Near surface properties derived from Phobos transits with HP RAD ³ on InSight, Mars

We use the surface temperature response to Phobos transits as observed by a radiometer on board of the InSight lander to constrain the thermal properties of the uppermost layer of regolith. Modeled transit lightcurves validated by solar panel current measurements are used to modify the boundary conditions of a 1D heat conduction model. We test several model parameter sets, varying the thickness and thermal conductivity of the top layer to explore the range of parameters that match the observed temperature response within its uncertainty both during the eclipse as well as the full diurnal cycle. The measurements indicate a thermal inertia of 103 +48-24 Jm -2 K -1 s -1/2 in the uppermost layer of 0.2 to 4 mm, signiﬁcantly smaller than the thermal inertia of 200 Jm -2 K -1 s -1/2 derived from the diurnal temperature curve. This could be explained by larger particles, higher density, or a very small amount of cementation in the lower layers.


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Observations of the brightness temperature in response to changes in insolation con-36 strain the thermophysical properties of the upper layer of planetary surfaces, most fre-37 quently reported as thermal inertia (TI) defined as the root of the product of volumet-38 ric heat capacity ρc and thermal conductivity k. The thermal conductivity constrains 39 the particle size of regolith (Presley & Christensen, 1997a, 1997b, 1997cPiqueux & Chris-40 tensen, 2009a) but is also highly sensitivity to cementation (Piqueux & Christensen, 2009b). 41 The depth of regolith that can be probed is approximately the diurnal skin depth d =  (Golombek, Warner, et al., 2020). If the diurnal temperature curve of the same location 46 is sampled at sufficiently separate local times, typically using in-situ observations (Fergason 47 et al., 2006;Hamilton et al., 2014) instead of sun-synchronous orbiters, it is also possi-48 bly to infer layering within the diurnal skin depth (Vasavada et al., 2017;Edwards et al., 49 2018;Piqueux et al., 2021). 50 Based on orbiter observations, Golombek et al. (2017) state that the TI of the In-51 Sight landing ellipse derived from orbit (200 J m −1 K −1 s −1/2 ) is "consistent with a sur-52 face composed of cohesionless sand size particles or a mixture of slightly cohesive soils 53 (cohesions of less than a few kPa)" covered by a coating of surface dust responsible for the high albedo of 0.24, which is too thin to affect the diurnal curve. The landing site 55 features TI in the lower range of the landing ellipse (Golombek, Warner, et al., 2020;Golombek, 56 Kass, et al., 2020;Piqueux et al., 2021), however the Heatflow and Physical Properties 57 Package (HP 3 ) mole (Spohn et al., 2018), although designed and tested for such mate-58 rial, failed to deploy its instrumented tether to the subsurface (Fig. 1). The mole is de-  This raises the question whether the particles are smaller than thought, since such ce-65 mentation has the potential to strongly increase the thermal conductivity (Piqueux & 66 Christensen, 2009b). 67 In addition to the derivation of thermophysical properties from the diurnal response, 68 it is also possible to use insolation changes with shorter timescales to probe shallower 69 depths of the material. Transits of the Martian moons, which eclipse a significant frac-  InSight's rocket assisted landing has reduced the albedo locally by removal of the 82 surficial dust layer (Golombek, Warner, et al., 2020). Though some dust might have been 83 shielded behind topographic highs (see Fig.19 in Golombek, Kass, et al., 2020), the re- material (Golombek, Warner, et al., 2020). Observing the temperature response to the 88 transit therefore has the potential to characterize material that is similar to the parti-89 cle size of bulk regolith, but is known to not be cemented.
The half space below has a thermal conductivity of 50 mW m −1 K −1 , matching the di-

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urnal TI. This model fits the diurnal curve similarly well as the homogeneous model, and 201 matches the amplitude of the observed temperature drop during the transit on sol 501.

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The shape of the transit response overall is however not a good match, the minimum oc-203 curs too early and the return to pre-transit temperatures is too fast. rial with sufficient strength to support a near vertical wall with embedded clasts (Fig. 1 b).

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The depth of the hardware footprints is at least several mm so that a thickness of 4 mm 208 is adopted. The temperature drop of the transit on sol 501 is fitted by choosing a ther-  (Fig. 3 b).  -0.39 Table 1. Eclipse parameters and derived thermal conductivity assuming a 1 mm top layer.
The observed transits differ in solar elevation e sol , duration t ecl , visible wavelength dust opacity τ , observed temperature difference between start and minimum during the transit ∆Tmin, as- Text S1. The numerical calculation is based on the finite difference scheme in the work of (Kieffer, 2013), but modified to solve the equations implicitly. The finite difference equation is: where T is temperature, i indicates the i th layer, t is time, a prime indicates the previous time-step, H i+.5 indicates the heat flow through the top of the i th layer,H i−.5 same through the bottom, B is layer thickness, ρ is density, and C is specific heat capacity.
The temperatures of the top and bottom of layer interface, T i−.5 and T i+.5 respectively, are: Calculating this interface starting from the layer below and substituting for T i+.5 provides the interface heat flow (same approach for top of layer interface): At the uppermost layer (i = 1) the heat flow through the upper interface (the surface) is the boundary condition: Here a is visible albedo, H vis incident visible band heat flux e is infrared emissivity, H ir is infrared incident heat flux, and σ b is the Stefan-Boltzmann constant. The incident heat fluxes are calculated using the KRC model (Kieffer, 2013) with the atmospheric opacity derived from imaging of the sky. At the lower boundary condition at i = N the geothermal heat flow is H N +0.5 = H geo . The geothermal heat flow is here assumed to be zero, since within the range of plausible values it is not significant for the observable temperature.
The solution of the set of non-linear implicit equations 1 is found by iteratively approaching the set of N + 2 temperatures [T .5 , T 1 , T 2 , T 3 , ...T N , T N +.5 ] that are the root of the function: using Broyden's method (Press et al., 1992). To reduce number of calculations per step this equation is simplified to: In the special case of the bottom and top layer centers (i = 1, i = N ) the finite difference is evaluated at the surface and bottom interface, so half a layer thickness and constant parameters are assumed for the heat flow calculation: Thus Eq. 7 is for these special cases: Further we define ∆t = t − t and G i = F i ∆t and the following coefficients that are calculated once per model time-step based on the previous temperature state: With the notation for the sake of simplicity of implementation: T 0 = T 0.5 and T N +1 = T N +0.5 the equation for root finding is then: : X -5 The top and bottom temperatures T 0 and T N +1 are determined by the heat flow boundary conditions: Broyden's method iterates solutions to Eq. 17 and Eq. 18 to find of N + 2 temperatures T i for which G i converges to zero. Uniform 0.2 mm very low k 1 4 mm low k 1 , high k 2 Heterogeneous 1 mm low k 1 Figure S4. The diurnal surface temperatures observed within 3 sols of each transit. The error bars are total uncertainty of the radiometer which is mostly related to calibration uncertainty with only a minor contribution from atmospheric noise. Also plotted are the same models as described in the text of the main article.