Direct Trajectory
(Brachistochrone)
Calculations

Reference:
Project Rho: Atomic Rockets by Winchell Chung

Note I: For Brachistochrone trajectories to work within the solar system, the acceleration must be significantly above 0.004 m/sec; which is roughly the gravitational pull that the Sun exerts on a body within the solar system.

Note II: The vehicle must also match the orbital velocity of it's target at the end of the trajectory. This is very important. To calculate orbital velocities, go HERE. For a list of precomputed Orbital Velocities, go HERE.

Continuous Thrust Trajectory Equations

TimeTravel = 2 * SQRT (Distance / Acceleration)

TimeTravel is in Seconds
Distance is in Meters
Acceleration is in M/Sec.

DeltaVee = 2 * SQRT (Distance * Acceleration)

DeltaVee is in M/sec
Distance is in Meters
Acceleration is in M/Sec.

Thrust-Coast Trajectory Equations

Timothy Charters worked out the following equations for spacecraft which must coast during the midpoint phase:

TimeTravel = ((D – (A * TimeAccelerate2)) / (A * TimeAccelerate)) + (2 * TimeAccelerate)

TimeTravel is total travel time in seconds
TimeAccelerate is time spent accelerating in seconds.
D is Distance in Meters
A is Acceleration in M/Sec.

NOTE: If you want to calculate the duration of the coast, use the following equation:

TimeCoast = TimeTravel – (2 * TimeAccelerate)

Rules of Thumb:

1 Astronomical Unit (AU) = 149,598,000,000 meters (or 1.49 E11)
1 Gee = 9.81 m/sec
1/10th Gee = 0.981 m/sec
1 Hour = 3,600 Seconds
1 Day = 86,400 Seconds
1 Month (30 days) = 2,592,000 Seconds
1 Year (365 Days) = 31,536,000 Seconds