Spherical geometry
Gammaray and neutron emissions from planetary objects are a consequence of galacticcosmicray (GCR) induced nuclear spallation reactions, as well as the decay of naturally radioactive elements (e.g., ^{40} K, ^{232}Th, and ^{238}U). On spherical objects, the planetary surface is uniformly illuminated by GCRs, and the resulting gamma rays and neutrons are emitted from the surface with an angular dependence that is proportion to (cosθ_{E})^{1/2} (Lawrence et al. 2006). All geometric parameters presented in this section, including θ_{E}, are based on the formalism of Prettyman et al. (2006) and defined in Fig. 2.
The total gammaray or neutron flux at an orbiting spacecraft is a sum of all of the measured emissions from each individual location i across the viewable surface, weighted by the distance from the spacecraft to that point (D_{i}) and the emissionangle dependence of (cosθ_{E})^{1/2}. For a spherical object, the viewable surface extends from θ_{S} = 0, the subnadir point to the horizon (θ_{S}^{max}), whose angle is calculated as:
$$\theta_{S}^{\max } \left( H \right) = \arcsin \left[ {\frac{R}{{\left( {R + H} \right)}}} \right],$$
(1)
where R is the radius of the sphere and H is the altitude of the spacecraft (above the subnadir point).
Calculating the gammaray flux measured by the spacecraft begins by segmenting the surface into individual gammaray and neutronemitting facets. For a spherical object, each of these facets is at location i and has an area A_{i}, calculated as:
$$A_{i} = R^{2} d\omega_{c} d\phi ,$$
(2)
where ω_{c} is cosθ_{c}, and ϕ is the azimuthal angle (see Fig. 2). Likewise, ω_{E} is cosθ_{E}. The gammaray or neutron current J(ω_{E}) from each of these facets is calculated using radiation transport modeling tools like MCNPX (e.g., McKinney et al. 2006; Prettyman et al. 2006) and Geant4 (e.g., Peplowski 2018; Mesick et al. 2018). The number of gamma rays or neutrons emitted from facet at i is denoted j_{γ}(ω_{E}, ϕ_{E}) or j_{n}(ω_{E}, ϕ_{E}), respectively. For spherical surfaces, j_{γ}(ω_{E}, ϕ_{E}) and j_{n}(ω_{E}, ϕ_{E}) are independent of azimuthal angle and can be derived from J(ω_{E}) values as:
$$j_{\gamma } \left( {\omega_{E} ,\varphi_{E} } \right) = \frac{{J_{\gamma } \left( {\omega_{E} } \right)}}{2\pi },$$
(3)
$$j_{n} \left( {\omega_{E} ,\varphi_{E} } \right) = \frac{{J_{n} \left( {\omega_{E} } \right)}}{2\pi },$$
(4)
where J_{γ}(ω_{E}) and J_{n}(ω_{E}) are the modelprovided gammaray and neutron surface currents, respectively.
The number of gamma rays per unit area at the spacecraft (N_{γ}) can then be calculated as the sum of all of the contributions from each visible facet as:
$$N_{\gamma } = \mathop \sum \limits_{i} \left[ {\frac{{A_{i} \left( {R,\omega_{c} ,\phi } \right)J_{i} \left( {\omega_{E} } \right)}}{{2\pi D_{i}^{2} }}} \right].$$
(5)
The number of neutrons at the spacecraft can be similarly calculated, but requires corrections for neutron decay (neutron mean life is ~ 880 s) and the fact that neutrons travel on ballistic trajectories. Details of these corrections are provided by Feldman et al. (1989).
Application to irregularly shaped objects
For spherical objects like planets or large moons, the only objectspecific geometric information that is required to solve Eq. 5 is the radius R. All other information stems from the use of spherical geometry (e.g., Fig. 2) and knowledge of the spacecraft location. A number of complications arise when considering a nuclear spectroscopy measurement of an irregularly shaped object like an asteroid or a small moon. For example:

1.
The subnadir point is not necessarily the closest point to the spacecraft (e.g., Point 1 vs. Point 2; Fig. 3),

2.
Points with θ_{S} < θ_{S}^{max} are not necessarily in the field of view of the spacecraft due to intervening topography (e.g., Point 3; Fig. 3).

3.
The emission angle θ_{E} is not related to θ_{S}, as the surfacenormal vector varies for each facet (e.g., Point 4; Fig. 3),
These are just some examples of the complications associated with irregular objects, and they highlight the need for precise knowledge of the threedimensional shape of the object, like that provided by the shape models used by the SBMT.
Most of the mathematical relations detailed in the “Spherical geometry” section no longer apply for irregularly shaped objects such as Phobos and Deimos. First, the portion of the surface within the field of view of the spacecraft can no longer be calculated using Eq. 1, but instead requires raytracing between the spacecraft and each location (facet) on a threedimensional shape model of the object. Second, Eq. 2 cannot be used to determine the area of each facet within the spacecraft field of view, again because the shape cannot be described in spherical coordinates. Fortunately, shape models inherently describe the shape of the object in terms of facets, and these facets can be adopted unaltered for the nuclear spectroscopy calculations so long as the area of the facet and the direction of its surfacenormal vector are known. Finally, the surfacenormal vector of each facet, required to determine the emission angle θ_{E}, must also be determined from the shape model. These quantities are calculated by the SBMT for each facet on Phobos and used in place of the spherical geometry values to solve Eq. 5. In the next section, we examine the fidelity of the shape model needed to be applied to MEGANE applications.
We validated Eq. 5 using data collected by the GRS on the Near Earth Asteroid Rendezvous (NEAR) spacecraft. NEAR orbited the asteroid 433 Eros, whose irregular shape (~ 34.4 × 11.2 × 11.2 km) makes it an excellent test of the application of Eq. 5. Peplowski (2016) reported NEAR GRS count rate data and noted that the poor correlation between count rate and altitude (Fig. 4A) could result from Eros’ highly irregular shape. We calculated the gammaray flux at NEAR (Eq. 5) using an Eros shape model (Gaskell 2008), along with the spacecraft ephemeris (altitude, latitude, longitude) for each NEAR GRS measurement. The resulting values, which are not corrected for cosmic ray flux, detector area, or detection efficiency, are shown in relative units in Fig. 4B. The excellent linear correlation between the measured and calculated count rates, highlighted by the dashed red line (Fig. 4B), provides realworld validation of our model for calculating gammaray fluxes from shape models.
Examination of shape model resolution
An important item to examine for computing the MEGANE footprint is the resolution required for the Phobos shape model. Higher resolution shape models capture finerscale topographic features but also increase the computational time required for the calculations. Currently in the SBMT, the shape model produced by Ernst et al., submitted to Earth Planets and Space is available in four standard resolutions: 49,152 plates, 196,608 plates, 786,432 plates, and 3,145,728 plates. The MEGANE footprint on the surface of Phobos is a function of the viewable surface as described in the “Application to irregularly shaped objects” section. Thus, the area of the surface contributing to the MEGANE signal can be quite large for orbital measurements, including covering nearly a hemisphere of the body at any given time, and smallscale topographic features are expected to produce only negligible contributions to the overall MEGANE signal. Consequently, it is anticipated that the 49,152plate Phobos shape model is sufficient for MEGANE’s calculations and that the use of the higherresolution Phobos shape models in SBMT are not required, but it is worthwhile to evaluate this expectation.
The first evaluation was to examine the effect of the resolution of the shape model on the calculation of the spacecraft’s altitude. MEGANE’s measurements are highly sensitive to spacecraft altitude, as shown in the “Spherical geometry” section. As a rule of thumb, the spatial resolution of a gammaray or neutron investigation is proportional to the altitude, i.e., an altitude of 10 km yields a fullwidth at halfmaximum spatial resolution of ~ 10 km. For this evaluation, the Ernst et al., submitted to Earth Planets and Space derived shape model with 49,152 plates was used as a basis to create lower resolution, 800, 3000, and 12,000plate shape models of Phobos. A 49,000plate ellipsoid based on Phobos IAU radius values of 13.0, 11.4, and 9.1 km (Archinal et al. 2018) was also produced. These different resolution shape models were then used along with a MMX trajectory file that covers the planned operations about Phobos from 1 April to 30 May 2026 (Nakamura et al. 2021) to calculate the spacecraft altitude over this duration. Figure 5 shows the altitude calculation results, which show that using the ellipsoid that does not account for Phobos’ irregular shape results in altitude calculations that differ from when a higherfidelity Phobos shape model is used. Additionally, the Phobos shape model with only 800 plates results in altitude calculations that differ from the higherresolution shape models, demonstrating that the 800plate model is insufficient for calculating the MEGANE footprint on Phobos. The 3000 and 12,000plate shape models yield similar altitude histogram results to the 49,152plate model (Fig. 5), demonstrating a shape model resolution of 3000 plates or better is sufficient for MEGANE’s altitude calculations.
A second evaluation was to examine the implementation of the MEGANE footprint described in the “Spherical geometry” and the “Application to irregularly shaped objects” sections to different resolution Phobos shape models, and those results are shown in Fig. 6. Including the irregular shape of Phobos affects the relative contributions to the MEGANE measurement of different surface locations. Even utilizing the lowresolution, 3000plate Phobos shape model results in noticeable differences in interpreting which portions of Phobos’ surface are contributing most strongly in the 60s footprint in comparison to the ellipsoid shown in Fig. 6. The higherresolution shape models of 12,000 and 49,152 plates provide finer details of the relative contributions of different surface locations, but those differences are small in comparison to the total spatial extent of the 60s footprint. Overall, the largescale pattern of the relative contribution to a MEGANE measurement is similar when using Phobos shape models with ≥ 3000 plates. This conclusion is further discussed and supported in the “MEGANE mapping of Phobos” section with mapping Phobos’ blue unit.
These evaluations indicate that a shape model of at least 3000 plates is required for interpretation of MEGANE measurements. Thus, the 49,152plate Phobos shape model in the SBMT is more than sufficient to include the effects of Phobos topography on MEGANE’s resulting measurements.
Viewable sky considerations for Phobos
The SBMT is also valuable for examining gammaray and neutron production on Phobos, as Phobos’ irregular shape has implications for the surfaceincidence GCR rate at the surface. Local topography like crater walls can reduce the field of view to space, meaning that these locations will experience a reduced flux of GCRs and thus have lower gammaray and neutron emissions. Additionally, Mars blocks an appreciable fraction of the sky on the Marsfacing hemisphere. To quantify this effect, we used the Phobos shape model to calculate the solid angle of the viewable sky (Ω_{sky}) at each location on Phobos’ surface (Fig. 7). Here Ω_{sky} quantifies the view to open space, meaning that the portion of the sky subtended by Mars is not included.
The maximum (unobstructed) view to space for an observer on a spherical surface is 2π sr (Ω_{sky} ~ 6.28 sr). While a few locally elevated points on Phobos have an unobstructed view to space that exceeds a hemispherical view, up to 6.38 sr, we found that on Phobos, Ω_{sky} can be as low as 3.7 sr, 57% of the maximum value, within the deepest craters. As a result, the gammaray and neutron emissions from within these craters will be reduced by up to 43%. If uncorrected for, this phenomenon would be mistaken for variations in surface composition during analysis of MEGANE data.
Phobos orbits Mars with a mean orbit radius of 9,376 km (2.76 Mars radii), in a tidally locked configuration with a small libration amplitude of 1.09° (Oberst et al. 2014); the subMars point is defined as 0º longitude. Accordingly, Mars’ location in the sky is essentially fixed, and it is observable only from the subMars hemisphere. Our Ω_{sky} calculation includes the effect of Mars, which subtends 0.42 sr (6.7% of the sky) at the subMars point. The value at any given position depends on the position and local topography. The effect of Mars is visible in Fig. 7 as a hemispherical dichotomy wherein Ω_{sky} is lower on the subMars side of Phobos. The consequence is lower cosmicray exposure on the Marsfacing hemisphere, and a corresponding lower gammaray and neutron flux.
The Ω_{sky} map provides a correction to Eq. 5. As noted in the “Spherical geometry” section, the gammaray and neutron current at the surface of Phobos is calculated using radiation transport models. Those models use a spherical target object, which greatly simplifies the setup of the model and minimizes the computational time required by leveraging the spherical symmetry of the geometry. Yet this approach introduces an assumption, that Ω_{sky} is always 2π sr. Corrections to Eqs. 3 and 4 are therefore needed to account for the actual Ω_{sky} value. These corrected equations take the form:
$$j_{\gamma } \left( {\omega_{E} ,\varphi_{E} } \right) = \frac{{J_{\gamma } \left( {\omega_{E} } \right)}}{2\pi }\frac{{{\Omega }_{sky} }}{2\pi },$$
(6)
and
$$j_{n} \left( {\omega_{E} ,\varphi_{E} } \right) = \frac{{J_{n} \left( {\omega_{E} } \right)}}{2\pi }\frac{{{\Omega }_{sky} }}{2\pi }.$$
(7)
These corrections shift the computational resources associated with the irregular shape of Phobos from the radiation transport models to the SBMT, which is better suited for this challenge.