Lower Phobos Tether

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 Mars-Phobos L1 that the Mars-ward newtons exceed the Phobos-ward 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 Phobos-ward or Mars-ward. 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 Mars-ward, 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.

Benefits

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

Benefits

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.

Benefits

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.

How Wolfe's tether spreadsheet works

I plan to do a series of posts examining elevators and tethers. I will link to them as posts are completed:

Lower Phobos Tether

Upper Phobos Tether

Deimos Tether

Mars Elevator

Liftport Lunar Elevator through EML1

Lunar Sky Hook

Trans Cislunar Railroad

LEO Rotovator

Pluto Charon elevator

They will be based on Chris Wolfe's spreadsheet for modeling tethers.

I'll try to explain how Wolfe's spreadsheet works.

Tensile strength

Density and tensile strength are important quantities for tether material. Tensile strength is measured in pascals.

A pascal is a newton per square meter, newton/(meter²). A newton is a unit of force, mass times acceleration.

Zylon has a tensile strength of 580 megapascals or 580 meganewtons per square meter. On earth's surface with it's 9.8 meter/sec² acceleration, it would take a 591,836,735 kilogram mass to exert that much force. It would take a zylon cord with a cross section of one square meter to support this force. But that's more than half a million tonnes!

10 tonnes is more plausible payload for space cargo. A much thinner cord could support this. Cross section of a Zylon cord need only be 1.72e-9 square meters. If a circular cross section, cord would be about 47 micrometers thick. Strands of hair can be anywhere from 17 to 181 micrometers thick.

So number of newtons determines tether cross sectional area.

How many newtons?

How to figure number of newtons at the tether foot? First we set maximum payload mass as well as foot station mass. The default in Wolfe's spreadsheet is a ten tonne payload mass and a foot station massing 100 kilograms. But how many newtons does this 1,100 kilogram mass exert?

The net acceleration on this foot mass is acceleration from planet's gravity minus centrifugal acceleration minus moon's gravity.

(Click on illustration to embiggen)

This spreadsheet sets the origin at the planet center.

Tether foot radius is the foot's distance from planet center.

Barycenter radius is Orbital Radius * mass planet / (mass moon/(mass planet + mass moon)

Tether anchor radius is Orbital Radius - Moon Radius. The tether anchor is assumed to be at the near point of a tide locked moon.

Distance from Barycenter to Tether Foot is Tether Food Radius - Barycenter Radius.

The three force equations:

Gravity Planet = G * Mplanet / Tether Foot Radius²

Centrifugal Acceleration = ω² * Distance from Barycenter to Tether Foot. ω is constant, it is the angular velocity of the orbit.

Gravity Moon = G * Mmoon / (Orbital Radius - Tether Foot Radius)²

Net acceleration is the sum of these three.

An illustration of the accelerations with net acceleration in red. Moon gravity is negative because it is pulling away from the planet. Centrifugal acceleration is also pulling away from the planet except left of the barycenter it is towards the planet. 

When a curve crosses the axis the value is zero. Centrifugal crosses the axis at the barycenter. In most cases barycenter will be beneath planet surface. The illustration above has an exceptionally large moon. 

Net acceleration crosses the axis at L1, at this point the three accelerations sum to zero. to the right of L1, net acceleration is towards the moon.

To approximate the tether we chop it into many small lengths:

To find tether volume in step 1, we multiply the cross section by length of step 1. (Recall cross sectional area is set by number of newtons coming from tether foot.) Multiplying this volume by tether density gives step 1 tether mass. Multiplying this mass by net acceleration gives us the newtons this length exerts.

Adding the newtons from step 1 to payload newtons means the next step has a thicker cross section. We multiply this new cross section by tether length * tether density * net acceleration to get newtons from the tether length along step two.

And so on.

Summing all the masses from each step gives us total tether mass.

This is an approximation. The finer we chop the tether, the closer the approximation. The spread sheets we'll be using cut the tether length into 1,000 parts.

Our sheet can be found here. It is a 1.7 megabyte file.

For an upper moon tether, anchor will be on the far side. Moon's gravity will be added instead of subtracted from planet's gravity. I'll label tether end "Tether Top" instead of "Tether Foot".  Otherwise, the spread sheet will be the same as the lower moon tether spreadsheet.