Spherical geometry
Gamma-ray and neutron emissions from planetary objects are a consequence of galactic-cosmic-ray (GCR) induced nuclear spallation reactions, as well as the decay of naturally radioactive elements (e.g., 40 K, 232Th, and 238U). 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 (cosθE)1/2 (Lawrence et al. 2006). All geometric parameters presented in this section, including θE, are based on the formalism of Prettyman et al. (2006) and defined in Fig. 2.
The total gamma-ray 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 (Di) and the emission-angle dependence of (cosθE)1/2. For a spherical object, the viewable surface extends from θS = 0, the sub-nadir point to the horizon (θSmax), 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 sub-nadir point).
Calculating the gamma-ray flux measured by the spacecraft begins by segmenting the surface into individual gamma-ray and neutron-emitting facets. For a spherical object, each of these facets is at location i and has an area Ai, calculated as:
$$A_{i} = R^{2} d\omega_{c} d\phi ,$$
(2)
where ωc is cosθc, and ϕ is the azimuthal angle (see Fig. 2). Likewise, ωE is cosθE. The gamma-ray or neutron current J(ωE) 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 jγ(ωE, ϕE) or jn(ωE, ϕE), respectively. For spherical surfaces, jγ(ωE, ϕE) and jn(ωE, ϕE) are independent of azimuthal angle and can be derived from J(ωE) 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 Jγ(ωE) and Jn(ωE) are the model-provided gamma-ray and neutron surface currents, respectively.
The number of gamma rays per unit area at the spacecraft (Nγ) 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 object-specific 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 sub-nadir point is not necessarily the closest point to the spacecraft (e.g., Point 1 vs. Point 2; Fig. 3),
-
2.
Points with θS < θSmax 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 θE is not related to θS, as the surface-normal 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 three-dimensional 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 ray-tracing between the spacecraft and each location (facet) on a three-dimensional 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 surface-normal vector are known. Finally, the surface-normal vector of each facet, required to determine the emission angle θE, 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 gamma-ray 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 real-world validation of our model for calculating gamma-ray 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 finer-scale 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 small-scale topographic features are expected to produce only negligible contributions to the overall MEGANE signal. Consequently, it is anticipated that the 49,152-plate Phobos shape model is sufficient for MEGANE’s calculations and that the use of the higher-resolution 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 gamma-ray or neutron investigation is proportional to the altitude, i.e., an altitude of 10 km yields a full-width at half-maximum 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 12,000-plate shape models of Phobos. A 49,000-plate 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 higher-fidelity Phobos shape model is used. Additionally, the Phobos shape model with only 800 plates results in altitude calculations that differ from the higher-resolution shape models, demonstrating that the 800-plate model is insufficient for calculating the MEGANE footprint on Phobos. The 3000- and 12,000-plate shape models yield similar altitude histogram results to the 49,152-plate 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 low-resolution, 3000-plate Phobos shape model results in noticeable differences in interpreting which portions of Phobos’ surface are contributing most strongly in the 60-s footprint in comparison to the ellipsoid shown in Fig. 6. The higher-resolution 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 60-s footprint. Overall, the large-scale 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 49,152-plate 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 gamma-ray and neutron production on Phobos, as Phobos’ irregular shape has implications for the surface-incidence 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 gamma-ray and neutron emissions. Additionally, Mars blocks an appreciable fraction of the sky on the Mars-facing hemisphere. To quantify this effect, we used the Phobos shape model to calculate the solid angle of the viewable sky (Ωsky) at each location on Phobos’ surface (Fig. 7). Here Ωsky 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 (Ωsky ~ 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, Ωsky can be as low as 3.7 sr, 57% of the maximum value, within the deepest craters. As a result, the gamma-ray 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 sub-Mars point is defined as 0º longitude. Accordingly, Mars’ location in the sky is essentially fixed, and it is observable only from the sub-Mars hemisphere. Our Ωsky calculation includes the effect of Mars, which subtends 0.42 sr (6.7% of the sky) at the sub-Mars 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 Ωsky is lower on the sub-Mars side of Phobos. The consequence is lower cosmic-ray exposure on the Mars-facing hemisphere, and a corresponding lower gamma-ray and neutron flux.
The Ωsky map provides a correction to Eq. 5. As noted in the “Spherical geometry” section, the gamma-ray 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 Ωsky is always 2π sr. Corrections to Eqs. 3 and 4 are therefore needed to account for the actual Ωsky 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.