TPM with Shape Models

More reliable results can be obtained by using a thermophysical model in which the heat diffusion equation is solved for facets distributed on a shape model rather than a sphere. There are two published shape models for Psyche. The first one is produced by the lightcurve inversion method (Kaasalainen et al., 2002) applied to optical disk-integrated data. This model was later scaled in size by stellar occultation measurements (Durech et al.ˇ ,2011) and disk-resolved images from the near infrared camera mounted on the Keck II telescope (Hanuˇs et al.,2013a). We downloaded this model from the public Database of Asteroid Models from Inversion Techniques (DAMIT7_{; Durech et al.ˇ , 2010). Moreover, Shepard et al. (2017)}

recently derived a shape model based on optical lightcurves and delay-Doppler images. This shape model was provided us by Dr. Shepard.

Using the lightcurve-based shape model of Kaasalainen et al.(2002) we have determined the rotational phase of Psyche as viewed from Spitzer during both observations. The uncer- tainty in the rotation phase is about ten degrees, which is small enough for our purposes and conclusions. The synthetic lightcurves and both shape model projections at the two epochs of observations by Spitzer are shown in Figures 4.3 and 4.4. The synthetic lightcurves were generated by collaborator Josef Hanuˇs.

(a) Synthetic lightcurve

(b) Lightcurve-inversion shape model (c) Radar shape model

Figure 4.3

At top (a), Psyche’s synthetic lightcurve for one rotational period beginning at the start of SL2 observations. At bottom (b and c), lightcurve-inversion and radar shape models, respectively, shown at a rotational phase of 225◦, corresponding to the observation geometry at the beginning of SL2 observations.

(a) Synthetic lightcurve

(b) Lightcurve-inversion shape model (c) Radar shape model

Figure 4.4

At top (a), Psyche’s synthetic lightcurve for one rotational period beginning at the start of SL1 observations. At bottom (b and c), lightcurve-inversion and radar shape models, respectively, shown at a rotational phase of 225◦, corresponding to the observation geometry at the beginning of SL1 observations.

As with the spherical models, we consider emissivity values of =0.9, 0.7, and 0.5 and three cases of surface roughness: [1] smooth (RMS surface slope = 0◦); [2] moderately rough (RMS surface slope = 20◦); [3] very rough (RMS surface slope = 66◦). We do not vary the effective radius as in the spherical TPM, and instead assume a fixed size as constrained by

the radar, occultation, and adaptive optics data, and assume a fixed geometric albedo of pv = 0.15.

With this high fidelity version of the thermophysical model, we consider only the SL1 data, as the SL2 data are not as reliable or constraining. The wavelength range covered by SL2 is less diagnostic of thermal properties than the broader and longer wavelength range covered by SL1 (Figure 4.5). The presence of fine-grained silicates, which are more transparent at wavelengths shorter than ∼ 8µm in this wavelength range, also complicates fitting to SL2. In the presence of a steep thermal gradient, the warmer subsurface is detected at wavelengths where silicates are transparent. This appears as an excess in thermal flux at those wavelengths, and makes the spectrum deviate from a blackbody curve. There is additional uncertainty in SL2 from our subtraction of modeled reflected light.

Figure 4.5: Six SEDs produced from our thermophysical model, over the 5–40µm range. We use the viewing geometry of Psyche’s SL2 observations. The wavelength ranges covered by SL2 and SL1 are denoted. At longer wavelengths, the differences in the SEDs become more apparent.

The differences between the radar shape model and the lightcurve-only shape model, as well as the reported ±10% uncertainties in the dimensions of the shape models, contribute some additional uncertainty to the thermal inertia determination. To constrain this uncer- tainty, we consider both the nominal shape models and also “flex” the dimensions of each shape model by ±10% (so that the surface of area of each facet increases or decreases by 19% of the nominal area). As shown in Figure 4.6, how much the shape model uncertainty affects the thermophysical model’s goodness of fit and best-fit thermal inertia depends upon the value of bolometric emissivity used.

Figure 4.6: The effects of shape model uncertainties on the goodness-of-fit for each ther- mophysical model as a function of thermal inertia. Each panel shows χ2 as a function of thermal inertia for six shape model cases: the radar shape model and the lightcurve-only shape model at its nominal sizes and at the extrema of their 10% size uncertainties. Each row represents a different value of bolometric emissivity (). Solid horizontal lines represent the 1-, 2- and 3-σ χ2 cutoff values. These curves are calculated for the SL1 data and use the smooth surface thermophysical model.

For all thermal models, the derived range of values for each free parameter is determined by computing cutoff values of the χ2 goodness-of-fit parameter for 1-, 2-, and 3-σ confidence levels. Each combination of model free parameter values that produces a fit with a χ2 less than or equal to a cutoff value will be considered acceptable within the confidence level for that cutoff value. The cumulative distribution function (CDF) for χ2_{, that is, the probability}

that χ2 _{for a model fit to a dataset with k degrees of freedom will be valued ≤ x is given by}

Fχ(x, k) =

γ k₂,x₂
Γfunc k₂

(4.3)

where γ is the incomplete gamma function and Γfunc is the gamma function. To compute

a cutoff value of χ2 for the 1-σ confidence level for a given number of degrees of freedom k, we want the χ2 value x1σ such that Fχ(x1σ, k) = 0.683; i.e., there is a 68.3% probability that

χ2 will be ≤ x1σ. For the 2-σ and 3-σ confidence levels, we want cutoff values x2σ and x3σ

such that Fχ(x2σ, k) = 0.954 and Fχ(x3σ, k) = 0.997.

The cutoff values are computed from the percent point function (PPF) for χ2_{, which is the}

inverse of the CDF. The PPF for χ2_{is computed numerically, and we use a Python routine in}

SciPy’s scipy.stats.chi2 package8 _{to calculate the cutoff values from the PPF. Because these}

cutoff values are based on a theoretical ideal χ2 distribution, we add to them the minimum computed χ2 value from our model fits. The number of degrees of freedom are the number of data points in the spectrum we are fitting minus the number of free model parameters. To ensure that spectral points are independent of each other and to decrease computation time when fitting models, the spectra are binned by at least a factor of five.