Planar reentry equations

The planar reentry equations are the equations of motion governing the unpowered reentry of a spacecraft, based on the assumptions of planar motion and constant mass, in an Earth-fixed reference frame.^[1]

${\begin{cases}{\frac {dV}{dt}}&=-{\frac {\rho V^{2}}{2\beta }}+g\sin \gamma \\{\frac {d\gamma }{dt}}&=-{\frac {V\cos \gamma }{r}}-{\frac {\rho V}{2\beta }}\left({\frac {L}{D}}\right)\cos \sigma +{\frac {g\cos \gamma }{V}}\\{\frac {dh}{dt}}&=-V\sin \gamma \end{cases}}$

where the quantities in these equations are:

$V$ is the velocity
$\gamma >0$ is the flight path angle
$h$ is the altitude
$\rho$ is the atmospheric density
$\beta$ is the ballistic coefficient
$g$ is the gravitational acceleration
$r=r_{e}+h$ is the radius from the center of a planet with equatorial radius $r_{e}$
$L/D$ is the lift-to-drag ratio
$\sigma$ is the bank angle of the spacecraft.

^ Wang, Kenneth; Ting, Lu (1961). "Approximate Solutions for Reentry Trajectories With Aerodynamic Forces" (PDF). PIBAL Report No. 647: 5–7.

[1] Wang, Kenneth; Ting, Lu (1961). "Approximate Solutions for Reentry Trajectories With Aerodynamic Forces" (PDF). PIBAL Report No. 647: 5–7.

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