EOE-006 Interstellar navigation

Some equations

For the Newtonian case, we are simply integrating ΣFj = m a in acceleration units, including the thrust acceleration vector f/m and the gravitational accelerations from N mass Mi objects located at xi:

.

For the anyspeed case using only proper forces (below) including gravity's proper-force approximation, we switch over to frame-invariant proper time τ instead of map time t as the independent variable [1]. These two time variables are related by:

.

We then integrate the 3-vector proper-force net-acceleration equation dw/dτ = γαo||w+αo⊥w = αo+(γ-1)αo||w to which gravity's contribution (using its associated proper force Σmg' = Σm(g||w+γg⊥w) where gravitational acceleration g = GM/r2) simplifies to γΣg:

.

Since at low speeds γ ≈ 1, this reduces to the Newtonian integration in that limit.

To see if we need to also incorporate gravitational time-dilation (or even geometric-force gravity), we might test with a "mean-field" gravitational time-dilation factor of the form:

.

When many objects contribute to making this quantity significantly larger than one, it may be easiest simply to factor it into a hybrid γ ≡ dt/dτ by multiplying it with the motion-related γ given above.  

This will give some clues to the size of gravitational time-dilation effects, but some of the detailed dynamics of orbits around dense objects (especially at high speed) will likely be left out. The strengths and weaknesses of this approximation remain to be examined closely.

To address the shortcomings when in the neighborhood of a single dense object which is making γr greater than 1, we might further decide to switch over to a 4-vector integration using the Schwarzschild (non-rotating object) or Kerr (rotating-object) metric. For details on that here including an on-line simulator [2], stay tuned...

Footnotes