Navigation Equations

For the distance d travelled with initial acceleration a1 and final deceleration a2 the values of interest for the journey are the maximum velocity at the turning point (where we change from acceleration to deceleration) vu, the distance d2 of the turning point from the target and the time td that is needed for travelling the distance d.

We assume that forces other than those produced by our own engines can be neglected and that we start and reach the target with zero velocity. Furthermore we assume that the engines are used all the time at full power.

If the main engine is used for acceleration <#179#>and<#179#> deceleration the equations are very simple:

#math158#
vu = #tex2html_wrap_indisplay2056#
, *1cm#tex2html_wrap_indisplay2057#, *1cm#tex2html_wrap_indisplay2058# 		(2)

This fastest mode of flight is used to accelerate up to the turning point velocity at the distance d2, turn the ship and decelerate until the target is reached.

If the autopilot is used instead, it will use the main engine for acceleration and the secondary thrusters (retro thrusters) for deceleration. In this case we get the equations:

#math159#
vu = #tex2html_wrap_indisplay2062#
, *1cm#tex2html_wrap_indisplay2063#, *1cm#tex2html_wrap_indisplay2064# 		(3)

The time td is always a lower bound for the really used time because the autopilot doesn't use the engines at full power all the time.

I approximated the real time consumption of autopilot flight with a least mean square fit of data from 20 undisturbed flights with a Cobra MK III. Measuring the distance in astronomical units [AU] and the acceleration in units of earth gravitational acceleration [g], I got for the time [d]:

#math160#
tdA = #tex2html_wrap_indisplay2070#  #tex2html_wrap_indisplay2071# - 0.001 (4)