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Physical Properties of Near-Earth Asteroid 2011 MD

We report on observations of near-Earth asteroid 2011 MD with the Spitzer Space Telescope. We have spent 19.9 h of observing time with channel 2 (4.5 {\mu}m) of the Infrared Array Camera and detected the target within the 2{\sigma} positional uncertainty ellipse. Using an asteroid thermophysical model and a model of nongravitational forces acting upon the object we constrain the physical properties of 2011 MD, based on the measured flux density and available astrometry data. We estimate 2011 MD to be 6 (+4/-2) m in diameter with a geometric albedo of 0.3 (+0.4/-0.2) (uncertainties are 1{\sigma}). We find the asteroid's most probable bulk density to be 1.1 (+0.7/-0.5) g cm^{-3}, which implies a total mass of (50-350) t and a macroporosity of>=65%, assuming a material bulk density typical of non-primitive meteorite materials. A high degree of macroporosity suggests 2011 MD to be a rubble-pile asteroid, the rotation of which is more likely to be retrograde than prograde.


Introduction
Little is known about the physical properties of near-Earth asteroids with diameters smaller than 100 m. Mainzer et al. (2014) measured the sizes and albedos of the smallest optically discovered near-Earth asteroids (d > 10 m) from NEOWISE data. Mommert et al. (2014) Mazanek et al. 2013). One of the proposed mission concepts for ARRM involves capturing an asteroid less than ∼10 m in size and guiding it into orbit about the Moon, where it could be visited and explored by astronauts. Candidate asteroids for this concept could have masses in the range of tens to hundreds of metric tons, but the maximum mass for each candidate would depend on its orbital parameters. The size and mass of 2011 MD were not known accurately enough to -5say whether it could be considered a more serious candidate for the proposed mission.
We utilize observations obtained by the Spitzer Space Telescope to constrain the physical properties of 2011 MD.

Spitzer Observations and Data Reduction
We observed 2011 MD with the Infrared Array Camera (IRAC, Fazio et al. 2004) on-board the Spitzer Space Telescope (Werner et al. 2004) in Program ID 10132 using a total of 19.9 h of observation time. Observations (astronomical observation request 49716480) started on 11 February, 2014, 20:30:47 UT, using the "Moving Single" object mode to track in the moving frame of the object. We performed the observations in full array mode with 100 s frames in IRAC channel 2 (4.5 µm) only, using a medium cycling dither pattern with 227 dither positions and 3 repeats, resulting in a total of 681 frames, or 18.3 h on-source exposure time.
At the time of the observations, 2011 MD was 1.09 au from the Sun and 0.14 au from Spitzer with a solar phase angle of 54 • . The observation window was selected based on Spitzer observing constraints.
The data were reduced using the method by Mommert et al. (2014). A mosaic of the field is constructed from the dataset itself and then subtracted from the individual basic calibrated data (BCD) frames. After subtraction of the background mosaic, residual background sources and bright cosmic ray artifacts are masked in the individual BCDs before being mosaicked in the reference frame of the moving object.
In the final co-move map we find a source within 2σ of the expected position of 2011 MD (see Figure 1, and Sections 5 and 3 for a discussion). We identify this source as 2011 MD and derive a flux density of (0.60±0.27) µJy. The uncertainty is derived as 15" N E 10"

Modeling
We constrain the physical properties of 2011 MD by combining an asteroid thermophysical model with a model of the nongravitational forces acting on the asteroid, similar to the approach taken by Mommert et al. (2014).
The thermophysical model approximates the surface temperature distribution of 2011 MD and is used to determine the thermal-infrared emission from its surface as a function of its physical properties, including spin axis orientation (represented by the obliquity, γ), rotational period, P , thermal inertia, Γ, and surface roughness. Surface roughness causes infrared beaming, an effect that focuses thermal emission radiated towards the observer, and is modeled as emission from spherical craters (see Mueller 2007, for more details). The model solves the heat transfer equation numerically for a large number of plane surface facets that form a sphere. The model we use is nearly identical to the one used by Mueller (2007). Since the single-band nature of our observation precludes a direct fit of the target's spectral energy distribution, we take a probabilistic approach in which we explore the parameter space by varying the individual input parameters.
Similar to Mommert et al. (2014), we model the nongravitational acceleration of the object as a result of the solar radiation pressure (using the approach by  and the Yarkovsky force . The model asteroid is assumed to be spherical and the heat transfer is solved analytically using the linearized heat transfer equation (Vokrouhlický 1998;Vokrouhlický & Farinella 1999).
By fitting all available astrometric data of 2011 MD, the model derives the bulk density, ρ, and the goodness-of-fit parameter χ 2 as a function of the asteroid's properties.

Ground
wherer andt are the radial and transverse directions, respectively, and r is the heliocentric distance. A 2 /r 2 translates into the transverse component of the Yarkovsky effect (Bottke et al. 2006), whereas A 1 /r 2 models the solar radiation pressure and the radial component of the Yarkovsky effect. We use this simplified model approach for ephemeris predictions and to investigate the detectability of nongravitational forces in the astrometric data. In order to fit the model to the astrometric data, we applied the Chesley et al. (2010) debiasing and weighting scheme. Since timing errors are more relevant when an object is observed at small geocentric distances, we relaxed the data weights for these observations.
In au/d 2 , 5.2σ). We ascribe our higher uncertainty to a less strict weighting used for some of the available astrometric data, and the fact that we have taken into account the Yarkovsky effect (A 2 ), which was neglected by Micheli & Tholen (2014), and leads to additional uncertainty, due to the correlation of A 1 and A 2 .

Results
We We reject synthetic model asteroids with unphysically high Bond albedos. The Bond albedo, A, describes the reflectivity integrated over the whole electromagnetic spectrum and can be approximated as A ∼ q · p v , with the phase integral q = 0.290 + 0.684 · G (Bowell et al. 1989). Hence, high geometric albedos p V can lead to A > 1, which contradicts the law of conservation of energy.
The final results of our analysis are depicted in Figures 2 to 4 and show that 2011 MD has a mean diameter of (6 +4 −2 ) m (1σ) and an albedo of 0.3 +0.4 −0.2 (1σ). Note that in the case of asymmetric uncertainties, the 1σ confidence interval refers to the 68.3% of values higher/lower than the median value. The 3σ confidence interval covers a range of (2-26) m in diameter and ≥0.02 in albedo. From the orbital model we find a most probable bulk density of (1.1 +0.7 −0.5 ) g cm −3 (1σ, 3σ interval: (0.2-5.0) g cm −3 ), which translates into a total mass of (110 +240 −60 ) t (1σ, 3σ interval: (10-2500) t). The measured albedo is compatible with a number of non-primitive taxonomic classes (Thomas et al. 2011).
Our model results favor a retrograde rotation of 2011 MD, which is suggested by the χ 2 distribution produced by the orbital model (see Figure 3), or the negative value of A 2 (compare to Farnocchia et al. 2013). Note that in case of a complex rotation of the object, our definition of obliquity is referenced to the rotational angular momentum vector rather than the spin axis. We are unable to constrain the thermal inertia of 2011 MD, given the low confidence in the measurement of A 2 .   Figure 2 for definitions). Both parameters are a strongly correlated to albedo. We adopt the median of each weighted distribution, yielding a most probable bulk density of (1.1 +0.7 −0.5 ) g cm −3 and total mass of (110 +240 −60 ) t.

Discussion
co-move map. The appearance of the source agrees with the IRAC point-spread function, which has a full width half maximum of 1.66 ′′ (1.4 px). Figure 1 shows that potential sources other than 2011 MD are constrained to only one high-signal pixel. We investigated the possibility that the source we identify as 2011 MD is a product of noise. For this reason, we created co-move maps from the BCDs with rates that are close, but not identical, to the consist of material comparable to ordinary chondrites, which has the lowest density of all non-primitive materials, we can derive a lower limit on the macroporosity of this object of 65% (see, Mommert et al. 2014;Britt et al. 2002). This high degree of macroporosity suggests a rubble-pile nature for 2011 MD, which is possible despite its fast rotation (Scheeres et al. 2010).
Our bulk density estimate (1.1 g cm −3 ) is nearly twice as high as the value found by Micheli & Tholen (2014). This difference is caused by their neglect of Yarkovsky forces and the assumption that 2011 MD's albedo follows the albedo distribution for small (10 < d < 100 m) asteroids (Mainzer et al. 2014). to values observed in some asteroids larger than 100 m (see Mommert et al. 2014).
Note that the diameter derived as part of this work is the effective diameter of a sphere with the same volume as the real shape of 2011 MD. The large lightcurve amplitude of 0.8 mag (Ryan & Ryan 2012), however, suggests a highly elongated shape of 2011 MD with an axis ratio of b/a ∼0.5, where (a,b,c) are the axes of a triaxial ellipsoid. The rotational period of 0.1939 h (Ryan & Ryan 2012) is significantly shorter than our observation duration (19.9 h); any optical lightcurve effects are hence averaged over our observation.
We investigate the impact of the temperature distribution of an ellipsoid with an axis ratio similar to that of 2011 MD compared to that of a sphere. We use a simplistic model of the shape of 2011 MD that is based on a triaxial ellipsoid with axis ratio (1,0.5,0.5); since there is no information on the c axis, we assume c = 0.5, which provides a principal axis rotation of the body. We approximate the measured lightcurve of 2011 MD with a step function: the observer is faced the long side of the asteroid (a × c) for 75% of the rotation period and the short side (b × c) for the remaining 25% (compare with Figure 2 by Ryan & Ryan 2012).
We realize this lightcurve behavior by using a composite flux density that consists to 75% of the flux density emitted by the long side and to 25% of the flux density emitted by the short side. We compare the diameter derived with this composite flux density with that of a spherical shape and find differences up to 20%, depending on the spin axis orientation and thermal inertia. This uncertainty, which is based on a coarse approximation of the real shape of 2011 MD, is well within the nominal 1σ diameter uncertainties used in our model approach. Also, the assumed ellipsoidal shape has a cross section that is different from that of a spherical shape, which affects the solar radiation pressure acting on the object, and hence changes its bulk density. We find that the average cross section of the ellipsoid is 10% larger than that of a sphere, forcing the same change in bulk density. Again, the nominal uncertainty in bulk density is significantly larger than this change.
Our observations have provided a determination of the physical properties of 2011 MD, and in particular, its size and mass. A final evaluation of 2011 MD as a candidate target for the proposed ARRM mission is beyond the scope of this work.