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.
formula_1|border colour=#0073CF|background colour=#F5FFFA}}where the quantities in these equations are:
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:
These assumptions are valid for hypersonic speeds, where the Mach number is greater than 5. Then the planar reentry equations for the spacecraft are:
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Rearranging terms and integrating from the atmospheric interface conditions at the start of reentry formula_17 leads to the expression:
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The term formula_19 is small and may be neglected, leading to the velocity:
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Allen and Eggers were also able to calculate the deceleration along the trajectory, in terms of the number of g's experienced formula_21, where formula_22 is the gravitational acceleration at the planet's surface. The altitude and velocity at maximum deceleration are:
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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 formula_24 at the stagnation point, with units of Watts per square meter, is assumed to have the form:
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where formula_26 is the effective nose radius. The constant formula_27 for Earth. Then the altitude and value of peak convective heating may be found:
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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:
From these two assumptions, we may infer from the second equation of motion that:
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