Lift, wing loading, stall
L = ½ ρ v² S CL, D = ½ ρ v² S CD, CD = CD0 + CL² / (π e AR)
vstall = √( 2 W / (ρ S CLmax) ), wing loading = W / S
A wing can only hold the animal up above vstall. Doubling linear size multiplies mass by 8 but area by only 4, so wing loading grows as m1/3 and the minimum flight speed as m1/6 (the "Mass vs vstall" graph). Big animals must be fast or have relatively huge wings.
Glide ratio and power
(L/D)max = ½ √( π e AR / CD0 ), sink rate = v CD/CL
Preq(v) = 2k W²/(π ρ b² v) + ½ ρ v³ S CD,pro + ½ ρ v³ SbCD,b
The U-shaped power curve (induced + profile + parasite, after Pennycuick) is compared with the muscle power available: pectoral-muscle mass (fraction of body mass x specific power) x ~220 W/kg for a burst of a few seconds, ~70 W/kg sustained. Anything below the curve cannot fly level, however good its wings.
Flapping as extra airspeed
While flapping, the wing moves through the air even when the body stands still. The model adds the mean-square flapping tip speed to the squared forward speed: veff² = v² + 0.7 U² with U = 0.7 Φ f b (stroke amplitude Φ, beat frequency f ∝ m3/8 g1/2 b-23/24 S-1/3 ρ-3/8). The share of that beat the muscle can really drive is limited by the available power relative to the minimum-power requirement.
The three launch hypotheses
- Ground-up: run, flap, leap. Leg force falls linearly to zero at the animal's top running speed; flapping adds thrust and unloads the feet once lift reaches ~80 % of weight.
- WAIR: flapping while running up a steep log presses the feet into the surface. Traction needs
μ(W cosθ + Fwing) ≥ W sinθ, giving a maximum slope θmax = atan μ + asin( μ r / √(1+μ²) ) with r = Fwing/W. The sim integrates the climb and the slope it finds agrees with this formula.
- Trees-down: step off a perch, gain speed in a dive, pull out at the best glide angle. The height needed is set by stall speed, lift-to-drag and how tightly the wing can pull out.
What is simplified - and what is uncertain
- The animal is a point mass in a vertical plane with a quasi-steady wing: no unsteady or leading-edge-vortex lift, no real wing-beat cycle, no tail or hind-limb aerodynamics beyond the lumped wing area (four-winged forms count hind wings in S).
- Masses, spans, wing areas, muscle fractions, stroke amplitudes and running speeds of extinct animals are illustrative estimates; published values differ by tens of percent, and the conclusions (who "could" fly) are debated. The sim shows how conclusions move when you change the assumptions - it does not settle the debate.
- Feather colours are reconstructions from melanosome studies where available (Anchiornis, Microraptor, Archaeopteryx) and guesses otherwise. Skeletons are stylised from the fossils, not scans.
- Included fossil taxa are shown at the ages given; Microraptor is a dromaeosaurid, not a bird. Avian traits (keel, pygostyle, U-shaped furcula, fused hand) appear gradually, in a mosaic.