7 kilometer upper Phobos tether  tether doesn't collapse but remains extended
I used Wolfe's spreadsheet to find location of tether top where tether length Phobos side of L2 balances the length extending beyond L2. This occurs 6.6 kilometers from the tether anchor. Having the tether extend 7 kilometers is sufficient to maintain tension.
Safety
Factor

Zylon
Taper
Ratio

Tether to
Payload
Mass Ratio

1

1.01

.04

2

1.03

.06

3

1.04

.09

Benefits
Docking with a facility at the L1 or L2 regions is easier than landing on Phobos. In the words of Paul451: "Instead of a tricky rocket landing at miniscule gravity on a loosely consolidated dusty surface, you just dock with the L1hub of the ribbon (same as docking with ISS), transfer the payload to the elevator car and gently lower it to the surface. Reverse trip to bring fuel from Phobos to your ship (Assuming ISRU fuel is available on Phobos.)"
Also this small tether can serve as scaffolding on which to add longer tether lengths.
937 kilometer upper Phobos tether  transfer to Deimos tether
Given tethers from two coplanar moons tidelocked to the same central body, it is possible to travel between the two moons using nearly zero reaction mass.
Above I attempt to show how periaerion and apoaerion of elliptical transfer orbit matches velocity of the tether points this ellipse connects. Tether Vs are red, transfer ellipse'sVs are blue.
Above I try to explain the math for finding the tether lengths from Deimos and Phobos.
Trip time between the two tethers is about 8 hours.
Safety
Factor

Zylon
Taper
Ratio

Tether to
Payload
Mass Ratio

1

1.02

.035

2

1.04

.070

3

1.05

.107

With a safety factor of three, one tonne of Zylon could accommodate about 9 tonnes of payload.
I look at the Deimos tether here.
I look at the Deimos tether here.
Benefits
Easy travel between Deimos and Phobos is a benefit in itself.
But this would be a huge help to ion driven Mars Transfer Vehicles.
I like the notion of reusable ion driven MTVs. Ion engines have have great ISP thus allowing a more substantial payload mass ratio. However they have pathetic thrust. Andy Weir's fictional Hermes spacecraft can accelerate at 2 millimeters/sec^2. Which actually is very robust ion thrust. However ithis is only medium implausible. Low thrust means little or no planetary Oberth benefit. Plus a looong time to climb in and out of planetary gravity wells.
300 km above Mars surface in low Mars orbit, gravitational acceleration is about 3 meters/sec^2. For a 300 km altitude low earth orbit, gravitational acceleration is about 9 meters/sec^2. 2 mm/s^2 acceleration is less than 10^3 of the gravitational acceleration at initial orbit velocity in both these case. However I will be kind and go with Adler's .856 * initial orbit velocity.
At 2 millimeters/s^2 it would take Hermes 38 days to spiral out of earth's gravity well from low earth orbit and 17 days to spiral out of Mars gravity well. Most of the slow spiral out of earth's gravity would be through the intense radiation of the Van Allen belts.
I was very disappointed when Neil deGrasse Tyson's trailer had Hermes departing from low earth orbit and arriving in Mars' orbit 124 days later.
Besides adding 10 km/s to the delta V budget, climbing in and out of gravity wells would add about two months to Hermes' trip time. Tyson's video describes an impossible trajectory. I wish he'd fact check himself with the same enthusiasm he applies to others.
It would be much better for Hermes to travel between the edges of each gravity well. At least as close as practical to the edge. In earth's neighborhood, Hermes could park at EML2 between trips. In Mars' neighborhood, parking at Deimos would save a lot of time and delta V. From Deimos, astronauts and payloads can transfer to Phobos and then to Mars surface. In this scenario, Hermes' 124 day trip from earth to Mars is plausible.
2345 kilometer upper Phobos tether  Mars escape
If anchor in a circular orbit, escape velocity can be achieved if tether top is at a distance 2^(1/3) anchor's orbital radius. I try to demonstrate that here. Phobos is in a nearly circular orbit. To achieve escape, the tether would need to be 2435 kilometers long.
Safety
Factor

Zylon
Taper
Ratio

Tether to
Payload
Mass Ratio

1

1.11

.204

2

1.22

.436

3

1.35

.700

A 7 tonne Zylon tether could deal with a 10 tonne payload, even with a safety factor of three.
Benefits:
Achieve mars escape.
6155 km kilometer upper Phobos tether  To a 1 A.U. heliocentric orbit
A tether this long can fling payloads to a 1 A.U. heliocentric orbit, in other words an earth transfer orbit.
Safety
Factor

Zylon
Taper
Ratio

Tether to
Payload
Mass Ratio

1

1.80

1.57

2

3.24

4.77

3

5.82

11.16

With a safety factor of three, an 11.2 tonne elevator could lift a one tonne payload. Not great, but it'd be worthwhile if we were tossing lots of payloads earthward.
Benefits
Catch/throw payloads to/from earth. Phobos is about 24º from Mars orbital plane. Mars orbit is about 1.5º from the ecliptic. So there may be some plane change expense.
7980 kilometer upper Phobos tether  to a 2.77 A.U. heliocentric orbit.
Safety
Factor

Zylon
Taper
Ratio

Tether to
Payload
Mass Ratio

1

2.5

3.1

2

6.4

12.5

3

16.2

39.5

With a safety factor of three, it would take a 40 tonne Zylon tether to handle a 1 tonne payload. We would need to be tossing many payloads for this to be worthwhile.
Benefits:
2.77 A.U. is the semi major axis of Ceres. A tether this long could catch/throw payload to/from Ceres. But this doesn't take into account plane change because of Ceres inclination.
Even with plane change expense, this tether could be very helpful for traveling to and from The Main Belt.
This could also throw payloads into a faster than Hohmann transfer orbit towards earth.
This could also throw payloads into a faster than Hohmann transfer orbit towards earth.
3 comments:
Did I really write "elevator card"? ("Car", obviously.)
Anyway, have you considered a rotating throwing arm for Phobos? Kind of like a horizontal rotovator.
You have a short (compared to the space elevator) tower, at the top a rotating head with two arms. On one side is a winchable tether, the other side a counterweight of dead rock or waste. You spin the arms until centripetal force allows you to unwinch the payload tether (and the counterweight) without touching the surface of Phobos. At the right tangential velocity, and at the right moment, you release the payload (and the counterweight).
Much shorter tether length for any given velocity. Although the strength requirements go up.
Catching payloads seems unlikely. However, when going "uphill", circularisation deltav becomes more and more trivial with distance. And for escape trajectory, irrelevant.
Paul, typo fixed, thanks.
I am going to look at rotovators. Using Moravec's model for rotovators, the taper ratio climbs fast. So tether to payload mass ratio gets high quickly. When you toss in gravity to the ω^2 * r (a.k.a centrifugal acceleration), taper ratio doesn't shoot up as fast. That's a big reason why I like vertical tethers.
However I am still thinking of rotovators attached to a vertical Phobos tether. If I remember right, Phobos orbit is 27º from the ecliptic. Not good for throwing stuff into the ecliptic plane. That off plane vector is somewhat softened when added to Mars' 24 km/s vector. But still it would be nice to be able to throw stuff from planes other than Mars' equatorial plane. A rotovator in a plane perpendicular to Mars' local vertical could be helpful.
I also hope to look at a LEO rotovator. LEO's high debris density is a big incentive to minimize cross sectional profile. A rotovator's short length can make for a smaller cross section and thus less vulnerable to debris impacts.
I believe difficulty of catching will be connected to net acceleration at rendezvous point. We're all familiar with making catches and doing landings in a 9.8 meter/second^2 acceleration field. If net acceleration is a fraction of that, it'd be like doing a catch in slow motion. The Phobos tether to Deimos tether toss tether is pretty mild and shouldn't be hard to catch. A fastball from Ceres would be a harder catch for the Phobos tether.
Very interesting posts about tethers!
Off topic. Do you, who are geometrically interested and talented, know about and have any view about Norman Wildberger's view of math and geometry? His way of describing it using only integers, a ruler and a compass. Hating infinities. I'm not the one to judge the new math in it, but at least it is a new (or ancient) pedagogical take on how to present it.
He starts every of his thousands of lectures on youtube with the phrase:
"Hi! I'm Normal.
Wild burgers!"
◔_◔
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