Single Impulse Deorbit (short paper)

4 Pages·1966·0.134 MB·English

Single Impulse Deorbit From a Circular Orbit

The single impulsive maneuver required to deorbit a spacecraft from an initial circular
orbit can be determined from the following expression

   
   
1  2(r(cid:4)−1)   2(r(cid:4)−1) 
∆V =V 1− =V 1− 
ce r(cid:4)   r(cid:4) 2  ci   r(cid:4) 2 
−1 −1
       
cosγ cosγ
  e     e  

where

h +r r
r(cid:4) = i eq = i = radius ratio
h +r r
e eq e
µ µ
V = = = local circular velocity at entry interface
ce (h +r ) r
e eq e
µ µ
V = = = local circular velocity at initial circular orbit
ci (h +r ) r
i eq i
γ = flight path angle at entry interface
e
h = altitude of initial circular orbit
i
h = altitude at entry interface
e
r = radius of initial circular orbit
i
r = radius at entry interface
e
r = Earth equatorial radius
eq

µ= Earth gravitational constant

This algorithm is described in the classic technical article, “Deboost from Circular
Orbits”, A. H. Milstead, The Journal of the Astronautical Sciences, Vol. XIII, No. 4, pp.
170-171, Jul-Aug., 1966. Additional information can be found in Chapter 5 of
Hypersonic and Planetary Entry Flight Mechanics by Vinh, Busemann and Culp, The
University of Michigan Press.
page 1
The true anomaly on the deorbit trajectory at the entry interface θ can be determined
e
from the following two equations:

r(cid:5) a (1−e2)
sinθ = d d
e e µ
d

a (1−e2) 1
cosθ = d d −
e e r e
d e d

and the following four quadrant inverse tangent operation

θ = tan−1(sinθ,cosθ)
e e e

where
e = eccentricity of deorbit trajectory
d
a = semimajor axis of deorbit trajectory
d
µ2a r −r2 −a2(1−e2)
r(cid:5)=−  d e e d d 
a r2
d e

The time of flight between perigee of the deorbit trajectory and the entry true anomaly θ
e
is given by

τ   1−e θ  e 1−e2 sinθ 
t(θ)= 2tan−1 d tan e− d d e
e 2π  1+ed 2  1+ed cosθe 

In this equation τ is the orbital period of the deorbit trajectory and is equal to 2π a3 µ.
d

Therefore, the flight time between the deorbit impulse and entry interface is given by

τ
∆t =t(θ)−t(180(cid:68))=t(θ)−
e e 2

Finally, the orbital speed at the entry interface V can be determined from
e

2µ µ
V = −
e r a
e d

page 2
The following is a graphics display of the single impulsive ∆V required to deorbit from a
range of circular Earth orbit altitudes and several entry interface flight path angles. The
entry interface altitude for this example is 400,000 feet.

Deorbit Delta-V versus Initial Circular Orbit Altitude
900 and Flight Path Angle at Entry Interface
800
altitude at entry interface = 400,000 feet
700
s)
V (fp 600 aftli gehntt rpya itnht earnfagclee
a-
elt -20
D
orbit 500 alt = 190.8 nm -1.50
De ∆v = 444.5 fps -10
400
alt = 158.9 nm
∆v = 334.1 fps
300
alt = 127.6 nm
∆v = 223.2 fps impulsive maneuver
200
50 100 150 200 250 300 350
Initial Circular Orbit Altitude (nm)

This plot is also labeled with the optimum deorbit altitude and corresponding ∆V for a
given flight path angle at the entry interface. An examination of the following plot of the
eccentricity of the deorbit trajectory reveals why these minima exist.

Eccentricity of Deorbit Trajectory versus Initial Circular Orbit
Altitude and Flight Path Angle at Entry Interface
0.07
-20
0.06
ectory 0.05 altitude at entry interface = 400,000 feet
aj
bit tr
deor 0.04 -1.50
of
city
ntri
Ecce 0.03
0.02 -10
flight path angle impulsive maneuver
at entry interface
0.01
50 100 150 200 250 300 350
Initial Circular Orbit Altitude (nm)
page 3
The next two plots illustrate the behavior of the impulse-to-entry interface mission time
and speed at the entry interface.

Impulse-to-Entry Time versus Initial Circular Orbit Altitude
and Flight Path Angle at Entry Interface
40
-10
-1.50
-20
30
flight path angle
at entry interface
n)
mi
y (
ntr
E
o
V t 20
∆
m
o
me fr altitude at entry interface = 400,000 feet
Ti
10
impulsive maneuver
0
50 100 150 200 250 300 350
Initial Circular Orbit Altitude (nm)

Entry Speed versus Initial Circular Orbit Altitude
26400 and Flight Path Angle at Entry Interface
26000
s)
p
e (f
c
a
erf
nt
y I 25600
ntr -10
E
at
d
e
pe altitude at entry interface = 400,000 feet
S -1.50
25200
-20
flight path angle impulsive maneuver
at entry interface
24800
50 100 150 200 250 300 350
Initial Circular Orbit Altitude (nm)
page 4