Lunar Trajectories

7/29/2019 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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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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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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(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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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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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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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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(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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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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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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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.

Lunar Trajectories

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