Icarus 305 (2018) 174–185

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The triaxial ellipsoid size, density, and rotational pole of asteroid (16) Psyche from Keck and Gemini AO observations 2004–2015 Jack D. Drummond a,∗, William J. Merline b, Benoit Carry c, Al Conrad d, Vishnu Reddy e, Peter Tamblyn b,f, Clark R. Chapman b, Brian L. Enke b, Imke de Pater g, Katherine de Kleer g, Julian Christou d, Christophe Dumas h a

Leidos, Starfire Optical Range, AFRL, Kirtland AFB, NM 87117, USA Southwest Research Institute, Boulder, CO 80302, USA Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France d Large Binocular Telescope Observatory, University of Arizona, Tucson, AZ 85721, USA e Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA f Binary Astronomy, Aurora, CO 80012, USA g University of California Berkeley, Berkeley, CA 94720, USA h TMT Observatory, Pasadena, CA 91124, USA b c

a r t i c l e

i n f o

Article history: Received 24 July 2017 Revised 30 December 2017 Accepted 10 January 2018 Available online 11 January 2018 Keywords: Asteroid Psyche Asteroid surfaces Adaptive optics Image processing

a b s t r a c t We analyze a comprehensive set of our adaptive optics (AO) images taken at the 10 m W. M. Keck telescope and the 8 m Gemini telescope to derive values for the size, shape, and rotational pole of asteroid (16) Psyche. Our fit of a large number of AO images, spanning 14 years and covering a range of viewing geometries, allows a well-constrained model that yields small uncertainties in all measured and derived parameters, including triaxial ellipsoid dimensions, rotational pole, volume, and density. We find a best fit set of triaxial ellipsoid diameters of (a,b,c) = (274 ± 9, 231 ± 7, 176 ± 7) km, with an average diameter of 223 ± 7 km. Continuing the literature review of Carry (2012), we find a new mass for Psyche of 2.43 ± 0.35 × 1019 kg that, with the volume from our size, leads to a density estimate 4.16 ± 0.64 g/cm3 . The largest contribution to the uncertainty in the density, however, still comes from the uncertainty in the mass, not our volume. Psyche’s M classification, combined with its high radar albedo, suggests at least a surface metallic composition. If Psyche is composed of pure nickel-iron, the density we derive implies a macro-porosity of 47%, suggesting that it may be an exposed, disrupted, and reassembled core of a Vesta-like planetesimal. The rotational pole position (critical for planning spacecraft mission operations) that we find is consistent with others, but with a reduced uncertainty: [RA;Dec]=[32°;+5°] or Ecliptic [λ; δ ]=[32◦ ; −8◦ ] with an uncertainty radius of 3°. Our results provide independent measurements of fundamental parameters for this M-type asteroid, and demonstrate that the parameters are well determined by all techniques, including setting the prime meridian over the longest principal axis. The 5.00 year orbital period of Psyche produces only four distinct opposition geometries, suggesting that observations before the arrival of Psyche Mission in 2030 should perhaps emphasize observations away from opposition, although the penalty then would be that the asteroid will be fainter and further than at opposition. Published by Elsevier Inc.

1. Background

mean diameter of D = 223 ± 7 km, and a rotation period of 4.2 h ˇ (Durech et al., 2011). It is the prime target for the Psyche Discov-

M-class asteroids have a wide range of radar albedos, and in the mean they have among the highest albedos in the Solar System (Ostro et al., 1985; Magri et al., 2007). As one of the defining members of the M-class (Chapman et al., 1975; Tholen, 1984), and the largest, asteroid (16) Psyche has a volume equivalent or

ery Mission (Elkins-Tanton et al., 2014), which is expected to rendezvous with the asteroid in 2030. Being classified as an M-asteroid in itself does not mean that it is metallic. For example, prior to the flyby of Rosetta in 2010, we used our adaptive optics (AO) observations to estimate the density of another M-class asteroid, (21) Lutetia, and argued in favor of an enstatite-chondrite composition for that object (Drummond et al., 2010; Chapman et al., 2010). That interpretation was supported by

∗

Corresponding author. E-mail address: [email protected] (J.D. Drummond).

https://doi.org/10.1016/j.icarus.2018.01.010 0019-1035/Published by Elsevier Inc.

J.D. Drummond et al. / Icarus 305 (2018) 174–185

post-flyby analysis of the Rosetta data for Lutetia, which showed that its surface composition is more like enstatite chondrite than iron meteorites (Vernazza et al., 2011). An enhanced radar albedo, however, is the most important characteristic indicating a metallic surface composition. Therefore, one possible interpretation of Psyche’s high radar albedo of 0.37 (Shepard et al., 2017) is that it is the remnant core (or an exposed, disrupted, and reassembled core) of a planetesimal that lost its mantle and crust during a catastrophic collision after its formation. Consistent with radar albedo is the visible-near-IR spectrum of Psyche, which is similar to nickel-iron meteorites (Chapman and Salisbury, 1973). Using S-band radar at Arecibo Observatory, Shepard et al. (2017) obtained observations to create a 3-D shape model of Psyche. This shape model coupled with NASA IRTF near- and mid-IR observations showed a complex world where some fraction of the surface was covered with possible exogenic carbonaceous material (Sanchez et al., 2017) rich in volatiles (Takir et al., 2017). Using the radar derived shape and published mass from Carry (2012), Shepard et al. (2017) estimated a bulk density ρ = 4.5 ± 1.4 g cm−3 , and concluded that if Psyche were made of nickel-iron metal (ρ = 7.8 g cm−3 ), it would have 40% macroporosity, and no porosity if its composition were similar to stony-iron meteorites such as mesosiderites. Rotationally resolved radar and near-IR spectra of Psyche show an inverse relationship between radar albedo and silicate abundance (Sanchez et al., 2016), which suggests a metallic surface with varying small amounts of silicates and exogenic carbonaceous materials.

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The ambiguity between nickel-iron and stony-iron analogs for Psyche, along with the enstatite chondrite interpretation for (21) Lutetia, the only M-class asteroid to be visited by a spacecraft, prompted us to revisit Psyche’s density estimate using an extensive set of AO images that we acquired with the 8 m Gemini and 10 m Keck II telescopes. Here, we derive a triaxial ellipsoid shape model, based on least squares fits to our images and compare the results with previous work, in particular the model of Psyche developed from radar (Shepard et al., 2017). We also continue the literature compilation of the size and mass of Psyche, finding a new and better determined mass for the asteroid. The radar shape model, which includes concavities and topographic highs, gives an estimate of Psyche’s volume enclosed by the surface of the polygon. Shepard et al. (2017) used 7 raw AO images from two oppositions to help constrain the model, but here we use 25 sets of deconvolved AO images from six different nights (265 images in total) in four oppositions over 14 years, and the sheer number of images leads to smaller systematic uncertainties in our triaxial ellipsoid dimensions and volume, as well as in the position of the pole. The pole position (critical for planning spacecraft mission operations) from Shepard et al. is consistent with our newly derived value and its reduced uncertainty. Although Shepard et al. (2017) used a few of our images, our results provide measurements of the size, shape, volume, density, and pole position of Psyche with an independent technique. We have lowered the uncertainties in all parameters and demonstrate that these fundamental parameters of this M-type asteroid are well determined by all methods.

to achieve the latter goal for objects exceeding about twice the diffraction limit of the 8–10 m telescopes used, in J-, H-, or K-bands (1.2 – 2.1 μm), or about 0.09  in angular size. The 10 m diameter mirror of Keck provides the largest filled aperture currently available for imaging, and with its NIRC2 detector having a pixel size of 0.009945  , the resolution of 0.043  is twice oversampled (2.2 times Nyquist) in the K-band. We also use imaging data that we acquired at the Gemini-North 8 m telescope using the Altair/NIRI AO system, primarily in the near-IR K-band, and with a pixel size of 0.022  it also oversamples, by 1.2, the 0.054  resolution of the 8 m telescope. This oversampling, plus acquiring images at multiple locations on the detector (effectively, dithering), and taking many images at a single epoch, have allowed us to determine shapes and sizes of asteroids with great accuracy. For all observations, Psyche was bright enough (and small enough, in angular size) that it could be used as the AO guide source. Therefore, Laser Guide Star (LGS) mode was not required and all observations here were acquired using Natural Guide Star (NGS) mode. For the best images of Psyche at Keck, we achieved about 21 pixels across its equatorial diameter, or 4.8 resolution elements (‘rexels’). For the largest apparent ellipse presented in 2015, there were 283 pixels or 15 rexels covering the disk area of Psyche. Over this program, we have made observations of many asteroids with the above instruments, and analyzed them both with the ellipsoidal fitting method used and described here, as well as the KOALA method (Carry et al., 2010a; Kaasalainen et al., 2011), which combines AO observations with lightcurve and occultation data. Among these are (511) Davida (Conrad et al., 2007); four objects from Drummond et al. (2009), (129) Antigone, (409) Aspasia, (532) Herculina, and (704) Interamnia; (3) Juno (Viikinkoski et al., 2015); (52) Europa (Merline et al., 2013); near-Earth asteroid 2005 YU55 (Merline et al., 2012) with a diameter determined to be only 308 ± 9 m; (2) Pallas (Drummond et al., 2009; Carry et al., 2010a); (19) Fortuna (Drummond et al., 2011); and (9) Metis (Drummond et al., 2012). Of particular note are measurements of two objects that were subsequently visited by spacecraft, and therefore provide ground truth for our techniques. In 2010, prior to the arrival of the Rosetta spacecraft at (21) Lutetia, we derived the size, shape, and pole position (Carry et al., 2010b; Drummond et al., 2010). Evaluation of the data from the spacecraft flyby showed our model to have triaxial dimensions that were good to 2%, with RMS deviations of the actual surface topography of only 2 km on this 100 km diameter body (Carry et al., 2012). We also determined the triaxial dimensions of Ceres prior to the arrival of the Dawn spacecraft, and were within 0.1–0.6% (our quoted errors were 0.5–1.0%) in each of the 3 axes (Carry et al., 2008; Drummond et al., 2014; Russell et al., 2016). Table 1 gives the log of the observation geometries from Keck and Gemini. A set of between 3 and 15 raw images was acquired and fit with parametric blind deconvolution (see Section 4), yielding mean projected ellipse parameters and deconvolved images for each epoch. These images, deconvolved by the Lorentzian point spread function (PSF), are shown in Fig. 1, and the orientation on the sky plane of the resulting ellipsoid model for each epoch are shown in Fig. 2. In Figs. 3 and 4 the mean parameters from fits of each image are plotted at the mean UT of each epoch for two dates. Most images were made at 2.1 μm (K or K -band), with a few at 1.6 μm (H-band).

3. Observations

4. Reductions to triaxial ellipsoid results

Our observations are part of a larger effort that we have made using AO on the world’s largest ground-based telescopes, to both search for satellites of asteroids and to determine shape, size, and pole positions for resolvable asteroids. We have been able

We use the Parametric Blind Deconvolution (PBD) method (Drummond, 1998; Drummond et al., 1998; 2009) to measure the apparent size and orientation of Psyche. The details of our technique have been extensively described in our studies of previous

2. Overview

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J.D. Drummond et al. / Icarus 305 (2018) 174–185 Table 1 Psyche Observation Log. In addition to the EQJ20 0 0 RA and Dec for each date, the asteroid-Sun and asteroid-Earth distances, the solar phase angle, the position angle of the Sun (NtS) at the asteroid, the km per arc second scale, and the observatory, where n is the number of epochs, are listed. Date

RA(°)

Dec(°)

Sun (AU)

Earth (AU)

Phase(°)

NtS(°)

km/ 

Tele (n)

2004 Jun 3 2009 Aug 17 2010 Oct 6 2010 Oct 30 2010 Oct 31 2015 Dec 25

319.6 313.4 82.2 82.8 82.8 72.8

−13.5 −16.3 +19.4 +18.9 +18.9 +18.0

2.794 2.687 2.608 2.633 2.634 2.702

2.204 1.692 2.085 1.850 1.841 1.756

19.1 5.0 21.1 15.9 15.6 7.1

71.4 259.2 88.4 91.4 91.5 250.8

1598 1227 1512 1342 1335 1274

Gem Keck Keck Gem Gem Keck

(1) (2) (8) (6) (4) (4)

Fig. 1. Deconvolved images of Psyche at each epoch. The order of the images, left to right, top to bottom is chronological. At the top of each image, the observatory, Keck or Gemini, and the two digit year are listed along with the rotational phase and sub-Earth latitude for the telescope/dates listed in Table 1. The black outline around each image is the measured ellipse obtained during the deconvolution process. The asteroid appears larger as the sub-Earth point gets closer to the pole and when the asteroid comes closer to the Earth. Each image is 400 km on a side.

asteroids (Conrad et al., 2007; Drummond, 2000; Drummond et al., 1998; 2009; 2010; 2014; Merline et al., 2013). Assuming that an asteroid is smooth and uniformly bright from limb to terminator, and that the AO PSF is Lorentzian in shape (Drummond, 1998), we perform a least squares fit of the FFT of an image to find the apparent major diameter α , the apparent minor diameter β , and the position angle (PA) of the long axis of both the asteroid and the PSF simultaneously in the plane of the sky. Limb-darkening at low solar phase angles or minor albedo variation on an asteroid have little impact on a fit made in the frequency domain, where the main signature in the FFT of the image is the characteristic ringing of a Bessel function from hard edges. See Drummond et al. (2014) for some further discussion. The three asteroid parameters that comprise the observables at each rotational phase are used in a non-linear least squares fit for six unknowns, the three ellipsoid dimensions and three angles, a rotational phase zero point ψ 0 and the two coordinates of the rotational pole, with the north or positive pole defined according to

the right hand rule. Where the zero longitude on the asteroid is the meridian over one end of the long axis, ψ = 270◦ − L. Thus, rotational phase increases with time while longitudes decrease as the asteroid spins. The rotational phase is defined to be zero at maximum area and maximum major apparent axis as shown in Figs. 3 and 4, and the location of the pole yields the latitude of the sub-Earth point (θ ) and the position angle of the line of nodes (the intersection of the equatorial plane and the plane of the sky) at each epoch. Altogether, these describe the size and orientation of the ellipse projected by a triaxial ellipsoid. Minimizing the differences between the measured and predicted apparent sizes and orientations as a function of rotational phase yields the full ellipsoid dimensions and the location of the spin axis. We only need a sidereal period to tie all of the observations together. Although we can solve for the period in many cases by adding one more unknown, ˇ here we use the 4.195948 h sidereal period of Durech et al. (2011), which has an uncertainty of 1 in the last digit. Any departures from our triaxial ellipsoid assumptions, including minimal limb

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Fig. 2. The view of our triaxial ellipsoid model at the time of each image in Fig. 1. Celestial north is up and east is counterclockwise from north. Lines of latitude are drawn every 30° and lines of longitude every 45°, with longitude 0° over the tip of a long axis shown with bold dots. In every case the south pole of Psyche is visible. The sub-Earth point drawn as a , and the sub-Sun point drawn with a , are connected by a line that may be distinguishable only at large solar phase angles. The longitude of the sub-Earth point decreases with rotation, which is in a right handed sense about the asteroid’s north pole (not visible) following the convention of Archinal et al. (2011).

darkening and albedo variation, is captured in the formal uncertainty of all parameters derived from the least squares fit of the data in toto. Although formal statistical uncertainties for the three diameters and the three angles come directly from the non-linear least squares fits, systematic effects can arise in the process of constructing a 3-D description of an asteroid from information limited to a 2-D plane (images). Therefore, one needs to be particularly vigilant regarding model assumptions, and their appropriateness for a particular situation. While the uncertainties derived for the parameters as fit by the model are straightforward, estimating the systematic biases that are present is not. Deriving realistic uncertainties, including systematics, that are directly applicable by other workers is the most challenging aspect of our work. In addition to the multiple ground-truth tests we have undertaken (see Section 3 above), we have also carefully calibrated some of these uncertainties by making observations of external sources of known size, e.g., the moons of Saturn (Drummond et al., 2008). One of the results of that work has shown that our systematic uncertainties are larger for objects of smaller angular diameter, until we reach a limit (at about 0.09" for a 10 m telescope) where we can no longer get reliable sizes. Aspect ratios of projected shapes are still possible, but absolute sizes break down. We have found that our systematics from those tests span about 1–4% per linear

dimension. For the angular size of the asteroid observed in this work (about 0.2  ), we estimate that our systematics are approximately 3% on each axis. We do not try to correct for any systematic offset of measured vs actual size in this scenario, but instead add this 3% per dimension quadratically with the model uncertainties. In fact, these systematics are larger than the model fit uncertainties, and so dominate our quoted errors. Because this methodology has worked well for us in our ground-truth tests, we have gained significant confidence in it, and continue to use the same process to estimate our overall uncertainties in Psyche’s triaxial dimensions. Our experience with estimating uncertainties in pole position allows us also to assign an uncertainty from possible systematic biases of 3° for the model pole error, again much larger than the formal uncertainty from the least squares fit of the data. 4.1. Triaxial ellipsoid diameters Table 2 gives the triaxial ellipsoid results from our global fit of the 25 epochs. The uncertainties in Table 2 are formal fitting errors, and in parentheses, the total estimated uncertainties are calculated as the quadratic sum of fit errors and the 3% (on each triaxial dimension) estimate for possible systematic biases. Fig. 3 shows our data from Keck in 2010 superimposed on the predictions from the fit, and from later in the month Fig. 4 shows the

178

J.D. Drummond et al. / Icarus 305 (2018) 174–185

UT 14

15

16

17

300

250

0.16 0.14

Arc Sec

Diameter (km)

0.18

200 0.12 150 -90

-45

0

45

90

135

180

0.1 270

225

Rotational Phase 14

15

16

17

PA(°) of Long Axis

240 210 180 150 120 90 60 -90

-45

0

45

90

135

180

225

270

Rotational Phase Fig. 3. Triaxial ellipsoid fit to measured projected ellipse parameters for Psyche on 2010 October 6 from Keck, when the solar phase angle was 21° and the sub-Earth latitude was −39◦ . Circles and squares are measured apparent axes diameters. The solid lines are the predictions for the projected ellipses from the triaxial ellipsoid parameters in Table 2 and the dashed lines are for the ellipse parameters for the terminator ellipse (Drummond et al., 1985; Drummond, 20 0 0). The data should lie approximately midway between the dashed and solid lines. The lower subplot shows the same information in the same way for the position angle of the long axis. The solid line across the lower subplot is the line of nodes where the asteroid’s equator intersects the plane of the sky. This figure is corrected for light time travel, i.e., the plot is in the body-centered time frame.

Table 2 Triaxial ellipsoid diameters from 25 AO epochs. a (km) b (km) c (km) D (km)

274 231 176 223

± 2 ( ± 9) ± 2 ( ± 7) ± 4 ( ± 7) ± 3 ( ± 7)

same from a two night run at Gemini. Fig. 5 shows the residuals from all nights as a function of rotational phase and sub-Earth latitude. Table 3 summarizes our triaxial ellipsoid results, the results from Shepard et al. (2017), and the two radius vector models, convex and non-convex, currently listed on the DAMIT ˇ web site1 (Durech et al., 2010). The radius vector model of Shepard et al. (2017) was derived mostly from radar, but because an independent spin period or rotational pole cannot be derived from radar alone, a pole and period was taken from lightcurve work, specifically from the DAMIT web site for the convex model. They made a small adjustment to the initial DAMIT pole and a final adjustment to their c dimension using our 2009 Keck

1

http://astro.troja.mff.cuni.cz/projects/asteroids3D/web.php.

Table 3 Triaxial ellipsoid models. Diameter

Max Extenta

DEEVEb

DAMITc

DAMITd

This paper

a b c D = (abc )1/3

279 ± 28 232 ± 23 189 ± 19 230 ± 23

268 ± 27 229 ± 23 189 ± 19 226 ± 23

267 208 169 211

293 234 162 223

274 ± 9 231 ± 7 176 ± 7 223 ± 7

a The maximum diameters observed on the radar radius vector model (Shepard et al., 2017). b The Dynamically Equivalent Equal Volume Ellipsoid (Shepard et al., 2017). c The DAMIT convex model (Hanuš et al., 2013). d The DAMIT non-convex model (Hanuš et al., 2017).

AO images and three stellar occultations. The DEEVE column in Table 3 refers to a Dynamically Equivalent Equal Volume Ellipsoid that Shepard et al. (2017) derived, one that has the same volume and moments of inertia as the radar radius vector model. Just as Drummond (2014) did, if we make a weighted triaxial ellipsoid fit of the radius to the faces (not the vertices) of the two DAMIT radius vector models for Psyche, where the weights are the volume of the individual tetrahedra, then like the DEEVE ellipsoid, these triaxial ellipsoids have the same volumes and moments of inertia as the radius vector models. The results from these fits are

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UT 8

9

10

11 0.22 0.2

250 0.18 0.16 200

Arc Sec

Diameter (km)

300

0.14 150 -90

-45

0

45

90

135

180

225

0.12 270

225

270

Rotational Phase 8

9

10

11

PA(°) of Long Axis

240 210 180 150 120 90 60 -90

-45

0

45

90

135

180

Rotational Phase Fig. 4. Same as Fig. 3, but for 2010 October 30 from Gemini, with data from October 31 folded back and shown as circles or squares with asterisks. Also, the first three open symbols are folded back from one rotation later. The solar phase angle was 16° and the sub-Earth latitude was −39◦ .

shown in the DAMIT columns in Table 3. For the record, Hanuš et al. (2017) list a mean diameter of 225 ± 4 km for this rugged and irregular non-convex model that they contributed to the DAMIT site. 4.2. Densities To calculate asteroid densities, Carry (2012) compiled diameter and mass estimates from the literature. Here we continue this effort for Psyche in Table 4 for diameters and in Table 5 for masses. The tables show his original list plus 6 additional diameters and 7 additional masses. Fig. 6 shows plots of both. Combining our latest mass estimate (the average listed in Table 5) of 2.43 ± 0.35 × 1019 kg with the volume from our work with 265 AO images, Table 6 lists our new density estimate for Psyche for comparison to the density from Shepard et al. (2017). For Carry (2012) the dominant source of error for his density of 3.38 ± 1.16 g cm−3 was the 28% mass uncertainty. However, the uncertainty of our new mass of 14% helps to substantially reduce the density uncertainty here. 5. Rotational pole In Fig. 7 we show the wedge uncertainty region around our pole from Table 7 from our global fit. We also show the radar pole from Shepard et al. (2017), the poles from lightcurves (without un-

´ et al., 2007), and certainties) as listed on a website2 (Kryszczynska the two DAMIT model poles. Because it is bright and has a short rotational period, many lightcurves of Psyche have been obtained since 1955, and in fact, Psyche has rotated some 125,350 times between the first lightcurve in 1955 (van Houten-Groeneveld and van Houten, 1958) and our 2015 AO images, exactly 60 years apart to the day. We use the sidereal period derived from these lightcurves of 4.195948 h ˇ ( ± 1 in the last digit) from Durech et al. (2011), but we can also fit for a sidereal period and find a period of 4.195951 h ( ± 2 in the last digit) from our 25 epochs, not significantly different. For ˇ the period of Durech et al. (2011), and our pole at [RAp Decp ]=[32° +5]°, we calculate W0 =273.6° (see Appendix A) at the zero epoch of 20 0 0 Jan 1.5 according to Archinal et al. (2011). By fitting the DAMIT models as triaxial ellipsoids we also find the angles, , necessary to rotate the models to principal axes. Thus their longitudes L can be converted to principal axes based longitudes L with L = L + . For the ellipsoid models in Table 3 we list in Table 7 the poles, W0 , and rotation angles for comparison. Our zero longitude is based is the meridian over the tip of the long axis that transits at minimum projected area, the left edge of each of Figs. 3–5. This is the same hemisphere that marks the zero longitude of Shepard et al. (2017), but there is a 2° difference between the transit time of the prime meridians of the two models

2

http://vesta.astro.amu.edu.pl/Science/Asteroids/.

20

20

10

10

0

0

-10

-20 -90

-10

-60

-30

0

30

60

90

120

150

180

210

240

Residuals (degrees)

J.D. Drummond et al. / Icarus 305 (2018) 174–185

Residuals (km)

180

-20 270

Rotational Phase 20 α β pa

10

10

0

0

-10

-20 -60

-10

Residuals (degrees)

Residuals (km)

20

-20 -45

-30

-15

0

Sub-Earth Latitude Fig. 5. Residuals between observed ellipses and projected triaxial ellipsoid model (O-C). Apparent ellipse major axes diameters are shown as filled circles, minor axes diameters as squares, and position angles of the long axes as triangles. The right scale in degrees is for this latter quantity, See Figs. 3 and 4 to disentangle the pile up of points at latitude −39◦ in the lower sub-plot.

in the sense that the radar model’s long axis transits 2° in rotation earlier than our long axis. After we fit the two DAMIT models, we can rotate their coordinate systems so that their zero longitudes correspond to the principal axes of inertia, finding that the convex and non-convex models’ prime meridians transit 5° later and 2° earlier than ours, respectively. Thus, the prime meridian of Psyche is set by all four models, to within a few degrees, as illustrated in Fig. 8. Since the obliquity, the angle between Psyche’s orbital and rotational pole, is so high (95°), the sub-Earth point ranges from one pole to the other as the asteroid travels along its orbit. It does so in regular steps along north-south lines on the asteroid in succeeding oppositions because the orbital period of 5.00 years is nearly commensurate with the Earth’s. Thus, Psyche’s oppositions occur at four distinct geometries. The dates of the next eight oppositions are given in Table 8 as well as the number of days since the previous opposition, and the RA, Dec, solar phase angle ω, distance from Earth r, and sub-Earth latitude θ at each opposition. However, the sub-Earth point will still range some ± 6° in latitude over the 10 weeks surrounding an opposition. 6. Interpretation and summary We have used a large number of AO images to determine the triaxial dimensions, volume, density, pole direction, and prime meridian of (16) Psyche. Over 265 images were incorporated, and we have shown once again that such a collection of AO images, taken from several vantage points, can produce a rotational pole

and accurate sizes and shapes independent of radar, lightcurve, or occultation data. Although Shepard et al. (2017) considered AO images too problematic for size/shape determination because of their indistinct edges, in fact asteroid edges are well-defined after or during deconvolution with, for example, PBD (Section 4). In the end, of course, data from all techniques, AO, radar, stellar occultations, and lightcurves should be directly incorporated to make a unified model for an asteroid. Psyche’s density provides us with important constraints on its macro-porosity, meteorite analog and ultimately its origin. Using our new mass of 2.43 ± 0.35 × 1019 kg, and the volume calculated from our AO triaxial ellipsoid dimensions in Table 3, we estimate a density for Psyche of 4.16 ± 0.64 g cm−3 . We agree with Shepard et al. (2017) that Psyche has significant macroporosity (47%) if it is composed mainly of nickel-iron. Thus, unless our adopted error bars on mass and/or volume are wrong, it cannot be a solid exposed iron core. If it is a core, then that core must have been disrupted somehow and reassembled into a rubble pile. Fragmental macroporosities can range from ∼ 20% to ∼ 50%, consistent with our result. The alternative that Psyche is a mesosiderite-like stony-iron, with low macroporosity, seems to be incompatible with the metal-rich composition of its surface since known mesosiderites have roughly equal proportions of stone and metal, or even larger proportions of stone. Also, the silicates in mesosiderites are similar to diogenites (iron-rich pyroxenes) compared to the iron-poor pyroxene detected on Psyche by Sanchez et al. (2016). Most Pallasites can have larger fractions of metal, but the silicates in Psyche are pyroxenes, not olivines, which

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Table 4 The diameter estimates of (16) Psyche collected from the literature. For each, the uncertainty, method, selection flag, and bibliographic reference are reported. The methods are hmag: Absolute magnitude, im-te: Ellipsoid from Imaging, lcimg: 3-D Model scaled with Imaging, lcocc: 3-D Model scaled with Occultation, neatm: Near-Earth Asteroid Thermal Model, radar: Radar Echoes, stm: Standard Thermal Model, tpm: Thermophysical Model. A check mark indicates that the value is used in calculating the weighted average, and an X that it is not, according to Carry (2012). #

D (Km)

σD (Km)

Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

247.00 253.16 213.00 262.80 222.58 269.69 211.00 209.00 207.22 244.00 288.29 213.00 226.00 225.00 223.00

24.70 4.00 21.00 4.10 5.58 11.50 21.00 29.00 2.98 8.00 4.63 15.00 23.00 4.00 7.00

stm stm hmag im-te stm neatm lcocc lcocc stm tpm neatm lcimg radar lcimg im-te

235.4

16.1

Weighted Average

Sel. √ √

Reference Morrison and Zellner (2007) Tedesco et al. (2004) Lupishko (2006) Drummond and Christou (2008) Ryan and Woodward (2010) Ryan and Woodward (2010) ˇ Durech et al. (2011) ˇ Durech et al. (2011) Usui et al. (2011) Matter et al. (2013) Masiero et al. (2012) Hanuš et al. (2013) Shepard et al. (2017) Hanuš et al. (2017) This work

✗ ✗ √ √ √ √ ✗ √ ✗ √ √ √ √

Table 5 The mass estimates of (16) Psyche collected from the literature. For each, the uncertainty, method, selection flag, and bibliographic reference are reported. The methods are defl: Deflection, ephem: Ephemeris. A check mark indicates that the value is used in calculating the weighted average, and an X that it is not, according to Carry (2012). #

Mass (kg)

Method

Sel.

Reference

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

(25.3 ± 3.6) × 1019 (1.73 ± 0.52) × 1019 (4.97 ± 0.20) × 1019 (6.72 ± 0.56) × 1019 (2.67 ± 0.44) × 1019 (2.19 ± 0.08) × 1019 (7.96 ± 2.78) × 1019 (3.17 ± 0.06) × 1019 (3.35 ± 0.34) × 1019 (2.23 ± 1.03) × 1019 (4.59 ± 1.93) × 1019 (3.22 ± 0.60) × 1019 (2.27 ± 0.08) × 1019 (2.47 ± 0.68) × 1019 (2.35 ± 0.39) × 1019 (2.46 ± 0.16) × 1019 (2.44 ± 0.16) × 1019 (2.02 ± 0.43) × 1019 (2.51 ± 0.36) × 1019 (2.51 ± 0.44) × 1019 (1.77 ± 0.42) × 1019 (2.54 ± 0.20) × 1019 (2.23 ± 0.36) × 1019 (2.33 ± 0.04) × 1019 (2.21 ± 0.05) × 1019 (2.54 ± 0.49) × 1019

defl defl defl defl defl defl defl ephem ephem ephem defl ephem defl ephem defl defl defl defl ephem ephem ephem ephem ephem defl defl ephem

✗ √

Vasiliev and Yagudina (1999) Viateau (20 0 0) Krasinsky et al. (2001) Kuzmanoski and Kovacˇ evic´ (2002) Kochetova (2001) Baer et al. (2008) Ivantsov (2008) Fienga et al. (2009) Fienga et al. (2008) Fienga et al. (2010) Somenzi et al. (2010) Pitjeva and Klioner (2010) Baer et al. (2011) Konopliv et al. (2011) Zielenbach (2011) Zielenbach (2011) Zielenbach (2011) Zielenbach (2011) Fienga et al. (2011) Fienga et al. (2012) Kuchynka and Folkner (2013) Pitjeva (2013) Fienga et al. (2014) Goffin (2014) Kochetova and Chernetenko (2014) Fienga (priv. comm)

(2.43 ± 0.35) × 1019

Weighted Average

Table 6 Volume and density.

Vol ( × 10 km ) Density (gm cm−3 ) 6

3

✗ ✗ √ √ ✗ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √

Table 7 Psyche rotational poles, W0 , and . DEEVE (Shepard et al., 2017)

This paper

6.074 ± 1.215 4.5 ± 1.4

5.845 ± 0.335 4.16 ± 0.64

are the chief silicates in pallasites. Of course, there remain the possibilities that the bulk composition of Psyche is more stony-rich than its surface or that Psyche’s composition differs from known meteorites.

RA° ; Dec° Ecl λ° ; δ ° W0◦ ° (= L − L)

This paper

Radar

Convex

Non-Convex

32 ; +5 32 ; −8 273.6 0

34 ; +6 34 ; −7 271.5 0

32 ; +5 32 ; −8 292.7 15.6

25; +1 23 ; −9 294.4 21.9

Although Psyche spins nearly perpendicular to its orbit (i.e., its spin axis lies in its orbital plane), which implies we would see it from one pole to another over several apparitions, the orbital pe-

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300

Diameters (km)

280 260 240 220 200 180 0

2

4

6

8

10

12

14

16

Masses (kg×10 19 )

10 8 6 4 2 0 0

5

10

15

20

25

Fig. 6. A plot of the estimates of Psyche’s diameters (top) from Table 4 and masses (bottom) from Table 5 (but without the first entry). The solid lines are the weighted average, weighted by the reciprocal of the listed uncertainties squared, wt = 1/σ 2 , and the dotted lines are the standard deviations about the mean. The X-axis on each sub-plot refers to the # in Table 4 or 5. Points marked with an X are not included in the fit.

Table 8 Future Psyche oppositions (see text). Date

Days

RA°

Dec°

ω°

r (AU)

θ°

2018-May-10 2019-Aug-07 2020-Dec-07 2022-Mar-03 2023-May-10 2024-Aug-06 2025-Dec-08 2027-Mar-03

433 454 488 451 433 454 489 450

228 316 75 164 228 316 75 164

−13 −15 +18 +8 −13 −15 +18 +8

1.4 0.5 1.7 0.3 1.4 0.5 1.7 0.3

2.24 1.70 1.69 2.23 2.24 1.71 1.69 2.23

+72 −12 −46 +41 +72 −12 −46 +41

riod of 5.00 y results in only four distinct oppositions, with subEarth latitudes of −46◦ , −12◦ , +41°, and +72°. Thus, further study of the asteroid before the arrival of the Psyche Mission in 2030 should perhaps emphasize observations away from opposition, although the penalty then would be that the asteroid will be fainter and further than at opposition.

tory was made possible by the generous financial support of the W.M. Keck Foundation. Some of the observations were obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnologa e Innovacin Productiva (Argentina), and Ministrio da Ciłncia, Tecnologia e Inovao (Brazil). The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. This research made use of NASA’s Astrophysics Data System and JPL’s Horizons ephemerides tool. We thank two anonymous referees for their helpful reviews. Appendix A. Calculating sub-Earth latitudes and longitudes

Acknowledgments Here we give formulae for predicting the asterocentric latitude

This work was supported, in part, by research grants to our group (Merline PI) from the NASA Planetary Astronomy Program and the NSF Planetary Astronomy Program. Some of the observations were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observa-

θ and longitude L for any date for an asteroid, following the recommendations of Archinal et al. (2011). As defined in Section 4, we measure asterocentric longitudes counterclockwise from one of the long tips of the asteroid, a principal axis of momentum, while looking down from above its North Pole according the right hand rule. The relationship between the longitude of the sub-Earth point and the rotational phase ψ as defined in Section 4 is L = 270◦ − ψ .

J.D. Drummond et al. / Icarus 305 (2018) 174–185

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Fig. 7. Celestial globe location of Psyche rotational poles from our AO (Table 7), the radar model of Shepard et al. (2017), and from lightcurve techniques. The pole determined from the global AO fit is at the center of the two shaded uncertainty regions, where the dark region represents the formal one sigma uncertainty from Table 2 and the lighter shaded region 3σ . The positions of Psyche for the 5 epochs are marked with asterisks (labeled with the observatory and year) at the apex of the dotted lines leading to the pole. The uncertainty in the direction to the pole is indicated by the spread of the dotted lines, and the sky positions of Psyche for each epoch show how far the asteroid is from the rotational pole on each date. When the asteroid is 90° from the pole, the sub-earth point is on the asteroid’s equator. The position of the radar model pole is in the center of the large circle with its 5° radius of uncertainty, and small circles denote poles from various lightcurve studies, with the synthesis of these lightcurve ´ poles according to Kryszczynska et al. (2007) shown as the small filled circle. The two filled squares are the two DAMIT poles; with the non-convex pole being the apparent outlier.

While rotational phases increase with time as the asteroid rotates, longitudes decrease. Times are body-centered; they should always be lighttime corrected back to the asteroid by subtracting 8.31 min/AU from an observation time. Let the pole location in celestial coordinates be given by α p and δ p , and the position of the asteroid by α a and δ a . The angular distance of the sub-Earth point on the asteroid from the asteroid’s South Pole is ζ

The sub-Earth latitude is then

ζ = cos−1 [sin δ p sin δa + cos δ p cos δa cos(α p − αa )],

L = −[W0 + 360(JD − 2451545 )/(P/24 )] + k, .

(A1)

and the longitude of the sub-Earth point in an inertial coordinate system is k, given by −1

k = sin

[(sin δ p cos ζ − sin δa )/(sin ζ cos δ p )];

if sin(α p − αa ) < 0, k = 180◦ − k.

(A2)

θ = ζ − 90◦ .

(A3)

To calculate an asterocentric longitude, Archinal et al. (2011) defines a zero time angle W0 for JD=2451545=20 0 0 January 1 12 UT. Conceptually, if the asteroid is placed at RA p − 90◦ and Dec=0°, W0 is the time in rotational phase since the zero longitude transited on 20 0 0 Jan 1.5. Asterocentric longitudes are then

(A4)

However, for the DAMIT radius vector models, the X and Y body axes are not necessarily aligned to the asteroid’s principal axes. Therefore, we fit the radius to the faces (not the vertices) of these models as a triaxial ellipsoid, weighting the fit by each volume element, while holding the pole direction constant. This yields an

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Fig. 8. Views of the four models of Psyche in Table 7 at four times in late 2015, all at the same scale. The first column is our model, the second is the radar DEEVE model, the third is the DAMIT convex model, and the last column is the non-convex DAMIT model. These can be directly compared to Fig. 8 in Shepard et al. (2017), who also show the radar radius vector model at these times. As in Fig. 2, celestial north is up and east is to the left. Since the southern pole of Psyche is visible, the asteroid appears to rotate clockwise. Its equator is indicated by a heavier dotted line, as is the long principal axis (prime meridian) which defines longitude zero, even for the DAMIT models using the from Table 7. The b axis is 90° clockwise from the prime meridian. All of the models show the prime meridian orientation during the times of the radar observations.

angle, , to rotate the X and Y axes to principal axes X and Y . Longitudes based on principal axes, L , are then

L = L + .

(A5)

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