Kick-orbit distribution ($f_c(c)=1/(\pi \sqrt{1-c^2})$)

Uniform planar kick angle $\theta$ gives $c=\cos \theta$ with probability density $1/(\pi \sqrt{1-c^2})$. This maps onto a one-dimensional curve in orbital eccentricity–semi-major axis space. The two radial-kick signs share that curve but have opposite tangential components of the eccentricity vector. The resulting density is enhanced near tangential-kick endpoints.