Instead of rendering the debris cloud in 3D space, this version plots every debris parcel in specific-orbital-energy space — the representation astrophysicists actually use to derive the fallback law. Each mass element at offset ξ from the star's center gets, in the "frozen-in" impulse approximation, its own energy:
ε(ξ) ≈ G·M_BH·ξ / r_p²
Because ξ is uniformly spread from −R★/2 to +R★/2, ε comes out uniformly spread too — split almost exactly 50/50 into bound (ε<0) and unbound (ε>0) halves, exactly the classic Rees (1988) result. Each parcel's own two-body Kepler orbit around the black hole (ellipse if bound, hyperbola if unbound) is solved exactly every frame — not integrated step-by-step like a naive N-body loop, so it can't accumulate numerical drift near pericenter. A bound parcel's orbital period sets its return time:
T(ε) = 2π·G·M_BH / (2|ε|)^(3/2)
Combining the (flat) mass-per-energy density with dT/dε gives the mass fallback rate analytically — no curve-fitting involved:
Ṁ(t) = M★ / (3·t_min) · (t / t_min)^(−5/3)
where t_min = T at the most-bound energy. That derivation is exactly the textbook t^−5/3 power law astronomers use to flag a real TDE. The top panel shows every parcel's live radius (log scale, vertical) against its fixed energy (horizontal) — bound parcels swing back through pericenter and vanish once accreted, unbound ones climb off the top of the chart forever. The bottom panel compares the numerically-binned return-time histogram against the analytic curve above.
- MBH — supermassive black hole mass; sets r_t and t_min.
- M★ — solar-type star mass (radius via R★ ∝ M★^0.8).
- β — r_t/r_p; below 1 the pericenter stays outside the tidal radius and the star survives intact (no disruption).