Planar reentry equations explained

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] where the quantities in these equations are:

V is the velocity\gamma > 0 is the flight path angleh is the altitude\rho is the atmospheric density\beta is the ballistic coefficientg is the gravitational accelerationr = 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.^[2] They made several assumptions:

1. The spacecraft's entry was purely ballistic

(L = 0 ) .

1. The effect of gravity is small compared to drag, and can be ignored.
2. The flight path angle and ballistic coefficient are constant.
3. 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} (dV)/(dt) & =- (\rho _{0})/(2 \beta) V ^{2} e ^{-h/H} \\ (dh)/(dt) & = - V \sin \gamma \end{cases} \implies (dV)/(dh) = (\rho _{0})/(2 \beta\sin\gamma) Ve ^{-h/H}

Rearranging terms and integrating from the atmospheric interface conditions at the start of reentry

(V _{atm} ,h _{atm} ) leads to the expression:(dV)/(V) = (\rho _{0})/(2 \beta\sin \gamma) e ^{-h/H} dh \implies log \left( (V)/(V _{atm}) \right) =- (\rho _{0} H)/(2 \beta \sin\gamma) \left( e ^{-h/H} - e ^{- h _{atm} /H} \right)

The term

\exp( - h _{atm} /H) is small and may be neglected, leading to the velocity:V(h) = V _{atm} \exp \left( - (\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 ^{-1} _{0} (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( (\rho _{0} H)/(\beta \sin \gamma) \right), V _{n _{max}} = V _{atm} e ^{-1/2} \implies n _{max} = (V ^{2} _{atm} \sin \gamma)/(2 g _{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^[3] is commonly chosen. The heat rate • q '' at the stagnation point, with units of Watts per square meter, is assumed to have the form:• q '' = k\left( (\rho)/(r _{n}) \right) ^{1/2} V ^{3} \sim W / m ^{2}

where

r _{n} is the effective nose radius. The constant k = 1 . 74153 x 10 ^{-4} for Earth. Then the altitude and value of peak convective heating may be found:h _{• q _{max} ''} = - H log \left( (\beta \sin \gamma)/(3 H\rho _{0}) \right) \implies • q _{max} '' = k \sqrt{ (\beta \sin\gamma)/(3 Hr _{n} e) }V ^{3} _{atm}

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:

\cos\gamma ≈ 1 , \sin\gamma ≈ \gamma .

1. The flight path angle changes very slowly, such that

d\gamma/dt ≈ 0 .

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

\left[ (1)/(r) + (\rho)/(2 \beta) \left( (L)/(D) \right) \cos \sigma \right]V ^{2} = g \implies V(h) = \sqrt{ (g r)/(1 + (\rho r)/(2 \beta) \left( (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 . 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. . NACA Technical Report 1381 . National Advisory Committee for Aeronautics.
3. Sutton . K. . Graves . R. A. . 1971-11-01 . A general stagnation-point convective heating equation for arbitrary gas mixtures . NASA Technical Report R-376 . en.
4. Eggers, Jr. . A.J. . Allen . H.J. . Niece . S.E. . 1958 . A Comparative Analysis of the Performance of Long-Range Hypervelocity Vehicles . NACA Technical Report 1382 . National Advisory Committee for Aeronautics.

Further reading

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