Solar Radiation Pressure

Solar Radiation Pressure (SRP) represents the momentum transfer from solar photons to a spacecraft's surface. When sunlight strikes a spacecraft, photons impart momentum that creates a small but persistent force. For spacecraft with large surface areas relative to their mass, or for missions requiring high precision, SRP can be a significant perturbing force.

Physical Description

Solar radiation pressure arises when electromagnetic radiation from the Sun interacts with spacecraft surfaces. The interaction depends on the surface material properties:

• Absorption: Photons are absorbed, transferring all momentum to the surface
• Specular reflection: Photons reflect like a mirror, reversing momentum direction
• Diffuse reflection: Photons scatter in all directions from the surface

The magnitude of SRP force depends on:

• Solar flux: Intensity of solar radiation at the spacecraft's location
• Cross-sectional area: Effective area exposed to solar radiation
• Surface properties: Reflectivity, absorptivity, and specularity coefficients
• Sun-spacecraft distance: Follows inverse square law with distance

The basic SRP acceleration is given by:

a_srp = -(Ψ/c) * (A/m) * C_R * (s/|s|)

Where:

• Ψ is solar flux at spacecraft distance
• c is speed of light
• A is cross-sectional area perpendicular to Sun direction
• m is spacecraft mass
• C_R is radiation pressure coefficient (1 ≤ C_R ≤ 2)
• s is unit vector from spacecraft to Sun

Shadow Effects

Spacecraft experience reduced or eliminated SRP when in shadow:

Eclipse Types

• Umbra: Complete shadow (no direct sunlight)
• Penumbra: Partial shadow (partially blocked sunlight)
• No shadow: Full illumination

Shadow Models

AstroForceModels implements three shadow models for eclipse calculations:

Conical Shadow Model: Sophisticated model accounting for Sun's finite size

• Models both umbra (complete shadow) and penumbra (partial shadow) effects
• Accounts for angular sizes of both Sun and Earth as seen from spacecraft
• Based on Montenbruck & Gill algorithm with geometric shadow calculations
• Most accurate shadow modeling available
• Computationally efficient despite sophistication

Cylindrical Shadow Model: Simplified model assuming parallel solar rays

• Sharp transition between sunlight (factor = 1.0) and complete shadow (factor = 0.0)
• Assumes Sun rays are parallel (infinite distance approximation)
• Faster computation than conical model
• Suitable for missions where high shadow accuracy isn't critical

SmoothedConical Shadow Model (Aziz et al.): Differentiable sigmoid-based model

• Uses a logistic sigmoid to smoothly transition between sunlight and shadow
• Fully compatible with automatic differentiation (no branching on shadow state)
• Parameterized by sharpness coefficient cs (default: 289.78) and transition coefficient ct (default: 1.0)
• Ideal for gradient-based optimization (e.g., Q-Law, trajectory optimization)
• Default parameters calibrated for geocentric orbits

NoShadow Model: Disables all eclipse effects

• Always returns shadow factor = 1.0 (full sunlight)
• Eliminates computational overhead of shadow calculations
• Useful for high-altitude missions where eclipses are rare
• Essential for comparative studies and debugging

Performance Comparison

*Perfect accuracy for missions without eclipses

Components

The SRP force model in AstroForceModels includes several components:

SRPAstroModel

The main struct that encapsulates SRP parameters:

• satellitesrpmodel: Spacecraft optical and geometric properties
• sun_data: Solar position calculation methods
• eop_data: Earth orientation parameters
• shadow_model: Eclipse/shadow calculation method
• R_Sun: Solar radius (default: 695,700 km)
• R_Occulting: Occulting body radius (default: Earth radius)
• Ψ: Solar flux constant (default: 1361 W/m²)
• AU: Astronomical unit for distance scaling

Satellite Shape Models

AstroForceModels provides two satellite shape models for SRP calculations:

CannonballFixedSRP: Spherical spacecraft model with fixed reflectivity coefficient

• Assumes spacecraft is a perfect sphere with radius r
• Cross-sectional area: A = π × r²
• Fixed reflectivity coefficient throughout mission
• Reflectivity ballistic coefficient: RC = C_R × A / m
• Most common model for preliminary analysis and small satellites

Surface Optical Properties

Key parameters defining surface interactions:

• Absorptivity (α): Fraction of incident energy absorbed
• Specularity (ρ_s): Fraction of reflected energy that reflects specularly
• Diffusivity (ρ_d): Fraction of reflected energy that reflects diffusely
• Emissivity (ε): Ability to emit thermal radiation

Relationship: α + ρs + ρd = 1

Usage Example

using AstroForceModels
using SatelliteToolboxCelestialBodies

# Define spacecraft optical properties using fixed reflectivity coefficient
satellite_model = CannonballFixedSRP(
    radius = 0.89,     # m (equivalent to 2.5 m² area)
    mass = 150.0,      # kg
    reflectivity_coeff = 1.3  # radiation pressure coefficient
)

# Load Earth orientation parameters  
eop_data = fetch_iers_eop()

# Create Sun position model
sun_model = ThirdBodyModel(; body=SunBody(), eop_data=eop_data)

# Create SRP models with different shadow models

# High-precision model with penumbra effects
srp_model_conical = SRPAstroModel(
    satellite_srp_model = satellite_model,
    sun_data = sun_model,
    eop_data = eop_data,
    shadow_model = Conical(),    # Most accurate shadow model
)

# Compute acceleration (typically called within integrator)
acceleration(state, parameters, time, srp_model_conical)

Implementation Details

The SRP acceleration computation involves:

1. Solar Position: Computing Sun's position relative to spacecraft
2. Shadow Check: Determining eclipse status using shadow model
3. Solar Flux: Calculating actual solar flux at spacecraft distance
4. Surface Interactions: Computing force based on optical properties
5. Area Calculation: Determining effective cross-sectional area
6. Force Vector: Computing acceleration vector in appropriate coordinate frame

Radiation Pressure Coefficient

The radiation pressure coefficient C_R characterizes overall surface interaction:

• C_R = 1: Perfect absorption (black body)
• 1 < C_R < 2: Mixed absorption and reflection (typical spacecraft)
• C_R = 2: Perfect specular reflection (ideal mirror)

Common values:

• Solar panels: C_R ≈ 1.3-1.5
• Spacecraft bodies: C_R ≈ 1.2-1.4
• Thermal blankets: C_R ≈ 1.1-1.3

Typical Magnitudes

SRP acceleration varies significantly with mission characteristics:

• Large GEO satellites: ~10⁻⁸ to 10⁻⁷ m/s²
• Small satellites/CubeSats: ~10⁻⁶ to 10⁻⁵ m/s²
• Solar sails: ~10⁻⁴ to 10⁻³ m/s²
• GPS satellites: ~10⁻⁸ m/s²

For comparison, other perturbations:

• Earth gravity (LEO): ~10⁻¹ m/s²
• Atmospheric drag (LEO): ~10⁻⁶ to 10⁻⁴ m/s²
• Third-body gravity: ~10⁻⁶ to 10⁻⁵ m/s²

References

[1] Vallado, David A. "Fundamentals of Astrodynamics and Applications." 4th ed., 2013. [2] Montenbruck, Oliver, and Eberhard Gill. "Satellite Orbits: Models, Methods and Applications." Springer, 2000. [3] AI Solutions. "Solar Radiation Pressure." FreeFlyer User Guide. [4] Milani, A., et al. "Non-gravitational perturbations and satellite geodesy." 1987. [5] Knocke, P. C., et al. "Earth radiation pressure effects on satellites." 1988.