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Analytical Assessment of Drag-Modulation Trajectory Control for Planetary Entry


Discrete-event drag-modulation trajectory control is assessed for planetary entry using the closed-form Allen-Eggers solution to the equations of motion. A control authority metric for drag-modulation trajectory control systems is derived. Closed-form analytical relationships are developed to assess range divert capability and to identify jettison condition constraints for limiting peak acceleration and peak heat rate. Closed-form relationships are also developed for drag-modulation systems with an arbitrary number of stages.

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  1. A section on mathematical notation is provided in the sequel.


a :

sensed acceleration magnitude, Earth g

b :


C :


C D :

hypersonic drag coefficient

e :

base of the natural logarithm

g :

acceleration due to gravity, m/s2

h :

altitude, m

H :

atmospheric scale height, m

i :

integer index

k :

stagnation-point convective heating constant, kg1/2/m

m :

mass, kg

n :


N :


\(\dot {Q}\) :

stagnation-point convective heat rate, W/cm2

r :

effective nose radius, m

R :

planetary radius, m

s :

range, km

S r e f :

aerodynamic reference area, m2

V :

velocity magnitude, m/s

V i :

velocity magnitude at jettison, m/s

β :

ballistic coefficient, kg/m2

γ :

flight-path angle, positive above local horizontal, rad

γ :

Allen-Eggers constant flight-path angle, rad

\(\bar {\gamma }\) :

Euler-Mascheroni constant

ρ :

atmospheric density, kg/m3

ρ i :

atmospheric density at jettison, kg/m3


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This work was supported in part by a NASA Space Technology Research Fellowship.

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Correspondence to Zachary R. Putnam.

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Putnam, Z.R., Braun, R.D. Analytical Assessment of Drag-Modulation Trajectory Control for Planetary Entry. J of Astronaut Sci 65, 470–489 (2018).

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  • Entry
  • EDL
  • Drag modulation
  • Mars