Geomagnetic Lorentz Force

The geomagnetic Lorentz force model computes the perturbative acceleration experienced by a charged spacecraft moving through Earth's magnetic field. When a spacecraft accumulates electrostatic charge (from the space plasma environment or intentional charging), it interacts with the geomagnetic field to produce a velocity-dependent force perpendicular to both the velocity and magnetic field vectors.

Physical Description

A spacecraft in Earth orbit naturally acquires surface charge through interactions with the ambient plasma environment (photoelectric emission, electron/ion collection). The resulting charge creates a Lorentz force when the spacecraft moves relative to the co-rotating geomagnetic field:

𝐅 = q(𝐯_rel × 𝐁)

where:

• q is the spacecraft charge [C]
• 𝐯_rel = 𝐯 - 𝛚×𝐫 is the velocity relative to the co-rotating magnetic field [m/s]
• 𝐁 is the geomagnetic field vector [T]
• 𝛚 is Earth's angular velocity vector

The acceleration is:

𝐚 = (q/m)(𝐯_rel × 𝐁)

Key Properties

• Perpendicular force: Always acts perpendicular to both the relative velocity and the magnetic field
• Non-dissipative: Does not add or remove energy in the rotating frame
• Charge-dependent: Linear in the charge-to-mass ratio, enabling propellantless control via active charging
• Strongest at poles: The Lorentz force effect is most pronounced in polar orbits where the magnetic field is strongest

Geomagnetic Field Models

Two geomagnetic field models are available:

IGRF (International Geomagnetic Reference Field)

The IGRF v14 model uses a spherical harmonic expansion up to degree 13 for high-fidelity geomagnetic field computation. Valid for dates between 1900 and 2035. This is the recommended model for quantitative analysis.

Simplified Dipole

A simplified dipole model that assumes Earth's magnetic field is a perfect tilted dipole. Less accurate than IGRF but computationally cheaper and sufficient for preliminary analysis where uncertainties are high.

Components

Spacecraft Charge Models

Two charge-to-mass ratio models are provided:

• FixedChargeMassRatio(q_over_m): Fixed charge-to-mass ratio [C/kg]. Use for constant-charge analysis.
• StateChargeModel(f): State-dependent model where f(u, p, t) returns q/m [C/kg]. Use for time-varying or active charge control scenarios.

MagneticFieldAstroModel

The main force model struct:

• spacecraft_charge_model: Model providing the charge-to-mass ratio q/m [C/kg]
• geomagnetic_field_model: IGRFField() (default: DipoleMagneticField()) for field computation
• eop_data: Earth Orientation Parameters for coordinate transformations
• max_degree: Maximum spherical harmonic degree for IGRF (default: 13, ignored for dipole)
• P, dP: Optional pre-allocated Legendre polynomial buffers for allocation-free IGRF evaluation

Typical Charge-to-Mass Ratios

Magnitude of Effects

For a spacecraft with q/m = 1e-3 C/kg in LEO (B ~ 30 μT, v_rel ~ 7.6 km/s):

|a| ~ (q/m) × v_rel × B ~ 1e-3 × 7600 × 3e-5 ≈ 2.3e-4 m/s² ≈ 2.3e-7 km/s²

This is a second-order perturbation for typical natural charging levels, but becomes significant for proposed active Lorentz-augmented orbit applications.

References

[1] Peck, M. A. (2005). "Prospects and Challenges for Lorentz-Augmented Orbits." AIAA Guidance, Navigation, and Control Conference. AIAA 2005-5995.

[2] Streetman, B. & Peck, M. A. (2007). "New Synchronous Orbits Using the Geomagnetic Lorentz Force." Journal of Guidance, Control, and Dynamics, 30(6), 1677-1690.

[3] Khalil, K. I. & Abdel-Aziz, Y. A. (2014). "Electromagnetic effects on the orbital motion of a charged spacecraft." Research in Astronomy and Astrophysics, 14(5), 589.