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