#### 2.2. Energetic Particle Radiation Environment Module

[17] EPREM traces individual nodes along magnetic field lines as they are carried out with the solar wind and solves the energetic particle transport equations in the comoving (Lagrangian) field aligned grid. Each grid node propagates out with the solar wind. In each time step Δ *t*, a given node's displacement is simply Δ**x** = Δ*t***V**(**x**) where **x** is the grid location and **V**(**x**) is the solar wind velocity. A snapshot of this evolving node mesh is shown in Figure 4. On the inner boundary, grid nodes are spawned and rotated by angles consistent with the solar rotation rate at each time step. Grid nodes, like magnetic field lines, are frozen into the solar wind flow. Hence, the configuration of grid nodes naturally follows a Parker spiral field configuration.

[18] At each time step, a shell of nodes is spawned at the inner boundary, the inner boundary is rotated, and the nested shells of nodes (node shells) are advanced outward, radially away from the inner boundary. If the solar wind is uniform, the node shells lie on nested spherical surfaces. However, if the flow is nonuniform the nested surfaces of node shells become distorted (nonspherical). On each shell, the location of a given node is where a field line pierces the shell, and a list of nodes traced through nested shells follows a magnetic field line.

[19] We have also developed a special class of nodes that links up to a given observer (a location at a planet, moon or a satellite, where energetic particle distributions are projected). We refer to these as observer nodes, and the magnetic field lines attached to them as observer-connected field lines. One set of observer-connected field lines can be seen in Figure 4 looping up and around slightly above the ecliptic on the right-hand side of the image. This group of field lines connects to an observer near the position of the Ulysses spacecraft. The nodes along observer-connected field lines also migrate outward with the solar wind flow, but their angular positions are continually adjusted so that a given observer-connected field line passes through the given observer.

[20] The energetic particle solver is updated on the observer-connected field lines in the same manner as on normal evolving field lines. This presents some issues with the time histories of MHD quantities on observer-connected nodes. However, energetic particle propagation timescales are short compared to large-scale plasma evolution. The observer-connected field lines approximate the conditions of energetic particle evolution as snapshots of the plasma conditions along the field line connected to the observer. Other mechanisms using interpolation of field lines near the observer were tried, but the proximity of nearest neighbor field lines varies with time introducing artificial temporal gradients in the predictions. The introduction of observer-connected field lines allows for much more accurate time histories.

[21] The grid has been set up to handle evolving flows. Interfaces have been constructed to both corotating MHD flow solutions (K. Kozarev et al., Streaming directionality and radiation effectiveness of a solar energetic proton event as modeled with the MHD-coupled EMMREM framework, manuscript in preparation, 2009) and to a new LFM-Helio MHD model under development by V. Merkin at Boston University (V. G. Merkin et al., LFM-helio: A new global heliosphere MHD model and initial results of coupling with EPREM, manuscript in preparation, 2009). In the simulations used in this paper, we assume a uniform solar wind flow, a density that falls off with the inverse square of radial distance from the Sun, and a nominal Parker spiral field configuration.

[22] Along each field line (a connected list of nodes) we solve for particle transport, adiabatic focusing, adiabatic cooling, convection, pitch angle scattering, and stochastic acceleration according to the formalism introduced recently [*Kóta et al.*, 2005]. A slightly modified form of the focused transport equation [*Skilling*, 1971; *Ruffolo*, 1995; *Tylka*, 2001; *Ng et al.*, 2003] is used to treat transport and energy change; however the coefficients are specified so they can be computed along nodes that move with the solar wind flow:

Here _{b} is the unit vector along the magnetic field, *μ* is the cosine of the pitch angle, *n* is the solar wind density, *B* is the magnetic field strength, *p* is the ion momentum, the pitch angle diffusion coefficient is given by

where the parallel mean free path at *R*_{1} = 1 AU is *λ*_{0}, and the coefficient associated with diffusion of particle speed (or equivalently, momentum) is

Here, the ensemble averaged square of the longitudinal field variations is *η*^{2} = 〈(*B* − *B*_{0})^{2}/*B*_{0}^{2}〉, where *B*_{0} is the mean magnetic field. The coefficient *D*_{0} characterizes the rate of stochastic acceleration. For reference, an average value of *η*^{2} = 0.05 is characteristic of the variations observed by Ulysses out near 4 AU in slow solar wind. This form for stochastic acceleration term was derived by *Schwadron et al.* [1996] based on Ulysses observations of interplanetary acceleration of pickup protons.

[23] The form of the stochastic acceleration term can be easily modified within the code. In the pickup ion application described here, we take a value of *D*_{0} = 4 × 10^{−6} s^{−1}. The distribution function is averaged over gyrophase and is a function of time, position, particle momentum, and pitch angle: *f* = *f*(*t*, **x**, *p*, *μ*).

[24] The formulation of the pitch angle diffusion coefficient is also highly flexible. The expression for the pitch angle scattering term (2) is based on previous applications; however, in the future when coupled to MHD solutions, it may become convenient to express the mean free path as a function of the plasma density. For example, if the diffusion coefficient scales with the plasma density, the scattering mean free path will become smaller near enhanced density structures such as foreshocks.

[25] The advantage of the focused transport formulation in (1) is that coefficients are expressed as time derivatives in the frame of reference moving with the plasma. Therefore, in the system of nodes that move with the plasma in EPREM (Figure 4), most of the focused transport coefficients are obtained simply by differencing the state quantities (e.g., density, field strength, plasma velocity) at each node between the updated values (at time *t*) and the values at the previous time step (at time *t* − Δ*t*). Since node lists follow field lines, the field line gradients needed in the second term of (1) are easily computed. This leaves the pitch angle and energy diffusion to be solved as a matrix inversion.

[26] Cross-field diffusion and drift are also solved for within EPREM. At each time step and at each node, we update the distribution based on equation (1). The isotropic portion of the distribution function is formed from the average of the distribution over pitch angle, *f*_{0}(*t*, **x**, *p*) = 1/2*dμf*(*t*, **x**, *p*, *μ*) and is then updated at each time step in a separate routine according to the following convection-diffusion equation [*Jokipii et al.*, 1977; *Lee and Fisk*, 1981]:

where

Note that diffusion has been neglected from (4) since it is already solved for in the focused transport equation (1). In both the case of drift and perpendicular diffusion we use an explicit differencing method applied to the node shells. We identify the North, East, West and South nearest neighbors of each node. We then find the gradients associated with the nearest neighbor differences applied to the isotropic distribution. The gradients are then projected perpendicular to the field to determine the diffusion term. The drift term is solved as the dot product of the drift velocity with the gradient operator based on nearest neighbor differences of the isotropic distribution function. In this case, it is relatively straightforward to solve this gradient operator to second order. The drift velocity is solved assuming that the local structure of the field is functionally similar to the Parker spiral. The ratio of the azimuthal field to the radial field, the field strength, and polarity are used to solve for the local structure of the field (making no assumption about distance from the Sun). The curl is then applied using the functional form of the Parker magnetic spiral, and the drift velocity is solved analytically. This avoids numerical difficulties in computing the curl operator, but approximates the drift according to a Parker spiral field structure, which has no azimuthal velocity component. In practice, even with large transient disturbances, the spiral structure is largely retained (becoming overwound or underwound depending on whether the local field is compressed or rarefied) and the drift velocity calculated using this technique retains the main features of the curvature and gradient drift.

[27] The energetic particle solver is broken up into five separate steps, in which we solve the change to the distribution function at each node due to (1) adiabatic change (the term involving ∂*f*/∂ ln *p* in (1)), (2) diffusive streaming (the *vμ*_{b} · ∇*f* and ∂/∂*μ*{[*D*_{μμ}/2]∂*f*/∂*μ*} and the adiabatic focusing terms involving the ∂*f*/∂*μ* operator), (3) shell diffusion (perpendicular diffusion), (4) drift (curl and gradient drift), and (5) diffusive acceleration (involving the {1/*p*^{2}}∂/∂*p*{*p*^{2}*D*_{pp}∂*f*_{0}/∂*p*}). There is a single macro–time step in which all sub–time steps are applied and the energetic particle solution for the distribution function at each node is advanced. In each of the sub–time steps, we solve for the time change needed to maintain numerical stability and then advance by a series of these smaller sub–time steps to achieve the macro–time step. The macro–time step must be small enough so that the individual substeps remain tightly coupled. In the runs described in this paper, we have used macro–time steps on the order of tens of seconds, and the sub–time steps can be as small as 1 s intervals. The smallest sub–time steps are regulated by diffusive streaming at the highest levels in the energy grid. In practice, we apply the criterion that the sub–time steps cannot be more than an order of magnitude smaller than the macro–time steps, insuring the substeps remain tightly coupled.

[28] The energetic particle solvers have been developed to work robustly over a wide range of energies. As such, we have applied the code for both relatively low energy pickup ions (500 eV to 10 keV) and much higher energy solar energetic particle events (1 MeV to 1 GeV). For example, *Hill et al.* [2009] applied EPREM to study the evolution of pickup He^{+} and suprathermal solar wind He^{++} as a function of distance from the Sun. In this application we used a lower-energy boundary near 500 eV/nucleon, which is substantially lower than the injection energy of pickup ions. The pickup ions were injected as a ring distribution. We took an upper energy boundary at 100 keV and an energy grid composed of 100 logarithmically spaced steps. The solar wind alphas were assumed to adopt the form of the kappa distribution beneath the energy boundary. The two lowest-energy steps near 500 eV were then updated with appropriate values from the kappa function at each time step. We validated our solution method at low energies by comparing numerical to analytic solutions.

[29] An example of a validation run is shown in Figure 5 for pickup He^{+}. We show 4 time steps as the code converges to a solution. The solution is compared to two analytic functions: the *Vasyliunas and Siscoe* [1976] solution (solid curve); and the analytic solution of *Schwadron et al.* [1996]. These solutions were taken at 4 AU with identical values for the neutral interstellar densities (0.01 cm^{−3}), spatial distribution (we use an exp(−*l*/*r*) spatial distribution with *l* = 1 AU), production rate (3.5 × 10^{−7} s^{−1} at 1 AU), and stochastic acceleration rate as given above. It can be seen that the numerical model converges, and slightly exceeds the analytic model at higher energies. This slight excess is not a numerical error. The numerical model includes terms that are neglected in the analytic model. Most importantly, the numerical model includes pitch angle diffusion with a scattering mean free path of 1 AU, and therefore diffusive streaming of pickup ions. There is an outward gradient in the pickup ions, which causes a slight excess in the distribution at higher energies where the pickup ions have higher mobility and stream in from beyond 4 AU. Overall, the agreement between the analytic model and the numerical model is excellent.

[30] The validation run in Figure 5 is one example of a number of validation runs that have been performed. We have also built the code so that modules for updating the energetic particles can be interchanged easily. As such, the code itself will likely be updated and improved continually.

[31] The EPREM code was designed for a wide array of energetic particle applications using a range of energies from suprathermal ions, pickup ions, low-energy energetic particles and up to GeV cosmic rays. This flexibility is afforded by the application of robust solvers applied in the comoving reference frame, and the ability to modify the energy range, step size, and the application of boundary conditions in energy space.

#### 2.4. Galactic Cosmic Ray Doses

[35] A 3-layer version of HZETRN 2005 that incorporates Mars atmosphere shielding effects has been configured to calculate GCR dose and dose equivalent for Martian surface and atmosphere scenarios (Townsend et al., manuscript in preparation, 2009). The code has been used to develop a lookup table of daily effective dose, organ doses and dose equivalents behind thicknesses of aluminum shielding relevant to habitat or surface rover configurations anywhere on the Martian surface. The Badhwar-O'Neill GCR model for interplanetary magnetic field potentials ranging from the most highly probable solar minimum (450 MV) to solar maximum conditions (1800 MV) in the solar cycle is used as input into the calculations. This model is the standard one used for space operations at the Space Radiation Analysis Group (SRAG) at NASA Johnson Space Center. A lookup table is used because the large spread in interplanetary magnetic field conditions, large numbers of GCR ion species and their many reaction product secondary particles must be transported through as much as 500 g/cm^{2} of shield materials. Calculations of such complex spectra at such depths take approximately half a day for each possible spectrum and cannot be carried out in near real time simulations. Also, since GCR intensities change very little from day to day, daily dose and dose equivalent estimates are sufficient and are consistent with the typical time frames for SEP event exposures, thereby enabling relative comparisons to be made between them. In addition, the HZETRN 2005 code itself is export controlled, which precludes it from being publicly available, although its use to generate dose data for this project is approved and licensed by NASA Langley Research Center. HZETRN 2005 was selected for use in the project, over earlier, publicly released versions of HZETRN, because it is the most up to date and complete version available.

[36] Figure 6 shows an example of the calculation of dose rate from GCRs. We show here the dose rates in 1 g/cm^{2} of water (Figure 6, top) and 10 g/cm^{2} of water (Figure 6, bottom), which are used as proxies for skin doses and blood forming organ (BFO) doses, respectively, behind various thicknesses of Al. The Al thicknesses serve as proxies for shielding of a nominal spacesuit (0.3 g/cm^{2}), a thick spacesuit (1 g/c^{2}), nominal spacecraft shielding (5 g/cm^{2}) and thick spacecraft shielding (a storm shield; 10 g/cm^{2}). More detailed organ doses have also been solved for using HZETRN 2005 and are detailed by Townsend et al. (manuscript in preparation, 2009).

[37] The GCR dose rates are shown as a function of the modulation potential, which is the approximate energy loss of a galactic cosmic ray on its entry through the heliosphere. The modulation potential varies with the solar cycle. During solar maximum, larger interplanetary field strengths and the presence of closed magnetic flux from coronal mass ejections [*Schwadron et al.*, 2008] causes stronger GCR modulation and decreases the flux of GCRs in the inner solar system (e.g., near the Earth, Moon and Mars). Therefore the modulation potential is larger near solar maximum. We show as vertical solid lines in Figure 6 the modulation potential derived from neutron fluxes at Earth [*O'Neill*, 2006] near solar minimum in 2008 and near solar maximum in 2002. The modulation potential is derived from the modulation parameter,

where the modulation potential is Φ = ∣*Ze*∣ϕ(*r*), the integral extends from the inner boundary at radius *r* to the outer modulation boundary *R*_{b}, the solar wind speed is *V*(*x*) and κ_{1}(*x*) is related to the radial diffusion coefficient, κ. In particular, the form for κ is based on a fit to the observed spectrum over time and species [*O'Neill*, 2006]: κ = κ_{1}(*r*)*Pβ* where *P* is the rigidity in GV, *β* is the particle speed over the speed of light, κ_{1}(*r*) ∝ 1+ (*r*/*r*_{0})^{2} and *r*_{0} = 4 AU. Equation (6) is solved explicitly so that the modulation potential at a given radial distance at time, Φ(*r*, *t*), can be determined from the modulation potential inferred at *r*_{1} = 1 AU, Φ_{1}(*t*):

The vertical dashed lines then show the modulation potential near Mars (at 1.5 AU) during the periods near solar max and min. The modulation potential near Mars is only slightly lower than that near 1 AU.

[38] The doses derived in Figure 6 are substantial compared to the career effective dose limits for 1 year missions [*NASA*, 2007, Table 3]. For example, a 40 year old male and female have a career dose limits for a 1 year mission of 80 cSv and 62 cSv, respectively. However, the use of proxies generally can lead to an overestimate of expected doses. Nevertheless, GCRs represent a significant issue for long-term exposure to the space environment.