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]

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

where the quantities in these equations are:

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

Simplifications

Allen-Eggers solution

Harry Allen and Alfred Eggers, based on their studies of ICBM trajectories, were able to derive an analytical expression for the velocity as a function of altitude. ^[2] They made several assumptions:

1. The spacecraft's entry was purely ballistic ${\displaystyle (L=0)}$.
2. The effect of gravity is small compared to drag, and can be ignored.
3. The flight path angle and ballistic coefficient are constant.
4. An exponential atmosphere, where ${\displaystyle \rho (h)=\rho _{0}\exp(-h/H)}$, with ${\displaystyle \rho _{0}}$ being the density at the planet's surface and ${\displaystyle H}$ being the scale height.

These assumptions are valid for hypersonic speeds, where the Mach number is greater than 5. Then the planar reentry equations for the spacecraft are:

${\displaystyle {\begin{cases}{\frac {dV}{dt}}&=-{\frac {\rho _{0}}{2\beta }}V^{2}e^{-h/H}\\{\frac {dh}{dt}}&=-V\sin \gamma \end{cases}}\implies {\frac {dV}{dh}}={\frac {\rho _{0}}{2\beta \sin \gamma }}Ve^{-h/H}}$

Rearranging terms and integrating from the atmospheric interface conditions at the start of reentry ${\displaystyle (V_{\text{atm}},h_{\text{atm}})}$ leads to the expression:

${\displaystyle {\frac {dV}{V}}={\frac {\rho _{0}}{2\beta \sin \gamma }}e^{-h/H}dh\implies \log \left({\frac {V}{V_{\text{atm}}}}\right)=-{\frac {\rho _{0}H}{2\beta \sin \gamma }}\left(e^{-h/H}-e^{-h_{\text{atm}}/H}\right)}$

The term ${\displaystyle \exp(-h_{\text{atm}}/H)}$ is small and may be neglected, leading to the velocity:

${\displaystyle V(h)=V_{\text{atm}}\exp \left(-{\frac {\rho _{0}H}{2\beta \sin \gamma }}e^{-h/H}\right)}$

Allen and Eggers were also able to calculate the deceleration along the trajectory, in terms of the number of g's experienced ${\displaystyle n=g_{0}^{-1}(dV/dt)}$, where ${\displaystyle g_{0}}$ is the gravitational acceleration at the planet's surface. The altitude and velocity at maximum deceleration are:

${\displaystyle h_{n_{\max }}=H\log \left({\frac {\rho _{0}H}{\beta \sin \gamma }}\right),\quad V_{n_{\max }}=V_{\text{atm}}e^{-1/2}\implies n_{\max }={\frac {V_{\text{atm}}^{2}\sin \gamma }{2g_{0}eH}}}$

Equilibrium glide condition

Another commonly encountered simplification is a lifting entry with a shallow, slowly-varying, flight path angle. ^[3] The velocity as a function of altitude can be derived from two assumptions:

1. The flight path angle is shallow, meaning that: ${\displaystyle \cos \gamma \approx 1,\sin \gamma \approx \gamma }$.
2. The flight path angle changes very slowly, such that ${\displaystyle d\gamma /dt\approx 0}$.

From these two assumptions, we may infer from the second equation of motion that:

${\displaystyle \left[{\frac {1}{r}}+{\frac {\rho }{2\beta }}\left({\frac {L}{D}}\right)\cos \sigma \right]V^{2}=g\implies V(h)={\sqrt {\frac {gr}{1+{\frac {\rho r}{2\beta }}\left({\frac {L}{D}}\right)\cos \sigma }}}}$

See also

• Atmospheric entry
• Hypersonic flight

References

1. Wang, Kenneth; Ting, Lu (1961). "Approximate Solutions for Reentry Trajectories With Aerodynamic Forces" (PDF). PIBAL Report No. 647: 5–7.
2. Allen, H. Julian; Eggers, Jr., A.J. (1958). "A study of the motion and aerodynamic heating of ballistic missiles entering the earth's atmosphere at high supersonic speeds" (PDF). NACA Technical Report 1381. National Advisory Committee for Aeronautics.
3. Eggers, Jr., A.J.; Allen, H.J.; Niece, S.E. (1958). "A Comparative Analysis of the Performance of Long-Range Hypervelocity Vehicles" (PDF). NACA Technical Report 1382. National Advisory Committee for Aeronautics.