Abstract
This paper is concerned with the optimization of trajectories for coplanar, aeroassisted orbital transfer (AOT) from a high Earth orbit (HEO) to a low Earth orbit (LEO). In particular, HEO can be a geosynchronous Earth orbit (GEO). It is assumed that the initial and final orbits are circular, that the gravitational field is central and is governed by the inverse square law, and that two impulses are employed, one at HEO exit and one at LEO entry. During the atmospheric pass, the trajectory is controlled via the lift coefficient in such a way that the total characteristic velocity is minimized.
First, an ideal optimal trajectory is determined analytically for lift coefficient unbounded. This trajectory is called grazing trajectory, because the atmospheric pass is made by flying at constant altitude along the edge of the atmosphere until the excess velocity is depleted. For the grazing trajectory, the lift coefficient varies in such a way that the lift, the centrifugal force due to the Earth's curvature, the weight, and the Coriolis force due to the Earth's rotation are in static balance. Also, the grazing trajectory minimizes the total characteristic velocity and simultaneously nearly minimizes the peak values of the altitude drop, the dynamic pressure, and the heating rate.
Next, starting from the grazing trajectory results, a real optimal trajectory is determined numerically for lift coefficient bounded from both below and above. This trajectory is characterized by atmospheric penetration with the smallest possible entry angle, followed by flight at the lift coefficient lower bound. Consistently with the grazing trajectory behavior, the real optimal trajectory minimizes the total characteristic velocity and simultaneously nearly minimizes the peak values of the altitude drop, the dynamic pressure, and the heating rate.
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Abbreviations
- C D :
-
drag coefficient
- C L :
-
lift coefficient
- D :
-
drag, N
- DP:
-
dynamic pressure, N/m2
- E :
-
lift-to-drag ratio modulus
- g :
-
local acceleration of gravity, m/sec2
- g a :
-
acceleration of gravity ath=h a , m/sec2
- h :
-
altitude, m
- h a :
-
thickness of the atmosphere, m
- HR:
-
heating rate, W/m2
- L :
-
lift, N
- m :
-
mass, kg
- r :
-
radial distance from the center of the Earth, m
- r a :
-
radius of the outer edge of the atmosphere, m
- r e :
-
radius of the Earth, m
- S :
-
reference surface area, m2
- t :
-
T/τ=dimensionless time
- T :
-
running time, sec
- V :
-
velocity, m/sec
- V a :
-
circular velocity atr=r a , m/sec
- V * :
-
reference velocity, m/sec
- γ :
-
path inclination, rad
- μ :
-
Earth's gravitational constant, m3/sec2
- ρ :
-
air density, kg/m3
- ρ a :
-
air density ath=h a , kg/m3
- ρ * :
-
reference air density, kg/m3
- τ :
-
final time, sec
- θ :
-
Earth's angular velocity, rad/sec
- ΔV :
-
characteristic velocity, m/sec
- 0:
-
entry into the atmosphere
- 1:
-
exit from the atmosphere
- 00:
-
exit from the initial orbit
- 11:
-
entry into the final orbit.
- •:
-
derivative with respect to dimensionless time
- ∼:
-
variable computed in an inertial system.
- AOT:
-
aeroassisted orbital transfer
- GEO:
-
geosynchronous Earth orbit
- HEO:
-
high Earth orbit
- LEO:
-
low Earth orbit
- SGRA:
-
sequential gradient-restoration algorithm
- TPBVP:
-
two-point boundary-value problem
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This research was supported by NASA Marshall Space Flight Center Grant No. NAG-8-820, by Jet Propulsion Laboratory Contract No. 956415, and by Texas Advanced Technology Program Grant No. TATP-003604020.
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Miele, A., Wang, T. & Deaton, A.W. Properties of the optimal trajectories for coplanar, aeroassisted orbital transfer. J Optim Theory Appl 69, 1–30 (1991). https://doi.org/10.1007/BF00940459
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DOI: https://doi.org/10.1007/BF00940459