Drag. Acceleration a = -1/2 rho B v_rel^2 with ballistic coefficient B = Cd A / m (m^2/kg) and v_rel the speed relative to the rotating air (the atmosphere co-rotates with Earth, omega x r). Heavy, compact objects (low B) last longer; light, large ones (high B) decay fast.
Decay per orbit. For a near-circular orbit, King-Hele gives da/dt = -rho B sqrt(mu a), i.e. Δa = -2π B rho a^2 per revolution (no rotation term). Drag lowers the orbit but speeds the satellite up (it falls deeper into the gravity well): the "satellite drag paradox".
Atmosphere. Density table (typical standard-model values, quoted from memory: spot-check before citing), interpolated log-linearly between nodes, i.e. locally rho ~ exp(-h/H) with a scale height H that grows with altitude. Solar activity multiplies thermosphere density above ~120 km (x0.35 quiet, x1 moderate, x3.2 active: illustrative; the real swing is up to about a factor of 10 at 400 km and depends on solar flux and geomagnetic activity). Day/night and latitude variation are not modelled.
Re-entry heating. Stagnation heat flux (Sutton-Graves) q = 1.7415e-4 sqrt(rho/R_n) v^3 W/m^2 with nose radius R_n = sqrt(A/pi). Lumped body: m_h c dT/dt = q A - 4 A eps sigma (T^4 - T0^4); at the melting point the latent heat m_h L_f must be absorbed before the part counts as demised. Simplifications: one uniform temperature, no fragmentation, no ablation of coatings, fixed attitude, no real structure (hollow thin walls heat much faster than a solid lump, so the "heated mass fraction" is an illustrative knob). Material properties are typical handbook values. This is a teaching model, not a design or risk tool.
Retro-burn. Vis-viva v^2 = mu (2/r - 1/a); lowering the perigee of a circular orbit r to r_p needs Δv = sqrt(mu/r) - sqrt(mu(2/r - 1/a)), a = (r + r_p)/2. The burn is treated as instantaneous.
Other simplifications: equatorial orbit, spherical Earth (no J2), constant Cd and mean area (a tumbling satellite has a mean area), no third bodies or radiation pressure. Integrator: RK4, step at most 20 s in orbit and 0.1 s in the entry phase. Educational orbital mechanics only. Models built in Blender; the Earth texture is procedural, not a photograph.