Condor Glide Polar and MacCready Speed-to-Fly: Thermal Hop Lab

An Andean condor crosses a 24 km valley without flapping: it glides between thermals, circles in the updraft, and chooses how fast to fly. Change the day and the bird's strategy, then see how the glide polar and the MacCready rule set the cross-country speed.

Scenario

Playback speed

Speed-to-fly

The day

The bird's strategy

Why those numbers

Glide polar: sink rate vs airspeed

Circling: climb rate vs bank angle

Altitude along the course, m a.s.l.

Experiment: which MacCready setting is fastest?

Press the button: the same day is flown 17 times with MC = 0 ... 4 m/s. Settings above what the thermals really give make the bird land short.

Missions

Independent numerical checks

Also run outside the browser: node src/check.js.

Educational model with typical, illustrative numbers [assumption]: mass 11 kg, wing area 0.95 m², span 3.05 m, CD0 0.034, still air, Gaussian thermals, ISA density. Real condors differ in size, wing shape and behaviour, and real thermals are far messier. Not a flight-planning tool.

Controls

Drag / one finger: orbit the camera; wheel / pinch: zoom. Chase, Wing, Thermal and Map buttons set the view.
Fly plays the flight; the time slider scrubs it exactly (the whole flight is computed in advance and replayed). Changing any slider recomputes the flight.
Bird size only scales the drawing so that a 3 m bird is visible from far away; it does not change the physics.

The science

Glide polar. In steady glide, lift balances weight: CL = 2 n m g / (ρ V² S) with load factor n = 1/cosφ. Drag is parabolic, CD = CD0 + k CL², k = 1/(π e AR), so the sink rate is w(V) = V CD / CL and the glide ratio is V / w. Its maximum is 1/(2 √(CD0 k)) at CL = √(CD0/k); minimum sink is at CL = √(3 CD0/k), a slower speed. Below the stall speed (CLmax = 1.8, slotted feathers [assumption]) the polar stops.

Thermals. The updraft is a Gaussian core w(r) = w0 exp(-(r/Rt)²). Circling at radius R = V²/(g tanφ) around a centre that is a distance d from the core, the mean lift is w0 exp(-(R²+d²)/Rt²) I0(2Rd/Rt²) (I0 = modified Bessel function). A steeper bank gives a tighter circle in stronger lift, but raises the load factor and the sink: the best bank is a compromise (see the climb-vs-bank chart). The bird needs time to centre the circle: the offset decays as d0 exp(-t/τ).

MacCready speed-to-fly. If the bird climbs at mean rate c and glides at V with sink w(V), the average cross-country speed is V c / (c + w(V)). Maximising over V gives the tangent construction: the best V is where the polar's slope equals (w + c)/V. At c = 0 this is the best-glide speed; stronger thermals mean flying faster between them. The MC setting is the climb rate the bird expects; it also sets when to leave a thermal (when the climb falls below MC).

What the sweep shows. If MC is set higher than the thermals really deliver, the bird leaves good lift early and flies fast into sinking air: it lands short. The fastest finishing setting is close to the mean climb actually achieved, and the measured average speed matches the MacCready formula (made altitude-neutral) within a fraction of a percent.

Not modelled: wind and shear, thermal drift, ridge lift, flapping, thermal-seeking by following other birds, turbulence, real feather aeroelasticity. Thermal strength, spacing and the bird's polar are typical values, not measurements of one individual.