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.{{Cite journal |last1=Wang |first1=Kenneth |last2=Ting |first2=Lu |date=1961 |title=Approximate Solutions for Reentry Trajectories With Aerodynamic Forces |url=https://apps.dtic.mil/sti/tr/pdf/AD0257618.pdf |journal=PIBAL Report No. 647 |pages=5–7}}

Definition

The equations are given by:{{Equation box 1|cellpadding|border|indent=:|equation= $\begin{cases} \frac{dV}{dt} &= -\frac{\rho V^{2}}{2\beta} + g \sin \gamma \\ \frac{d\gamma}{dt} &= -\frac{V \cos\gamma}{r} - \frac{\rho V}{2\beta} \left( \frac{L}{D} \right) \cos \sigma + \frac{g \cos \gamma}{V} \\ \frac{dh}{dt} &= -V\sin \gamma \end{cases}$ |border colour=#0073CF|background colour=#F5FFFA}}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.{{Cite journal |last1=Allen |first1=H. Julian |last2=Eggers, Jr. |first2=A.J. |date=1958 |title=A study of the motion and aerodynamic heating of ballistic missiles entering the earth's atmosphere at high supersonic speeds. |url=https://ntrs.nasa.gov/api/citations/19930091020/downloads/19930091020.pdf |journal=NACA Technical Report 1381 |publisher=National Advisory Committee for Aeronautics}} 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 $\rho(h) = \rho_{0}\exp(-h/H)$ , 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:

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

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

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

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

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

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

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

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{{Cite journal |last1=Sutton |first1=K. |last2=Graves |first2=R. A. |date=1971-11-01 |title=A general stagnation-point convective heating equation for arbitrary gas mixtures |url=https://ntrs.nasa.gov/citations/19720003329 |journal=NASA Technical Report R-376 |language=en}} 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 \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.{{Cite journal |last1=Eggers, Jr. |first1=A.J. |last2=Allen |first2=H.J. |last3=Niece |first3=S.E. |date=1958 |title=A Comparative Analysis of the Performance of Long-Range Hypervelocity Vehicles |url=https://ntrs.nasa.gov/api/citations/19930092363/downloads/19930092363.pdf |journal=NACA Technical Report 1382 |publisher=National Advisory Committee for Aeronautics}} 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:

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

Planar reentry equations

Definition

Simplifications

= Allen-Eggers solution =

= Equilibrium glide condition =

See also

References

Further reading