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.

Definition

The equations are given by: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
* $r = r_ + h$ is the radius from the center of a planet with equatorial radius $r_$
* $L/D$ 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:

# The spacecraft's entry was purely ballistic $(L = 0)$.
# The effect of gravity is small compared to drag, and can be ignored.
# The flight path angle and ballistic coefficient are constant.
# An exponential atmosphere, where $\rho(h) = \rho_\exp(-h/H)$, with $\rho_$ 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:

:$\begin \frac &= -\fracV^e^ \\\frac &= -V \sin \gamma \end \implies \frac = \fracVe^$

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

:$\frac = \frace^dh \implies \log \left( \frac \right) = -\frac \left( e^ - e^ \right)$

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

:$V(h) = V_ \exp \left( -\frac e^ \right)$
Allen and Eggers were also able to calculate the deceleration along the trajectory, in terms of the number of g's experienced $n = g_^ (dV/dt)$, where $g_$ is the gravitational acceleration at the planet's surface. The altitude and velocity at maximum deceleration are:

:$h_ = H\log\left( -\frac \right), \quad V_ = V_e^ \implies n_ = -\frac$
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 $\dot''$ at the stagnation point, with units of Watts per square meter, is assumed to have the form:

:$\dot'' = k\left( \frac \right)^V^ \sim \text/\text^$

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

:$h_ = H \log \left( -\frac \right) \implies \dot_'' = k \sqrtV_^$

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:

# The flight path angle is shallow, meaning that: $\cos\gamma \approx 1, \sin\gamma\approx \gamma$.
# 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:

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

See also

*Atmospheric entry
*Hypersonic flight

References

Further reading

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

Atmospheric entry
Differential equations
Aerospace engineering
Classical mechanics