Please support my efforts. I just finished a conic sections and orbital mechanics coloring book. I need help with printing costs. Through this Kickstarter you can preorder a signed coloring book. I look at conic sections, Kepler's laws, Hohmann transfer orbits, the Oberth effect, space tethers, Tsiolkovsky's rocket equation and lots of other space stuff. The coloring book is $5 plus $5 shipping and handling ($10 shipping and handling if you're outside the U.S.).
Kickstarter for this coloring book ends 4:30 a.m. April 13, 2020.
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A Phobos tether can be built in increments, it is useful in the early stages. So there's no pressing need to build a huge structure overnight. I will look at various stages of a Phobos tether, examining mass requirements and benefits each length confers. To model the tether I am using Wolfe's spreadsheet. I will use Zylon with a tensile strength at 5,800 megapascals and density of 1560 kilograms per cubic meter. Here is the version of the spreadsheet with Phobos data entered.
7 kilometer lower Phobos tether  tether doesn't collapse but remains extended
At a minimum, the lower Phobos tether must extend far enough past MarsPhobos L1 that the Marsward newtons exceed the Phobosward newtons. This will maintain tension and keep the elevator from falling back to Phobos.
I used Wolfe's spreadsheet to find location of tether foot where tether length Mars side of L1 balances tether length from Phobos to L1. That occurs when tether foot is about 6.6 kilometers from tether anchor:
So going past that a ways will give a net Marsward force.
Safety
Factor

Zylon
Taper
Ratio

Tether to
Payload
Mass Ratio

1

1

.000003

2

1

.000006

3

1

.000009

Even with a safety factor of three, a tenth of a kilogram tether (about 3 ounces) can handle a 10 tonne payload.
Benefits
Escape velocity of Phobos is about 11 meters/sec or about 25 miles per hour. A small rocket burn would be needed for a soft landing. This burn could kick up dust and grains of sand, some of which could achieve orbit. This would create an annoying debris cloud.
However a spacecraft could dock with a station at Mars Phobos L1 much the same way we dock with the I.S.S. Payloads could then descend the tether and arrive at Phobos without kicking up debris.
It would also allow low thrust ion engines to rendezvous with Phobos.
It would also serve as a foundation which can be added to.
It would take a Mars Ascent Vehicle about 5 km/s to leave mars and rendezvous with this tether. Trip time would be about two hours, so the MAV could be small.
From this Phobos tether, a .55 km/s burn can send drop a lander to an atmosphere grazing periapsis. Aerobraking can circularize to a low Mars orbit moving about 3.4 km/s. If Phobos is capable of providing propellent, much of that 3.4 km/s could be shed with reaction mass.
In contrast, a lander coming from earth will enter Mars atmosphere at about 6 km/s. Since it takes about 14 km/s to reach this point, the lander will not have reaction mass to shed the 6 km/s. For more massive payloads like habs or power plants, shedding 6 km/s in Mars atmosphere is a difficult Entry Descent Landing (EDL) problem.
87 kilometer lower Phobos tether  copper pulls it's own weight
It would be nice to have power to the elevator cars. However copper only has a tensile strength of 7e7 pascals and density of 8920 kilograms per cubic meter. Have copper wire along the length of the Zylon tether would boost taper ratio. Using the spreadsheet, I set tensile strength and density to that of copper and lowered the tether foot until I got a taper ratio of 1.1. That gives a length of about 87 kilometers.
Benefits
Along this length of the tether, copper pulls it's own weight, as well as supports the payload. A massive power source can be placed at L1  at L1 there are no newtons either Phobosward or Marsward. A copper only tether of this length would be about .2 times that of payload mass.
Elevator cars can ascend this length without having to carry their own solar panels and battery.
If descending from L1 Marsward, Mars' gravity can provide the acceleration and no power source is needed.
Of course copper wires can be extended further but this would boost taper ratio as well as tether mass to payload mass ratio.
From this tether foot, it takes .54 km/s to drop to an atmosphere grazing orbit. Trip time is about two hours.
1,400 kilometer lower Phobos tether  release to an atmosphere grazing orbit
Safety
Factor

Zylon
Taper
Ratio

Tether to
Payload
Mass Ratio

1

1.05

.1

2

1.1

.21

3

1.15

.33

Even with a safety factor of three, a 1 tonne zylon elevator could handle 3 tonnes of payload.
Releasing from the foot of this tether will send a payload to within a 100 kilometers of Mars' surface. Skimming through Mars upper atmosphere each periapsis will shed velocity and lower apoapsis.
Low Mars orbit velocity is about 3.5 km/s. The payload arrives at 4.1 km/s.
4,300 kilometer lower Phobos tether  payload enters atmosphere at 3 km/s.
Safety
Factor

Zylon
Taper
Ratio

Tether to
Payload
Mass Ratio

1

1.9

2.6

2

3.4

8.3

3

6.3

20.4

At 4,300 kilometers from Phobos, dropping a payload will have an atmospheric entry of 3 km/s, about .5 km/s less than low Mars orbit.
5800 kilometer lower Phobos tether  maximum length
Phobos orbit has an eccentricity of .0151. It bobs up and down a little. Mars' tallest mountain is about 25 kilometers tall. Given these considerations, tether can't be more than 5800 kilometers. Else the foot might crash into the top of Olympus mons.
Safety
Factor

Zylon
Taper
Ratio

Tether to
Payload
Mass Ratio

1

4.4

16.1

2

19.2

114

3

83.8

638

Given a reasonable safety factor of three, it would take a nearly 640 tonne elevator to lift a one tonne payload. I don't think a Zylon elevator from Phobos to Mars' upper atmosphere is practical.
The tether foot will be moving about .57 km/s with regard to Mars. Mars Entry, Descent and Landing (EDL) is far simpler with .57 km/s. If Phobos is a source of propellent, much of that .57 km/s can be taken care of with reaction mass.
For an ascent vehicle, only a small suborbital hop is needed to rendezvous with the tether foot.