The basis for the analysis is the assumption, that at the arrival of the lunar probe at the Moon the Moon must lie in the orbital plane of the probe at launch.
Beta = eastward angle along
the equator from the ascending node to the meridian through the launch
site.
Fi = latitude of the launch
site
i = inclination of the trajectory
From spherical trigonometry we know that:
sin(Beta) = tan(Fi)/tan(i);
For a launch from Baikonur (Fi = 45.9235 oN) into a i = 51.6o trajectory, Beta becomes = 54.94o
The assumption is that for launches from the northern hemisphere the ascending node of the transfer orbit must be on the opposite side of the earth from where the Moon is at the time of arrival. So, the approximate launch time (LT) can be computed:
LT = (Lunar Age at launch + flight time to the Moon)/(duration in days for one lunation) + (Beta-Launch Site Longitude)/360;
With launch site longitude lo= 63.3392 oE for Baikonur and with i = 51.6o the formula reduces to
LT = (AGE+FT)/29.5 + (54.94-63.3392)/360.0 = (AGE+FT)/29.5 - 0.02333;
Thus, we obtain:
| Mission | Flight time to the Moon | LAUNCH TIME (UT fraction of day) |
|---|---|---|
| 3 rd gen Lunas | 4.3 days | AGE/29.5+0.1224 |
| Zond | 3.3 days | AGE/29.5+0.0885 |
If one fits the real launch
times to the AGE it turns out that these equations have to be amended slightly
- a constant of 0.0529 days should be added. This may be caused by the
fact that the trajectories were not aimed straight at the Moon, but in
front of the Moon. For a very long flight time to the Moon, the motion
at transfer orbit apogee would be very slow and therefore maybe the aiming
point was further in front of the Moon? Anyway, the following empirical
equations can be derived:
| Mission | Flight time to the Moon | LAUNCH TIME (UT fraction of day) |
|---|---|---|
| 3 rd gen Lunas | 4.3 days | AGE/29.5+0.1753 |
| Zond | 3.3 days | AGE/29.5+0.14795 |
Figure 1. Launch time
as function of Lunar Age for Zond-type flights to the Moon.
Copyright © 1996
Sven Grahn
