Which interaction rule keeps a thin flock together, and how fast does a turn travel?
Press an experiment. It runs several hidden simulations, then reports numbers.
The science
Model. Each bird i has velocity vi with speed kept between 0.75 and 1.35 of the cruise speed (10 m/s). Its acceleration is the sum of ws ∑ (xi - xj) / (rij² + 0.05) (separation, only closer than 1.6 m), wa (⟨vj⟩ - vi) (alignment), wc (⟨xj⟩ - xi) (cohesion), a weak pull toward a slowly moving roost point, random turning noise and, near a falcon, a flee push. The averages run over the neighbour set: the k nearest birds (topological) or all within R (metric). Integration is explicit Euler with a fixed step of 1/60 s.
Why 7 neighbours. Field measurements of starling flocks (Ballerini et al., 2008, typical value from the literature, please verify) found that each bird interacts with about 6 to 7 neighbours, regardless of how far apart they are. A fixed-radius rule loses its neighbours when the flock thins out and the group breaks into pieces; a topological rule keeps the same number of links at any density. The thin-flock experiment reproduces this qualitatively.
Order and groups. The order parameter is φ = |∑ vi| / ∑ |vi|: 1 means everyone flies the same way, and for N random headings ⟨φ²⟩ = 1/N (checked on this page). Groups are connected clusters of birds closer than 8 m.
Honest simplifications. Birds are point masses with a speed clamp: no aerodynamics, no vision cone, no reaction delay, no gusts. The falcon is a pursuit curve that never touches a bird. The turning-wave speed is an emergent number from this toy model; real flocks have been reported to pass turns at tens of metres per second (typical order of magnitude, not a calibration). Bird and falcon sizes are drawn larger than life by the display-size slider; physics uses point positions. Nothing here predicts a real flock.