Thermal Infrared Observations of Asteroid (99942) Apophis with Herschel
T. G. Müller, C. Kiss, P. Scheirich, P. Pravec, L. O'Rourke, E. Vilenius, B. Altieri
AAstronomy & Astrophysics manuscript no. apo c (cid:13)
ESO 2018October 8, 2018
Thermal Infrared Observations of Asteroid (99942)Apophis with Herschel (cid:63)
T. G. M¨uller , C. Kiss , P. Scheirich , P. Pravec , L. O’Rourke , E. Vilenius , and B.Altieri , Max-Planck-Institut f¨ur extraterrestrische Physik, Postfach 1312, Giessenbachstraße, 85741Garching, Germany Konkoly Observatory, Research Center for Astronomy and Earth Sciences, HungarianAcademy of Sciences; Konkoly Thege 15-17, H-1121 Budapest, Hungary Astronomical Institute, Academy of Sciences of the Czech Republic, Friˇcova 1, CZ-25165Ondˇrejov, Czech Republic European Space Astronomy Centre (ESAC), European Space Agency, Apartado de Correos78, 28691 Villanueva de la Ca˜nada, Madrid, SpainReceived ; accepted
ABSTRACT
The near-Earth asteroid (99942) Apophis is a potentially hazardous asteroid. We obtained far-infrared observations of this asteroid with the Herschel Space Observatory’s PACS instrument at70, 100, and 160 µ m. These were taken at two epochs in January and March 2013 during a closeEarth encounter. These first thermal measurements of Apophis were taken at similar phase anglesbefore and after opposition. We performed a detailed thermophysical model analysis by using thespin and shape model recently derived from applying a 2-period Fourier series method to a largesample of well-calibrated photometric observations. We find that the tumbling asteroid Apophishas an elongated shape with a mean diameter of 375 + − m (of an equal volume sphere) and a ge-ometric V-band albedo of 0.30 + . − . . We find a thermal inertia in the range 250-800 Jm − s − . K − (best solution at Γ =
600 Jm − s − . K − ), which can be explained by a mixture of low conductivityfine regolith with larger rocks and boulders of high thermal inertia on the surface. The ther-mal inertia, and other similarities with (25143) Itokawa indicate that Apophis might also have arubble-pile structure. If we combine the new size value with the assumption of an Itokawa-likedensity and porosity we estimate a mass between 4.4 and 6.2 · kg which is more than 2-3times larger than previous estimates. We expect that the newly derived properties will influenceimpact scenario studies and influence the long-term orbit predictions of Apophis. Key words.
Minor planets, asteroids: individual – Radiation mechanisms: Thermal –Techniques: photometric – Infrared: planetary systems 1 a r X i v : . [ a s t r o - ph . E P ] A p r ¨uller et al.: Herschel observations of (99942) Apophis
1. Introduction
The near-Earth asteroid 99942 Apophis was discovered in 2004 (Minor Planet Supplement 109613)and found to be on an Aten-type orbit crossing the Earth’s orbit in regular intervals. At that time,the object raised serious concerns following the discovery that it had a 2.7% chance of strikingthe planet Earth in 2029 . Immediate follow-up observations to address these concerns took placeand provided predictions that eliminated the possibility of collision in 2029, although it does enterbelow the orbit of the geostationary satellites at that time. However there did remain the possibilityof Apophis passing through a precise region in space (gravitational keyhole) which could set it upfor an impact in the mid-term future (Farnocchia et al. 2013). Apophis remains an object with oneof the highest statistical chances of impacting the Earth among all known near-Earth Asteroids.The studies performed to determine the impact probability require a clear set of physical prop-erties in order to understand the orbital evolution of this asteroid ( ˇZiˇzka & Vokrouhlick´y 2011;Farnocchia et al. 2013; Wlodarczyk 2013). The lack of availability of such properties (albedo, size,shape, rotation, physical structure, thermal properties) is a major limiting factor which leads to un-certainties in the role played by non-gravitational e ff ects on that orbit. The Yarkovsky e ff ect due tothe recoil of thermally re-radiated sunlight is the most important of these non-gravitational e ff ects.Besides the input to the orbit evolution, the physical properties serve also to address the possibleimplications if an impact were to occur. A solid body of 300 m versus a rubble pile hitting the Earthimplies di ff erent levels of severity as regards its ability to pass through the atmosphere unscathedto create regional versus grandscale damage.Delbo et al. (2007a) determined from polarimetric observations an albedo of 0.33 ± = ± ±
60 m, slightlysmaller than earlier estimates in the range 320 to 970 m, depending on the assumed albedo. Binzelet al. (2009) described the results of observations they performed in the visible to near infrared(0.55 to 2.45 µ m) of Apophis where they compared and modeled its reflectance spectrum withrespect to the spectral and mineralogical characteristics of likely meteorite analogs. Apophis wasfound to be an Sq-class asteroid that most closely resembled LL ordinary chondrite meteorites interms of spectral characteristics and interpreted olivine and pyroxene abundances. They found thatcomposition and size similarities of Apophis with (25143) Itokawa suggested a total porosity of40% as a current best guess for Apophis. Applying these parameters to Apophis yielded a massestimate of 2 · kg with a corresponding energy estimate of 375 Megatonnes (Mt) TNT for itspotential hazard. Substantial unknowns, most notably the total porosity, allowed uncertainties inthese mass and energy estimates to be as large as factors of two or three.Up to the time of our own observations, there were no thermal infrared measurements existingon this asteroid. Observations from the Spitzer Space Telescope were not possible as Apophis wasnot in the Spitzer visibility region during the remainder of its mission. Due to the fact that therewas no close encounter with Earth between discovery and now, there are also no groundbasedN- / Q-band observations, no Akari and also no WISE observations available. (cid:63)
Herschel is an ESA space observatory with science instruments provided by European-led PrincipalInvestigator consortia and with important participation from NASA. The current orbit’s perihelion is at 0.746 AU, aphelion at 1.0985 AU, with a = = ◦ , e = http://neo.jpl.nasa.gov/riskhttp://newton.dm.unipi.it/neodys We observed this near-Earth asteroid with the Herschel Space Observatory’s (Pilbratt et al.2010) PACS (Photodetector Array Camera and Spectrometer) instrument (Poglitsch et al. 2010)at far-infrared wavelengths (Section 2). We present our thermophysical model (TPM) analysis(Section 3) and discuss the results (Section 4).
2. Far-infrared observations with Herschel-PACS
Fig. 1.
The object-centered images of the target in the 3 PACS filters for the first visit on Jan. 6,2013. Top: blue (70 µ m), middle: green (100 µ m), bottom: red (160 µ m).The far-infrared observations with the Herschel Space Observatory were performed in severalstandard PACS mini scan-map observations in tracking mode. The observations took place on Jan.6, 2013 (four individual observations) and on Mar. 14, 2013 (one individual observation). Eachindividual observation consisted of several repetitions of a mini scan-map. The observational cir-cumstances are listed in Table 1. During the first epoch all three PACS filters at 70 (blue), 100(green), and 160 µ m (red band) were used, while in the second epoch we concentrated only on the / µ m filter setting due to observing time limitations. Each measurement consisted of a miniscan-map with 10 scan-legs of 3 arcmin length and separated by 4 arcsec, the scan direction was70 ◦ (along the diagonal of the detector arrays), and the scan-speed was 20 (cid:48)(cid:48) / s. Each scan-leg iscentered on the true object position at scan mid-time. The PACS photometer takes data frames with40 Hz, but binned onboard by a factor of 4 before downlink. The total duration of our Herschel-PACS observations was about 2 h during the first epoch, split in 4 measurements of about 30 mineach: 2 × × / red filter setting. In this case we split thedata into 6 individual datasets with 3 repetitions each.Figure 1 shows the object-centered images of the first visit in January 2013. They are producedby stacking all frames of a given band on the source position in the first frame. The backgroundstructures in these figures are not real and related to background source artefacts caused by the re-centering of images on the rapidly changing Apophis position. During the first visit Apophis wasmoving in a clean part of the sky without any significant sources along the object’s path. During thesecond visit the source moved over faint objects located in a field of di ff use background emissionwhich we could not entirely eliminate in the reduction process. We followed the object’s flux (inthe background-subtracted images) and noticed a 1-2 mJy residual background emission in parts ofthe object’s trajectory (see footnote in Table 1). In addition to the six sub-images we also combinedall background-free and clean images (repetitions 4-9, 16-18) to obtain a final object-centered mapfor high-quality photometry.We performed aperture photometry on the final calibrated images and estimated the flux errorvia photometry on artificially implemented sources in the clean vicinity around our target. Thefluxes were finally corrected for colour terms due to the di ff erences in spectral energy distributionbetween (99942) Apophis and the assumed constant energy spectrum ν F ν = const. in the PACScalibration scheme. The calculated colour-corrections for our best Apophis model solution are1.005, 1.023, 1.062, at 70.0, 100.0, and 160.0 µ m respectively. These values agree with the expectedcorrections for objects with temperatures around 250 K. The absolute flux calibration error is 5%in all three bands. This error is based on the model uncertainties of the fiducial stars used in thePACS photometer flux calibration scheme (Nielbock et al. 2013; Balog et al. 2014). Since this erroris identical for all our observations, we consider it at a later stage in the discussion about the qualityof our derived properties. The final monochromatic flux densities and their flux errors at the PACSreference wavelengths 70.0, 100.0 and 160.0 µ m are listed in Table 1.
3. Radiometric analysis
Pravec et al. (2014) found that Apophis has a non-principal axis rotation and it is in a moderatelyexcited Short Axis Mode state. The strongest observed lightcurve amplitude is related to a retro- PACS technical report PICC-ME-TN-038, v1.0: http://herschel.esac.esa.int/twiki/pub/-Public/PacsCalibrationWeb/cc report v1.pdf The full (peak-to-trough) amplitude of the strongest lightcurve frequency 2 P − = P − φ − P − ψ ) is 0.59 ± φ = ± ψ = ± grade rotation with P = ± λ ecl , β ecl ) = (250 ◦ ,-75 ◦ ). The relevant parameters for our radiometric analysis are (i) the orientation of the object atthe time of the Herschel observations, given by the object’s z axis which is connected to the largestmoment of inertia in the asteroid’s co-rotating coordinate frame, and the angle φ which specifiesthe rotation angle of the body at the given julian date. (ii) the rotation history of the object to ac-count for thermal inertia e ff ects (the thermal inertia is responsible for ”transporting” heat to thenon-illuminated parts of the surface).Pravec et al. (2014) were also able to reconstruct the physical shape model (see Figure 3) ofApophis following the work by Kaasalainen (2001; 2001a) and Scheirich et al. (2010). The convexshape model with the non-principal axis rotation was determined by Pravec et al. to be the best-fitsolution to the observed lightcurves from December 2012 to April 2013. The available photometricobservations were found to cover our Herschel measurements in January 2013 very well. In March2013 the situation is less favorable and the photometric points are sparsely distributed in the daysbefore and after the Herschel observations (see Fig. 6 in Pravec et al. 2014).The lightcurve-derived shape model does not have absolute size information. A dark (lowalbedo) and large object could explain the observed lightcurves equally well as a bright (highalbedo) but much smaller object. For our analysis we used the physical shape model and the rota-tional properties presented in Pravec et al. (2014), the relevant coordinates and angles connectedto our thermal measurements are listed in Table 2 and shown in the context of a full rotation inFigure 2. Thermophysical model (TPM) techniques are very powerful in deriving reliable sizes and albe-dos. In cases where enough thermal data are available and if there is already information aboutthe object’s shape and spin axis then this technique also allows to solve for thermal propertiesof the surface (e.g., Harris & Lagerros 2002; M¨uller et al. 2005). Here the radiometric analy-sis was done via a thermophysical model which is based on the work by Lagerros (1996; 1997;1998). This model is frequently and successfully applied to near-Earth asteroids (e.g., M¨uller etal. 2004; 2005; 2011; 2012; 2013), to main-belt asteroids (e.g., M¨uller & Lagerros 1998; M¨uller& Blommaert 2004), and also to more distant objects (e.g. Horner et al. 2012; Lim et al. 2010).The TPM takes into account the true observing and illumination geometry for each observationaldata point, a crucial aspect for the interpretation of our Apophis observations which cover before-and after-opposition measurements . The TPM allows to use any available convex shape model incombination with spin-axis properties. The heat conduction into the surface is controlled by thethermal inertia Γ , while the infrared beaming e ff ects are calculated via a surface roughness model,implemented as concave spherical crater segments on the surface and parameterized by the rootmean square (r.m.s.) slope angle. We performed our radiometric analysis with a constant emissiv-ity of 0.9 at all wavelengths, knowing that the emissivity can decrease beyond ∼ µ m for someobjects (e.g., M¨uller & Lagerros 1998; 2002), but our measurements are all at shorter wavelengths. Before opposition: object is leading the Sun, positive phase angle in Table 1; after opposition: object istrailing the Sun, negative phase angles 5¨uller et al.: Herschel observations of (99942) Apophis
Table 1.
Observing geometries (Herschel-centric) and final calibrated flux densities (FD). r helio isthe heliocentric distance, ∆ obs the object’s distance from Herschel, and α is the phase-angle, withnegative values after opposition. OD is Herschel’s operational day, OBSID: Herschel’s observationidentifier. The repetitions specify the number of scan-maps performed and / or used to derived thegiven flux and error. The Herschel-centric apparent motions of Apophis were 205 (cid:48)(cid:48) / h (first visit inJanuary 2013) and 58 (cid:48)(cid:48) / h (second visit in March 2013). Julian Date λ ref FD FD err r helio ∆ obs α OD / repeti- durationmid-time [ µ m] [mJy] [mJy] [AU] [AU] [deg] OBSID tions [s]first visit a on Jan. 6, 2013:2456298.50745 70.0 36.3 1.1 1.03593 0.096247 + / + / + / + / + / + / + / + / + +
59 all 36582456298.55394 100.0 22.56 1.17 1.03604 0.096220 + +
60 all 40242456298.54375 160.0 9.41 1.29 1.03602 0.096226 + b on Mar. 14, 2013:2456365.77802 70.0 12.6 c / / / c / c / / / < / Notes. ( a ) Light-travel time is 48.0 s; ( b ) light-travel time is 115.9 s; ( c ) Photometry is still a ff ected by 1-2 mJyresiduals from the background elimination process, not used for the final photometry on the combined mea-surement. We used a mean absolute magnitude of H V = ± = ± The average observed Apophis flux at 70 µ m changed from 36.7 mJy on Jan. 6 to 11.2 mJy on Mar.14, 2013, resulting in a flux ratio FD epoch / FD epoch of 3.2. This ratio is driven by (i) the change in The mean absolute magnitude corresponds to the mean observed cross-section.6¨uller et al.: Herschel observations of (99942) Apophis
Table 2.
The Apophis orientation for the nominal rotation model during the Herschel observations.The angular velocity was 5.00 radians / day during the first visit and 5.02 radians / day during thesecond visit (around the true spin axis at the given times). Numbers are given in the Apophis-centric frame. Julian Date z-axis [deg] rot. angle Rotation axis [deg]mid-time λ ecl β ecl φ [deg] λ ecl β ecl first visit on Jan. 6, 2013:2456298.50745 19.9 -58.5 243.6 234.53 -75.512456298.53059 11.5 -57.5 241.2 234.39 -75.992456298.55258 3.7 -56.4 239.3 234.44 -76.452456298.57455 356.4 -55.2 237.8 234.74 -76.92second visit on Mar. 14, 2013:2456365.77802 294.3 -72.9 96.2 233.35 -70.322456365.78760 293.1 -72.1 97.7 232.58 -70.532456365.79719 291.9 -71.3 99.2 231.81 -70.762456365.80677 290.6 -70.6 100.6 231.04 -71.012456365.81635 289.3 -69.8 101.9 230.38 -71.252456365.82594 287.9 -69.0 103.2 229.78 -71.49 observing geometry (r, ∆ , α ); (ii) the change in cross section due to the object’s non-spherical shapeand the di ff erent rotational phase; (iii) thermal e ff ects which transport heat to non-illuminated partsof the surface.Assuming a spherical object in instantaneous equilibrium with solar insolation (thermal inertiaequals zero) would produce a very di ff erent 70 µ m flux ratio FD epoch / FD epoch of 6. This calcu-lation was the baseline for our planning of the Herschel observations in March (second epochmeasurement) where we expected to see approximately 6 mJy instead of the observed 11.2 mJy.The discrepancy between expectations and observations shows that changes in the observed crosssection and thermal e ff ects, in addition to the changes in observing geometry, play a significant roleand are key elements for our radiometric analysis. With the availability of Apophis’ shape model and rotational properties it is also possible to calcu-late the influence of the apparent cross section on the observed flux. Apophis was showing a 1.21times larger cross section during the second epoch as compared to the first epoch. The combinedgeometry and cross-section change would result in a 70 µ m flux ratio FD epoch / FD epoch of about3.7 which is still significantly larger than the observed ratio of 3.2. This is a strong indication thatthermal e ff ects play an important role. The e ff ect can also nicely be seen in Figure 3: before oppo-sition we see the object under a phase angle of about + ◦ with a cold morning terminator, whilein the second epoch we have seen Apophis at about -61 ◦ with a warm evening side which has justrotated out of the Sun. In both cases thermal e ff ects play a strong role: during the first epoch a substantial part of the surface heat is transported to the non-visible side while in the second epochthe heat transport to the non-illuminated part remains visible. Looking at Figure 3 (left side), we find that the observed flux from the first epoch data taken on Jan.6, 2013 is dominated by the illuminated / heated part of the surface and the cold morning side doesnot contribute in a significant manner. However, depending on the thermal inertia of the top-surfacelayer there is some heat transported to the non-visible rear side. The conversion of the observedflux into a size and albedo solution depends therefore on the thermal inertia and larger values forthe thermal inertia lead to smaller size and larger albedo estimates (see Fig. 4). We applied the ra-diometric method to all epoch-1 data (see first part of Table 1) simultaneously and derived the size(of an equal-volume sphere) and the geometric albedo (in V-band). For the calculations we used thetrue, Herschel-centric observing geometry together with the correct orientation of the object at thetime of the measurements (see Table 2 and Fig. 3, left side). For signal-to-noise reasons we usedthe combined 100 µ m flux (S / N = µ m flux (S / N = µ m fluxes (S / N = for a wide rangeof thermal inertias and surface roughness settings. Only solutions connected to thermal inertiasbelow ∼
250 Jm − s − . K − can be excluded due to high χ -values above 1.8. Figure 4 (top) showsthe derived size and geometric albedo values for the full range of thermal inertias. Larger valuesfor the thermal inertia cause more heat transport to the non-visible rear side and require thereforesmaller sizes to explain the observed fluxes. For the albedo there is an opposite e ff ect and largerthermal inertias are connected to higher albedo values. The minor influence of roughness is shownby the dashed (low r.m.s. slope angle of 0.2) and dotted-dashed (high r.m.s. slope angle of 0.9)lines. The errorbars indicate the standard deviations at each thermal inertia for the size and albedovalues derived from each of the four individual flux measurements. These errorbars indicate thereproducibility of the result: the sizes and albedos connected to each of the independent measure-ments are inside the shown errorbars. The 5% absolute calibration error of the PACS photometery(Balog et al. 2014) is considered later in the discussion section (Section 4).The thermal inertia changes the shape of the far-IR lightcurve considerably at the time of ourobservations (see Figure 5). At 70 µ m (top) and at 100 µ m (bottom) there is a flat part or even asecondary maximum developing for the higher thermal inertias. The low thermal inertia lightcurveshows a steady decrease in flux during the two hours of Herschel measurments. This is not seenin our time-separated observations at 70 µ m and at 100 µ m. The completely independent measure-ments in both bands seem to follow the curves for the higher thermal inertia values. At 160 µ m theerrorbars are too large to see a similar trend. The repeated 3-band high-SNR measurements fromJan. 6, 2013 are therefore best fit by an object with a size of 355 to 385 m (the diameter of a spherewith the volume equal to the asteroid), a geometric albedo of 0.28 to 0.33, and a thermal inertialarger than 250 Jm − s − . K − . Good fit solutions in the sense of a weighted least-squares parameter estimation require χ reduced (cid:46) The size of an equal-volume sphere.8¨uller et al.: Herschel observations of (99942) Apophis
Figure 3 (right side) illustrates nicely that the observed flux is influenced by the non-illuminated,but still warm part of the surface which just rotated out of the Sun. The thermal inertia influencesthe temperature distribution on the surface and very little heat is transported to the non-visiblerear side. The conversion of the observed flux into a size and albedo solution depends thereforemuch less on the thermal inertia. But here we are encountering some problems: (1) The SNR ofthe second-epoch measurement is much lower due to the 2.4 times larger Herschel-centric distanceand a significant background contamination which could not be eliminated entirely (see Table 1).(2) We only obtained a single-band detection at 70 µ m and an upper limit at 160 µ m, but no 100 µ mpoint was taken. (3) The coverage in optical photometric points around epoch 2 is much poorerresulting in a less accurate model at the given orientation (see Pravec et al. 2014). The syntheticmodel lightcurve of the best-fit solution shows a local maximum on the decreasing branch and thereliability of the calculated cross-section is not clear.We calculated for each thermal inertia the radiometric size and albedo solutions together withthe corresponding uncertainty range. The 13% error in the observed 70 µ m flux translates into a6% error in diameter and 12% error in albedo, the 160 µ m detection limit unfortunately does notconstrain the solution in a noticable way. As a consequence, the full range of thermal inertias iscompatible with our epoch 2 data. The corresponding size and albedo values range from 370 m to430 m and from 0.30 to 0.22, respectively. As a final step, we combined the radiometric results of Sections 3.2.3 and 3.2.4 while consideringthe derived errors. We calculated for each thermal inertia the weighted mean size and albedo so-lution and used our TPM setup (considering also the changing orientation state of the object) tomake flux predictions for the four epoch-1 and one epoch-2 data points. Figure 6 shows the re-duced χ values together with the 1- σ confidence level for five independent measurements whichis around 1.7. The three di ff erent levels of surface roughness are shown as dashed line ( ρ = ρ = ρ = − s − . K − , which isabout mid-way inside the ∼ − s − . K − formal acceptance range. The connected sizeand albedo values are 368-374 m and 0.30-0.31, respectively, with this solution being dominatedby the high-quality epoch-1 data. Giving a stronger weight to the epoch-2 observations shifts the χ -minima to lower thermal inertias: if we weight the epoch-1 and epoch-2 solutions simply bythe number of independent measurements (here 4:1) then we find the χ -minima at a thermal in-ertia of around 300-350 Jm − s − . K − and values above 800 Jm − s − . K − would be excluded. Thecorresponding sizes are about 10 m larger and the albedo is around 0.29, but the overall match tothe observations is degraded with reduced χ values just below 1.7. This kind of “weighting bynumber of observations” is somewhat arbitrary, but it shows how a better balanced (higher S / N)second epoch measurement could have influenced our results. In Section 4 we continue with thecorrect weighting of the observations taking into account the observational errorbars.
4. Discussions
The radiometric method has been found to work reliably for objects where shape and spin prop-erties are known (e.g., O’Rourke et al. 2012 or M¨uller et al. 2014). The application to tumblingobjects is more complex and requires the knowledge of the object’s orientation and its spin axis atthe times of the thermal measurements. For our epoch-1 data set, the tumbling is not critical sincethe observed flux is clearly dominated by the illuminated part of the surface. The observed flux isnot influenced by the path of the heat transport to the non-visible rear side, independent whether theobject rotates around the moment of inertia or the true spin axis. For our epoch-2 data, the situationis slightly di ff erent since the temperature distribution on the warm evening side contributes to theobserved disk-integrated flux. In this case the tumbling causes a slight spatial displacement of thecontributing warm region close to the terminator. It may be that our epoch-2 flux is slightly influ-enced by this e ff ect and that our model predictions are therefore too low. A careful investigationshowed us that the temperature of a very small region close to the rim and outside the direct sunillumination might in reality be higher than in our TPM calculations. But the impact on the disk-integrated flux is well below 5% and the consequences for our radiometric results are negligible.The error bars in the epoch-2 observation are simply too large.The final uncertainties of the derived size and albedo solutions depend mainly on the qualityof the thermal measurements. A 10% flux error translates typically in a 5% error in equivalent sizeand about 10% in geometric albedo. With several independent measurements the errors can reduceto even smaller values. But this is only the case when the H-magnitude is precisely known and thethermal inertia is well constrained by the available observational dataset. Our dataset has a goodcoverage in thermal wavelengths, as well as phase angles before and after opposition which is suf-ficient to determine the thermal inertia reliably. However, due to the above mentioned problemswith epoch 2 the situation is not perfect. The epoch-1 data indicate thermal inertias larger thanabout 250 Jm − s − . K − , while the combined data set excludes only the largest values above about1500 Jm − s − . K − . Delbo et al. (2007b) found an average thermal inertia of 200 ±
40 Jm − s − . K − for a sample of km-sized near-Earth objects with a maximum derived value of 750 Jm − s − . K − .We investigated the e ff ects of very high thermal inertia values above 800 Jm − s − . K − in the con-text of phase-angle and wavelength trends (as shown in Fig. 7). Although statistically still possible,these high values produce a trend in the observation-to-model ratios with phase angle and causealso a poor match to our most reliable 70 µ m fluxes. The most likely range for the thermal inertiais therefore 250-800 Jm − s − . K − , with our best solution connected to 600 Jm − s − . K − . Thesehigh values for the thermal inertia can be explained by a mixture of (very little) low conductivityfine regolith with larger rocks and boulders of high thermal inertia on the surface (see also discus-sions in M¨uller et al. 2012, 2013, 2014). If we take our best solution for the thermal inertia andassume a surface density of lunar regolith (1.4 g cm − ), and that the heat capacity is somewherebetween lunar regolith (640 J kg − K − ) and granite (890 J kg − K − ) then the thermal conductiv-ity κ would be 0.3-0.4 W K − m − . This is compatible with Itokawa’s 0.3 W K − m − (M¨uller et al.2005) whereas the typical value for near-Earth asteroids is 0.08 W K − m − (Mueller M. 2007). Ifwe take the full range of uncertainties into account ( Γ = − s − . K − , heat capacity 450-1200 J kg − K − , and surface density 1.3-2.0 g cm − ) then the range for thermal conductivity wouldbe 0.03-1.1 W K − m − , which is a range of two orders of magnitude. The size range corresponding to our thermal inertia solution is 371 to 385 m (best solution375 m) with a statistical error of about 6 m only. The smallest radiometric size solutions are pro-duced by the high-roughness and high-inertia settings in the TPM, while the largest sizes are relatedto low-roughness / low-inertia settings (see also Rozitis & Green 2011 for a discussion on the degen-eracy between roughness and thermal inertia). Since the PACS photometric system is only accurateon a 5% level (Balog et al. 2014), we have to consider it also in the context of our size solution .The final size value is therefore 375 + − m.Our derived albedo range of 0.28 to 0.31 (larger values for high-roughness, high-inertia case)has a very small statistical error below 3%. But here we have to include the influence of the abso-lute flux calibration (5%), as well as the H-magnitude error of ± + . − . . This valueis in nice agreement with the Delbo et al. (2007a) of 0.33 ± ±
60 m by Delbo et al. (2007a) was mainly related to theirH-magnitude which is very di ff erent from the value by Pravec et al. (2014) which we used here.We can now also determine the bolometric Bond albedo A. The uncertainty in G translates into anuncertainty in the phase integral q (Bowell et al. 1989), combined with a 5% accuracy of the q-Grelation (Muinonen et al. 2010), we obtain a Bond albedo of A = q · p V = + . − . .Figures 7 and 8 show our best model solution at intermediate roughness level in di ff erent rep-resentations. In Figure 7 we present the observations divided by the corresponding model solutionsas a function of phase angle (top) and as a function of wavelength (bottom). No trends with phaseangle or wavelength can be seen. Figure 8 shows the observations and the model solution on anabsolute scale. Here we also show the 160 µ m upper limit from epoch 2 which is in nice agreementwith the model solution.Binzel et al. (2009) found compositional similarities between 99942 Apophis and25143 Itokawa. They both are in a similar size range, have similar albedos and similar thermalinertias. The measured density of Itokawa is 1.9 ± / cm (Fujiwara et al. 2006; Abe et al.2006 found a slightly higher density of 1.95 ± / cm ). Using Itokawa’s density and our newsize estimate gives a mass estimate of 5.2 + . − . · kg. Both Itokawa and Apophis have been in-terpreted to be analoguous to LL chondrite meteorites (Fujiwara et al. 2006; Binzel et al. 2009).The bulk density of meteorites of that type is 3.21 ± / cm (Britt & Consolmagno 2003).A larger uncertainty is in the macro-porosity of Apophis. Britt et al. (2002) report that asteroids’macro-porosities may be up to 50%. The porosity of Itokawa is 41% (Abe et al. 2006). Assuminga porosity range of 30-50% for Apophis implies a mass between 4.4 and 6.2 · kg.The comparison with Itokawa is interesting in many aspects: The rubble-pile near-Earth aster-oid 25143 Itokawa has an e ff ective size of 327.5 ± ± + . − . for Apophis. Also the thermal inertias compare verywell: M¨uller et al. (2014) found 700 ±
200 Jm − s − . K − for Itokawa, well within the derived rangefor Apophis. Itokawa has a SIV-type taxonomic classification (Binzel et al. 2001) and the Hayabusadata revealed an olivine-rich mineral assemblage silimar to LL5 or LL6 chondrites (Abe et al. 2006;Okada et al. 2006). Apophis is characterised as an Sq-type that most closely resembles LL ordinary We added quadratically the statistical size error with a 2.5% size error related to the 5% in absolute fluxcalibration. 11¨uller et al.: Herschel observations of (99942) Apophis chondrite meteorites (Binzel et al. 2009). The high thermal inertia indicates a lack (or only verysmall amounts) of low-conductivity fine regolith on the surface. The formation of a thick regolith(typically with Γ -values below 100 Jm − s − . K − ) might have been hampered by frequent seismicinfluences. Such processes can reorganise the body’s interior and surface over short time scales ifthe object has a rubble-pile structure. Apophis is also in the size range predominated by asteroidswith cohesionless structures (Pravec et al. 2007). On the other hand, the density of S-type aster-oids is distributed in a very narrow density interval, slightly below the density of their associatedmeteorites, the ordinary chondrites (Carry 2012). The macroporosity for this type of asteroids isgenerally smaller than 30% and pointing to coherent interiors, with cracks and fractures, but notrubble piles. Interestingly, the four S-type asteroids listed by Carry (2012) with sizes below a fewkilometres and with high quality density information (quality codes A, B, or C) all have densi-ties below 2 g cm − and a porosity of 40% or above, indicative of a rubble-pile structure. Overall,Apophis’ size, the surface characteristics related to a relatively high thermal inertia, and the com-parison with similar-size objects, make a cohesionless structure more likely.The newly derived properties will influence the long-term orbit predictions. Chesley et al.(2003; 2008) and Vokrouhlick´y et al. (2008) found that the Yarkovsky e ff ect which is due to therecoil of thermally re-radiated sunlight is acting on many near-Earth asteroids. It is the most sig-nificant non-gravitational force to be considered for risk analysis studies (e.g., Giorgini et al. 2002,2008; Chesley 2006). The calculation of the Yarkovsky orbit drift requires -in addition to the spinstate which was determined by Pravec et al. (2014)- also some knowledge about the object’s size,bulk density, and surface thermal inertia. Our work will contribute with information about size andthermal inertia (Vokrouhlick´y et al., in preparation). The bulk density can be estimated from theYarkovsky-related orbit change, expected to be detected by radar observations during the next closeEarth approach in September 2021 (Farnocchia et al. 2013). ˇZiˇzka & Vokrouhlick´y (2011) showedthat also the solar radiation pressure has a small, but relevant e ff ect on Apophis’ orbit which mightbe noticable after the very close Earth encounter in 2029. Here it is mainly the size and bulk den-sity which play a role. The combined non-gravitational forces -Yarkovsky e ff ect and solar radiationpressure- cause small orbit drifts up to a few kilometers per decade in case of Apophis (Farnocchiaet al. 2013). In comparison, the extension of the keyholes associated with Earth-impacts after the2029 close encounter are in the order of a 100 m or smaller. The studies of the non-gravitationalorbit perturbations are therefore important to estimate the distance between the true trajectory andthe locations of the dangerous keyholes.
5. Conclusions
The shape and spin properties of Apophis presented by Pravec et al. (2014) were the key elementsfor our radiometric analysis. The interpretation of the ∼ e f f = + − m; this is the scaling factor for the shape modelpresented in Pravec et al. (2014) and corresponds to the size of an equal volume sphere.
2. The geometric V-band albedo was found to be p V = + . − . , almost identical to the valuefound for the Hayabusa rendezvous target 25143 Itokawa; the corresponding bolometric Bondalbedo A is 0.14 + . − . .3. A thermal inertia of Γ = + − Jm − s − . K − explains best our combined dataset comprisingthree di ff erent bands and two di ff erent epochs.4. Using either Itokawa’s bulk density information or a rock density of 3.2 g / cm combined with30-50% porosity, we calculate a mass of (5.3 ± · kg which is 2 to 3 times higher thanearlier estimates.5. No information about surface roughness can be derived from the radiometric analysis of ourmeasurements due to the lack of observations at shorter wavelengths and smaller phase anglesclose to opposition. But Apophis’ thermal inertia is similar to the value derived for Itokawa andthis might point to a surface of comparable roughness.6. Apophis’ size, the surface characteristics related to the high thermal inertia, and the comparisonwith similar-size objects, make a cohesionless structure more likely.The interior structure -rubble pile or coherent body- is relevant in the context of impact scenariostudies. In case of a rubble-pile structure (which is the more likely option) pre-collision encounterswith planets could disrupt the body by tidal forces while a more solid interior would leave the objectintact. We also expect that the newly derived properties will a ff ect the long-term orbit predictions ofApophis which is influenced by the Yarkovsky e ff ect and in second order also by the solar radiationpressure. In this context, the radiometrically derived size and thermal inertia will play a significantrole in risk-analysis studies beyond Apophis’ close encounter with Earth in 2029. Acknowledgements.
We would like to thank the Herschel operations team which supported the planning and schedulingof our time-constrained observations. Without their dedication and enthusiasm these measurements would not have beenpossible. The first-visit data are part of the Herschel GT1 MACH-11 project (PI: L. O’Rourke), while the second-visit datawere obtained via a dedicated DDT project (PI: T. M¨uller). The work of P.S. and P.P. was supported by the Grant Agencyof the Czech Republic, Grant P209 / / References
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200 250 300 350ecliptic longitude [deg]-90-80-70-60-50 e c li p t i c l a t i t ude [ deg ] z-axis orientationspin-vector orientation Herschel epochs 2 Fig. 2.
Variations of the object’s z-axis and spin-vector orientation during a full rotation of 30.56 h,starting about 26 h before the first Herschel measurement and ending about 2 h after the last mea-surement. The epochs of the Herschel observations (from Table 1) are shown as diamonts. The zaxis (dashed line) is connected to the largest moment of inertia in the asteroid’s co-rotating coor-dinate frame, the solid line shows how the orientation of the object’s spin axis changes with time.Top: covering the first observations on Jan. 6, 2013; bottom: covering the second observation onMar. 14, 2013.
Fig. 3.
Viewing geometry during the two Herschel observing epochs at phase angles of roughly60 ◦ angle before (left) and after opposition (right). Top: calculated observing geometry on basisof the nominal solution in Pravec et al. (2014). L is fixed vector of angular momentum, the Ariessign is the X axis of the ecliptical frame, S is a direction to the Sun, and x, y, z are the axes of theasteroid co-rotating coordinate frame (corresponding to the smallest, intermediate and the largestmoment of inertia of the body, respectively). Middle: The solar insolation in [W / m ]. Bottom: TPMtemperature calculations assuming a Itokawa-like thermal inertia of 600 Jm − s − . K − . Fig. 4.
The radiometrically derived size (top) and albedo (bottom) as a function of thermal inertia.The influence of model surface roughness is shown as dashed (low roughness) and dotted-dashed(high roughness) lines. The errorbars indicate the standard deviation of observation-to-model ratiosfor our epoch-1 measurements. no r m . F l ux a t µ m [ m J y ] µ m, Epoch 1 Γ = 25 Γ = 100 Γ = 350 Γ = 1000 no r m . F l ux a t µ m [ m J y ] µ m, Epoch 1 Γ = 25 Γ = 100 Γ = 350 Γ = 1000 Fig. 5.
The TPM lightcurves at 70 µ m (top) and at 100 µ m (bottom) together with the observedfluxes and their errorbars, all normalised at mid-time. The influence of thermal inertia on thelightcurve is clearly visible and the measurements seem to follow the higher-inertia curves. Fig. 6.
Reduced χ -values calculated for the radiometric analysis of the combined epoch-1 andepoch-2 dataset. The dashed line shows the low roughness case, while the dashed-dotted line rep-resents the very high roughness case. Good-fit solutions are found below the dashed horizontal linerepresenting the reduced χ threshold for five measurements at 1.7. Fig. 7.
The calibrated PACS observations divided by the best TPM solution as a function of phaseangle (top) and as a function of wavelength (bottom).
Fig. 8.