Shard Orbit (2D): debris cloud after a satellite break-up

1. LEO collision: watch the X-shaped Gabbard diagram form, then the cloud shear into a ring (graph 4).
2. Raise Typical fragment kick to 2.8 km/s at 600 km: the fast fragments exceed 10.9 km/s and leave (graph 3 shows them above the red escape curve).
3. Break up at perigee vs apogee of a GTO orbit (true anomaly 0 or 180) and compare the period spread.
4. Switch drag off and on at 330 km with 3,000x warp: low-perigee fragments re-enter first.
5. Turn J2 off: the node stops precessing and the cloud stays in one plane.

What is simulated

A satellite on a Keplerian orbit breaks up (an accidental collision or the explosion of a spent upper stage). Every fragment receives a small velocity change dv in a random direction; each one then follows its own orbit around the Earth. Orbits are integrated numerically with a 4th-order Runge-Kutta step (adaptive, about 1.2 % of the local dynamical time) in an Earth-centred inertial frame, in km and seconds. The 2D page and the 3D page run exactly the same engine.

Governing equations

Two-body gravity: a = -mu r / |r|^3, mu = 398600.4418 km^3/s^2. Vis-viva: v^2 = mu (2/r - 1/a). Period: T = 2 pi sqrt(a^3/mu). Escape speed: v_esc = sqrt(2 mu / r) = sqrt(2) v_circ; a fragment is unbound when its specific energy eps = v^2/2 - mu/r is positive.

Oblateness (J2 = 1.08263e-3): a_J2 = (3/2) J2 mu R^2 / r^4 [ (x/r)(5z^2/r^2 - 1), (y/r)(5z^2/r^2 - 1), (z/r)(5z^2/r^2 - 3) ]. It makes the orbital plane precess at dOmega/dt = -(3/2) J2 n (R/p)^2 cos i, which is why a compact cloud smears into a shell (graph 4).

Drag: a = -(1/2) rho B v_rel^2 v_rel/|v_rel| with B = Cd A / m (m2/kg) and the velocity relative to the co-rotating atmosphere. Density comes from a piecewise exponential table (moderate solar activity). A circular orbit loses da/rev = -2 pi B rho a^2. Light, flat fragments (large B) fall first.

Gabbard diagram

Plotting each fragment's apogee and perigee altitude against its period gives the classic X-shaped Gabbard diagram: a fragment pushed forward gets a higher apogee and a longer period, one pushed backward a lower perigee and a shorter period; the two branches cross at the parent's period. Fragments whose perigee falls below 100 km are removed as re-entered.

Break-up model and honest simplifications

• Kicks are isotropic with a log-normal magnitude whose median is the slider value; lighter (larger B) fragments get slightly larger kicks, following the trend of the NASA standard break-up model. Equal-mass fragments, mean kick removed so the centre of mass keeps the parent velocity. Real clouds also have a mass distribution and non-isotropic ejection.
• Gravity is Earth-only: no Moon, Sun, solar-radiation pressure or harmonics above J2. Fragments do not collide with each other.
• Atmosphere: static spherical exponential model, altitude measured from the equatorial radius, no solar-cycle variation. Re-entry is declared at 100 km. Escape is declared when eps > 0 beyond 120,000 km.
• Satellite and fragment sizes are not to scale; they are drawn at a constant on-screen size.

Numerical checks run in this page

Five independent checks run at load (below the controls): the Kepler period closes the orbit, energy is conserved over 10 revolutions of an eccentric orbit, the closed-form apoapsis of a radial launch (and v_esc = sqrt(2) v_circ), the J2 nodal regression rate, and the drag decay per revolution.

This is an educational orbital-mechanics model of generic debris; it is not a tracking, prediction or targeting tool.