# Entry Deceleration and Mass Change of an Ablating Body

## Abstract

For many years ballisticians have been concerned with the motion of bodies in the Earth’s atmosphere—bodies of constant shape, size, and mass. With the advent of higher flight speeds with large rates of aerodynamic heating, the body may suffer a loss in mass due to surface ablation and a consequent change in shape and size. This effects a change in the ballistic coefficient with a consequent change in the body’s dynamics and a corresponding change in the subsequent aerodynamic heating and mass loss. This coupling between the dynamics and ablation introduces a number of complications into the solution of the equations for the dynamics and mass loss. While machine solutions are generally necessary, analytic solutions have been possible in a few cases. Notable among these is the case of a body in a free-molecule flow (the typical meteor case) which is simplified by the lack of dependence of the aerodynamic drag and heat-transfer coefficients on the size and shape of the body. The solution to the meteor problem was first obtained by Hoppe^{1} and Levin.^{2} As Bronshten^{3} has recently pointed out, larger meteoric bodies penetrate deeper into the atmosphere and consequently into the continuum flow regime.

## Keywords

Mass Loss Drag Coefficient Aerodynamic Drag Luminous Efficiency Aerodynamic Heating## Symbols

- A
cross-sectional frontal area of body;

- C
_{D} aerodynamic drag coefficient;

- C
_{Di} initial value of drag coefficient;

- C
_{D}̄ drag-coefficient ratio =

*C*_{ D }*/C*_{ Di };- C
_{H} dimensionless heat-transfer coefficient;

- C
_{Hi} initial value of heat-transfer coefficient;

_{H}̄ratio =

*C*_{ H }*H*_{ i }*/C*_{ Hi }*H);*- D
characteristic dimension of body, changing dimension of body;

- D
_{i} initial body dimension;

- D̄
dimensionless body size;

- E
base of natural logarithms;

- h
altitude;

- H
effective heat of ablation;

- H
_{i} initial value of H;

- I
luminous intensity;

- M
mass of body;

- m
_{i} initial mass of body;

- m
*̄* dimensionless mass ratio =

*m/m*_{ i };- n
geometry parameter, see Eq. (7–6);

- q
heat-transfer rate to body;

- u
velocity of body;

- n
_{i}; initial velocity of body;

- ū
dimensionless velocity ratio =

*u/u*_{ i };- x
distance along path.

- Α
reciprocal of density scale height (

*α*= d ln*ϱ/*d*h)*;- Λ
dimensionless mass of air encountered per unit cross-sectional area= ∫ρ

*u*d*t(m/C*_{ D }*A)*_{ i; }- Φ
dimensionless factor =

*C*_{ Hi }*u*^{2}_{ i }*(*2*C*_{ Di }*H*_{ i }*)*;- Ρ
atmospheric density;

- ϱ
_{SL} atmospheric density at sea level;

- σ
density ratio,

*ϱ*/*ϱ*_{SL}- θ
flight path angle with respect to local horizontal (positive for descent, negative for ascent;)

- τ
luminous efficiency.

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## References

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