“TNOs are Cool”: A survey of the trans-Neptunian region

“TNOs are Cool”: A survey of the trans-Neptunian region

VI. Herschelthanks: Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA./PACS observations and thermal modeling of 19 classical Kuiper belt objects
E. Vilenius Max-Planck-Institut für extraterrestrische Physik, Postfach 1312, Giessenbachstr., 85741 Garching, Germany
vilenius@mpe.mpg.de
   C. Kiss Konkoly Observatory of the Hungarian Academy of Sciences, 1525 Budapest, PO Box 67, Hungary    M. Mommert Deutsches Zentrum für Luft- und Raumfahrt e.V., Institute of Planetary Research, Rutherfordstr. 2, 12489 Berlin, Germany    T. Müller Max-Planck-Institut für extraterrestrische Physik, Postfach 1312, Giessenbachstr., 85741 Garching, Germany
vilenius@mpe.mpg.de
   P. Santos-Sanz LESIA-Observatoire de Paris, CNRS, UPMC Univ. Paris 06, Univ. Paris-Diderot, France    A. Pal Konkoly Observatory of the Hungarian Academy of Sciences, 1525 Budapest, PO Box 67, Hungary    J. Stansberry Stewart Observatory, The University of Arizona, Tucson AZ 85721, USA    M. Mueller SRON LEA / HIFI ICC, Postbus 800, 9700AV Groningen, Netherlands UNS-CNRS-Observatoire de la Côte d’Azur, Laboratoire Cassiopeé, BP 4229, 06304 Nice Cedex 04, France    N. Peixinho S. Fornasier Center for Geophysics of the University of Coimbra, Av. Dr. Dias da Silva, 3000-134 Coimbra, Portugal Astronomical Observatory of the University of Coimbra, Almas de Freire, 3040-04 Coimbra, Portugal LESIA-Observatoire de Paris, CNRS, UPMC Univ. Paris 06, Univ. Paris-Diderot, France Univ. Paris Diderot, Sorbonne Paris Cité, 4 rue Elsa Morante, 75205 Paris, France    E. Lellouch LESIA-Observatoire de Paris, CNRS, UPMC Univ. Paris 06, Univ. Paris-Diderot, France    A. Delsanti Laboratoire d’Astrophysique de Marseille, CNRS & Université de Provence, 38 rue Frédéric Joliot-Curie, 13388 Marseille Cedex 13, France    A. Thirouin Instituto de Astrofísica de Andalucía (CSIC), Camino Bajo de Huétor 50, 18008 Granada, Spain    J. L. Ortiz Instituto de Astrofísica de Andalucía (CSIC), Camino Bajo de Huétor 50, 18008 Granada, Spain    R. Duffard Instituto de Astrofísica de Andalucía (CSIC), Camino Bajo de Huétor 50, 18008 Granada, Spain    D. Perna INAF – Osservatorio Astronomico di Roma, via di Frascati, 33, 00040 Monte Porzio Catone, Italy INAF - Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, 80131 Napoli, Italy    N. Szalai Konkoly Observatory of the Hungarian Academy of Sciences, 1525 Budapest, PO Box 67, Hungary    S. Protopapa University of Maryland, College Park, MD 20742, USA    F. Henry LESIA-Observatoire de Paris, CNRS, UPMC Univ. Paris 06, Univ. Paris-Diderot, France    D. Hestroffer IMCCE, Observatoire de Paris, 77 av.  Denfert-Rocherea, 75014, Paris, France    M. Rengel Max-Planck-Institut für Sonnensystemforschung, Max-Planck-Straße 2, 37191 Katlenburg-Lindau, Germany    E. Dotto INAF – Osservatorio Astronomico di Roma, via di Frascati, 33, 00040 Monte Porzio Catone, Italy    P. Hartogh Max-Planck-Institut für Sonnensystemforschung, Max-Planck-Straße 2, 37191 Katlenburg-Lindau, Germany
Received December 26, 2011; accepted March 6, 2012
Key Words.:
Kuiper belt – Infrared: planetary systems – Techniques: photometric
Abstract

Context:Trans-Neptunian objects (TNO) represent the leftovers of the formation of the Solar System. Their physical properties provide constraints to the models of formation and evolution of the various dynamical classes of objects in the outer Solar System.

Aims:Based on a sample of 19 classical TNOs we determine radiometric sizes, geometric albedos and beaming parameters. Our sample is composed of both dynamically hot and cold classicals. We study the correlations of diameter and albedo of these two subsamples with each other and with orbital parameters, spectral slopes and colors.

Methods:We have done three-band photometric observations with Herschel/PACS and we use a consistent method for data reduction and aperture photometry of this sample to obtain monochromatic flux densities at 70.0, 100.0 and . Additionally, we use Spitzer/MIPS flux densities at 23.68 and when available, and we present new Spitzer flux densities of eight targets. We derive diameters and albedos with the near-Earth asteroid thermal model (NEATM). As auxiliary data we use reexamined absolute visual magnitudes from the literature and data bases, part of which have been obtained by ground based programs in support of our Herschel key program.

Results:We have determined for the first time radiometric sizes and albedos of eight classical TNOs, and refined previous size and albedo estimates or limits of 11 other classicals. The new size estimates of 2002 MS and 120347 Salacia indicate that they are among the 10 largest TNOs known. Our new results confirm the recent findings that there are very diverse albedos among the classical TNOs and that cold classicals possess a high average albedo (). Diameters of classical TNOs strongly correlate with orbital inclination in our sample. We also determine the bulk densities of six binary TNOs.

Conclusions:

1 Introduction

The physical properties of small Solar System bodies offer constraints on theories of the formation and evolution of the planets. Trans-Neptunian objects (TNO), also known as Kuiper Belt objects (KBO), represent the leftovers from the formation period of the outer Solar System (Morbidelli2008 ()), and they are analogues to the parent bodies of dust in debris disks around other stars ((Wyatt2008, ; MoroMartin2008, , and references therein)).

In addition to Pluto more than 1400 TNOs have been discovered since the first Kuiper belt object in 1992 (Jewitt1993 ()), and the current discovery rate is 10 to 40 new TNOs/year. The dynamical classification is based on the current short-term dynamics. We use the classification of Gladman (Gladman et al. (2008),  Myr time-scale): classical TNOs are those non-resonant TNOs which do not belong to any other TNO class. The classical TNOs are further divided into the main classical belt, the inner belt () and the outer belt (). The eccentricity limit for classicals is , beyond which targets are classified as detached or scattered objects. The classification scheme of the Deep Eplictic Survey Team (DES, Elliot2005 () 2005) differs from the Gladman system in terms of the boundary of classical and scattered objects, which do not show a clear demarcation in their orbital parameters. Some of the classicals in the Gladman system are scattered-near or scattered-extended in the DES system. Another division is made in the inclination/eccentricity space. Although there is no dynamical separation, there seems to be two distinct but partly overlapping inclination distributions with the low-i “cold” classicals, limited to the main classical belt, showing different average albedo (Grundy2005 (); Brucker2009 () 2009), color (Trujillo2002 ()), luminosity function (Fraser2010 () 2010), and frequency of binary systems (Noll2008 ()) than the high-i “hot” classicals, which has a wider inclination distribution. Furthermore, models based on recent surveys suggest that there is considerable sub-structure within the main classical belt (Petit2011 ()). To explain these differences more quantitative data on physical size and surface composition are needed.

The physical characterization of TNOs has been limited by their large distance and relatively small sizes. Accurate albedos help to correctly interpret spectra and are needed to find correlations in the albedo-size-color-orbital parameters space that trace dynamical and collisional history. The determination of the size frequency distribution (SFD) of TNOs provides one constraint to formation models and gives the total mass. The SFD of large bodies is dominated by accretion processes and they hold information about the angular momentum of the pre-solar nebula whereas bodies smaller than 50 to are the result of collisional evolution (Petit2008 ()). The SFD can be estimated via the luminosity function (LF), but this size distribution also depends on assumptions made about surface properties such as albedo. Consequently, ambiguities in the size distributions derived from the LFs of various dynamical classes are one significant reason why there is a wide uncertainty in the total TNO mass estimate ranging from 0.01  (Bernstein2004 ()) to 0.2  (Chiang1999 ()). Among the formation models of our Solar System the “Nice” family of models have been successful in explaining the orbits of planets and the formation of the Kuiper belt (Tsiganis2005 (), Levison2008 ()), although they have difficulties in explaining some of the details of the cold and hot distributions and the origin of the two sub-populations (e.g. Fraser2010 () 2010, Petit2011 (); Batygin2011 ()).

Only a few largest TNOs have optical size estimates based on direct imaging and assumptions about the limb darkening function (e.g. Quaoar, Fraser2010b () 2010). The combination of optical and thermal infrared observations gives both sizes and geometric albedos, but requires thermal modeling. Earlier results from Spitzer and Herschel have shown the usefulness of this method (e.g. Stansberry2008 () 2008, Muller2010 () 2010) and significantly changed the size and albedo estimates of several TNOs compared to those obtained by using an assumed albedo.

In this work we present new radiometric diameters and geometric albedos for 19 classical TNOs. Half of them have no previously published observations in the wavelength regime used in this work. Those which have been observed before by Spitzer now have more complete sampling of their SEDs close to the thermal peak. The new estimates of the 19 targets are based on observations performed with the ESA Herschel Space Observatory (Pilbratt2010 ()) and its Photodetector Array Camera and Spectrometer (PACS; Poglitsch2010 ()). Other Herschel results for TNOs have been presented by Muller2010 () (2010), Lellouch2010 () (2010) and Lim2010 () (2010). New estimates of 18 Plutinos are presented in Mommert2012 () (2012) and of 15 scattered disc and detached objects in SantosSanz2012 () (2012).

This paper is organized in the following way. We describe our target sample in Section 2.1, Herschel observations in Section 2.2 and Herschel data reduction in Section 2.3. New or re-analyzed flux densities from Spitzer are presented in Section 2.4. As auxiliary data we use absolute V-band magnitudes (Section 2.5), which we have adopted from other works or data bases taking into account the factors relevant to their uncertainty estimates. Thermal modeling is described in Section 3 and the results for individual targets in Section 4. In Section 5 we discuss sample properties of our sample and of all classicals with radiometric diameters and geometric albedos as well as correlations (Section 5.1) and the bulk densities of binaries (Section 5.2). Finally, the conclusions are in Section 6.

2 Observations and data reduction

Our sample of 19 TNOs has been observed as part of the Herschel key program “TNOs are Cool” (Muller2009 ()) mainly between February and November 2010 by the photometry sub-instrument of PACS in the wavelength range 60–.

2.1 Target sample

The target sample consists of both dynamically cold and hot classicals (Table 1). We use a cut-off limit of in illustrating the two subsamples. Another typical value used in the literature is . The inclination limit is lower for large objects (Petit2011 ()) which have a higher probability of belonging to the hot population. Targets 119951 (2002 KX), 120181 (2003 UR) and 78799 (2002 XW) are in the inner classical belt and are therefore considered to belong to the low-inclination tail of the hot population. The latter two would be Centaurs in the DES system and all targets in Table 1 with would belong to the scattered-extended class of DES.

The median absolute V-magnitudes (, see Section 2.5) of our sample are 6.1  mag for the cold sub-sample and 5.3  mag for the hot one. Levison2001 () (2001) found that bright classicals have systematically higher inclinations than fainter ones. This trend is seen among our targets (see Section 5.1.3). Another known population characteristics is the lack of a clear color demarcation line at (Peixinho2008 ()), which is absent also from our sample of 10 targets with known colors.

Target KBO IAU Minor Planet Center, http://www.minorplanetcenter.net/iau/Ephemerides/Distant/, accessed July 2011. IAU Minor Planet Center, http://www.minorplanetcenter.net/iau/Ephemerides/Distant/, accessed July 2011. IAU Minor Planet Center, http://www.minorplanetcenter.net/iau/Ephemerides/Distant/, accessed July 2011. Color Taxonomic class from Fulchignoni et al. (2008) unless otherwise indicated. Spectral slope
location (AU) (AU) (°) (% / 100 nm) (mag)
119951 (2002 KX$_14$) inner 38.9 37.1 0.4 0.05 RR-IRBarucci2011 () (2011). Fornasier et al. (2009). Rabinowitz et al. (2007).
(2001 XR$_254$) main 43.0 41.7 1.2 0.03 Benecchi et al. (2009). Grundy et al. (2009).
275809 (2001 QY$_297$) main 44.0 40.4 1.5 0.08 BR footnotemark: Benecchi et al. (2009).
(2001 RZ$_143$) main 44.4 41.3 2.1 0.07 footnotemark: SantosSanz2009 () (2009).
(2002 GV$_31$) main 43.9 40.0 2.2 0.09 IAU Minor Planet Center / List of Transneptunian Objects at http://www.minorplanetcenter.net/iau/lists/TNOs.html, accessed June 2011. See Section 2.5 for conversion to .
79360 Sila (1997 CS$_29$) main 43.9 43.4 2.2 0.01 RR Fornasier et al. (2009). footnotemark:
88611 Teharonhiawako (2001 QT$_297$) main 44.2 43.2 2.6 0.02   Benecchi et al. (2009). Benecchi et al. (2009).
120181 (2003 UR$_292$) inner 32.6 26.8 2.7 0.18 Stephen Tegler, priv. comm. IAU Minor Planet Center / List of Transneptunian Objects at http://www.minorplanetcenter.net/iau/lists/TNOs.html, accessed June 2011. See Section 2.5 for conversion to .
(2005 EF$_298$) main 43.9 40.1 2.9 0.09 RR IAU Minor Planet Center / List of Transneptunian Objects at http://www.minorplanetcenter.net/iau/lists/TNOs.html, accessed June 2011. See Section 2.5 for conversion to .
138537 (2000 OK$_67$) main 46.8 40.0 4.9 0.14 RR footnotemark: footnotemark:
148780 Altjira (2001 UQ$_18$) main 44.5 41.8 5.2 0.06 RR footnotemark: footnotemark:
(2002 KW$_14$) main 46.5 37.3 9.8 0.20 Perna et al., in prep.
(2001 KA$_77$) main 47.3 42.8 11.9 0.10 RR footnotemark: Doressoundiram et al. (2005).
19521 Chaos (1998 WH$_24$) main 46.0 41.1 12.0 0.11 IR footnotemark: footnotemark:
78799 (2002 XW$_93$) inner 37.6 28.3 14.3 0.25 IAU Minor Planet Center / List of Transneptunian Objects at http://www.minorplanetcenter.net/iau/lists/TNOs.html, accessed June 2011. See Section 2.5 for conversion to .
(2002 MS$_4$) main 41.7 35.6 17.7 0.15   Stephen Tegler, priv. comm. IAU Minor Planet Center / List of Transneptunian Objects at http://www.minorplanetcenter.net/iau/lists/TNOs.html, accessed June 2011. See Section 2.5 for conversion to .
145452 (2005 RN$_43$) main 41.8 40.6 19.2 0.03 RR-IRBarucci2011 () (2011). Fornasier et al. (2009). Perna et al., in prep.
90568 (2004 GV$_9$) main 41.8 38.7 22.0 0.07 BRPerna et al. (2010). Stephen Tegler, priv. comm. DeMeo et al. (2009).
120347 Salacia (2004 SB$_60$) main 42.2 37.9 23.9 0.10 Fornasier et al. (2009). Benecchi et al. (2009).
111The horizontal line marks the limit of dynamically cold and hot classicals according to our dynamical analysis using the Gladman system. * denotes a known binary system (Noll2008 ()). References. $h$$h$footnotetext: Calculated using the technique of Hainaut2002 () (2002). $j$$j$footnotetext: Values from the Minor Bodies in the Outer Solar System database, http://www.eso.org/~ohainaut/MBOSS. $o$$o$footnotetext: Peixinho et al. (2004). $q$$q$footnotetext: Tegler and Romanishin (2000).
Table 1: Target sample. Semimajor axis , perihelion distance , inclination , eccentricity , color taxonomy, spectral slope, and the average absolute visual magnitudes (V or R-band) of 19 TNOs ordered according to increasing inclination. For used in our analysis, see Table 5.

2.2 Herschel observations

PACS is an imaging dual band photometer with a rectangular field of view of with full sampling of the -telescope’s point spread function (PSF). The two detectors are bolometer arrays, the short-wavelength one has 64  32 pixels and the long-wavelength one 32  16 pixels. In addition, the short-wavelength array has a filter wheel to select between two bands: 60 –  or 85 – , whereas the long-wavelength band is 125 – . In the PACS photometric system these bands have been assigned the reference wavelengths , and and they have the names “blue”, “green” and “red”. Both bolometers are read-out at continuously and binned by a factor of four on-board.

We specified the PACS observation requests (AOR) using the scan-map Astronomical Observation Template (AOT) in HSpot, a tool provided by the Herschel Science Ground Segment Consortium. The scan-map mode was selected due to its better overall performance compared to the point-source mode (Muller2010 () 2010). In this mode the pointing of the telescope is slewed at a constant speed over parallel lines, or “legs”. We used 10 scan legs in each AOR, separated by . The length of each leg was , except for Altjira where it was , and the slewing speed was . Each one of these maps was repeated from two to five times.

To choose the number of repetitions, i.e. the duration of observations, for our targets we used the Standard Thermal Model (see Section 3) to predict their flux densities in the PACS bands. Based on earlier Spitzer work (Stansberry2008 (), 2008) we adopted a geometric albedo of 0.08 and a beaming parameter of 1.25 for observation planning purposes. The predicted thermal fluxes depend on the sizes, which are connected to the assumed geometric albedo and the absolute V-magnitudes via Equation (3). In some cases the absolute magnitudes used for planning purposes are quite different (by up to 0.8 mag) from those used for modeling our data as more recent and accurate visible photometry was taken into account (see Section 2.5).

The PACS scan-map AOR allows the selection of either the blue or green channel; the red channel data are taken simultaneously whichever of those is chosen. The sensitivity of the blue channel is usually limited by instrumental noise, while the red channel is confusion-noise limited (PACSrelnote ()). The sensitivity in the green channel can be dominated by either source, depending on the depth and the region of the sky of the observation. For a given channel selection (blue or green) we grouped pairs of AORs, with scan orientations of and with respect to the detector array, in order to make optimal use of the rectangular shape of the detector. Thus, during a single visit of a target we grouped 4 AORs to be observed in sequence: two AORs in different scan directions and this repeated for the second channel selection.

The timing of the observations, i.e. the selection of the visibility window, has been optimized to utilize the lowest far-infrared confusion noise circumstances (Kiss2005 ()) such that the estimated signal-to-noise ratio due to confusion noise has its maximum in the green channel. Each target was visited twice with similar AORs repeated in both visits for the purpose of background subtraction. The timing of the second visit was calculated such that the target has moved 30- between the visits so that the target position during the second visit is within the high-coverage area of the map from the first visit. Thus, we can determine the background for the two source positions.

The observational details are listed in Table 2. All of the targets observed had predicted astrometric uncertainties less than at the time of the Herschel observations (David Trilling, priv. comm.).

Target OBSIDs Duration Mid-time
(min) (AU) (AU) (°)
119951 (2002 KX) 1342205144-5147 59.8 26-Sep-2010 21:52:54 39.3993 39.8605 1.30
119951 (2002 KX) 1342205175-5178 59.8 27-Sep-2010 15:35:54 39.3993 39.8716 1.29
(2001 XR) 1342205184-5187 78.6 27-Sep-2010 18:48:10 44.1004 44.4032 1.25
(2001 XR) 1342205264-5267 78.6 28-Sep-2010 15:48:21 44.1004 44.3886 1.25
(2001 QY) 1342209492-9495 97.4 18-Nov-2010 10:06:17 43.2452 43.3599 1.31
(2001 QY) 1342209650-9653 97.4 19-Nov-2010 20:50:15 43.2455 43.3850 1.31
(2001 RZ) 1342199503-9506 97.4 01-Jul-2010 00:52:31 41.3006 41.6947 1.31
(2001 RZ) 1342199614-9617 97.4 01-Jul-2010 20:28:08 41.3005 41.6819 1.32
(2002 GV) 1342198847-8850 59.8 20-Jun-2010 20:07:56 40.2818 40.5238 1.42
(2002 GV) 1342198897-8900 59.8 21-Jun-2010 21:55:30 40.2817 40.5412 1.41
79360 SilaObservation in the chopped and nodded point-source mode and only at the blue and red channels (Muller2010 () 2010). 1342187073 94.4 18-Nov-2009 14:24:02 43.5090 43.2410 1.27
79360 Sila 1342196137-6140 59.8 09-May-2010 01:29:23 43.5057 43.6530 1.33
79360 Sila 1342196137-6140 59.8 10-May-2010 09:15:12 43.5057 43.6753 1.32
88611 Teharonhiawako 1342196099-6102 78.6 09-May-2010 22:19:55 45.0856 45.3612 1.24
88611 Teharonhiawako 1342196145-6148 78.6 10-May-2010 18:30:58 45.0857 45.3475 1.25
120181 (2003 UR) 1342199618-9621 59.8 01-Jul-2010 21:49:00 26.7872 27.1477 2.04
120181 (2003 UR) 1342199646-9649 59.8 02-Jul-2010 12:43:03 26.7872 27.1379 2.05
(2005 EF) 1342208962-8965 97.4 03-Nov-2010 20:07:09 40.7207 41.1758 1.25
(2005 EF) 1342208999-9002 97.4 04-Nov-2010 10:08:12 40.7207 41.1667 1.25
138537 (2000 OK) 1342197665-7668 78.6 03-Jun-2010 00:55:51 40.3008 40.3581 1.46
138537 (2000 OK) 1342197717-7720 78.6 04-Jun-2010 13:55:32 40.3006 40.3319 1.46
148780 Altjira 1342190917-0920 76.0 21-Feb-2010 23:32:17 45.5387 45.5571 1.25
148780 Altjira 1342191120-1123 76.0 24-Feb-2010 01:31:06 45.5390 45.5940 1.25
(2002 KW) 1342204196-4199 59.8 09-Sep-2010 11:11:36 40.8385 41.0710 1.38
(2002 KW) 1342204282-4285 59.8 10-Sep-2010 09:37:01 40.8389 41.0870 1.38
(2001 KA) 1342205962-5965 78.6 06-Oct-2010 21:09:43 48.1789 48.6245 1.07
(2001 KA) 1342206013-6016 78.6 07-Oct-2010 14:32:45 48.1787 48.6354 1.07
19521 Chaos 1342202285-2288 41.0 08-Aug-2010 19:40:03 41.6914 42.1566 1.24
19521 Chaos 1342202316-2319 41.0 09-Aug-2010 11:34:49 41.6913 42.1464 1.25
78799 (2002 XW) 1342190913-0916 57.8 21-Feb-2010 22:20:43 44.6271 44.3163 1.22
78799 (2002 XW) 1342191116-1119 57.8 24-Feb-2010 00:19:32 44.6279 44.3518 1.23
(2002 MS) 1342204140-4143 41.0 08-Sep-2010 21:12:31 47.1673 46.9025 1.19
(2002 MS) 1342204292-4295 41.0 10-Sep-2010 15:03:37 47.1670 46.9307 1.20
145452 (2005 RN) 1342195583-5586 41.0 25-Apr-2010 22:21:44 40.6885 41.1522 1.26
145452 (2005 RN) 1342195600-5603 41.0 26-Apr-2010 21:43:12 40.6885 41.1378 1.28
90568 (2004 GV) 1342202869-2872 41.0 11-Aug-2010 18:35:31 39.1876 39.3841 1.46
90568 (2004 GV) 1342202921-2924 41.0 12-Aug-2010 15:22:58 39.1877 39.3984 1.46
120347 Salacia 1342199133-9136 41.0 22-Jun-2010 01:17:20 44.1464 44.0169 1.33
120347 Salacia 1342199133-9136 41.0 22-Jun-2010 18:57:52 44.1466 44.0058 1.32
222, and are from the JPL Horizons Ephemeris System (Giorgini1996 ()).
Table 2: Individual observations of the sample of 19 TNOs by Herschel/PACS. OBSIDs are the observation identifiers, duration is the total duration of the four AORs (see text), mid-time is the mean UT time, is the mean heliocentric distance, is the mean Herschel-target distance, and is the mean phase angle. Each line corresponds to one visit and all three channels were observed at each visit.

2.3 Data reduction

The data reduction from level 0 (raw data) to level 2 (maps) was done using Herschel Interactive Processing Environment (HIPE333Data presented in this paper were analysed using “HIPE”, a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia members, see http://herschel.esac.esa.int/DpHipeContributors.shtml.) with modified scan-map pipeline scripts optimized for the “TNOs are Cool” key program. The individual maps (see Fig. 1 for examples) from the same epoch and channel are mosaicked, and background-matching and source-stacking techniques are applied. The two visits are combined (Fig. 2), in each of the three bands, and the target with a known apparent motion is located at the center region of these maps. The detector pixel sizes are in the blue and green channels, and in the red channel whereas the pixel sizes in the maps produced by this data reduction are / / in the blue / green / red maps, respectively. A detailed description of the data reduction in the key program is given in Kiss et al. (in prep.).

Figure 1: Individual maps of 120347 Salacia. Each map is the product of one observation (AOR). The first row (M1-M6) is from the first visit and the second row (M7-M12) from the follow-on visit. The first two columns (M1-M2, M7-M8) are observations in the or “green” channel and the others in the or “red” channel. The two scan angles are 110°(odd-numbered maps) and 70°. The source is clearly seen in the map center in the green channel whereas the red channel is more affected by background sources and confusion noise. Orientation: north is up and east is to the left.

Once the target is identified we measure the flux densities at the photocenter position using DAOPHOT routines (Stetson1987 ()) for aperture photometry. We make a correction for the encircled energy fraction of a point source (PSF2010 ()) for each aperture used. We try to choose the optimum aperture radius in the plateau of stability of the growth-curves, which is typically 1.0-1.25 times the full-width-half-maximum of the PSF ( / / in the blue / green / red bands, respectively). The median aperture radius for targets in the “TNOs are Cool” program is 5 pixels in the final maps (pixel sizes 1.1”/1.4”/2.1” in the blue / green / red maps). For the uncertainty estimation of the flux density we implant 200 artificial sources in the map in a region close to the source () excluding the target itself. A detailed description of how aperture photometry is implemented in our program is given in SantosSanz2012 () (2012).

In order to obtain monochromatic flux density values of targets having a spectral energy distribution different from the default one color corrections are needed. In the photometric system of the PACS instrument flux density is defined to be the flux density that a source with a flat spectrum (constant, where is the wavelength and is the monochromatic flux) would have at the PACS reference wavelengths (Poglitsch2010 ()). Instead of the flat default spectrum we use a cool black body distribution to calculate correction coefficients for each PACS band. The filter transmission and bolometer response curves needed for this calculation are available from HIPE, and we take as black body temperature the disk averaged day-side temperature calculated iteratively for each target (, using STM assumptions from Section 3, the Lambertian emission model and the sub-solar temperature from Eq. 2.) This calculation yields on the average 0.982 / 0.986 / 1.011 (flux densities are divided by color correction factors) for the blue / green / red channels, respectively, with small variation among the targets of our sample.

The absolute flux density calibration of PACS is based on standard stars and large main belt asteroids and has the uncertainties of 3% / 3% / 5% for the blue / green / red bands (PACScal2011 ()). We have taken these uncertainties into account in the PACS flux densities used in the modeling, although their contribution to the total uncertainty is small compared to the signal-to-noise ratio of our observations.

The color corrected flux densities are given in Table 3. They were determined from the combined maps of two visits, in total 4 AORs for the blue and green channels and 8 AORs for the red. The only exceptions are 19521 Chaos and 90568 (2004 GV), whose one map was excluded from our analysis due to a problem in obtaining reliable photometry from those observations. The uncertainties in Table 3 include the photometric and absolute calibration uncertainties. 17 targets were detected in at least one PACS channel. The upper limits are the noise levels of the maps, including both the instrumental noise and residuals from the eliminated infrared background confusion noise. 79360 Sila has a flux density which is lower by a factor of three in the red channel than the one published by Muller2010 () (2010). We have re-analyzed this earlier chopped/nodded observation using the latest knowledge on calibration and data reduction and found no significant change in the flux density values. As speculated in Muller2010 () (2010) the 2009 single-visit Herschel observation was most probably contaminated by a background source.

Target
(mJy) (mJy) (mJy)
119951 (2002 KX)
(2001 XR)
275809 (2001 QY)
(2001 RZ)
(2002 GV)
79360 Sila
88611 Teharonhiawako
120181 (2003 UR)
(2005 EF)
138537 (2000 OK)
148780 Altjira
(2002 KW)
(2001 KA)
19521 Chaos Observation 1342202317 excluded.
78799 (2002 XW)
(2002 MS)
145452 (2005 RN)
90568 (2004 GV) Observation 1342202923 excluded.
120347 Salacia
444
Table 3: Color corrected Herschel flux densities of the sample of 19 classical TNOs from coadded images of two visits. F, F and F are the monochromatic flux densities of the PACS blue / green / red channels.
Figure 2: Combined maps of 120347 Salacia from the individual maps (Fig. 1) in the green (left) and red channels. Orientation: north is up and east is to the left.

2.4 Complementary Spitzer observations

About 75 TNOs and Centaurs in the “TNOs are Cool” program were also observed by the Spitzer Space Telescope (Werner2004 ()) using the Multiband Imaging Photometer for Spitzer (MIPS; Rieke2004 ()). 43 targets were detected at a useful signal-to-noise ratio in both the and bands of that instrument. As was done for our Herschel program, many of the Spitzer observations utilized multiple AORs for a single target, with the visits timed to allow subtraction of background confusion. The MIPS band, when combined with 70-160 data, can provide very strong constraints on the temperature of the warmest regions of a TNO.

The absolute calibration, photometric methods and color corrections for the MIPS data are described in Gordon et al. (2007), Engelbracht et al. (2007) and Stansberry et al. (2007). Nominal calibration uncertaintes are 2% and 4% in the and bands respectively. To allow for additional uncertainties that may be caused by the sky-subtraction process, application of color corrections, and the faintness of TNOs relative to the MIPS stellar calibrators, we adopt uncertainties of 3% and 6% as has been done previously for MIPS TNO data (e.g. Stansberry2008 () 2008, Brucker2009 () 2009). The effective monochromatic wavelengths of the two MIPS bands we use are and . With an aperture of the telescope-limited spatial resolution is and in the two bands.

Spitzer flux densities of 13 targets overlapping our classical TNO sample are given in Table 4. The new and re-analyzed flux densities are based on re-reduction of the data using updated ephemeris positions. They sometimes differ by or more from those used to point Spitzer. The ephemeris information is used in the reduction of the raw data, for generating the sky background images, and for accurate placement of photometric apertures. This is especially important for the Classical TNOs which are among the faintest objects observed by Spitzer. Results for four targets are previously unpublished: 275809 (2001 QY), 79360 Sila, 88611 Teharonhiawako, and 19521 Chaos. As a result of the reprocessing of the data, the fluxes for 119951 (2002 KX), 148780 Altjira, 2001 KA and 2002 MS differ from those published in Brucker2009 () (2009) and Stansberry2008 () (2008).

The bands of PACS and MIPS are overlapping and the flux density values agree typically within for the six targets observed by both instruments with SNR .

Target Duration Starting day Previous works This work
(min) of observing MIPS F MIPS F MIPS F MIPS F
epoch (mJy) (mJy) (mJy) (mJy)
119951 128.10 2005-08-26 39.61 39.59 1.5 Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties. Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties.
119951 37.27 2006-03-30 39.58 39.20 1.4 Stansberry2008 () (2008), uncertainty calculated from SNR and includes quadratically added calibration uncertainties, upper limits shown here are . Stansberry2008 () (2008), uncertainty calculated from SNR and includes quadratically added calibration uncertainties, upper limits shown here are .
275809 594.87 2008-11-21 42.76 42.39 1.3
(2001 RZ) 203.30 2004-12-26 41.38 40.98 1.3 Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties. Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties.
79360 494.70 2008-05-18 43.52 43.30 1.3
88611 398.42 2004-11-04 45.00 44.67 1.2
138537 257.45 2004-11-04 40.57 40.11 1.3 Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties. Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties.
148780 400.92 2006-02-16 45.33 45.08 1.2 Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties. Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties.
(2002 KW) 213.05 2005-08-26 40.04 40.03 1.5 Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties. Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties.
(2001 KA) 400.90 2006-03-31 48.56 48.35 1.2 Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties. Color corrected MIPS flux density with measurement uncertainty from Brucker2009 () (2009) together with quadratically added calibration uncertainties.
19521 66.06 2004-09-24 42.05 41.68 1.3
(2002 MS) 82.04 2006-03-31 47.39 47.48 1.2 Stansberry2008 () (2008), uncertainty calculated from SNR and includes quadratically added calibration uncertainties, upper limits shown here are . Stansberry2008 () (2008), uncertainty calculated from SNR and includes quadratically added calibration uncertainties, upper limits shown here are .
90568 57.00 2005-01-28 38.99 39.01 1.5 Stansberry2008 () (2008), uncertainty calculated from SNR and includes quadratically added calibration uncertainties, upper limits shown here are . Stansberry2008 () (2008), uncertainty calculated from SNR and includes quadratically added calibration uncertainties, upper limits shown here are .
120347 227.62 2006-12-03 43.82 43.39 1.2 Stansberry et al. (2012). Stansberry et al. (2012).
555The Spitzer program IDs are: 55 (for the observations of Chaos), 3229 (Teharonhiawako), 3283 (2002 KX14 in 2006, 2002 MS4, 2004 GV9), 3542 (2002 KX14 in 2005, 2001 RZ143, 2000 OK67, Altjira, 2002 KW14, 2001 KA77), 30081 (Salacia), 40016 (Sila), and 50024 (2001 QY297). References.
Table 4: Complementary Spitzer observations. The duration includes the total time of several visits. Observing epochs lasted 1-14 days. The other quantities are as in Table 2. Targets below the horizontal line have .

2.5 Optical photometry

The averages of V-band or R-band absolute magnitude values available in the literature are given in Table 1. The most reliable way of determining them is to observe a target at multiple phase angles and over a time span enough to determine lightcurve properties, but such complete data are available only for 119951 (2002 KX) in our sample. For all other targets assumptions about the phase behavior have been made due to the lack of coverage in the range of phase angles.

The IAU (H,G) magnitude system for photometric phase curve corrections (Bowell1989 ()), which has been adopted in some references, is known to fail in the case of many TNOs (Belskaya2008 () 2008). Due to the lack of a better system we prefer linear methods as a first approximation since TNOs have steep phase curves which do not deviate from a linear one in the limited phase angle range usually available for TNO observations. The opposition surge of very small phase angles (, Belskaya2008 () 2008) has not been observed for any of our 19 targets due to the lack of observations at such small phase angles. All of our Herschel and Spitzer observations are limited to the range .

We have used the linear method (, where is the apparent V-magnitude in the Johnson-Coussins or the Bessel V-band, and other symbols are as in Table 2, and is the linearity coefficient) to calculate the values from individual V-magnitudes given in the references (see Table 5) for 2001 XR, 275809 (2001 QY), 79360 Sila, 88611 Teharonhiawako, 148780 Altjira and 19521 Chaos. In order to be consistent with most of the values in the literature we have adopted calculated from Sheppard2002 () (2002). The effect of slightly different values of or assumptions of its composite value used in previous works is usually negligible compared to uncertainties caused by lightcurve variability (an exception is 90568 (2004 GV)).

For five targets with no other sources available we take the absolute magnitudes in the R-band ( in Table 5) from the Minor Planet Center and calculate their standard deviation, since the number of V-band observations for these targets is very low, and use the average (V-R) color index for classical TNOs (Hainaut2002 () 2002) to derive . While the MPC is mainly used for astrometry and the magnitudes are considered to be inaccurate by some works (e.g. Benecchi2011 () 2011, Romanishin2005 () 2005), for the five targets we use the average of 9 to 16 R-band observations and adopt the average R-band phase coefficient (calculated from Belskaya2008 () 2008).

The absolute V-magnitudes used as input in our analysis (the “Corrected ” column in Table 5) take into account additional uncertainties from known or assumed variability in . The amplitude, the period and the time of zero phase of the lightcurves of three hot classicals (145452 (2005 RN), 90568 (2004 GV) and 120347 Salacia) are available in the literature, but even for these three targets the uncertainty in the lightcurve period is too large to be used for exact phasing with Herschel observations. The lightcurve amplitude or amplitude limit is known for half of our targets and for them we add 88% of the half of the peak-to-peak amplitude ( i.e. 68% of the values of a sinusoid are within this range) quadratically to the uncertainty of . According to a study of a sample of 74 TNOs from various dynamical classes (Duffard2009 ()) 70% of TNOs have a peak-to-peak amplitude mag, thus we quadratically add 0.09 mag to the uncertainty of for those targets in our sample for which lightcurve information is not available.

Target -phase method L.c. L.c. period Corrected Corrections
ref. ([]=mag/°) (mag) () (mag) included
119951 (2002 KX) (a) phase study assumed
(2001 XR) (b) (1) ( a few 0.1:s)Benecchi2011 () (2011).
275809 (2001 QY) (c,d,e) (1) 0.49–0.66Thirouin2012 () (2012).
(2001 RZ) (f) SantosSanz2009 () (2009). assumed
(2002 GV) (g) (2)
79360 Sila (h,i,j,k) (1) 0.08Romanishin2005 () (2005).
88611 Teharonhiawako (l) (1) 0.2Osip et al. (2003), lightcurve caused by secondary component, amplitude calculated for the combined system. Osip et al. (2003), lightcurve caused by secondary component, amplitude calculated for the combined system.
120181 (2003 UR) (g) Calculated from MPC data from 16 observations.
(2005 EF) (g) (2)
138537 (2000 OK) (m,n) Doressoundiram et al. (2002). / Delsanti et al. (2001). assumed
148780 Altjira (o,p,d) (1) 0.3Barucci et al. (2000).
(2002 KW) (q) (1) 0.21 / 0.26Thirouin2012 () (2012).
(2001 KA) (o) (1) assumed
19521 Chaos (h,i,n,r) (1) 0.10Sheppard2002 () (2002).
78799 (2002 XW) (g) (2)
(2002 MS) (g) (2)
145452 (2005 RN) (q) (1) Thirouin2010 () (2010).
90568 (2004 GV) (s) Sheppard (2007). Dotto et al. (2008). Dotto et al. (2008). ,
120347 Salacia (q) (1) Thirouin2010 () (2010). Thirouin2010 () (2010).
666* denotes a known binary system (Noll2008 ()). (1) V-band linear phase coefficient (Sheppard2002 () 2002, calculated from values therein). (2) R-band linear phase coefficient (Average from Belskaya2008 () 2008). References. $e$$e$footnotetext: Doressoundiram et al. (2007). $g$$g$footnotetext: IAU Minor Planet Center / List of Transneptunian Objects at http://www.minorplanetcenter.net/iau/lists/TNOs.html, accessed June 2011. $h$$h$footnotetext: Boehnhardt et al. (2001). $i$$i$footnotetext: Davies et al. (2000). $j$$j$footnotetext: Barucci et al. (2000). $k$$k$footnotetext: Grundy et al. (2012). $l$$l$footnotetext: Benecchi et al. (2009). $m$$m$footnotetext: Benecchi2011 () (2011). $n$$n$footnotetext: Doressoundiram et al. (2002). $o$$o$footnotetext: Doressoundiram et al. (2005). $q$$q$footnotetext: Perna et al., in prep. $r$$r$footnotetext: Tegler and Romanishin (2000). $s$$s$footnotetext: DeMeo et al. (2009).
Table 5: Overview of optical auxiliary data. The average absolute V-band (or R-band) magnitudes from literature are given in Table 1, and this table gives the absolute V-magnitudes with uncertainties which take into account the lightcurve (L.c.) amplitude (either in the UBVRI system R-band or Sloan’s r’-band). The -phase method (see text) tells how the extrapolation to zero phase was done (either in the reference or in this work). The last column indicates which corrections are significant in contributing to the corrected .

3 Thermal modeling

The combination of observations from thermal-infrared and optical wavelengths allows us to estimate various physical properties via thermal modeling. For a given temperature distribution the disk-integrated thermal emission observed at wavelength is

(1)

where is the emissivity, the observer-target distance, Planck’s radiation law for black bodies, the temperature distribution on the surface and u the unit directional vector toward the observer from the surface element . The temperature distribution of an airless body depends on physical parameters such as diameter, albedo, thermal inertia, and surface roughness.

There are three basic types of models to predict the emission of an asteroid with a given size and albedo assuming an equilibrium between insolation and re-emitted thermal radiation: the Standard Thermal Model (STM), the Fast-rotating Isothermal Latitude thermal model (Veeder1989 ()), and the thermophysical models (starting from Matson (1971), e.g. Spencer1989 (); Lagerros1996 ()). While originally developed for asteroids in the mid-IR wavelengths these models are applicable for TNOs, whose thermal peak is in the far-IR.

The STM (cf. Lebofsky1986 () and references therein) assumes a smooth, spherical asteroid, which is not rotating and/or has zero thermal inertia, and is observed at zero phase angle. The subsolar temperature is

(2)

where is the Bond albedo, is the solar constant, is the beaming factor, is the Stefan-Boltzmann constant and is the heliocentric distance. In the STM does not depend on wavelength. The beaming factor adjusts the subsolar temperature. The canonical value is based on calibrations using the largest few main belt asteroids. The STM assumes an average linear infrared phase coefficient of 0.01 mag/degree based on observations of main belt asteroids.

In this work we use the Near-Earth Asteroid Thermal Model NEATM (Harris1998 ()). The difference to the STM is that in the NEATM is fitted with the data instead of using a single canonical value. For rough surfaces takes into account the fact that points on the surface radiate their heat preferentially in the sunward direction. High values () lead to a reduction of the model surface temperature, mimicking the effect of high thermal inertia, whereas lower values are a result of surface roughness. Furthermore, the phase angle is taken into account by calculating the thermal flux an observer would detect from the illuminated part of assuming a Lambertian emission model and no emission from the non-illuminated side.

Whenever data quality permits we treat as a free fitting parameter. However, in some cases of poor data quality this method leads to values which are too high or too low and therefore unphysical. In these cases we fix it to a canonical value of derived by Stansberry2008 () (2008) from Spitzer observations of TNOs. The physical range of values is determined by using NEATM as explained in Mommert2012 () (2012) to be .

Throughout this work we assume the surface emissivity , which is based on laboratory measurements of silicate powder up to a wavelength of (Hovis1966 ()) and a usual approximation for small bodies in the Solar System. At far-IR wavelengths the emissivity of asteroids may be decreasing as a function of wavelength (Muller1998 ()) from 24 to , but the amount depends on individual target properties. Thus, a constant value is assumed for simplicity. The variation of emissivity of icy surfaces as a function of wavelength could in principle provide hints about surface composition. HO ice has an emissivity close to one with small variations (Schmitt1998 ()) whereas other ices may show a stronger wavelength dependence (e.g. Stansberry1996 ()). Near-IR spectroscopic surface studies have been done for five of our targets but none of them show a reliable detection of ices, even though many dynamically hot classicals are known to have ice signatures in their spectra (Barucci2011 () 2011).

For the Bond albedo we assume that since the V-band is close to the peak of the solar spectrum. The underlying assumption is that the Bond albedo is not strongly varying across the relevant solar spectral range of reflected light. The Bond albedo is connected to the geometric albedo via , where is the phase integral and the geometric albedo in V-band. Instead of the canonical value of (Bowell1989 ()) we have adopted (Brucker2009 () 2009), which they used for a Spitzer study of classical TNOs. From the definition of the absolute magnitude of asteroids we have:

(3)

where is the area projected toward the observer, is the distance of one astronomical unit, is the apparent V-magnitude of the Sun, and is the absolute V-magnitude of the asteroid. We use mag (Bessell1998 (); Hayes1985 ()). An error of 0.02 mag in means a relative error of in the product (3).

We find the free parameters , and in a weighted least-squares sense by minimizing the cost function

(4)

where is the number of degrees of freedom, is the number of data points in the far-infrared wavelengths, is the observed flux density at wavelength with uncertainty , and is the modeled emission spectrum. When only upper flux density limits are available, they are treated as having zero flux density with a uncertainty equal to the upper limit flux density uncertainty (). The cost function (4) does not follow the statistical distribution since in a non-linear fit the residuals are not normally distributed even if the flux densities had normally distributed uncertainties.

3.1 Error estimates

The error estimates of the geometric albedo, the diameter and the beaming parameter are determined by a Monte Carlo method described in Mueller et al. (2011). We generate 500 sets of synthetic flux densities normally distributed around the observed flux densities with the same standard deviations as the observations. Similarly, a set of normally distributed values is generated. For those targets whose we do a rescaling of the errorbars of the flux densities before applying the Monte Carlo error estimation as described in SantosSanz2012 () (2012) and illustrated in Mommert2012 () (2012).

The NEATM model gives us the effective diameter of a spherical target. 70-80% of TNOs are known to be MacLaurin spheroids with an axial ratio of 1.15 (Duffard2009 (), Thirouin2010 () 2010). When the projected surface has the shape of an ellipse instead of a circle, then this ellipse will emit more flux than the corresponding circular disk which has the same surface because the Sun is seen at higher elevations from a larger portion of the ellipsoid than on the sphere. Therefore, the NEATM diameters may be slightly overestimated. Based on studies of model accuracy (e.g. Harris2006 ()) we adopt uncertainties of 5% in the diameter estimates and 10% in the estimates to account for systematic model errors when NEATM is applied at small phase angles.

4 Results of individual targets

In this Section we give the results of model fits using the NEATM to determine the area-equivalent diameters (see Section 3) as well as geometric albedos and beaming factors. We note that our observations did not spatially resolve binary systems, we therefore find area-equivalent diameters of the entire system rather than component diameters; this will be further discussed in Section 5.2.

In cases where also Spitzer/MIPS data are available for a target we determine the free parameters for both PACS only and the combined data sets. The solutions are given in Table 6. A floating- solution is only adopted if its is not much greater than unity. The exact limit depends on the number of data points. For this limit is . For the PACS-only data set we may adopt floating- solutions only if there are no upper-limit data points. 138537 (2000 OK) has only upper limits from PACS, therefore only the solution using combined data is shown. 2001 RZ has five data points, three of which are upper limits. The PACS flux density at is approximately a factor of three higher than the MIPS upper limit (see Tables 3 and 4). For this target we adopt the model solution determined without the MIPS channel. The best solution for each target is shown in Fig. 3.

The radiometric diameters determined with data from the two instruments are on the average close to the corresponding results using PACS data alone, but there are some significant differences as well, most notably 275809 (2001 QY) and 2002 KW. The former has a PACS-only solution, which is within the error bars of the green and red channel data points, but above the PACS blue channel data point. When the two upper limits from MIPS are added in the analysis the model solution is at a lower flux level below the PACS green channel data point but compatible with the other data (see Fig. 3). 2002 KW has upper limits in the PACS green and red channels as well as in the MIPS channel. Without this upper limit in the shortest wavelength the PACS-only solution is at higher flux levels in short wavelengths and gives a lower flux at long wavelengths.

When choosing the preferred solution we are comparing two fits with different numbers of data points used, therefore in this comparison we calculate for the PACS-only solution using the same data points as for the combined solution taking into account the different observing geometries during Herschel and Spitzer observations. In all cases where MIPS data is available the solution based on the combined data from the two instruments is the preferred one.

2002 GV is the only non-detection by both PACS and MIPS in our sample of 19 targets. The astrometric uncertainty at the time of the PACS observation was (semimajor axis of the confidence ellipsoid777Asteroids Dynamic Site by A. Milani, Z. Knezevic, O. Arratia et al., http://hamilton.dm.unipi.it/astdys/, accessed August 2011, calculations based on the OrbFit software.), which is well within the high-coverage area of our maps.

The error bars from the Monte Carlo error estimation method may sometimes be too optimistic compared to the accuracy of optical data and the model uncertainty of NEATM (see Section 3). We check that the uncertainty of geometric albedo is not better than the uncertainty implied by the optical constraint (Eq. 3) due to uncertainties in and . The lower uncertainty of four targets is limited by this (see Table 6).

Target Instruments Bin- Rabinowitz et al. (2007).
(km) ary?
119951 (2002 KX) PACS no
PACS, MIPS
(2001 XR) PACS yes (fixed)
275809 (2001 QY) PACS yes (fixed)
PACS, MIPS (fixed)
(2001 RZ) PACS yes (fixed)
PACS, MIPSGrundy et al. (2009).
(2002 GV) PACS no (fixed)
79360 Sila PACS yes (fixed)
PACS, MIPS
88611 Teharonhiawako PACS yes (fixed)
PACS, MIPS
120181 (2003 UR) PACS no (fixed)
(2005 EF) PACS no (fixed)
138537 (2000 OK) PACS, MIPS no (fixed)
148780 Altjira PACS yes (fixed)
PACS, MIPS
(2002 KW) PACS no (fixed)
PACS, MIPS
(2001 KA) PACS no (fixed)
PACS, MIPS
19521 Chaos PACS no
PACS, MIPS
78799 (2002 XW) PACS no
(2002 MS) PACS no (fixed)
PACS, MIPS Error estimate limited by the adopted diameter uncertainty of 5% of the NEATM model.
145452 (2005 RN) PACS no (fixed)
90568 (2004 GV) PACS no
PACS, MIPS Error estimate limited by the adopted diameter uncertainty of 5% of the NEATM model.
120347 Salacia PACS yes
PACS, MIPS Error estimate limited by the adopted diameter uncertainty of 5% of the NEATM model. Grundy et al. (2011).
888 $d$$d$footnotetext: Error estimate limited by the adopted uncertainty of 10% of the NEATM model.
Table 6: Solutions for radiometric diameters and geometric albedos (see text for explanations). For binary systems is the area-equivalent system diameter. In case of two solutions the preferred one is based on data from both PACS and MIPS instruments.
Figure 3: Adopted model solutions from Table 6. The black data points are from PACS (70, 100 and ) and the gray points are from MIPS (24 and ) normalized to the geometry of Herschel observations by calculating the NEATM solution (for given , and ) at both the epochs of the Herschel and Spitzer observations and using their ratio as a correction factor.
Figure 3: continued.

5 Discussion

Our new size and geometric albedo estimates improve the accuracy of previous estimates of almost all the targets with existing results or limits (Table 7). Two thirds of our targets have higher albedos (Table 6) than the used in the planning of these observations, which has lead to the lower than expected SNRs and several upper limit flux densities (Table 3). Previous results from Spitzer are generally compatible with our new estimates. However, two targets are significantly different: 119951 (2002 KX) and 2001 KA, whose solutions are compatible with the optical constraint (Eq. 3), but the new estimate has a large diameter and low albedo (target 119951) instead of a small diameter and high albedo, or vice versa (target 2001 KA). It can be noted that there is a significant difference in the re-processed Spitzer flux densities at compared to the previously published values (see Table 4), which together with the PACS data can explain the significant change in the diameter and geometric albedo estimates. Our new estimates for 148780 Altjira differ from the previous upper and lower limits based on Spitzer data alone (Table 7). This change can be explained by the addition of the and PACS data points (Table 3) to the earlier MIPS value and the MIPS upper flux density limit (Table 4).

This work Previous work
Target D (km) p () D (km) p Reference
119951 (2002 KX) 24, 71 Brucker2009 () (2009)
(2001 XR) B (binary) 130–208Lower uncertainty limited by the uncertainty of for 275809 / PACS-MIPS, 2005 EF, 78799, and 2002 MS / both solutions. 0.09–0.23 Grundy et al. (2011)
275809 (2001 QY) B (binary) 128–200Diameter of the primary component. 0.13–0.32 Grundy et al. (2011)
(2001 RZ) B 24 Brucker2009 () (2009)
79360 Sila B 70, 160 250–420 0.06-0.14 Muller2010 () (2010)
88611 Teharonhiawako B (binary) 114–180Diameter of the primary component. 0.13–0.32 Grundy et al. (2011)
138537 (2000 OK) 24 Brucker2009 () (2009)
148780 Altjira B 24 Brucker2009 () (2009)
(binary) 128–200Diameter of the primary component. 0.06–0.14 Grundy et al. (2011)
(2002 KW) 71 Brucker2009 () (2009)
(2001 KA) 24, 71 Brucker2009 () (2009)
19521 Chaos 1200 Altenhoff et al. (2004)
(2002 MS) 24, 71 Brucker2009 () (2009)
90568 (2004 GV) 24, 71 Brucker2009 () (2009)
120347 Salacia B (binary) 720–1140Diameter of the primary component. 0.01–0.03 Grundy et al. (2011)
24, 71 Stansberry et al. (2012)
(2001 QD) 24, 71 Brucker2009 () (2009)
(1996 TS) 24, 71 Brucker2009 () (2009)
50000 Quaoar B 24, 71, (direct) Diameter of the primary component. Fraser2010b () (2010)
(2002 GJ) 24, 71 Brucker2009 () (2009)
20000 Varuna 71 Brucker2009 () (2009)
55637 (2002 UX) B 24, 71 Stansberry2008 () (2008)
55636 (2002 TX) (occultation) Elliot et al. (2010)
55565 (2002 AW) 24, 71 Brucker2009 () (2009)
999
Table 7: Adopted physical properties in comparison with previous works. Binary systems are indicated by ’B’. Column lists the wavelengths used in the reference for radiometric diameters. If no radiometric result is available then binary system mass is used to give the diameter of the primary component assuming equal albedos. The second part of this table lists all other classical TNOs with radiometric or other reliable size estimates. The size of 50000 Quaoar is derived from both Spitzer and from direct imaging by Hubble. All targets in the lower part are dynamically hot.

The diameter estimates in our sample are ranging from 100 km of 120181 (2003 UR) up to 930 km of 2002 MS, which is larger than previously estimated for it. 2002 MS and 120347 Salacia are among the ten largest TNOs with sizes similar to those of 50000 Quaoar and 90482 Orcus. The size distribution of hot classicals in our sample is wider than that of the cold classicals, which are limited to diameters of 100- km (Fig. 4). The diameters of eight hot classicals from literature data (Table 7) are within the same size range as the hot classicals in our sample. The cumulative size distribution of this extended set of 20 hot classicals (Fig. 5) shows two regimes of a power law distribution with a turning point between 500 and 700 km. The slope of the cumulative distribution is for the (N=11) objects. There are not enough targets for a reliable slope determination in the regime. The size distribution is an important property in understanding the processes of planet formation. Several works have derived it from the LF using simplifying assumptions about common albedo and distance. Fraser2010 () (2010) reported a slope of the differential size distribution of for a dynamically hot TNO population (38 AU heliocentric distance 55 AU and ). Our based on a small sample of measured diameters of intermediate-size hot classicals is compatible with this literature value. The high- tail at in Fig. 5 indicates a change of slope when the population transitions from a primordial one to a collisionally relaxed population. Based on LF estimates, this change in slope for the whole TNO population was expected at (Kenyon2008 () based on data from Bernstein2004 ()) or at somewhat larger diameters (Petit2006 ()). For Plutinos a change to a steeper slope occurs at 450 km (Mommert2012 () 2012).

Figure 4: Distribution of diameters from this work (upper left), the cold classicals of this work (upper right), the hot classicals of this work (lower left), and all hot classicals including literature results from Table 7 (lower right). The last plot includes only dynamically hot classicals. The bin size is 100 km.
Figure 5: Cumulative size distribution of dynamically hot classicals from this work and literature (Table 7). The power law has a change between 500 and 700 km. The intermediate size classicals have a slope parameter of .

The dynamically cold and hot sub-populations are showing different geometric albedo distributions (Fig. 6) with the dynamically cold objects having higher geometric albedos in a narrower distribution. The average geometric albedo of the six cold classicals is (un-weighted average and standard deviation). The highest-albedo object is 88611 Teharonhiawako with , or possibly 2002 GV with the lower limit of 0.22. These findings are compatible with the conclusions of Brucker2009 () (2009) based on Spitzer data that cold classicals have a high albedo, although we do not confirm their extreme geometric albedo of 0.6 for 119951 (2002 KX).

The darkest object in our sample is the dynamically hot target 78799 (2002 XW) with a geometric albedo of 0.038. The highest-albedo hot classicals are found in the low- part of the sub-sample (see Fig. 7): 138537 (2000 OK) at has and the inner belt target 120181 (2003 UR) has a geometric albedo of 0.16. The 12 hot classicals in our sample have, on the average, lower albedos than the cold ones: . The average of the combined hot classical sub-population of this work and literature is if 55636 (2002 TX) is excluded.

Figure 6: Distribution of geometric albedos from this work (upper left), the cold classicals of this work (upper right), the hot classicals of this work (lower left), and all hot classicals including literature results from Table 7 (lower right). The bin size is 0.05. The Haumea family member 55636 (2002 TX) with is beyond the horizontal scale.

From the floating- solutions of eight targets with data from both instruments included in the fitted solution (see Table 6 and Section 4) we have the average (un-weighted). Most of these eight targets have implying a noticeable amount of surface thermal inertia. It should be noted, though, that inferences about surface roughness and thermal conductivity would require more accurate knowledge of the spin axis orientation and spin period of these targets. Our average is consistent with our default value of for fixed fits. For comparison with other dynamical classes, the average beaming parameter of seven Plutinos is (Mommert2012 () 2012) and of seven scattered and detached objects (SantosSanz2012 () 2012). The difference of using a fixed- instead of fixed- is that diameters would increase, on the average, by 10% and geometric albedos decrease by 16%. These changes are within the average relative uncertainties (19% in diameter and 57% in geometric albedo) of the fixed- solutions.

5.1 Correlations

We ran a Spearman rank correlation test (Spearman1904 ()) to look for possible correlations between the geometric albedo , diameter , orbital elements (inclination , eccentricity , semimajor axis , perihelion distance ), beaming parameter , visible spectral slope, as well as B-V, V-R and V-I colors101010From the Minor Bodies in the Outer Solar System database, http://www.eso.org/~ohainaut/MBOSS, accesses Nov 2011.. The Spearman correlation is a distribution-free test less sensitive to outliers than some other more common methods (e.g. Pearson correlation). We use a modified form of the test, which takes into account asymmetric error bars and corrects the significance for small numbers statistics. The details of our method are described in Peixinho et al. (2004) and SantosSanz2012 () (2012). The significance of a correlation is the probability of getting a higher or equal correlation coefficient value if no correlation existed on the parent population, from which we extracted the sample. Therefore, the smaller the P the more unlikely would be to observe a if it was indeed equal to zero, i.e. the greater the confidence on the presence of a correlation is. The 99.7% confidence interval (), or better, corresponds to , or smaller. We consider a ’strong correlation’ to have , and a ’moderate correlation’ to have . Selected results from our correlation analysis are presented in Table 8 and discussed in the following subsections.

Variables sub-sample Number of Correlation Significance Confidence
data points coefficient limit ()
, this work 18 0.002 3.2
this work / cold 6 0.8 0.2
this work / hot 12 0.0011 3.3
this work and previous works 26 0.0004 3.6
this work / hot and previous works 20 0.005 2.8
, this work 18 3.7
this work / cold 6 0.4 0.8
this work / hot 12 2.2
this work and previous works 26 3.2
this work / hot and previous works 20 2.1
, this work 18 0.015 2.4
this work / cold 6 1.0 0.03
this work / hot 12 2.7
spectral slope, this work / hot and previous works 15 0.011 2.6
Table 8: Selected correlation results (see text).

5.1.1 Correlations with diameter

We detect a strong size-inclination correlation in our target sample (see Fig. 7 and Table 8). When literature targets, all of whom are dynamically hot, are included in the analysis we get a correlation of similar strength. Previously this presumable trend has been extrapolated from the correlation between intrinsic brightness and inclination ((Levison2001, , 2001)). We see this strong size-inclination correlation also among the hot classicals sub-sample, but not among our cold classicals where we are limited by the small sample size.

Other orbital parameters do not correlate with size. We find no correlation between size and colors, or spectral slopes, nor between size and the beaming parameter . The possible correlation between size and geometric albedo is discussed in Section 5.1.2.

Figure 7: Radiometric diameter as well as geometric albedo vs inclination. The cold classicals of our sample are marked with blue squares, hot classicals with red crosses and hot classicals from literature with gray color. The high-albedo target 55636 (2002 TX) is beyond the scale.

5.1.2 Correlations with geometric albedo

We find evidence for an anti-correlation between diameter and geometric albedo, both in our sample and when combined with other published data of classical TNOs (see Fig. 8 and Table 8). Other dynamical populations with accurately measured diameters/albedos show a different behavior: there is no such correlation seen among the Plutinos (Mommert2012 () 2012) and a combined sample of 15 scattered-disc and detached objects show a positive correlation between diameter and geometric albedo at level (SantosSanz2012 () 2012).

As it might be suggested visually by the distribution of diameters of classical TNOs (see Figs. 4 and 8), we have analyzed the possibility of having two groups with different size-albedo behaviors, separating in size at km regardless of their dynamical cold/hot membership. We have found no statistical evidence for it.

With our method of accounting for error bars, which tend to ‘degrade’ the correlation values, geometric albedo does not correlate with , orbital parameters, spectral slopes, colors, nor beaming parameters . Also when the literature targets are added we find no evidence of correlations.

Figure 8: Geometric albedo vs radiometric diameter (blue triangles = cold classicals, red crosses = hot classicals from our sample, gray = other hot classicals from literature, see Table 7).

5.1.3 Other correlations

The known correlations between surface color/spectral slope and orbital inclination (Trujillo2002 (), (Hainaut2002, , 2002)), and between intrinsic brightness and inclination (Levison2001 () 2001), usually interpreted as a size-inclination correlation, might lead us to conclude there was a consequent color/slope-size correlation. Our analysis with measured diameters does not show a correlation neither with spectral slope nor visible colors, as one might expect. Note, however, that we do not possess information on the surface colors/slopes of of our targets ( when complemented with other published data) leading to the non-detection of the color/slope-inclination trend, which is known to exist among classicals. Thus, a more complete set of color/slope data would be required for our targets. Only when combining the hot sub-sample from this work and literature we see a non-significant anti-correlation between slope and inclination (see Table 8). We do not find any correlations of the B-V, V-R and V-I colors with other parameters.

The apparent vs anti-correlation in our target sample mentioned in Section 2.1 is almost significant () for our hot sub-population (see Table 8).

5.2 Binaries

Binary systems are of particular scientific interest because they provide unique constraints on the elusive bulk composition, whereas all other observational constraints of the composition only pertain to the surface of the object. The sizes of binaries can be constrained based on the relative brightness difference of the primary and the secondary components, but only if suitable assumptions about the relative geometric albedo are made. Alternatively, geometric albedos can be constrained under certain assumptions about the relative sizes. The ranges given in the literature are usually based on the following assumptions: i) the primary and secondary objects are spherical, ii) the primary and secondary have equal albedos, and iii) objects have densities within a limited assumed range.

Six of our targets are binaries with known total mass and brightness difference between the two components (see Table 9). Assuming the two components to have identical albedos, the latter can be converted into an area ratio and, assuming spherical shape, a diameter ratio (with component diameters and ). Component diameters follow from the measured NEATM diameter (see Table 6) and : (since is the area-equivalent system diameter). This leads to a “volumetric diameter” . The mass densities are given in Table 9. Within the uncertainties, the measured bulk densities scatter around roughly 1 g cm, consistent with a bulk composition dominated by water ice, as expected for objects in the outer Solar System. Significant mass contributions from heavier materials, such as silicates, are not excluded however, and would have to be compensated by significant amounts of macroporosity. The largest object, 120347 Salacia, has a bulk density g cm. This could indicate a lower amount of macroporosity for this object, which is subject to significantly larger gravitational self-compaction than our other binary targets.

Target Adopted VGrundy et al. (2011). MassGrundy et al. (2011). Bulk density
(mag) ( kg) (g cm)
(2001 XR) 0.43
275809 (2001 QY) 0.20
79360 Sila 0.12MIPS channel excluded.
88611 Teharonhiawako 0.70
148780 Altjira 0.23
120347 Salacia 2.32
111111References. $b$$b$footnotetext: Grundy et al. (2012).
Table 9: New density estimates for binaries.

6 Conclusions

The number of classical TNOs with both the size and the geometric albedo measured radiometrically is increased by eight from 22 to 30. Four other targets, which previously had estimated size ranges from the analysis of binary systems, now have more accurate size estimates. The number of targets observed and analysed within the “TNOs are Cool” program (Muller2010 () 2010, Lellouch2010 () 2010, Lim2010 () 2010, SantosSanz2012 () 2012, Mommert2012 () 2012) is increased by 18 and the observation of 79360 Sila disturbed by a background source in Muller2010 () (2010) has been re-observed and analyzed. Furthermore, three targets which earlier had upper and lower limits only based on Spitzer data alone (148780 Altjira, 138537 (2000 OK) and 2001 RZ) now have accurately estimated diameters and albedos. The new Altjira solution is outside of the previous limits based on Spitzer data alone. The three PACS data points near the thermal peak are providing reliable diameter/albedo solutions, but in some cases adding Spitzer data, especially the data point in the lower-wavelength regime, constrains the solution and allows smaller error bars and more reliable estimates of the beaming parameter. Compared to previous works the size estimates of 119951 (2002 KX), and 2002 MS have increased. 2002 MS (934 km) is similar in size to 50000 Quaoar and the refined size of 120347 Salacia (901 km) is similar to that of 90482 Orcus. We find a diameter for 2001 KA, which is approximately half of the previous estimate (Brucker2009 () 2009), and a geometric albedo approximately 4 times higher. The largest change in estimated geometric albedo is with 119951 (2002 KX) from 0.60 to 0.097.

The main conclusions based on accurately measured classical TNOs are:

  1. There is a large diversity of objects’ diameters and geometric albedos among classical TNOs.

  2. The dynamically cold targets have higher (average ) and differently distributed albedos than the dynamically hot targets () in our sample. When extended by seven hot classicals from literature the average is .

  3. Diameters of classical TNOs strongly correlate with orbital inclination in the sample of targets, whose size and geometric albedo have been accurately measured, i.e. low inclination objects are smaller. We find no clear evidence of an albedo-inclination trend.

  4. Our data suggests that geometric albedos of classical TNOs anti-correlate with diameter, i.e. smaller objects possess higher albedos.

  5. Our data does not show evidence for correlations between surface colors, or spectral slope, of classical TNOs and their diameters nor with their albedos.

  6. We are limited by the small sample size of radiometrically measured accurate diameters/albedos of dynamically cold classicals (N=6) finding no statistical evidences for any correlations.

  7. The cumulative size distribution of hot classicals based on the sample of measured sizes in the range of diameters between 100 and 600 km (N=11) has a slope of .

  8. We determine the bulk densities of six classicals. They scatter around g cm. The high-mass object 120347 Salacia has a density of g cm.

Acknowledgements.
We thank Chemeda Ejeta for his work in the dynamical classification of the targets of the “TNOs are Cool” program. We acknowledge the helpful efforts of David Trilling in the early planning of this program. Part of this work was supported by the German DLR project numbers 50 OR 1108, 50 OR 0903, 50 OR 0904 and 50OFO 0903. M. Mommert acknowledges support trough the DFG Special Priority Program 1385. C. Kiss and A. Pal acknowledge the support of the Bolyai Research Fellowship of the Hungarian Academy of Sciences. J. Stansberry acknowledges support by NASA through an award issued by JPL/Caltech. R. Duffard acknowledges financial support from the MICINN (contract Ramón y Cajal). P. Santos-Sanz would like to acknowledge financial support by the Centre National de la Recherche Scientifique (CNRS). JLO acknowledges support from spanish grants AYA2008-06202-C03-01, P07-FQM-02998 and European FEDER funds.

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