Direct
Trajectory |
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
Time_{Travel} = 2 * SQRT (Distance / Acceleration)
Time_{Travel}
is in Seconds
Distance
is in Meters
Acceleration
is in M/Sec.
Delta_{Vee} = 2 * SQRT (Distance * Acceleration)
Delta_{Vee} 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:
Time_{Travel} = ((D – (A * Time_{Accelerate}^{2})) / (A * Time_{Accelerate})) + (2 * Time_{Accelerate})
Time_{Travel}
is total travel time in
seconds
Time_{Accelerate} 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:
Time_{Coast} = Time_{Travel} – (2 * Time_{Accelerate})
Rules of Thumb:
1 Astronomical Unit
(AU) =
149,598,000,000 meters (or 1.49 E11)
1
Gee = 9.81
m/sec
1/10^{th}
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