lunar trajectories
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Lunar Trajectories
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Earth & Moon Common center of mass
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Earth-Moon Orbit
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Lunar time of flight and Injection Velocity
The lunar orbit has a circular radius of 384,400
km.
The transfer ellipse is in the lunar orbit plane.
The gravitational effect of the moon is negligible.
The injection point is at the perigee of thetransfer ellipse.
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Lunar time of flight and Injection Velocity
Unlike planetary launches, a lunar departure orbit is
elliptical rather than hyperbolic. In below diagram, orbit 1 is a minimum energy trajectory
(an ellipse).
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Lunar time of flight and Injection Velocity
Assuming a transfer ellipse with perigee of 275 km, the
nominal shuttle orbit, produces a minimum energy transferellipse with a time of flight of 119.5 h and an injection
speed of 10.853 km/s.
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Sphere of Influence
Sphere of Influence a major consideration for a
lunar trajectory and have to make a trajectory patchat its boundary.
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Lunar Patched Conic
Fig.1 Geometry of geocentric departure orbit
0 phase angle at departure
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Designing a lunar mission
(1) Set initial conditions To define the transfer
ellipse, it is necessary to pick injection altitude (orradius), velocity and flight path angle ( ). (If
injection is made at perigee, the flight path angle is
zero).
In addition, it is necessary to define the location
of the arrival point at the sphere of influence the
most convenient method is to set the angle l1, as
shown in previous figure (Fig.1).
0 0 0, ,r V
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Designing a lunar mission
(2) Define the transfer ellipse given r, Vand at the
point of injection using the energy/momentumtechnique.
If the initial velocity is not high enough, the
departure ellipse will not intersect the moon sphereof influence and a second set of initial conditions
must be chosen.
(3) Find the radius to the sphere of influence, in
Fig.1, from trigonometry.1
r
2 2
1 12 cosS Sr D R D R l
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Designing a lunar mission
The energy and angular momentum of the
departure orbit can be determined from
;
(4) The speed and flight path angle at arrival can be
determined from
2
00 0 0
0
cos2
Vh r V
r
1 1 11 1
2 ; cosh
V rr V
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Designing a lunar mission
Finally, the phase angle at arrival can be
determined from the geometry:
The time of flight t1
t0, from injection to arrivalat the lunar sphere of influence can be computed
once q0 and q1 are determined. Before the true
anomalies can be found we must determine a and
e of the geocentric trajectory from
1 1
1
sin sinS
R
r l
2
; ; 1
2
hp a e p a
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Designing a lunar mission
Then q0 and q1 follow from the polar equation of
a conic:
After determining the eccentric anomalies, thetime of flight can be obtained from
(5) Then define inside the sphere of influence
at the arrival point. The radius (r2) is the radius of
the sphere of influence, ie. 66,183 km.
0 1
0 1
1 1cos ; cos
p p
r e e r e eq q
3
1 0 1 1 0 0sin sina
t t E e E E e E
2 2,V
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Designing a lunar mission
(6) Given r2, inside the sphere of influence,
define the arrival point.
(7) If the arrival point is satisfactory, find the launch
day using the time of flight calculated earlier andaverage orbital velocity.
(8) If the arrival point is not satisfactory, adjust the
initial conditions and start over at beginning.
2 2,V
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Lunar patched conic Example
Assume the lunar orbit is circular with radius
384,400 km and is coplanar with the transferellipse. Define a lunar trajectory with the following
initial conditions:
Injection at perigee = 0
Injection radius = 6700 km
Injection velocity = 10.88 km/s
Arrival angle l1 = 60 deg
0r
0V
0
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Lunar patched conic Example
The specific energy and the specific momentum
on the transfer ellipse are
and
2
00 0 0
0
2
2 2 3
cos2
10.88 398600.4418 6700 10.88 cos 02 6700
0.3054 72896
Vh r V
r
km s km s
22
398600.4418652587.5
2 2 0.3054
728961 1 0.98973
398600.4418 652587.5
a km
he
a
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Lunar patched conic Example
Defining arrival conditions (intersection with SOI)
Using trigonometry cosine law, andphase angle at arrival can be found as,
With these values, the following parameters can be
calculated:
Time of flight,
1 355953r km1 9.266deg.
1
1
1
1 0
1.276
80.764deg
166.45deg
49
V km s
t t hours
q
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Lunar patched conic Example
The Moon moves through an angle
between injection and arrival at the lunar sphere ofinfluence, where is the angular velocity of the
Moon in its orbit . Based on the Simplified model of
the Earth-Moon system,
The phase angle at departure is then determinedfrom
1 0m t t
m
62.649 10 / secm rad
0 1 0 1 1 0m t t q q
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Reference(s)
Roger R. Bate, Donald D. Mueller, Jerry E. White,
Fundamentals of Astrodynamics, Dover Publications, 1971.
Charles D. Brown, Elements of Spacecraft Design, AIAA
Publications, 2002.