Physical Model

Overview

AsteroidThermoPhysicalModels.jl is a comprehensive toolkit for thermophysical modeling of asteroids. This package allows you to simulate the temperature distribution on an asteroid, and predict the non-gravitational force (i.e., Yarkovsky and YORP effects).

The thermophysical model (TPM) considers the following physical processes:

1. Heat Conduction: Solves a one-dimensional heat conduction equation to model heat transfer from the surface into the interior of the asteroid.
2. Self-Shadowing: Accounts for local shadows cast by topography.
3. Self-Heating: Considers re-absorption of scattered light and thermal radiation from surrounding surfaces.
4. Mutual Shadowing: For a binary asteroid, accounts for eclipses between the primary and secondary bodies.
5. Mutual Heating: For a binary asteroid, considers thermal exchange between the primary and secondary bodies.

Heat Conduction Equation

Heat conduction within the asteroid is modeled by the following one-dimensional heat diffusion equation:

\[\rho C_p \frac{\partial T}{\partial t} = k \frac{\partial^2 T}{\partial z^2}\]

where:

• $T(z,t)$ is the temperature at depth $z$ and time $t$
• $\rho$ is the density
• $C_p$ is the specific heat capacity at constant pressure
• $k$ is the thermal conductivity

Numerical Solvers

AsteroidThermoPhysicalModels.jl provides three numerical methods to solve the heat conduction equation:

1. Explicit Euler Method (ExplicitEulerSolver)

   • Forward difference in time
   • Conditionally stable: requires $\lambda = \alpha \Delta t / \Delta z^2 < 0.5$
   • First-order accurate in time
   • Fast for small time steps

2. Implicit Euler Method (ImplicitEulerSolver)

   • Backward difference in time
   • Unconditionally stable for any time step
   • First-order accurate in time
   • Requires solving a tridiagonal system

3. Crank-Nicolson Method (CrankNicolsonSolver)

   • Average of forward and backward differences
   • Unconditionally stable
   • Second-order accurate in both time and space
   • Best balance of accuracy and stability

The solver can be specified when creating the thermophysical model:

# Example: Using Crank-Nicolson solver
stpm = SingleAsteroidTPM(shape, thermo_params;
   SELF_SHADOWING = true,
   SELF_HEATING   = true,
   SOLVER         = AsteroidThermoPhysicalModels.CrankNicolsonSolver(thermo_params),
   BC_UPPER       = AsteroidThermoPhysicalModels.RadiationBoundaryCondition(),
   BC_LOWER       = AsteroidThermoPhysicalModels.InsulationBoundaryCondition(),
)

Boundary Conditions

Upper Boundary Condition (Surface)

At the surface, a radiative equilibrium boundary condition is applied:

\[-k \frac{\partial T}{\partial z}\bigg|_{z=0} = (1-R_\text{vis})(F_\text{sun} + F_\text{scat}) + (1-R_\text{ir})F_\text{rad} - \varepsilon \sigma T^4\]

where:

• The left side represents the heat flux from the surface into the interior
• The right side represents the energy balance at the surface (absorbed solar radiation, scattered light, and thermal radiation minus emitted thermal radiation)

Lower Boundary Condition

At the lower boundary, an insulation condition is typically applied:

\[\frac{\partial T}{\partial z}\bigg|_{z=z_\text{max}} = 0\]

Thermal Inertia

Thermal inertia is a physical quantity that represents the ability of a material to resist temperature changes, defined by:

\[\Gamma = \sqrt{k \rho C_p}\]

The unit is tiu (thermal inertia unit) or J·m⁻²·K⁻¹·s⁻¹/².

Thermal Skin Depth

The thermal skin depth is a characteristic length that represents how far a periodic thermal wave penetrates into a material:

\[l = \sqrt{\frac{4\pi P k}{\rho C_p}}\]

where $P$ is the period of the thermal cycle (typically the rotation period of the asteroid).

Non-Gravitational Effects

Yarkovsky Effect

The Yarkovsky effect is an orbital perturbation caused by the asymmetric thermal emission resulting from the day-night temperature difference due to the asteroid's rotation. This effect primarily affects the semi-major axis of the asteroid's orbit.

YORP Effect

The YORP effect (Yarkovsky-O'Keefe-Radzievskii-Paddack effect) is a rotational perturbation resulting from thermal emission due to the asymmetric shape of the asteroid. This effect influences the rotation rate and the orientation of the asteroid's spin axis.

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