Gravity-Gradient Satellite Libration Lab (2D)

Presets

Live readouts

Graphs

Pitch angle from the local vertical vs time

Phase portrait: pitch vs pitch rate

Gravity-gradient potential well

Numerical checks

running...

How to use

Start with Free libration 40 deg: the boom swings about the local vertical with period T_orbit/√(3k). Increase the boom length or tip mass and k rises towards 1.
Damped to vertical adds hysteresis damping; the pitch graph decays, the phase portrait spirals in.
Near the separatrix and Tumbling show the boundary in the phase portrait (dashed orange curve); the potential graph shows the two wells.
Eccentric orbit forcing shows the pumped oscillation at the orbital frequency; Disturbance offset shifts the equilibrium.
The buttons swap the main view between the rotating orbit frame and the inertial orbit view. Space pauses, R resets.

The science

A body in orbit feels slightly different gravity at different distances from the planet. For a satellite with principal moments of inertia I_x, I_y, I_z this tidal field exerts a torque. In the orbital plane, with θ the angle of the body x axis (the boom) from the local vertical and n the orbital rate, the pitch equation is

I_y (θ¨ + ν¨) = −(3μ/2r³)(I_z − I_x) sin 2θ ⇒ θ¨ = −(3/2) n² (a/r)³ k sin 2θ − ν¨, k = (I_z − I_x)/I_y

The axis of least inertia (the boom) is stable along the vertical. On a circular orbit (ν¨ = 0) the equation is a pendulum in ψ = 2θ with ω₀² = 3n²k, so the small-angle libration frequency is √(3k) n, √3 n for a thin dumbbell (k = 1). The libration period for amplitude θ₀ is exactly 4K(sin θ₀)/ω₀, with K the complete elliptic integral. The energy ½θ̇² − (3/4) n² k cos 2θ is conserved; the separatrix θ̇ = ±√(3k) n |cos θ| splits libration from tumbling; there is a second stable equilibrium with the boom pointing straight down (θ = 180°).

Eccentric orbits. The orbital angular rate changes along the orbit, ν¨ ≈ −2e n² sin M, and drives a forced libration of amplitude about 2e/(3k − 1) (it is resonant when k = 1/3). A real boom satellite also needs a damper (hysteresis rods or a magnetic or viscous damper) to dissipate the libration energy; here that is a damping ratio ζ on the libration, and a constant disturbance torque (solar pressure, residual magnetic moment) shifts the equilibrium.

Simplifications. Planar pitch motion only (the out-of-plane roll and yaw motions are not simulated), rigid body, point-mass tips, symmetric boom, Earth treated as a point mass for the torque (no J2 or magnetic field), one linear damper. The Earth map is a procedural stylised texture. In-page checks: the √3 frequency of a dumbbell; libration periods at 30, 60 and 80 degrees against the elliptic-integral solution; conservation of the libration energy; the separatrix (release at 85 deg stays bounded, at 95 deg it librates about the inverted vertical, a kick above the separatrix tumbles); and the forced-libration amplitude on an eccentric orbit.