Teaching model. Hover only. Textbook-level equations with honest simplifications; not a design tool. The vehicle is a generic coaxial rotorcraft of Ingenuity-like size (public mission figures: about 1.8 kg, 1.2 m rotor span, roughly 2400-2900 rpm on Mars). It carries no agency marks.
Momentum (actuator-disc) theory. A rotor of disc area A giving thrust T accelerates air to an induced velocity vi at the disc:
vi = sqrt( T / (2 rho A) ), Pideal = T vi = T1.5 / sqrt(2 rho A)
Real rotors need more: the figure of merit FM = Pideal / Pactual is 0.6-0.7 for good full-scale rotors and lower at low Reynolds number. For two equal coaxial rotors the ideal power of the pair, each carrying T/2, is multiplied by an interference factor kappa (about 1.16-1.28 in the literature; 1.2 is used here).
Thrust coefficient. T = CT rho A (Omega R)2. With density 1 % of Earth's, the same thrust needs about ten times the tip speed, hence the high rpm.
Blade-element momentum theory (BEMT). The blade is cut into 18 radial stations. At each, local inflow ratio lambda is found by equating blade-element thrust and momentum thrust (with Prandtl tip loss F):
dCT/dr = (sigma/2)(r2+lambda2)(Cl cos phi - Cd sin phi) = 4 F lambda2 r
with phi = atan(lambda/r), angle of attack alpha = theta(r) - phi, theta(r) = theta75 - twist (r - 0.75). Induced and profile power are integrated along the span. The solver is checked below: with no drag, no tip loss, linear lift and ideal twist it must equal momentum theory.
Section model (approximate). Cl = 5.5 (alpha + 3.5 deg) / sqrt(1 - M2) up to a soft stall, Cl,max about 0.9 at Re = 104 rising to about 1.15 at 106 and reduced as the local Mach number M passes 0.55. Profile drag Cd0 = 0.007 + 2.3 / sqrt(Re) + 0.02 Cl2, so thin Martian-air blades (Re about 104) have several times the drag coefficient of a full-scale rotor. Wave drag adds 6 (M - Mcrit)2 above Mcrit = 0.72 (adjustable) and lift fades. These are plausible engineering fits, not wind-tunnel data for one airfoil.
Worlds. Mars (Jezero-like): rho about 0.017 kg/m3 (roughly 1 % of Earth, varies with season and altitude), g = 3.71 m/s2, speed of sound about 240 m/s in cold CO2, mu = 1.1e-5 Pa s. Titan: rho about 5.4 kg/m3, g = 1.35 m/s2, a about 195 m/s in cold nitrogen, mu = 6e-6: a rotorcraft needs very little power there, which is the idea behind NASA's Dragonfly mission (public mission information; launch planned for the late 2020s). Earth: sea-level standard air.
Tip Mach. Mtip = Omega R / a. On Mars the speed of sound is low and the density is low, so lift needs speed and speed hits the compressibility drag rise first: the design is squeezed between stall (too little speed) and Mach drag rise (too much).
Battery. Electrical power = shaft power / eta + avionics. Flight time = capacity x usable fraction / power (reserve is kept). Forward flight, heating and wind are not modelled; the sortie animation uses the hover power throughout, which is conservative.
Vehicle limit. The practical thrust ceiling is 80 % of the stall peak. The collective is trimmed so that thrust equals weight; the wake animation uses actuator-disc contraction (far wake speed 2 vi, radius 0.71 R) and a ground wall-jet that lifts dust.
Simplifications. Rigid blades (no flapping, only a visual cone), uniform atmosphere, hover only, constant chord and twist, coaxial interference as one factor, no ground effect in the power numbers, and a single generic section model for all worlds.