Function brachistochrone_times 

pub fn brachistochrone_times(
    distance: Km,
    accel_km_s2: f64,
) -> (Seconds, Seconds)

Compute proper time for a brachistochrone (constant-acceleration) transfer.

For a flip-at-midpoint brachistochrone at constant proper acceleration a, covering distance d (coordinate distance), the proper time experienced by the crew is:

τ = (2c/a) × arcsinh(a × t_half / c)

where t_half is the coordinate time for the acceleration phase (half the total coordinate time), computed from the relativistic kinematics.

For a << c²/d (non-relativistic), this approaches the classical result.

Parameters:

• distance: total transfer distance in km
• accel_km_s2: constant proper acceleration in km/s²

Returns (coordinate_time, proper_time) both in seconds.