Computing Orbital Velocities

Reference:
Project Rho: Atomic Rockets by Winchell Chung

Computing orbital velocities is done via the following equation:

OV = SQRT[(G * M) / R]

Where:

OV = Orbital Velocity of the object in m/sec.
G = Gravitational Constant (0.00000000006673)
M = Mass of the System Primary in kg (The sun's mass is 1.989e30)
R = Distance between the object and the system primary in meters (semi-major axis or orbital radius)

If you just want to calculate Orbital velocities for our Solar System, it becomes:

OV = SQRT(1.33e20 / R)

Where

R = Distance between the object and the sun in meters (semi-major axis or orbital radius).

If you are moving from one orbit to another in a non-Hohmann (Brachistochrone) trajectory, you need to match orbital speeds with the target. You can figure out the delta vee you need to match orbits via computing the orbital velocities for the target and destination then subtracting the smaller from the larger. That gives you the D/V you need to shed/gain.

Example: We are burning from Earth to Mars in a torchship. How much delta vee do we need to have in excess to match orbital velocities?

Mars Orbital Velocity: SQRT(1.33e20 / 227,939,100,000) = 24,156 m/sec
Earth Orbital Velocity: SQRT(1.33e20 / 149,598,261,000) = 29,817 m/sec

24,156 m/sec – 29,817 m/sec = 5,661 m/sec

Thus, we need to have 5,661 m/sec of delta vee in order to match orbits in a Brachistochrone trajectory.