Delta-v budgets and modified gravity
I'm building a soft sci-fi world where gravity works a bit differently: the acceleration due to gravity drops off at a rate much greater than our inverse-square; and the acceleration due to gravity "flips" and oscillates at regular intervals, becoming repulsive rather than attractive and vice versa (similar I guess to MOND, but at smaller, planetary scales).
The acceleration $g$ experienced by an object of negligible mass can be modeled with:
$g\approx Se^{-\lambda du}\cos(du+\phi )$
where
* $S$ - gravitational acceleration at a planetary surface1 (in $m/s^{2}$),
* $\lambda$ - dimensionless decay constant (describes how gradual force drop-off is),
* $d$ - distance from the surface in meters,
* $u$ - scalar constant2 (converts meters to radian angle measure),
* $\phi$ - phase angle, almost always zero and can be ignored.
1If you follow the math through, the gravitational forces below the surface oscillate with exponentially greater magnitudes, which would obviously rip a planet to bits. So, the formula is only good for gravity above the surface. Below, a different (and unimportant) set of rules apply.
2$u=0.5\pi /d_0$, where $d_0$ is the distance between two zero-g points or $1/2$ the cosine wavelength.
$e^{-\lambda du}$ and $\cos(du)$ are dimensionless, leaving the result in the units of $m/s/s$. Multiplying the result by the object's mass gives force applied to the object, which may be positive or negative.
Because this is Worldbuilding, a lot of these parameters are toggleable and don't (seem to) arise from fundamental interactions. This is okay, I think. I don't need to fully flesh out and justify why these constants are what I choose them to be (or even why gravity is different), I can simply tuck them away to a black corner and work with their output.
There are some strange properties with this form of gravitation. Here's a graph:
$\lambda =0.5, S=1$
As an object rises above the surface, gravitational acceleration drops until, at some distance $\pi /u$, acceleration becomes zero and inverts, turning repulsive. The force vectors point away from the planet surface and the object is accelerated away until it crosses the recurring zero-g point and begins decelerating.
However, because the distance over which acceleration/deceleration occurs is constant and the magnitude of acceleration/deceleration always decreases, a non-propulsive object will always accelerate away from the planet--forever--once it has crossed the initial zero-g point above the planet's surface. To even approach the planet, an object must have some minimum "entry velocity."
What do orbits look like?
Gravity oscillates between positive and negative, attractive and repulsive. 512
(In the graph, attractive regions are solid.) Stable orbits seem to come in two varieties: orbits that remain solely inside attractive regions (largely circular), and orbits that oscillate between attractive and repulsive regions (largely sinusoidal). Highly elliptical orbits are impossible; if an object has enough orbital energy to cross more than one zero-g point, it'll be ejected from the system. Trajectories that nearly cross a second zero-g point are "critical."
(Attractive regions in black, repulsive in dark grey. Paths traced w.r.t non-rotating body at the center.)
Here we see an object (second image) on a critically oscillating trajectory, falling toward the planet though not having enough kinetic energy to overcome the repulsive region and being accelerated back out--though not great enough to enter the second repulsive region--ad infinitum. Tangential velocity seems almost independent of radial, meaning two objects can share an orbital region and yet have wildly different velocities compared to one another. Hence the thicker band of traced orbits (representing faster motion) within the larger, slower, more sparse trajectory.
Stable orbits only exist inside attractive regions, where the force vectors point toward the planet. Circular trajectories in these regions can have wildly different orbital velocities due to the large gravity gradient. At the peak of the region, where the force vector is strongest, orbital velocity is highest, whereas to either side of the peak, velocities decrease and approach zero. Geosynchronous trajectories are not limited to the space of the zero-g regions, but may also oscillate between attractive and repulsive regions. 246