Planar reentry equations

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:

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

It is also possible to compute the maximum stagnation point convective heating with the Allen-Eggers solution and a heat transfer correlation; the Sutton-Graves correlation^[3] is commonly chosen. The heat rate ${\displaystyle {\dot {q}}''}$ at the stagnation point, with units of Watts per square meter, is assumed to have the form:

${\displaystyle {\dot {q}}''=k\left({\frac {\rho }{r_{n}}}\right)^{1/2}V^{3}\sim {\text{W}}/{\text{m}}^{2}}$

where ${\displaystyle r_{n}}$ is the effective nose radius. The constant ${\displaystyle k=1.74153\times 10^{-4}}$ for Earth. Then the altitude and value of peak convective heating may be found:

${\displaystyle h_{{\dot {q}}_{\max }''}=-H\log \left({\frac {\beta \sin \gamma }{3H\rho _{0}}}\right)\implies {\dot {q}}_{\max }''=k{\sqrt {\frac {\beta \sin \gamma }{3Hr_{n}e}}}V_{\text{atm}}^{3}}$

Equilibrium glide condition

Another commonly encountered simplification is a lifting entry with a shallow, slowly-varying, flight path angle.^[4] 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:

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