Entry Deceleration and Mass Change of an Ablating Body

  • Carl GazleyJr.
Part of the Applied Physics and Engineering book series (APPLIED PHYS, volume 2)

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 Hoppe1 and Levin.2 As Bronshten3 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Symbols

A

cross-sectional frontal area of body;

CD

aerodynamic drag coefficient;

CDi

initial value of drag coefficient;

C D̄

drag-coefficient ratio = C D /C Di ;

CH

dimensionless heat-transfer coefficient;

C Hi

initial value of heat-transfer coefficient;

H̄

ratio = C H H i /C Hi H);

D

characteristic dimension of body, changing dimension of body;

Di

initial body dimension;

dimensionless body size;

E

base of natural logarithms;

h

altitude;

H

effective heat of ablation;

Hi

initial value of H;

I

luminous intensity;

M

mass of body;

m i

initial mass of body;

m̄

dimensionless mass ratio = m/m i ;

n

geometry parameter, see Eq. (7–6);

q

heat-transfer rate to body;

u

velocity of body;

ni;

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/dh);

Λ

dimensionless mass of air encountered per unit cross-sectional area= ∫ρudt(m/C D A) i;

Φ

dimensionless factor = C Hi u 2 i (2C Di H i );

Ρ

atmospheric density;

ϱSL

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

  1. 1.
    Hoppe, J.: Die physikalischen Vorgänge bein Eindringen meteoritischer Körper in die Erdatmosphäre, Astron. Nachr. 262, (1937) 169–198.CrossRefGoogle Scholar
  2. 2.
    Levin, B. J.: Elements of the physical theory of meteors, Dokl. Akad. Nauk SSSR 25, (1939) 372–375Google Scholar
  3. 2a.
    Levin, B. J.: Elements of the physical theory of meteors, See also Astron. Zh. 17, 12–41 (1940)Google Scholar
  4. 2b.
    Levin, B. J.: Elements of the physical theory of meteors, Astron. Zh. 18, (1941) 331–342.Google Scholar
  5. 3.
    Bronshten, V. A.: Problems of the Motion of Large Meteoric Bodies in the Atmosphere, Academy of Sciences of the USSR, Moscow (1963).Google Scholar
  6. 4.
    Gazley, Carl Jr.: Heat-transfer aspects of the atmospheric re-entry of long-range ballistic missiles, RAND Report R-273, The RAND Corporation (1954).Google Scholar
  7. 5.
    Allen, H. J. and A. J. Eggers Jr.: A study of the motion and aerodynamic heating of missiles entering the atmosphere at high supersonic speeds, NACA RM A23D28 (1953).Google Scholar
  8. 6.
    Gazley, Carl Jr.: Meteoric interaction with the atmosphere; theory of drag and heating and comparison with observations, Section 6 in Aerodynamics of the Upper Atmosphere compiled by D. J. Masson, RAND Report R-339, The RAND Corporation (1959).Google Scholar
  9. 7.
    Gazley, Carl Jr.: Atmospheric entry of manned vehicles, RAND Research Memorandum RM-2519. The RAND Corporation (1960). See also Aerosp. Engng 19, (1960) 22–23.Google Scholar
  10. 8.
    Gazley, Carl Jr.: Atmospheric Entry, Chapter 10 of Handbook of Astronautical Engineering (ed. by H. H. Koelle). McGraw-Hill, New York (1961).Google Scholar
  11. 9.
    Ceplecha, Zd.: Multiple fall of Pribram meteorites photographed, Bull. Astron. Inst. Czech. 12, (1961) 21–47.Google Scholar
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    Millman, R. A. and A. F. Cook: Photometric analysis of a very slow meteor, Astroph. J. 130, (1959) 648–662.CrossRefGoogle Scholar
  13. 11.
    U.S. Standard Atmosphere, 1962. U.S. Government Printing Office (1962).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1968

Authors and Affiliations

  • Carl GazleyJr.
    • 1
  1. 1.The RAND CorporationUSA

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