Orbital Manoeuvers

To enter an orbit of a planet from an interplanetary trajectory, a spacecraft slows until it reaches the velocity of this orbit. Since it must compensate for the added velocity due to free fall in the gravity well, the vessel must provide an extra Delta-V equal to 40% of the circular velocity. For this reason, spacecraft aim at high orbits. Similarly, the same extra Delta-V is needed to leave a planet, thus it is also referable to depart from high orbits.

Circular and escape velocities

In addition to the orbital velocities given in pp. TS29-50, the GM who wants to calculate them in any circular orbit may use the formula:

V = 440 mps × square root of M / (D+2H),

where M is the mass of the planet in Earth masses, D is the planet's diameter and H is the altitude of the spacecraft, in miles. The period of revolution in these orbits is

T = (D+2H) / (V×1150) hours

The escape velocity from orbit is 1.4 times the circular velocity.

The rotation axes of the planets are tilted with respect to the ecliptic plane (where most of the objects in the Solar System orbit), thus after reaching orbital velocity, a spacecraft is in this plane rather than in the equatorial plane of the planet. Many satellites and stations orbit in the equatorial plane, and a plane change is necessary to reach them. The cost is small, usually around 1 mps, and should be considered for slow spacecraft only. The Delta-V required for transfer between two orbital planes is equal to

Orbital velocity × angle between the planes / 60.

For example, a spacecraft in GEO (orbital velocity 2 mps) going from the Earth to Luna must first transfer from the Earth equatorial plane to the ecliptic plane (containing Luna). Going from the equatorial plane (23.4° tilt) to the ecliptic costs only 2 × 23.4 / 60 = 0.8 mps. At 200 mi altitude (4.8 mps), the cost is 4.8 × 23.4 / 60 = 1.9 mps.
Planets' axial tilts with respect to the ecliptic:
Mercury 0.0° Jupiter 3.1°
Venus 2.6° Saturn 26.7°
Earth 23.4° Uranus 97.8°
Mars 25.2° Neptune 28.3°

Transfer orbits

Spacecraft trajectories are hyperbolas or ellipses that intersect the departure and destination planetary orbits. But the cheapest way to travel between orbits is to use a Hohmann transfer orbit: an ellipse tangent to the departure and to the destination orbits. If the departure orbit is larger than the destination orbit, the spacecraft departs from the apoapsis of the Hohmann orbit and arrives at the periapsis; and vice versa if the destination orbit is larger. The launch window of a Hohmann orbit must be carefully calculated to find the correct relative configuration of the bodies; which happens at most once in every period of revolution of the fastest body. However, it is a reasonable approximation to consider slow elliptical orbits as Hohmann orbits, and to allow spacecraft to use them 50% of the time, that is when the destination is on the opposite side of the orbited body with respect to the departure point.

The transfer time between two circular orbits of altitudes H1 and H2 about a planet of radius D (in miles) and mass M (in Earth masses)

square root of (D+H1+H2)3 / (M × 1,000,000) hours

For a transfer orbit about the Sun, the formula is

65 × square root of (R1+R2)3 days

with R1 and R2 the orbital distances in AUs. The total delta-V is approximately the difference between the circular velocities of the departure and destination orbits.

For example, a transfer from GEO or HEO to LEO (1240 miles) requires 1 to 1.3 mps for injection into the transfer orbit (possibly with the fusion drive), and 1 to 2 mps (provided by a tow vessel or auxiliary engines) for injection in the low orbit, for a travel time of 5h20m. The ascent is symmetric.

Nemtos