1. Lunar Trajectories

2. Earth & Moon – Common center of mass

3. Earth-Moon Orbit

4. Variation of Earth-Moon System
Solar perturbations change the rotation period by as much as 7 hours.
As the moon’s orbit around the Earth is elliptical, the Earth-moon distance changes slightly with true anomaly.
The semi-major axis changes with time as the tides about Earth take energy from the moon orbit and slow its orbital velocity.
Small changes in eccentricity occur with a period of 31.8 days (the effect called ‘evection’).
The average orbit inclination with respect to the ecliptic is ± 0.15 deg varying with a period of 173 days.

5. 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 the transfer ellipse.

6. 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) – Longest possible time of flight and lowest injection velocity.

7. Lunar time of flight and Injection Velocity
Assuming a transfer ellipse with perigee altitude of 275 km, the nominal shuttle orbit, produces a minimum energy transfer ellipse with a time of flight of hrs and an injection speed of km/s.

8. Sphere of Influence (SOI)
Sphere of Influence – a major consideration for a lunar trajectory and have to make a trajectory patch at its boundary. (Discussed Earlier)

9. Lunar Patched Conic Method
Figure 1: Geometry of geocentric departure orbit

10. Designing a lunar mission
(1) Set initial conditions – To define the transfer ellipse, it is necessary to pick injection altitude (or radius), 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).

11. Designing a lunar mission
(2) Define the transfer ellipse given r, V and f at the point of injection using the energy/momentum technique.
If the initial velocity is not high enough, the departure ellipse will not intersect the moon sphere of 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.

12. 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 on the sphere on influence can be determined from

13. 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 arrival at 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

14. Designing a lunar mission
Then q0 and q1 follow from the polar equation of a conic:
After determining the eccentric anomalies, the time 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.

15. Designing a lunar mission
(6) Given r2, inside the sphere of influence, define the arrival orbit.
(7) If the arrival orbit is satisfactory, find the launch day using the time of flight calculated earlier and average orbital velocity.
(8) If the arrival orbit is not satisfactory, either the injection conditions ( ) or the arrival angle l1 should be adjusted and start over at beginning (trial and error) until the trajectory is acceptable.

16. Lunar patched conic – Example
Assume the lunar orbit is circular with radius 384,400 km and is coplanar with the transfer ellipse. Define a lunar trajectory with the following initial conditions:
Injection at perigee = 0
Injection radius = 6700 km
Injection velocity = km/s
Arrival angle l1 = 60 deg

17. Departure trajectory (transfer ellipse)
The specific energy and the specific momentum on the transfer ellipse are
and

18. Arrival at Moon’s Sphere of Influence
Defining arrival conditions (intersection with SOI)
Using trigonometry cosine law, and phase angle at arrival can be found as,
With these values, the following parameters can be calculated:
Time of flight, (from point of injection to arrival at the boundary of moon’s sphere of influence (patch point))

19. Defining the lunar orbit (inside the SOI)
Conditions at the patch point (defining the lunar orbit)
Now we need to determine the trajectory inside the Moon's sphere of influence where only lunar gravity is assumed to act on the spacecraft. Since we must now consider the Moon as the central body, it is necessary to find the speed and direction of the spacecraft relative to the center of the Moon.
In Figure 2, the geometry of the situation at arrival is shown in detail.

20. Defining the lunar orbit
Figure 2: Lunar arrival Geometry (NOT TO SCALE)

21. Defining the lunar orbit
If we let the subscript 2 indicate the initial conditions relative to the Moon's center, then the selenocentric radius, r2 , is = 66,183 km.
Here, = velocity of the spacecraft relative to
the center of the Moon,
and = velocity of the Moon relative to the
center of the Earth (1.023 km/s).

22. Defining the lunar orbit
The selenocentric arrival speed, may be obtained by applying the law of cosines to the vector triangle in Figure 2 (yellow coloured region):
From the arrival geometry (Figure 2), the flight path angle,
The angle b can be obtained by applying a vector triangle formula (shown in next slide) to the vector triangle shown in Figure 2 (yellow coloured region).

23. Arrival vector diagram
The vector triangle from Figure 2 and its corresponding values are shown below:
To find a from
vector triangle
formula:
a = deg
Therefore,
b = 180–( )
= deg
Therefore, = 180 – ( ) = deg

24. Arrival trajectory (hyperbola)
From , the orbital elements of the arrival
hyperbolic trajectory can be obtained as follows:
and

25. Arrival trajectory (hyperbola)
From the obtained values it is clear that the arrival trajectory (lunar orbit) is a hyperbola. The resulting lunar orbit is the relatively flat hyperbola as shown below (not to scale).

26. Arrival trajectory (hyperbola)
The periapsis (periselenium) radius can be obtained as:
and the periapsis speed can be obtained from
Recall the initial assumption that the trajectory was in the lunar plane. If a non-coplanar trajectory is desired, the inclination of the transfer plane can be incorporated into the calculations using the methods which was already discussed in realistic Venus mission.

27. Evaluation of the lunar trajectory
Once the lunar trajectory (transfer ellipse and arrival hyperbola) has been defined, it is essential to verify whether the obtained trajectory (orbit) is suitable or not for the required mission.
There are a number of terminal conditions of interest to evaluate the lunar orbit depending on the nature of the mission as follows:
For landing or impact at moon’s surface, the periselenium radius should be less than the radius of the moon (1738 km).
For reaching lunar orbit (orbit around the moon), we have to compute the delta-v required to produce a lunar satellite at the periselenium altitude.
For circumlunar flight, we have to compute both the periselenium conditions and the conditions upon exit from the lunar sphere of influence.

28. 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.
Second Year Higher Secondary Text Book of Mathematics, Vol 1.