A Family of Conic Sections
Below is a general vertical space elevator. The conic sections are the paths payloads would follow if released from a point on the tether a distance r from body center.
We choose our units so radius of the balance point is 1. Centrifugal acceleration matches gravity at the balance point and net acceleration is zero. For tether locations above the balance point, centrifugal force exceeds gravity and net acceleration is up (away from the planet). For locations below the balance point, gravity is greater than centrifugal acceleration and the net acceleration is down.
This family of conic sections are coplanar, coaxial and confocal. Eccentricity is r3-1, setting r = 1 at the circular orbit of the balance point. (See this stack exchange answer for the math).
In the yellow region are hyperbolic orbits. In the blue region are are elliptical orbits higher than the circular orbit at the balance point. In the orange region, the tether drops payloads into elliptical orbits lower than the circular orbit at the balance point.
A circle of eccentricity zero separates the orange and blue regions, radius of circle = 1.
A parabola of eccentricity 1 separates the blue and gold regions, radius of parabola's periapsis = 21/3
Here is the same graphic zoomed in:
Here is the graphic as a Scalable Vector Graphic. I am hoping science fiction writers and illustrators will download this resource and use it.
Scaling this graphic for a variety of scenarios:
The numbers are in kilometers. In the case of earth, the circular orbit is the geosynchronous orbit at an altitude of about 36,000 kilometers.
In general, radius of a synchronous orbit can be described as:
r = (Gm / ω2)1/3
Where ω is the body's angular velocity in radians, 2 pi radians/sidereal day.
We usually think of an a space elevator anchored at the body's equator. An elevator can also be in a non synchronous orbit. Here the template is scaled to match the orbits of Phobos or Deimos:
Notice Phobos' tether foot is above Mars surface. The foot is moving about .5 km/s with regard to Mars surface and therefore can't be anchored to Mars. Neither could a Deimos elevator be attached to Mars.
Orbital radius of Phobos is about 40% that of Deimos. So I cloned and shrunk Deimos' tether conics by 40%. I rotated the cloned family of conics by 180º. The result is an interesting moiré pattern:
It was this pattern that led me to search for a common ellipse.
Eccentricity of the common ellipse:
e = (1 - (ωDeimos/ωPhobos)1/2) / (1 + ωDeimos/ωPhobos)1/2)
Periapsis and apoapsis of the common ellipse:
rperiapsis = (1 + e)1/3 rPhobos
rapoapsis = (1 - e)1/3 rDeimos
This is an example of a Zero Relative Velocity Transfer Orbits (ZRVTO) - a term coined by Marshall Eubanks. In Marshall's words: "locations (and times, say for a Lunar and Terrestrial space elevator) where you drop things from one space elevator and they approach and hang motionless (for an instant) at a location on the other elevator. ... what you would want for large scale movement of material."
Eubanks goes on to say "In practice, you might need a little bit of course correction delta-V to make up for radiation pressure, etc."
Also it would be rare for the elevators playing catch to be perfectly coplanar. So a small plane change delta V expense will be the rule rather than the exception. Still the delta V budgets would be a small fraction of what it would take for normal lift off and insertion to Hohmann transfers.
Not just Phobos and Deimos
To be an anchor for a vertical elevator, a moon needs to be in a near circular orbit and tide locked to its planet. This describes most of the moons in our solar solar system. For two moons to share an ellipse, they need to be nearly coplanar. Again, most the moons in our solar system.
Here are the common ellipses between the moons of Saturn:
Judging by the two gas giants and two ice giants in our solar system, families of coplanar, tidelocked moons are common.
Mini Solar Systems
Earlier I had looked at Mini Solar Systems, a notion I stole from Retrorockets. In our solar system Hohmann trip times between planets are on the order of months or years. Launch windows are typically years apart. But for a system of moons around a gas giant, trip times and launch windows are days or weeks. So a Flash Gordon paced story could take place without wildly improbable engineering.
GIELO and ELM
GIELO - Giant In Earth Like Orbit. ELM - Earth Like Moon. I have long been infatuated with this setting. Here is a painting I had done in 2001:
ELM the earth like moon is in the upper right. In the foreground a generation star ship is sending quad pod scout probes to investigate an artifact at the GIELO-ELM L4 region.
James Cameron's Avatar uses such a setting. Pandora is an ELM. I believe this setting could be developed a lot more. If ELM had sister moons and they were all tide locked, it would be a nice mini-solar system setting.
Icey moons with hospitable interiors.
Gas giants in Goldilocks zones aren't the only possibility. Temperature and pressure rise as we burrow deeper into a body. Earth might not be the only location in the solar system that has liquid water at a livable pressure. Thus the icey moons of our own solar system might eventually become "mini solar systems".
Planets of red dwarfs
And recently an approximately earth sized planet was found in the goldilocks zone of Proxima Centauri. Proxima Centauri is a small red dwarf star. The possibly earth like planet has an orbital radius of about 7.5 million kilometers and an orbital period of about 12 days. Planets about small red dwarfs are yet another possible "mini solar system" setting. Planets so close are likely tide locked to the star. Would atmospheric convection mitigate the temperature extremes between the night side and day side? I'm not sure. In any case, I believe there would be a comfortable region hugging the planet's frozen terminator. (By "frozen" I mean stationary).
Delta V and the rocket equation
The Retrorockets guy took a second look at mini solar systems. While trip times are short and launch windows frequent, it still takes a lot of delta V to insert to a Hohmann transfer. I was annoyed he used the incorrect Tom Murphy method of patching conics. But most his math is sound. He is correct that Tsiolkovsky's rocket equation would be a major pain in the mini solar system just as it is in ours.
This is where a tether system comes into play. Given elevators on tide locked bodies and assuming most the bodies are nearly coplanar, travel between bodies could be done with very little reaction mass. It'd still take a lot of energy to move stuff up and down the elevators. But the difficult mass fractions imposed by the Tsiolkovsky's equation would no longer be a consideration.
Similar mathematical models and drawings can be used for a wide range of vertical tethers.
A popular misconception is that elevators are only good for getting off the ground. So it's a waste to build an elevator from a small body. But an elevator not only gets the payload off the ground, it can fling a payload towards a destination. The hyperbolic orbits portrayed in this post are especially interesting.
Space elevators would be especially useful in a system of tide locked moons. Or tide locked planets about a small star.
So far the only elevators I see portrayed in science fiction are from major planets. Like Kim Stanley Robinson's elevators in his Mars trilogy. Or Clarke's earth elevator in Fountains of Paradise. There are far more plausible elevators that could be very useful. These doable elevators could also provide many interesting settings.