the Ref PSF with normalized kernel matrices. Fitting 5×5 normalized kernel elements to the individual images improved the Uranus PSF subtraction near the centre of the PSF, making even the inner moons visible in some cases.
The PSF difference between the Uranus and its moons were clearly visible by leaving doughnut artefacts at the residual images of the moons. Using a simple 3×3 sharpening kernel for the moon PSFs completely eliminated this issue, originated clearly from their PSF size differences. The moons and Uranus had the same small distortions on the same image, therefore we applied the sharpening kernel to the (already PSF-matched) Uranus component of a given image, instead of the Ref PSF.
The Moon component of an image was generated by shifting the Moon PSF to the moon positions at a given epoch and multiplied by the relative flux parameter of each moon.
The optimal sizes of the kernels change with wavelength. To have the same number of free parameters and constraints for all wavelengths, we implemented a spatial scale factor for the kernels. In this scaled kernel image convolution the kernel values were used to weight the -2d, -1d, 0d, 1d, 2d distance units shifted Ref PSF instances around Uranus in X and Y-direction. Where the d units were d 70 = 1.5, d 100 = 1.25 and d 160 = 1 map pixels for the 70, 100, and 160 µm images, respectively. Finally all shifted elements were added together and were multiplied by a relative flux parameter. Note here, that this scaled kernel image convolution becomes a traditional image convolution with d = 1 pixel shift distance unit. Fig. 3 shows an example of the kernels. The 5×5 Uranus kernel is in blue and the 3×3 moon kernels are in black. See an example of a fitted Moon component at the middle and residual image at the right of Fig 4. The major part of the iteration loop was to fit these kernels and flux parameters to each individual image. For the fitting parameters the crucial point was to find a good balance between constraints and free parameters. The constraints were: 1.) Until the very last iteration loop the flux of each moon was set constant for all observation epochs. This was crucial, because with this constraint the flux of a given moon was fitted dominantly to those epochs where it was farther away from the centre of Uranus, due to the higher SNR of the image at those pixels. The noise estimate was taken from the associated standard deviation map of the image product. 2.) Although the optimal kernels were not symmetric for all individual images, it was crucial to impose symmetry on the kernels. The PSFs of the nearby moons were overlapping with some image convolution elements, making the fit redundant for their kernel elements. See e.g. Fig. 3 where the kernels of Uranus and Oberon are overlapping. This redundancy would incorrectly elevate some of the kernel components of Oberon, reducing the Uranus kernel values proportionally. Implementing rotational symmetry for the kernels solved these redundancies. The Uranus kernel therefore was an average of two 5×5 kernels with 180 o and 120 o rotation symmetric elements. 3.) The more point-like Moon PSF was generated by a convolution of the Uranus component with the simplest (2 parameter) 90 o rotation symmetric 3×3 normalized sharpening kernel. These fitted kernel elements were constant for all the epochs and the same for all moons, as the relative diameter ratios of Uranus and its moons can be considered as constant. 4.) The last free parameters to be fitted were the X and Y spatial offsets of the images to improve the relative positions of the individual PSFs. The PSF subtraction is very sensitive to any offset. An ≈100 mas uncertainty of the Uranus centre would result in a quite significant residual pattern. 5.) In the last iteration loop all previously fitted parameters were fixed, but the constant moon flux constraint was released. This last fit showed the variability of the moon fluxes from their averages for each epoch.

PSF subtraction
After fitting of all parameters to all individual images at the same time, two intermediate outputs were generated. After the last iteration loop the Uranus and Moon components were saved into the FITS files of the final moon map products. Subtracting both the Uranus and Moon components give the residual image. A residual image seen at the right of Fig. 4 clearly proves the correctness of the fit parameters and the right balance of free fit parameters and constraints.

Maps of the Uranian moons
All data products with the PSF subtracted maps and including the convolved Uranus PSF and the moon PSFs in additional extensions will be available in FITS format as Herschel Highly Processed Data Products (HPDPs) 3 in the Herschel Science Archive.
Figs. A.1 to A.2 show the final actual maps of the Uranian moon constellations with the Uranus PSF subtracted for the 5 observation epochs. The corresponding scan and cross-scan maps have been averaged. It is obvious that there is an inner area where the PSF subtraction does not work perfectly. This area is quantified by the results illustrated in Fig. 5.

Photometry of the Uranian moons
The PSF photometry of the moons is the side product of our PSF subtraction itself, as we have to fit and subtract the moons to get a moon-cleared image for the Ref PSF generation. In comparison with aperture photometry the constraint of knowing the exact PSF shape gives extra information into the PSF photometry, providing better results in crowded fields for overlapping sources. To get additional confidence in our PSF photometry, we have also performed standard aperture photometry, whenever any moon was well separated from Uranus.

PSF photometry
An example for PSF photometry fit results is shown in Fig. 4 for the combined scan and cross-scan map of OBSIDs 1342211117+18 from which 70 µm photometry of all 5 moons can be obtained. The PSF images of Oberon and Miranda are disturbed in the residual map due to imperfect Uranus PSF subtraction in this central area, nevertheless a significant fraction of the moon PSF is available to recover the total flux and reconstruct the intensity distribution. As already mentioned earlier, the fitting algorithm weights the pixels with their sigma value using the associated standard deviation map of the image product. In the case of Miranda the PSF is fitted dominantly to this outer part of the Uranus PSF, where the SNR of the pixels are higher than the ones closer to the Uranus centre. Of course the uncertainty of the PSF fit worsens, if only part of the PSF is available.
The unitless PSF flux fit parameters were relative fluxes, used to weight the Ref PSF. To get the flux in Jy from these weights, they have to be multiplied with the aperture photometry of the Ref PSF, in other words the average flux of Uranus over the measurements. The flux uncertainties were calculated the same way from the unitless 1-sigma parameter error values of the PSF fit parameters. This is the second component in our photometric uncertainty assessment. The related value is listed under σ par .

Aperture photometry
On the Uranus subtracted products we have performed standard aperture photometry too, as described in the PACS Handbook (Exter et al. 2018), Sect. 7.5.2. Subtracting all other moons from the product (except the one we were measuring) clearly enhanced the aperture photometry results. Still it was possible when a given moon was well separated from Uranus at a given epoch. This is mainly the case for Oberon and Titania, while unfortunately the number of comparison cases for Umbriel and Ariel is quite limited, in particular at 160 µm (70 µm: 8 cases, 100 µm: 4 cases, 160 µm: 0 cases).
The detailed comparison of PSF photometry with aperture photometry has been compiled in Table A.7. A statistical overview is given in Table 2. From this it can be seen that the consistency of the two photometric methods is very good (within 3-4%), thus confirming the principal quality of our PSF photometry procedure. This does, however, not exclude that individual fits may be unreliable or even fail, in particular in areas with high PSF residuals or confusion by close sources. The uncertainty of the fit gives then already good advice on the reliability.
The aperture photometry shows on average a systematic 3-4% negative flux offset with regard to the PSF photometry. This flux loss was a result of the small apertures and sky radii to achieve good residual rejection.

Photometry results
In Fig. 5 we have plotted the PSF photometry fluxes and their uncertainties and the corresponding signal-to-noise ratios (S/N) of the individual measurements depending on distance of the Uranian moon from the Uranus position for each filter. As a general feature one notices that uncertainties increase and hence S/N ra-tios degrade noticeably inside a certain radius, which is ≈ 7 . 8, 11 . 1, and 17 . 8 for 70, 100, and 160µm, respectively (these radii scale with λ c of the filter). This is due to PSF residuals as seen in Figs. A.1 to A.3. It should be noted that negative fluxes and hence negative S/N ratios do not occur, since the PSF fit produces either positive fluxes or fails. For the photometry of the individual moons the following can be concluded: • The S/N ratios of all Titania measurements are >10, so that all measurements should be very reliable. • The S/N ratios of the Oberon measurements for epochs 2 -5 are all >10, so that all these measurements should be very reliable. Regarding the measurements of the first epoch the moon is inside the critical radius. Nevertheless S/N at 70 and 100 µm are still 10, so that their quality should be medium. At 160µm the S/N ratios are <10, so that this photometry is less reliable. • For Umbriel the S/N ratios of the 70 and 100 µm measurements of epochs 1 and 5, which are outside the critical radius, are of very high quality. The corresponding 160 µm fluxes have S/N ratios 10, so that they are less reliable. The S/N ratios for the 70 and 100 µm measurements of epochs 2 -4 are between 10 -50, so that their quality should be still medium. However, the corresponding 160 µm fluxes have S/N ratios 2 -5, so that this photometry is less reliable. • For Ariel the S/N ratios of the 70 and 100 µm measurements of epochs 1 to 3 have medium to high quality ( 10 -<100). The S/N ratios of the 70 and 100 µm measurements of epochs 4 and 5 are 10, so that they are less reliable. The S/N ratios of all 160 µm measurements are 3, so that they are likely quite inaccurate. • For Miranda, which is considerably fainter than the other four moons and always close to Uranus, the S/N ratios of the 70 µm measurements of epochs 1 and 3 are in the range of 1 -3. These measurements indicate the order of flux, but they are not very reliable. All other measurements at 70 and 100µm have S/N ratios 1, so that individual measurements are not reliable at all. At 160 µm S/N ratios are <<1. Corresponding signal-to-noise ratios (S/N = fmoon σtot ). The dashed vertical line at ≈ 7 . 8, 11 . 1, and 17 . 8, respectively (scaling with λ c of the filter) indicates a radius inside which the uncertainty increases and the S/N degrades noticably due to PSF residuals. Small S/N ratios mean some restriction in the subsequent analysis. One should bear in mind that this is not a deficiency of the observation design, since the original design with just one modular mini scan map was meant to observe Uranus, so that signal-to-noise ratios for the moons are naturally not optimal.
The results of photometry from the individual scan maps are given in Tables A.2 to A.6 in Appendix A.5. For completeness we compile the Uranus photometry in Table A which is needed in generating the PSF reference. The determination of the distance corrected Uranus flux is described below. Table 3 provides an overview of the Uranian moon photometry with mean fluxes. For it a weighted mean moon-to-Uranus flux ratio was calculated from the individual photometry results listed in Tables A.1 using the σ i,tot of the moon photometry as weights. For the calculation of the mean moon fluxes a weighted mean Uranus flux at a mean distance of all Herschel observations is used. The mean Uranus distance is derived from the ∆ obs,i of the 20 individual observations (∆ obs,mean = 20 i=1 ∆ obs,i 20 = 20.024 AU; for the ∆ obs,i cf.
using the σ i,tot of the individual Uranus measurements as weights. These mean distance corrected fluxes of Uranus are listed in Table 3, too. The mean moon flux is then calculated as The combined flux contribution of the four largest moons relative to the Uranus flux is 5.7×10 −3 , 4.8×10 −3 and 3.4×10 −3 at 70, 100, and 160 µm, respectively. Hence, earlier published photometry of Uranus (Müller et al. 2016) not subtracting the moon contribution is not invalidated by our new results. We rather specify more precisely the systematic error of the Uranus flux by its moons, when using Uranus as a far-infrared prime flux calibrator. The fluxes in column f total of Table A.1 are very consistent with those in Table B No dependence of the moon fluxes on the distance to Uranus is expected, since all moons have orbits with small eccentricity. The variation of the angular separation of the moons to Uranus is a pure projection effect due to the inclination of the Uranian system.
Another plausibility check of the PACS photometry can be obtained from FIR two-colour diagrams. In Fig. 6 we show individual two-colour diagrams for the Uranian moons. The PACS fluxes are not colour corrected and refer to the PACS standard photometric reference SED ν × f ν = const. Modified blackbody functions ν β ν β 0 B ν (T b ) are good first order approximations for dust emission. Emission from the surface regolith of satellites is usually well approximated by pure blackbody emission, i.e. β should be zero or small. We calculated the two PACS colours of modi- Table 3. Mean fluxes of the Uranian moons (Eqn. 3) calculated from a weighted mean moon-to-Uranus flux ratio (Eqn. 1) and and a mean Uranus flux (Eqn. 2) over the Herschel observation campaign. σ tot of the individual moon photometry was used as weight. The applied mean distance (20.024 AU) normalized Uranus flux is given in the last line. n 70 , n 100 and n 160 give the number of reliable measurements used in the determination of fmoon f Uranus .
Object n 70 The modified blackbody fluxes have been colour corrected (cc λ ) to the PACS photometric reference SED for a homogeneous comparison with the moon colours, cf. PACS Handbook (Exter et al. 2018), formula 7.20 for the calculation for any SED shape. We checked which combination of β and T b best matched the measured colours. Fig. 6 shows the line for the best matching β value and a range of T b which crosses the measured combination of colours. For Titania and Oberon the approximation of the measured colours by pure blackbodies is quite good, since the match yields β = 0.10±0.06, T b = 73.0 K±2.0 K and β = 0.22±0.04, T b = 69.5 K±1.5 K, respectively. For Umbriel we find β = 0.85±0.25, T b = 54.7 K±5.2 K, which shows that the 160 µm flux is somewhat too low, so that the log( f 100 f 160 ) value is too high, thus requiring higher β values. For Ariel the fit gives β = 5.9±0.8, T b = 20.1 K±2.0 K, which is a completely unphysical spectral energy distribution solution for this moon. We conclude that the mean 160 µm flux is far too low (about a factor of 2) and unreliable, as suggested by the S/N analysis above. On the other hand, the mean 70 and 100µm photometry appears to be okay for all four moons, since log( f 70 f 100 ) values are all similar. Because of the partial deficiency or incompleteness of the measured SEDs we have derived colour correction factors for the PACS photometry from the best fitting models, see Table 7.

Auxiliary thermal data
In addition to the new PACS measurements, we searched in the literature to find more thermal data for the Uranian satellites. Brown et al. (1982) presented standard broad-band Q filter measurements taken by the 3-m IRTF 3 telescope. We re-calibrated the Q-band magnitudes (after applying the listed monochromatic correction factors and taking the specified -3.32 mag for αBoo) with the template flux of 185.611 Jy at 20.0 µm (Cohen et al. 1996). The resulting flux densities are given in Table 4.
An important set of measurements was taken by Spitzer-IRS (14-37 µm) (Houck et al. 2004). We took the reduced and calibrated low-resolution spectra (Lebouteiller et al. 2011) and 3 Infrared Telescope Facility on Mauna Kea, Hawaii  Table 4. Flux densities and uncertainties at 20.0 µm based on measured Q-band magnitues from Brown et al. (1982) and re-calibrated via the reference standard star αBoo. Data were taken in May 1982 with the IRTF (Miranda was not part of the study). r helio is the light-time corrected heliocentric range, ∆ obs is the range of target centre wrt. the observer, i.e. IRTF, α is the phase angle and "ang-sep" is the apparent angular separation from Uranus. The aspect angle during the measurements was around 163.5 • which means that IRTF saw mainly the South-pole region of Uranus and the four satellites. Uranus and its four satellites. An overview of these observations is given in Table 5.
The "optimal" extraction of the spectra from the CASSIS data assumes a perfect point-like source which is certainly the case for the Uranian satellites at 18-20 AU distance from the 0.85 m Spitzer Space Telescope. We also looked into the highresolution scans, but they only cover the longer wavelength range (19.5-36.9 µm) and from the comparison with the lowresolution spectra we concluded that they do not add any new information. At longer wavelengths (>22 µm for Umbriel, and >30 µm for Oberon) the CASSIS spectra (both, low-and highresolution ones) show significant additional fluxes, probably originating from the Uranus PSF (cf. Figs. 9 and 7, respectively). The Ariel spectra have fluxes which are at least a factor of 2-3 too high and it seems the data are still affected by the influence of Uranus (cf. Fig. 10). We eliminated those parts which show a strong deviation from a typical satellite thermal emission spectrum. We rebinned the spectra down to 10-15 wavelength points and added 10% to the measurement errors to account for absolute flux calibration uncertainties in the close proximity of a very bright source. Cartwright et al. (2015) presented IRTF/SpeX (∼0.81 -2.42 µm) and Spitzer/IRAC (3.6, 4.5, 5.8, and 8.0 µm) measurements. But even at 8 µm the measured fluxes are dominated by reflected sunlight. In the most favorable case, the thermal contribution was still well below 10%. We therefore excluded these measurements from our radiometric studies. Hanel et al. (1986) studied the Uranian system with infrared observations obtained by the infrared interferometer spectrometer (IRIS) on Voyager 2. The measurements were taken for Miranda and Ariel and cover the range between 200 and 500 cm −1 (20-50 µm). The South polar region was seen for both targets (under phase angles of 38 • for Miranda and 31 • for Ariel). They measured a maximum brightness temperature near the subsolar point, T S S , of 86±1 K and 84±1 K for Miranda and Ariel, respectively. We tested our final model solutions against these two brightness temperatures.

Thermophysical modelling of the Uranian moons
For the interpretation of the available thermal IR fluxes, we use the thermophysical model (TPM) by Lagerros (1996Lagerros ( , 1997Lagerros ( , 1998 and Müller & Lagerros (1998, 2002. The calculations are based on the true observer-centric illumination and observing geometry for each data point (topocentric for IRTF, Herschel-/Spitzer-centric). The model considers a one-dimensional heat conduction into the surface, controlled by the thermal inertia. The surface roughness is implemented via segmented hemispherical craters where the effective r.m.s. of the surface slopes is controlled by the crater depth-to-radius ratio and the surface coverage of the craters (Lagerros 1998). Additional input parameters are the object's thermal mid-/far-IR emissivity (assumed to be 0.9), the absolute V-band magnitudes H V of the Uranian satellites, the phase integrals q, the measured sizes D e f f and albedos p V . H V is only relevant in cases where we solve for radiometric size-albedo solutions. In cases where we keep the size fixed, H V is not used. Table 6 summarizes these values. For the satellites' rotation properties we assume a spin-axis orientation perpendicular to Uranus' equator (orbital inclinations are below 0.5 • , only for Miranda it is 4.2 • ), and a (presumed) synchronous rotation.
Using the above properties (and their uncertainties) allows us now to determine the moons' thermal properties. We vary the surface roughness from very smooth (r.m.s. of surface slopes <0.1) up to very rough surfaces (r.m.s. of surface slopes >0.7). In addition, low-conductivity surfaces can have very small thermal inertias (here we use a lower limit of 0.1 Jm −2 s −0.5 K −1 ) and compact solid surfaces have high conductivities (we consider thermal inertias up to 100 Jm −2 s −0.5 K −1 ).
One problem of radiometric studies in general is related to objects seen pole-on, or very close to pole-on (especially for distant objects where Sun and observer face the same part of the surface). In these cases, there is no significant heat transfer to the night side and it is much more difficult to constrain the object's thermal properties. A pole-on geometry is connected to an aspect angle of 0 • (North pole) or 180 • (South pole), while an equatoron geometry has 90 • . During the 1980s (including the IRTF measurements, but also the time of the Voyager 2 flyby) mainly the South pole region (of Uranus and also the synchronous satellites) was visible, the 2004/2005 Spitzer measurements were taken at aspect angles between about 97 • and 105 • , the 2010-2012 Herschel observations saw the Uranus system under aspect angles between about 70 • and 81 • , i.e. both were close to an equator-on view. The phase angles are typically small (below 3 • ) and the measured signals are in all cases related to almost fully illuminated objects.
Representative examples of our thermophysical models for the epoch 2011-07-12 (period 2) covering the wavelength range 5 -300 µm are shown for Oberon, Titania, Umbriel and Ariel in Figs. 7 to 10, including the corresponding surface temperature maps. All derived (and approved) flux densities for the 5 satellites, as well as the corresponding best TPM SEDs will be made available by the Herschel Science Centre through "User Provided Data Products" in the Herschel Science Archive 6 . Our Herschel flux densities and the auxiliary photometry shall also be imported into the "Small Bodies: Near and Far" (SBNAF) Table 5. Overview of the Spitzer-IRS CASSIS spectra. r helio is the light-time corrected heliocentric range, ∆ obs is the range of target centre wrt. the observer, i.e. Spitzer, α is the phase angle and "ang-sep" is the apparent angular separation from Uranus. The observations were designed to observe either the leading or the trailing hemisphere, which is indicated in column "hemisp". "UsefulSpectrum" indicates the wavelength range not affected by Uranus stray-light; in the case of Ariel the whole spectrum is affected.  (Veverka et al. 1991). This means that under an illumination geometry close to pole-on the satellite surface is hotter than under an illumination geometry close to equator-on, when heat transport to the night side results in a colder surface temperature. Therefore, a simple scaling of photometric measurements taken under different illumination geometry by just correcting for different ranges of target centre with regard to the observer will not allow a direct comparison. This has to be kept in mind for Figs. 7 to 10, where there seems to be some flux inconsistency between the IRTF fluxes and the thermophysical model fluxes matching the PACS observations. When taking the illumination geometry at the time of the IRTF measurements into account for the TPM, the consistency is very good, e.g. for Titania f IRTF f TPM(t IRTF ) = 0.250 Jy 0.233 Jy . As part of the analysis we also looked into differences between the leading (LH) and trailing hemispheres (TH) of the satellites. The tidally-locked and large satellites display stronger H 2 O ice bands on the leading hemispheres, but this effect decreases with distance from Uranus. In addition, Titania and Oberon show spectrally red material on their leading hemisphere. Cartwright et al. (2018) discuss the possible origin of the hemispherical differences and speculate that inward-migrating dust from the irregular satellites might be the cause of the ob-7 https://ird.konkoly.hu/ served H 2 O ice bands and red material differences in the two hemispheres. Since the IRTF measurements viewed only the South-pole regions, the measured fluxes did not allow such a separation. The Spitzer-IRS measurements were aiming for epochs were either the leading or trailing faces were seen. The measurements were timed for maximum elongation from Uranus, which are close to the epochs of the minimum and maximum heliocentric range-rate values. This was possible since the Uranus system was seen almost equator-on. The Herschel measurements were not timed to catch the objects at their range-rate maxima. Therefore, we consider Herschel observations as leading and trailing cases, if the apparent (heliocentric) range-rates were larger than 2/3 of the maximum possible. In all other cases, the observed signals are attributed to both hemispheres (labelled LH, TH, or BH in column˙r helio |ṙ max helio | of Tables A.2 to A.6).

Oberon
A standard radiometric analysis of the combined Herschel-PACS, IRTF, and Spitzer-IRS measurements leads to a range of size-albedo-thermal solutions with reduced χ 2 -values close to or below 1.0. However, the optimum solutions resulted in an effective diameter which is about 3-5% above the object's true size (and a geometric albedo of 0.29), connected to a thermal inertia in the range 20 -40 Jm −2 s −0.5 K −1 .
When we keep the diameter fixed to 1522 km (with p V =0.31) we can still find acceptable solutions (with reduced χ 2 -values close to 1): for an intermediate level of surface roughness (r.m.s. of surface slopes between 0.3 and 0.7) and thermal inertias between 9 and 33 Jm −2 s −0.5 K −1 (higher thermal inertias are connected to higher levels of surface roughness and vice versa).
The best solution is found for a thermal inertia of around 20 Jm −2 s −0.5 K −1 and an intermediate level of surface roughness (r.m.s. = 0.5). We confirmed the solution by using a modified input data set where the close-proximity PACS data (at only 6 apparent separation from Uranus) were eliminated. Nevertheless, the IRTF flux appears to be too high with regard to the model, because the IRTF measurement was done under close to pole-on illumination, when the moon was hotter, while the model reflects more a viewing geometry close to equator-on, when the moon was colder due to heat transport to the night side. The insert shows the resulting TPM surface temperature map of Oberon for the range 40 -85 K.
The leading (PACS 2 nd epoch, IRS-1) / trailing (PACS 4 th and 5 th epoch, IRS-2) analysis did not show any clear differences: both data subsets led to the same thermal properties (thermal inertia of 20 Jm −2 s −0.5 K −1 ) with very similar reduced χ 2 values. From the available measurements we cannot distinguish the leading and trailing hemispheres. The IRS spectra confirm this finding: both spectra (in comparison with the corresponding optimum TPM prediction) agree within 5%, except at the shortest end below 16 µm where the difference is about 10%.
The overall consistency ( f moon,cc f model ) of the models with colour corrected PACS fluxes for all 5 periods is 0.95±0.02, 0.94±0.06, and 1.00±0.02 at 70, 100, and 160 µm, respectively. Excluding the first epoch photometry, where Oberon was at only 6 apparent separation from Uranus, gives ratios of 0.95±0.01, 0.96±0.01, and 1.00±0.02. An illustrated comparison for epoch 2 is shown in Fig. 7.

Titania
The standard thermal analysis (PACS, IRTF, IRS) led to reduced χ 2 values close to 1 for a radiometric size which is again 2-4% larger than the true value, and a thermal inertia of 9-31 Jm −2 s −0.5 K −1 . All data are taken at sufficient separation from Uranus (>14 ), but the IRS spectra still seem to be contaminated around 30 µm. Adding the constraints from Titania's known size and albedo (see Table 6), leads to thermal inertia values of 5-15 Jm −2 s −0.5 K −1 , with optimum values of 7-11 Jm −2 s −0.5 K −1 , again for an intermediate level of surface roughness (r.m.s. = 0.4). We explicitly tested also other solutions for the thermal inertia, but a value of 20 Jm −2 s −0.5 K −1 , as found for Oberon, caused already severe problems in fitting our thermal measurements. The TPM predictions for the PACS measurements would decrease by 5-15% and the match to the observations would not be acceptable (outside 3-σ). The higher thermal inertia predictions would fit the IRS spectrum at short wavelength below 22 µm and beyond 35 µm, but not inbetween. Overall, we can exclude a thermal inertia larger than about 15 Jm −2 s −0.5 K −1 and smaller than about 5 Jm −2 s −0.5 K −1 for Titania. The Herschel-PACS measurements of Titania cover mainly the trailing hemisphere (1 st , 2 nd , and 5 th epoch) and a clean leading/trailing analysis is not possible. However, we ran our analysis on these trailing hemisphere measurements (three PACS epochs and IRS-2) and compared the results with the leading hemisphere IRS-1 measurement. The trailing data give a very consistent (reduced χ 2 of 0.7) solution with a thermal inertia between 5 and 9 Jm −2 s −0.5 K −1 . But this solution overestimates the fluxes from the IRS-1 spectrum. A higher thermal inertia of 9 -15 Jm −2 s −0.5 K −1 is needed to explain the leading hemisphere data. There are no PACS data to confirm this finding and due to the reduction/straylight residuals in the IRS spectra so close to Uranus, this can only be considered as an indication for differences between both hemispheres.
The overall consistency ( f moon,cc f model ) of the models with colour corrected PACS fluxes for all 5 epochs are 0.97±0.02, 0.97±0.03, and 0.99±0.03 at 70, 100, and 160 µm, respectively. An illustrated comparison for epoch 2 is shown in Fig. 8.

Umbriel
The standard radiometric search for the object's best size, albedo and thermal properties led to an unrealistically small thermal inertia below 5 Jm −2 s −0.5 K −1 and a diameter of just below 1100 km (p V ≈0.30), with reduced χ 2 values close to 1.0. The size and albedo values are in clear contradiction to the published values of 1170 km and p V =0.26 (Karkoschka 2001). Taking the larger size requires a higher thermal inertia to fit all observered fluxes. Intermediate levels of surface roughness, combined with thermal inertias between 5 and 15 Jm −2 s −0.5 K −1 seem to fit best (reduced χ 2 values just below the 1.7 threshold).
However, if we look at the observation-to-model ratios we can identify a few observations which suffer from low signal-tonoise ratios (all 5 PACS measurements at 160 µm and both IRS 15 µm spectral parts have SNR≤3), but our radiometric weighted solutions handle correctly the proper flux errors. More problematic are the long-wavelengths fluxes when Umbriel had only a small apparent separation (below 7 ) from Uranus: the PACS 100 and 160 µm measurements from 26-Dec-2011 and also the long-wavelength parts of both IRS spectra beyond about 22 µm seem to be affected by residual Uranus PSF features. Excluding these problematic measurements, we obtained reduced χ 2values close to 1.0 (for the fixed size of 1170 km), with preference for a lower surface roughness (around 0.3) than for Oberon and Titania, and a thermal inertia in the range between 5 and 12 Jm −2 s −0.5 K −1 . The Umbriel data have a well-balanced coverage of the leading (PACS 1 st epoch, IRS-1) and trailing (PACS 5 th epoch, IRS-2) hemispheres. Separate fits to the data for the two hemispheres led to the following results: the fits to the trailing hemisphere data are excellent (reduced χ 2 well below 1.0) with a thermal inertia at the lower end (around 5 Jm −2 s −0.5 K −1 ). The leading hemisphere data show an indication for a slightly higher thermal inertia closer to 10 Jm −2 s −0.5 K −1 . However, within the error bars, both sets can be fit with an intermediate solution.
The overall consistency ( f moon,cc f model ) of the models with colour corrected PACS fluxes for all 5 epochs is 1.02±0.05, 1.05±0.08, and 1.16±0.12 at 70, 100, and 160 µm, respectively. Excluding the second and third epoch, where Umbriel is at less than 9 apparent separation from Uranus, slightly improves the 70 µm ratio 1.01±0.04), but not the 100 and 160 µm ratios (1.05±0.08 and 1.23±0.13, respectively). An illustrated comparison for epoch 2 is shown in Fig. 9.

Ariel
Ariel was also seen by IRTF, Spitzer-IRS (leading and trailing) and by Herschel-PACS. However, the thermal IR fluxes are even lower than for Umbriel, and the apparent distances to Uranus are smaller. Two PACS measurement sequences (08-Jun-2012 and 14-Dec-2012) were taken with Ariel below 6 separation and had to be skipped. None of the IRS spectra are usable: the fluxes are too high by factors of 3 -45, cf. Fig. 10. A first radiometric analysis (just PACS 70/100 µm fluxes and the IRTF flux) produced sizes between about 1100 and 1400 km and only a very weak constraint on the thermal inertia (values below 100 Jm −2 s −0.5 K −1 ). Using the size constraint of 1159 km (a × b: 581 km × 578 km (Karkoschka 2001)) requires a thermal inertia between 6 and 25 Jm −2 s −0.5 K −1 for an intermediate surface roughness. But the reduced χ 2 is larger than 2.0 and a closer inspection shows a clear separation in the fits to the leading and trailing hemispheres. Taking the PACS measurements for the leading hemisphere (2011-Jul-12) and the trailing hemisphere (2010-Dec-13, and2011-Dec-26) separately, gave much better fits (reduced χ 2 close to 1.0), indicating a lower thermal inertia (5 -13 Jm −2 s −0.5 K −1 ) for the leading hemisphere, and a higher thermal inertia (13 -40 Jm −2 s −0.5 K −1 ) for the trailing hemisphere. Although the IRS spectra cannot be used for the radiometric studies, the flux levels for the leading hemisphere are about 5-10% higher. This also points to a lower thermal inertia for the leading side compared to the trailing side. With our final solution we calculated a maximum brightness temperature of about 86 K for the South-pole viewing geometry in early 1986. This compares very well with the maximum brightness temperature of 84±1 K seen by Voyager-2/IRIS (Hanel et al. 1986).
The overall consistency ( f moon,cc f model ) of the models with colour corrected PACS fluxes for all 5 epochs is 1.07±0.12, 0.94±0.29, and 2.09±0.48 at 70, 100, and 160 µm, respectively. The high 160 µm ratio of 2 is due to the fact that the measured values are all, except one, far too low. Excluding the fourth and fifth epoch, where Ariel is at less than 6 apparent separation from Uranus, improves the consistency at 70 and 100 µm considerably with ratios of 0.99±0.04, 1.00±0.17, respectively. However, due to the generally low 160 µm fluxes, this ratio (2.20±0.11) does not improve. An illustrated comparison for epoch 2 is shown in Fig. 10.

Miranda
For Miranda we have only the PACS measurements, but neither IRTF nor Spitzer-IRS data. The object was always within 10 from Uranus and the contamination problems are severe. We eliminated all 160 µm fluxes which are clearly completely off. In addition, we skipped the second-and fourth-epoch data, when Miranda was only at 3 . 4 and 4 . 4 apparent distance, respectively. For the last epoch, we also had to take out the 100 µm data.
In the end, only very few data points remained and the required coverage (in aspect angles, wavelengths, leading/trailing geometries, etc.) is missing for a robust radiometric analysis. With the size (474 km) and albedo (p V = 0.45) we only obtained an upper limit of about 50 Jm −2 s −0.5 K −1 for Miranda's thermal inertia. Larger values would force the TPM calculations to smaller fluxes which are not compatible with the highest SNR detections by PACS (the upper limit goes down to 20 Jm −2 s −0.5 K −1 , if we consider only the best 70 µm fluxes). The corresponding TPM calculations (with a thermal inertia below 20 Jm −2 s −0.5 K −1 ) for the Voyager-2/IRIS measurements in January 1986 produce a maximum temperature of about 87 K, in excellent agreement with the 86±1 K by Hanel et al. (1986). Table 7 provides an overview of the derived model parameters. Using these model SEDs we also calculated the colour correction factors to be applied to the measured PACS fluxes (cf. Tables A.2 to A.6).

Discussion
How do the derived properties for the Uranian satellites compare with thermal inertias of other satellites and distant TNOs? Lellouch et al. (2013) analysed a large sample of TNOs and found a Γ = 2.5±0.5 J m −2 s −1/2 K −1 for objects at heliocentric distances of r helio = 20 -50 AU (decreasing values for increasing heliocentric distance). The Uranian system is at about 20 AU and therefore one would expect (under the assumption of TNO-like surfaces) to find low values, maybe up to 5 J m −2 s −1/2 K −1 .
However, looking at dwarf planets, these general TNOderived values are usually exceeded: Haumea is at r helio = ∼51 AU and it was found to have a thermal inertia of around 10 J m −2 s −1/2 K −1 (Müller et al. 2019). The thermal inertia of Pluto and Charon (at r helio > 30 AU) are even larger: Γ Pluto = 16-26 J m −2 s −1/2 K −1 and Γ Charon = 9-14 J m −2 s −1/2 K −1 (Lellouch et al. 2011(Lellouch et al. , 2016. And putting the Pluto-Charon system closer to the Sun would increase the values significantly (assuming that the T 3 term dominates in the thermal conductivity, then the thermal inertia scales with ∝ r −3/4 ; see e.g. Delbo et al. 2015). In case of Pluto-Charon, the high Γ-values are attributed to a large diurnal skin depth due to their slow rotation (∼ P 1/2 dependence; see also discussion in Kiss et al. (2019)). In summary, the Uranian satellites Oberon, Titania, Umbriel, Ariel, and Miranda have thermal inertias which are higher than the very low values found for TNOs and Centaurs at 30 AU heliocentric distance. It seems that the thermal properties of the icy satellite surfaces are closer to the properties found for the TNO dwarf planets Pluto and Haumea.

Conclusions
We have successfully demonstrated an image processing technique for PACS photometer data allowing to remove the bright central point spread function of Uranus, and reconstructing source fluxes of its five major satellites in the order of 10 −3 of Uranus. We have obtained reliable moon fluxes outside radii of 7 . 8, 11 . 1, and 17 . 8 at 70, 100, and 160 µm, respectively, which corresponds to ≈3× the HWHM of the standard PSF (FWHM PSF = 5 . 6, 6 . 8 and 10 . 7, respectively). For Titania and Oberon we have established full sets of 70, 100, and 160 µm PSF photometry for all five observing epochs. For Umbriel there are two epochs (1 & 5) with high quality 70 and 100 µm photometry and for Ariel there are three epochs (1 -3). The 160 µm photometry of these two moons is either of low quality (Umbriel) or unreliable (Ariel). For Miranda 70 µm flux estimates could be obtained for two epochs (1 & 3).
This new FIR photometry and auxiliary photometry at shorter wavelengths compiled from the literature and retrieved from data archives has allowed to establish improved thermophysical models of the five major Uranus satellites with regard to thermal inertia and surface roughness.