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.

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

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.

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. They made several assumptions:

1. The spacecraft's entry was purely ballistic $(L=0)$.
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 $\rho (h)={\rho }_0\exp (-h/H)$, with ${\rho }_0$ being the density at the planet's surface and $H$ 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:

$\{\begin{array}{cc}\frac{dV}{dt}& =-\frac{{\rho }_0}{2\beta }{V}^2{e}^{-h/H}\\ \frac{dh}{dt}& =-V\sin \gamma \end{array}⟹\frac{dV}{dh}=\frac{{\rho }_0}{2\beta \sin \gamma }V{e}^{-h/H}$

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

$\frac{dV}{V}=\frac{{\rho }_0}{2\beta \sin \gamma }{e}^{-h/H}dh⟹\log \left(\frac{V}{V_{\text{atm}}}\right)=-\frac{{\rho }_{0}H}{2\beta \sin \gamma }(e^{-h/H}-e^{-h_{\text{atm}}/H})$

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

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

Allen and Eggers were also able to calculate the deceleration along the trajectory, in terms of the number of g's experienced $n=g_0^{-1}(dV/dt)$, where $g_0$ is the gravitational acceleration at the planet's surface. The altitude and velocity at maximum deceleration are:

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

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 is commonly chosen. The heat rate ${\dot{q}}^{″}$ at the stagnation point, with units of Watts per square meter, is assumed to have the form:

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

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

$h_{{\dot{q}}_{{\max }^{″}}}=H\log (-\frac{\beta \sin \gamma }{3H{\rho }_0})⟹{\dot{q}}_{{\max }^{″}}=k\sqrt{-\frac{\beta \sin \gamma }{3H{r}_{n}e}}{V}_{\text{atm}}^3$

Equilibrium glide condition

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

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

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

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

See also

• Atmospheric entry
• Hypersonic flight

Further reading

• Regan, F.J.; Anandakrishnan, S.M. (1993). Dynamics of Atmospheric Re-Entry. AIAA Education Series. pp. 180-184.