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.
10 comments:
Have you looked at spectra at all? It's already used in space missions such as the airbags on the Mars Exploration Rovers and the tether on Curiosity's rocket crane.
https://www.honeywellspectra.com/products/fibers/
Peter, I haven't. I want to but only have so much time and energy.
Do you know the tensile strength and density? If so you can input those and set various locations for tether foot. The spreadsheet is linked to above. I am hoping readers of this blog will use the spreadsheet to explore different scenarios.
I will post a spread sheet for the upper Phobos tether soon. For most of these blog posts I'll be looking at Zylon. Someone has mentioned I should also look at basalt fiber as that is a resource locally available at many interesting sites.
This link has a handy chart listing GPa and density for various kinds of Spectra:
https://www.honeywellspectra.com/products/fibers/
Another material that might be locally sourced would be silica nanofibre. When stretched very thin, optical fibers become ductile and quite strong. From my conversation with Dr. Brambilla, the silica nanofibre they produced had a tensile strength between 10 and 20GPa and a density of ~2.2g/cm3 (theoretical maximum strength is above 30 GPa).
Hmm.. I see you use a method with small tether increments, I have a tool for calculating the same tether properties using exact integrals. I can send it to you if you want. It use the the graphical interface of GeoGebra, so you must install that.
BTW, isn't the tensile strength for Zylon 5.8 GPa, not 580 MPa?
Sigvart, yes, I'd like to see the exact integrals.
Thanks for catching that typo. On my spreadsheet the tensile strength is 5.80e9 pascals. I will correct the blog post typo after posting this comment.
Dear David,
As a space enthusiast (as opposed to actually understanding anything) I was wondering of a variant that I hadn't seen discussed amongst the elevator/skyhook/rotorvator concept.
I was thinking of a "skipping rope" extending from (at or near) the poles of a body and meeting over the equator. I was trying to think how it would curve, and wonder if it would bow "outward" minimising the length in the atmosphere. In that scenario, you might be able to have it in a fasterthangeosynchronous orbit, e.g. an 8hour orbit, with the apex being fast enough to launch further. The motion relative to the surface at the pole would be small, although it would increase towards the equator.
Depending on the maths I suspect the takeoff angle might enable the use of airbreathing motors for a significant part of the ascent
On Earth, Location would be less difficult than circumequitorial elevators as one would be in Antarctica (which is technically not owned by signatories of the Antarctica conventions, and boundaries around the pole itself are vague) and the other would be over the arctic, which has a depth of about 1km  not insurmountable with modern construction techniques.
If atmospheric drag effects were permitting, you might even have it significantly lower? Meaning you could lift it in sections at a lower equitorial orbit, capture secondary segments from a more excentric orbit (e.g. extend the initial cable at LEO parrallel with the equator, send secondary segments in more eccentric orbits, rotating to match velocities, use aerobraking to control rotation) and extend from the ends once connected to ground, with the orbital period lengthening as the cable lengthens?
An added benefit of the curve might be using an active system to prop up the weight (e.g. a space fountain) with less exotic science than particle accelerators, and you might be able to use a relatively large mass secondary cable of nearly half the length that swings from one side to the other giving a "gravity train" effect  an antigravity train, essentially. Thrust equivalent to payload could be given at lower altitudes; You just have to move the payload up far enough that an equivalent mass of the train passes the apex, and apart from friction losses, the rest is "free" (once constructed, at least)
Basically the two biggest possible advantages I could think of are 1) the gravitytrain effect  especially in a tidallocked body like the moon of mars's moons, and 2 that the initial ramp is likely to be close to perpendicular to gravity, meaning atmosphere can be used to increase motor output / decrease required thrust in the initial stages.
Of course there would be a hazard to LEO but that is true of all space elevator concepts.
I of course understand if this is stupid, and of course the material requirements for earth would remain similar. The gravitytrain side of things would make lunar or martian orbit operations easier, though.
And one last thing; although I don't believe in "free energy" isn't there a power generation possibility simply by being a long conductor in the earths magnetic field and van allen belts?
Apologies again for the random ideas. I'm just a (medical) doctor, not a doorstop.
Lauren,
Thanks for your comment! No apologies needed. My own credentials: I graduated in the upper 60% of the Ajo High School class of 1973. So your background is more impressive than mine. But I love space  I Google a lot, participate in space forums when I have time, and buy textbooks. Sometimes I buy textbooks new but my favorite text is a thick calculus book I bought for 50¢ at a yard sale.
While encouraging you to keep on commenting and studying, I hope you won't take offense at my critique of your ideas.
So called centrifugal force is ω^2 * r. ω is angular velocity. For example on earth's surface ω is a full revolution every ~24 hours. Expressed in radians per second that would be 2 pi radians/(24 * 60 * 60 seconds).
Second part of that expression is r. That's the distance from a point to axis of rotation. At the surface of earth's poles r would be zero. At high latitudes it remains small but gets bigger as a point on earth's surface approaches the equator.
On earth ω^2 * r cancels earth gravity at altitude of ~36,000 kilometers about the equator. At lower orbits gravity exceeds ω^2 * r. At higher latitudes, gravity greatly exceeds ω^2 * r.
If the ends of the jumping rope were on earth, it would have to moving a lot faster than a revolution per day for the rope to remain aloft. And the higher latitude portions where gravity greatly exceeds ω^2 * r would induce a great deal of stress.
So I don't think this scheme is viable.
As for your idea of harvesting energy by dragging tethers through a magnetic field? That's interesting. However Lorentz force would sap orbital momentum over time bringing the tether's orbit down. Eventually it'd fall into the atmosphere and burn up. Restoring the orbit would take as much energy as was harvested.
If the tethers were anchored to a large momentum bank, this would be a good energy source. I believe Charles Stross used this scheme in some of his stories where tethers from Jovian moons harvested energy from Jupiter's powerful magnetic field.
Lauren:
I'm not clear on whether you are talking about something like this:
https://en.wikipedia.org/wiki/Launch_loop
the near polar elevator in this:
https://en.wikipedia.org/wiki/Lunar_space_elevator
or a third idea that is quite different from either.
Could you link to something with diagrams to make it clear what you are discussing?
Hop, I graduated in the bottom 10% of Stonewall Jackson class of 1968 (Manassas, VA) and I think it's a miracle I can follow the math you present. You have a real talent for clearly explaining stuff I figure was beyond me. Btw, I found a twovolume copy of Ramsay's Dynamics at a yard sale decades ago, and have used it ever since.
Indeed Hop, your illustrations and explanations make the subject matter come to life.
I especially enjoyed seeing the relative accelerations come together in such a neat diagram; it was clear from the numbers but the visual depiction is so much more intuitive. In a broader sense, your writings and diagrams helped me understand orbital mechanics in a way that physics courses didn't.
@Sigvart Brendberg, there are published formulae for exact solutions and a couple of ways to get results. I used the method of thin slices because my calculus skills are terribly atrophied, because I wasn't sure which of several formulae to choose and because I was able to step through each part of the algebraic process both in math and in words and be confident that the result was accurate to the limit of the approach. Even so, I made some mistakes in the original that Hop caught and corrected.
@Peter McArthur, Spectra is an excellent engineering polymer. The main advantages are excellent strengthtoweight and a very simple chemical composition of {CH2} with no aromatic rings. The main disadvantages are fairly high elasticity and poor UV / radiation resistance. For use as a tether, the fibers need to be coated either in a metal film or a UVopaque layer; it also adds some complication to motion of payloads since it can stretch quite a bit.
Hop might be interested to see the performance of a coppercoated Spectra tether. If I have a chance I'll look into that, but to some extent it would depend on how much current would be carried by the sheath for powering climbers.
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