Lunar and Solar Tides
Trajectories from the Launch Loop (and the destinations it aims at) will be modified by lunar and solar tides. All these objects, and the Earth they orbit, are in the gravitational fields of the Moon and Sun. If there was no difference in the gravitational acceleration from Moon and Sun, they would all be accelerated in unison and show no relative acceleration. Instead, the relative acceleration varies with tidal forces, which can be approximated as a perturbing cube law force in the direction of the perturber, and proportional to the dot product of the earth-object vector and the earth-perturber angle.
Let's take that apart and see what it means. We will start with the Earth-Moon and object system; assume the object is much smaller than Earth or Moon, and affects neither noticably.
The gravitational acceleration of the Earth towards the Moon is G M_m / R^2 , where \mu_m = G M_m is the Moon's gravitational parameter. Here are six objects and gravitational parameters we will be considering, and their typical distances from an object in geostationary orbit:
Object |
Gravit. Param. |
Distance |
Accel |
tidal |
GEO tide |
|
|
\mu m3/s2 |
R meters |
m/s2 |
1/s2 |
m/s2 |
|
Earth |
3.9860e14 |
4.22e07 |
to GEO |
2.2e-01 |
5.3e-09 |
|
Moon |
4.9028e12 |
3.85e08 |
Average |
3.3e-05 |
8.6e-14 |
3.6e-06 |
Sun |
1.3271e20 |
1.49e11 |
Average |
6.0E-03 |
4.0E-14 |
1.7e-06 |
Jupiter |
1.2669e17 |
5.9e11 |
Closest |
2.3E-07 |
5.9E-18 |
2.5e-10 |
Venus |
3.2486e14 |
3.8e10 |
Closest |
3.6E-07 |
6.2E-19 |
2.6e-11 |
Mars |
4.2828e13 |
5.6e10 |
Closest |
1.4E-08 |
2.4E-19 |
1.0e-11 |