Terminal velocity. Weight balances drag when m g = ½ ρ v2 CdA, so vt = √(2 m g / (ρ CdA)). The vehicle is a cylinder of diameter D and length L. Nose-on the drag area is Cd,a·πD2/4; broadside (belly-first) it is Cd,c·D·L. Their ratio is (4/π)(L/D)(Cd,c/Cd,a), about 10 here, so belly-first vt is about 3 times lower – that is why the manoeuvre exists: the belly stance is a huge air brake that costs no propellant.
Altitude. vt ∝ 1/√ρ. With the ISA atmosphere (ρ = 1.225 kg/m3 at sea level, 0.414 at 10 km, 0.0889 at 20 km) the vehicle first falls far faster than vt, is slowed by thickening air, and settles on the local vt(h) curve (see the "Speed vs vt(h)" chart). Note q = ½ρv2 = m g / CdA on that curve: about 2.4 kPa, independent of altitude.
Force model. Each body axis uses its own quadratic drag: Faxial = −½ρ Cd,aAa va|va|, Fnormal = −½ρ Cd,cDL vn|vn| (cross-flow principle), with va, vn the air-relative velocity components along and across the body. At angle b to the flow the resultant area is AR(b) = √((Cd,aAacos2b)2 + (Cd,cDL sin2b)2). The cross-flow force acts about 1 m behind the centre of mass, which makes the broadside attitude statically unstable: without control the vehicle tumbles. A mild transonic drag bump is included.
Attitude control. A PD law on the pitch angle (ωn = 0.6 rad/s, ζ = 0.9) holds the body about 10° off exactly broadside to the flow. Torque is allocated first to the four control flaps (torque = q Sf l Cn,max sin 2δ, so it scales with dynamic pressure and the fin authority slider limits δ), then to always-on thrusters (0.4 MN m). Flaps also add drag area when deflected, so they trim speed as well as attitude. At high altitude q is tiny and only thrusters work.
The flip. At the flip altitude a time-optimal-style rotation to vertical starts: θ̇ follows √(2 αmax |e|) with αmax = (τ + τfin)/I, capped at 0.6 rad/s. I = m(3r2 + L2)/12 ≈ 2.8×107 kg m2. A bang-bang rotation of angle Δθ takes t = 2√(Δθ I/τ). While the body turns the drag area collapses from belly to nose-on, so the vehicle speeds up: the altitude spent is the price of the flip. Torque comes from engine gimbal plus thrusters.
Landing burn. After the flip the engine (thrust/weight 2.4) lights at the last moment and commands the constant deceleration a = v2/(2h) (plus g). Touchdown is soft if vertical speed ≤ 4 m/s, sideways ≤ 3 m/s and tilt ≤ 8°. If the flip is late the required deceleration exceeds what the engine can give and the sim reports why. See the landing-burn lab for the burn itself.
Heating glow. The windward glow uses a Sutton–Graves-style index ∝ √ρ v3 mapped to brightness; it is a visual cue, not a thermal model.
Model and assumptions. 2-D vertical plane, RK4 at 1/120 s, constant mass 130 t (no propellant burn), flat Earth, ISA atmosphere extended isothermally above 85 km, horizontal wind profile peaking near 9 km, thrust along the body axis (gimbal enters only as torque), no aerodynamic pitch damping, no heat-shield physics. Experiment 1 holds a perfect attitude to the flow; the closed-form estimate in Experiment 2 is deliberately crude (it ignores in-flip drag history and burn-time drag).
Reference ideas: terminal velocity and cross-flow drag of cylinders (Hoerner, Fluid-Dynamic Drag, 1965), ISA (ISO 2533), scaling of re-entry heating (Sutton & Graves, 1971). This is a generic, civil/educational vehicle – all dimensions, masses, coefficients and times are illustrative, not data of any real vehicle.