# Trajectory effects of 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 LEO tide \mu m3/s2 R meters m/s2 1/s2 m/s2 m/s2 Earth 3.9860e14 4.22e07 to GEO 2.2e-01 1.1e-08 2.2e-01 Earth 3.9860e14 6.48e06 to LEO 9.5e-00 2.9E-06 9.5e-00 Moon 4.9028e12 3.85e08 Average 3.3e-05 1.7e-13 7.3e-06 1.1e-06 Sun 1.3271e20 1.49e11 Average 6.0E-03 8.0E-14 3.4e-06 5.2e-07 Venus 3.2486e14 3.8e10 Closest 3.6E-07 1.2E-17 5.0e-10 7.7e-11 Jupiter 1.2669e17 5.9e11 Closest 2.3E-07 1.2E-18 5.2e-11 8.0e-12 Mars 4.2828e13 5.6e10 Closest 1.4E-08 4.9E-19 2.1e-11 3.2e-12

The tidal acceleration (last two columns) is small compared to the gravitational acceleration - the tidal effect is proportional to the radius, while gravity is the inverse square of radius, so the proportions are the inverse cube of the distance. The tidal effects drop off to 50% at 80% of the radius, and the radius itself is decreasing as approximately the square law of the time, so about 90% of the tidal effect occurs above 90% of the radius, and 40% of the transfer orbit time.

This suggests the best time to apply corrective ΔV is near launch. For a GEO to LEO transfer, that is during the first two hours, and the corrections will be on the order of 10cm/sec (out of 1500 m/s launch velocity).

Tides (last edited 2013-01-15 20:41:25 by KeithLofstrom)