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.
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 L1-hub 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.)"
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 peri-aerion and apo-aerion 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.
Zylon taper ratio for a 937 kilometer tether length is 1.02. Tether to payload mass ratio is .0448. Or the tether is about 1/20 the mass of the payload.
I'll look at the Deimos tether in a later post.
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 lo-o-o-ng 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.
Zylon taper ratio: 1.11. Tether to payload mass ratio: .23. A little more than 1/5 of the payload mass.
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.
Taper ratio: 1.8. Tether to payload mass ratio 1.6. The tether mass is nearly double payload mass.
Catch/throw payload to/from earth.
7980 kilometer upper Phobos tether - to a 2.77 A.U. heliocentric orbit.
Zylon aper ratio: 2.53. Tether to payload mass ratio 3.21. A little more than triple payload mass.
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.