Prospects and Challenges for Lorentz-Augmented OrbitsorElectrodynamic Propulsion without a Tether:

Prospects and Challenges for Lorentz-Augmented Orbits or Electrodynamic Propulsion without a Tether Mason A. Peck
Cornell University
Sibley School of Mechanical and Aerospace Engineering

The lorentz force accelerates charged particles traveling in a magnetic field. Can it be used to control the motion of a spacecraft?
For example, Burns, Schaeffer, et al.; Cassini, Voyager data
Lorentz resonances determine structures in the rings of Jupiter and Saturn
Micron-size particles
A few volts of potential
It's not an electrodynamic tether
Current in a tether interacts with the geomagnetic field: J×B
Electrons traveling at cm/sec through a conductor
This spacecraft's charge is a current (high charge at high speed): qv×B The Idea: a Lorentz-Augmented Orbit (LAO) 17 e- q(t)

Dynamics:

Dynamics Electrodynamics in a rotating frame
Lorentz Force, as you’ve seen it before:
B rotates with the planet
Rotating frame E; inertial frame N
Position and angular velocity of E w.r.t N
Classically, orbital velocity is inertial:
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This distinction matters because the rotating B acts as an electric field in N (the "co-rotational" field).

Dynamics:

Dynamics Electrodynamics in a rotating frame
We're interested in the case of E=0
Debye shielding makes E from neighboring bodies (e.g. spacecraft) negligible--more later.
The co-rotational field in the rotating frame is zero
Time-varying B (due to solar wind) causes an E which may matter and will be addressed in future work
The Lorentz force becomes 15

Dynamics:

Dynamics "Magnetic fields do no work"
Since , the Hamiltonian in the rotating
frame is constant:
Integrate between arbitrary t1 and t2:
Hamiltonian in is constant in E… 14 r,

Dynamics:

Dynamics …but an LAO's energy is not constant in N
Consider an equatorial elliptical orbit, 13 An LAO steals a little energy from a planet's rotation, like a flyby steals some from its orbit

Dynamics:

Dynamics Sometimes you get nothing
Line of apsides aligned with magnetic poles:
Geostationary satellite with
time-invariant B in E
In fact, non-dipolar
terms in B complicate
these simple results.
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Designing LAO-capable Spacecraft:

Designing LAO-capable Spacecraft Important Questions
How much charge do you need?
How much power does it take?
How big is the spacecraft?
How is charge established and maintained?
What is the impact of this charge on spacecraft subsystems?
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Designing LAO-capable Spacecraft:

Designing LAO-capable Spacecraft How much charge do you need?
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Designing LAO-capable Spacecraft:

Designing LAO-capable Spacecraft How much power does it take?
If it weren't for ionospheric plasma, no power--all that would be required is to set the charge and forget it.
However, discharge into the plasma means a constant current is necessary to maintain an arbitrary potential.
From SPEAR I, assuming , 0.06 W/V at 200-350 km altitude. Much less at higher altitudes (0.001 - 0.0001 W/V at MEO)
The equivalent RC circuit has a time constant of about 1.4 seconds.
Consider charge & discharge cycles (rather than maintaining constant charge) in resonance with the orbit
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Designing LAO-capable Spacecraft:

Designing LAO-capable Spacecraft How big is the spacecraft?
The key sizing parameter is charge per unit mass (q/m), which is proportional to the acceleration (Dv) available
Charge depends on capacitance: q=CV (although capacitors per se are not useful)
For a spherical spacecraft surrounded by plasma,
8 Where is the Debye length, the thickness of an oppositely charged sheath that surrounds the charged body.
+ - Sheath thickness depends on plasma temperature (i.e. altitude)

Designing LAO-capable Spacecraft:

Designing LAO-capable Spacecraft How big is the spacecraft?
To maximize q/m, we seek a conductor on which high charge can reside with minimal discharge, but we want low mass.
Material stress from the charge will pressurize a sphere used as a conductor. The material's tensile strength therefore determines the Dv performance:
7 Cubesat Example
high-strength polypropylene balloon, 10m radius
< 1 mil thick => 0.6 kg
q/m=0.1 C/kg (1 MV charge) for a 1 kg cubesat. 0.6 kg

Designing LAO-capable Spacecraft:

Designing LAO-capable Spacecraft How is charge established and maintained?
Natural charging (due to plasma interactions and/or photoelectric effect) can't offer more than a few kV.
Requires hundreds of kV
Emit ions or electrons via a plasma contactor
Electron emission is a little easier and lighter, and it requires no propellant
Overcome discharge into the plasma
Power required depends on altitude, area, space weather…
Probably a few thousand Watts for a system of interest 6 e- q(t) 400 kV Van de Graaff Generator L'Garde Conductive Sphere

Designing LAO-capable Spacecraft:

Designing LAO-capable Spacecraft What is the impact on other subsystems?
Structural and Mechanical Requirements for the Sphere:
Conductive
Acts as a Faraday cage, shielding components from differential charging
Transparent for solar power (unless nuclear power is possible)
Deployable (note that the charge inflates it)
Is there a better way?
Payload Options
Has to work through a conductive shell
Maybe off until the spacecraft is in its operational orbit
T&C Options
Lasercomm through the sphere
Antenna protrudes through sphere (ESD issues)
Attitude Control
Little direct impact (the Lorentz force is independent of attitude)
Differential charge acts like a gravity-gradient effect, offering a means of attitude control (that's another paper…)
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Some Applications:

Some Applications Rendezvous
The potential function for an LAO alters Kepler's equation.
There are two solutions for the mean motion, depending on the sign of q and the orientation of B.
Charge one spacecraft, or each of a pair, and one will catch up to the other at the same altitude.
Retrograde orbits catch up faster because the velocity in E is greater. 4 Time to Rendezvous (years) in Circular Orbit;
q/m=0.001 C/kg

Some Applications:

Some Applications 3 Vertical Spacing for Circular Prograde Orbits: q/m=0.001 C/kg LAO Formations
LAO spacecraft in a formation do not have to interact through Coulomb forces
Most cannot because of the Debye sheath
Spacecraft with different electrical potentials and orbital altitudes can orbit with the same period.

Some Applications:

Some Applications 2 Earth Escape
It takes about 1 year for a q/m=5 C/kg spacecraft to escape earth orbit, with appropriate phasing of charge with true anomaly.
This level of charge is out of reach for the spherical-shell architecture, but its prospect might inspire other technical solutions.

Some Applications:

Some Applications 1 Jupiter Capture
Jupiter's magnetic field is about 20,000 times more powerful than earth's.
Its faster rotation (once every 9 hours) means that the co-rotational field can contribute energy to an LAO quickly.
Altitude above Jupiter (Rj=71,492 km) during 472 Day Orbit Insertion
For q/m=0.01 C/kg, a spacecraft can transition from a parabolic orbit at Jupiter to the orbit of Ganymede in a little over a year. Time, seconds Altitude, km

Future Directions:

Future Directions 0 Though difficult to achieve, an LAO seems feasible
q/m=0.1 C/kg is an optimistic result for a small spacecraft
q/m=0.01 - 0.001 C/kg can lead to useful maneuvers
Perform more detailed hardware trade studies
The spherical conductive shell concept solves many problems, including ESD, but it does not efficiently store charge.
Derive general, optimal orbit control (probably bang-bang)
Fuse LAO, Coulomb-based formation flying, and EM formation flying?

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