Orbital Momentum as Commodity

To The Moon, Alice

A tether needs to be substantially more massive than it's payloads. Else a catch/throw would wreck the tether orbit.

In my last post I described how near earth asteroids parked in retrograde lunar orbits might be used as anchors and momentum banks for lunar orbital tethers.

What could we use to anchor a vertical tether in earth orbit?

When a satellite in geosynchronous orbit satellite dies, it's sent to a graveyard orbit about 300 km above geosynch. According to IO9 there are more than a hundred sats in this graveyard. Here is an estimate of 371 dead GEO sats totaling 668 tonnes. There will be more as time goes on.

Boost one of these dead sats 25 km higher than the graveyard orbit and dangle a 25 km tether. The tether foot will be moving about 10 kilometers/hour or about 5.5 mph with regard to the sat graveyard. As satellites are caught, momentum can be added via ion engines. With an ion engine's high ISP, the satellite collector's reaction mass can be small with regard to the mass of the satellites collected.

There's little plane change delta V since most the GEO sats are equatorial.

This would cleanse the geosynch neighborhood of orbital debris. When the dead sats are collected into a single mass, cross section is much smaller than a multitude of satellites. Thus likelihood of debris generating impacts is much smaller.

Some parts of these dead sats may be salvageable. Many may still have working solar arrays, for example. Although it is possible salvage issues could be a road block to this scheme. Those parts not salvageable would still have value just as a source of orbital momentum.

Boost this ball of dead sats to an orbit 10,000 km higher than GEO and extend a tether 10,000 km up and 7,000 km down. Now we have a well anchored 17,000 km tether. Tether foot is 3,000 km above geosynch so there's no chance the foot will hit an GEO sat. There is also less chance of tether damage from an impact with orbital debris.

A payload released from this tether top will reach the moon.

If we use Kevlar with a tensile strength of 3.6 giga pascals and density of 1.44 grams per cubic centimeter, taper ratio for this tether is 1.18. Tether to payload mass ratio is less than 1.

The tether center is moving about 2.78 km/s. Grave yard sats move about 3.06 km/s. So delta V to raise the ball of sats is only .28 km/s.

From LEO it takes around 3.2 km/s to rendezvous with this tether foot. About what it takes for a normal Trans Lunar Insertion from LEO. This was disappointing to me.

However this tether could receive mass from an asteroid parked in a lunar orbit. It could then send asteroidal mass to lower orbits.

If lower tethers were receiving mass from above as well as from earth, it might be less costly to give the lower tethers a substantial anchor mass.

Here are three possible tethers: the super GEO tether already described, a sub MEO tether and a super LEO tether:

Each of these tethers is positioned to avoid the high satellite/debris density areas: GEO (Geosynchronous Earth Orbit), MEO (Middle Earth Orbit, home of GPS and other global positioning sats), and LEO (Low Earth Orbit).

Tether to payload mass ratio is also less than one for both the Super LEO and sub MEO tethers.

The apogees and perigees of these red ellipses match the tether velocities at rendezvous. So almost no propellent would be used for catches and throws.

From LEO it takes .5 km/s to reach the Super LEO Tether. From there the relay of tethers can deliver spacecraft to the moon.

Once In The Moon's Neighborhood...

The Super GEO tether tosses payloads to a 384,400 km apogee. 384,400 km is the moon's distance from earth. Velocity of this ellipse's apogee is about .54 km/s while the moon's speed is about 1.01 km/s. Vinf with regard to the moon is about .48 km/s.

In my earlier post I described an asteroid anchored tether that could toss stuff from the moon with a Vinfinity of 1.6 km/s. Tether to payload mass ratio would be around 3. If we only need a Vinfinity of .48 km/s, a less massive tether would do:

Tether center's 24,200 km from the moon's center. Tether foot 13740 km, tether top 39000 km. If we use Kevlar with 3.6 giga pascal tensile strength and 1.44 g/cm^3 density, the tether to payload mass ratio is less than 1/4.

Dropping from this tether foot, a payload would impact the moon at 2.23 km/s.

It'd also be somewhat easier to park a near earth asteroid in this higher lunar orbit. See the Keck Report for a proposed method to park rocks in lunar orbit. For asteroids with orbital energy just above zero it is doable to park them in an high lunar orbit.

Up Momentum as a Valuable Commodity

Catching from a lower orbit and throwing upwards will sap a tether's orbital momentum. Earlier I had mentioned that ion engines using xenon as reaction mass could restore momentum.

But asteroidal mass from above is a source of up momentum.

With two way traffic, the need for xenon is reduced. Momentum boosting maneuvers could be balanced with momentum sapping catches and throws. The need for reaction mass would be largely eliminated.

An asteroid anchored tether in lunar orbit would be helpful in capturing more asteroids to lunar orbit. There's a lot of near earth asteroids in accessible orbits, so there's a massive source of up momentum.

Electrodynamic Tethers

How about using Lorentz force to change orbital momentum? As an electron moves up (away from earth's center) it passes through earth's magnetic field and the tether is pushed east, boosting momentum. Sending electrons down will give a westward push, reducing momentum.

But this relies on one way current. If the circuit is closed, electrons move up as well as down and there's no net Lorentz force.

The ionosphere can close a circuit in low earth orbit. The tether can pick up electrons from the ionosphere as well as discharge electrons into the ionosphere.

But the tethers described here are above the ionosphere. Using Lorentz force doesn't seem to be an option.

But I don't know that much about electrodynamic tethers, I could be wrong. If someone corrected me, it'd be a pleasant surprise.

Why I Don't Like Rotovators

Why my obsession with vertical tethers? Yes, rotovators could be shorter. But there's several reasons I don't like rotovators.

Taper Ratio and Tether to Payload Mass Ratio

Tension in rotovators comes from so called centrifugal force, ω²r. At the Space Stack Exchange 2012rcampion takes a look at a sling's tether taper ratios. Campion's results look a lot like Moravec's equations.

As tether tip speed grows, taper ratio soars. Short tethers capable of a good throw would be quite massive. According to this Tether.com pdf, a LEO to GTO rotovator capable of tossing 5 tonne  payloads would need to mass 50 tonnes (3rd paragraph, page 4). A 10 to 1 tether to payload mass ratio.

In contrast, a vertical tether's tension comes from centrifugal force and gravity, ω²r - μ/r². Gravity mitigates stress from centrifugal force and taper ratio grows a lot more slowly. All the vertical tethers described in this post have a tether to payload mass ratio less than 1.

Vertical tethers do need an anchor mass. But anchor mass can be very useful. I'd like to see lots of solar arrays at the tether centers. Solar arrays could power electrolysis plants to crack water into bi-propellent, move elevator cars up and down, and occasionally power Hall Thrusters to adjust the tether's orbit.

Timing

A rotovator is good for catches and throws when it's vertically aligned. But it's well aligned only for a very brief time during its spin. The rotovator must be correctly positioned when a launch window occurs. Ditto for catching from a Hohmann orbit.

Catches are harder. As a payload approaches a rotovator, the tip remains on its path for a brief time and then either zooms up or down (from the payload's point of view). In contrast, the tips of a vertical elevator remain at constant altitude.

Attitude Adjustment

A vertical tether stays vertical due to tidal acceleration gradient. It seems to me a rotovators attitude would need to be adjusted from time to time.

Possible Imports to Earth and LEO

Water

I believe the most important import from asteroids will be water.

My last two blog posts are more or less in response to Jon Goff's The Slings and Arrows of Outrageous Lunar Transportation Schemes: Part 1 - Gear Ratios. Goff pointed out that only a small fraction of propellent mined at the lunar poles could be delivered to LEO.

Ever since Goff wrote that, I've been trying to think of ways to deliver extraterrestrial propellent to where it's needed.

"Wait a minute," you might be thinking, "This guy has just described a transportation system using momentum exchange and ion engines. Why is he still stuck on stone age chemical propellent?"

I believe the biggest obstacle to fully reusable spacecraft is an upper stage's 8 km/s re-entry. Hall Thrusters decelerate too slowly to help with that plunge.

A tether foot low enough to drop payloads into low velocity suborbital paths would be vulnerable to collision. Below 1000 km, space is full of sats and debris.

What the Super LEO tether could do is deliver propellent to LEO. An ellipse from the Super LEO foot would take .5 km/s to circularize at perigee. And some of that .5 km/s might be accomplished by aerobraking.

An upper stage refueled at LEO could do a healthy burn to lose most the 8 km/s. The upper stage might also beef up it's dry mass with structural support and TPS, also from asteroids. Given these options, upper stage re-use is very doable. If upper as well as booster stages can be economically re-used, the dream of cheap space access is realized.

Metals

Many asteroids have high concentrations of the platinum group metals. Right now these are precious due to rarity. But should they become more available, there are numerous ways they can be used.

Rare earth metals have many uses. Rare earths actually aren't rare. But they're hard to mine in an eco-friendly way. I would much rather see them mined on a lifeless, barren rock than in earth's biosphere. An asteroid in lunar orbit would make the moon's KREEP more accessible. Besides rare earths, KREEP also contains uranium and thorium, possible sources of energy.

Energy

As mentioned earlier, imported uranium and thorium could fuel terrestrial power plants.

The earth intercepts only .45 billionths of the sun's light. We're using only a tiny fraction of possible solar energy.

If we use solar energy to refine extra-terrestrial ore and import commodities to earth and LEO, in a sense we're importing energy.

Besides energy for refining, imported commodities would also require energy for transportation. To get a kilogram from just above C3 to LEO takes about 512 mega-watts. Tether momentum exchange could accomplish most of this. But it's energy used, regardless of source.

Summary

A source of up momentum would be a major game changer. A first step towards acquiring this source would have been the early version of the Asteroid Redirect Mission (ARM). This is doable. But for now it looks like popular opinion will keep this from ever being funded.

I'll continue singing the praises of asteroids and ARM. God willing, my small voice will have some influence.

Orbital Tethers as Momentum Capacitors

Orbital tethers as momentum capacitors

Carmack to Musk: What about rotovators?

Carmack's game set on Phobos

Zero Relative Velocity Transfer Orbit (ZRVTO) between Phobos and Deimos tethers

ZRVTO mass savings

ZRVTO equations

End of life sats to use as a momentum bank anchors

Propellant mass as cost driver

Re-entry propellent imported by tethers

The Heteroclinic Zone

Phobos help with Mars EDL

Park trans-planetary vehicles at the edge of gravity wells

Some years ago I was knocked off my chair when I was soldering a flash for our camera. The flash was powered by two AA batteries. How could such a dinky power source pack such a wallop?

It was because of the flash's capacitors. A capacitor will build up a charge over time and then release the accumulated charge suddenly. In this case the flash would deliver a very bright and brief flash of light when the camera shutter was open.

The orbital tether as a capacitor for momentum.

An ion engine can have an exhaust velocity of ~30 kilometers per second. That is nearly eight times that of the best chemical exhaust, around 4 kilometers per second. That means a much smaller exponent in the rocket equation. When we're taking exponents, scaling by 1/8 can make a huge difference in delta v delivered per kilogram of propellent.

The problem is the ion engine's dinky thrust. A chemical rocket can slam you back in your seat with 4 or 5 g's. But an ion engine's delicate push is barely perceptible, like the push of a feather. It takes a long time to build up delta V which makes it difficult to enjoy an Oberth benefit. It can also mean a trip lasting months for a trip that would take hours via a chemical rock

But a tether with an ion engine can take months or weeks between catches or throws to build up momentum.  So while it takes a long to build up momentum, it can release or impart it suddenly with a Catch or a throw.

So a tether can impart a brief and powerful change in momentum even with the ion rocket's barely perceptible thrust. It is like a capacitor but for momentum instead of electricity.

More tether stuff

It's been a long time since I did a post on tethers. So I'm going to toss in some other random bits that have accumulated in my head over the years.

Musk and Carmack on orbital tethers.

Back in 2016, shortly after SpaceX had landed a booster on an earth platform, John Carmack tweeted:

Elon replied:

I was delighted to see this exchange. I've been hanging around spaceflight forums since the 90s and Carmack has long been a big name in new space. Carmack and Armadillo Aerospace were X-Prize winners in 2006. I don't think I need to review what Musk has been doing.

Carmack, Fear and Dread

Carmack and ID software made a very successful computer game set on the Martian moon Phobos. The names of the Martian moons (Phobos and Deimos) means Fear and Dread. Which is very appropriate for the computer game Doom.

I love the idea of using the Martian moons as settings for science fiction stories. I believe they will be great assets in humanity's effort to settle the solar system. I've done a number of blog posts on Phobos and Deimos:

Phobos, Panama Canal of the Inner Solar System.

Upper Phobos Tether

Lower Phobos Tether

Deimos Tether.

But if I'm trying to sell Phobos and Deimos maybe I shouldn't be mentioning Doom. Oh well.

ZRVTO between Phobos and Deimos

Thinking about elevators anchored on Phobos and Deimos it occurred to me there would be a Zero Relative Velocity Transfer Orbit (ZRVTO) between the moons' tethers.

The transfer orbit's velocity at periapsis matches the speed of Phobos' tether top. Velocity at apoapsis matches the foot of the Deimos tether. Thus passengers and cargo could be exchanged between the moons using very little propellent.

I hope this idea will eventually be used.

ZRVTOs in other settings

There can be ZRVTOs in other settings. The tether anchor masses need to be tide locked in circular, coplanar orbits. Which describes a lot of the moons of Jupiter, Saturn, Uranus and Neptune. I look at this in Mini Solar Systems.

In Trans Cislunar Railroad I look at ZRVTOs between possible tethers in earth orbit:

In some ways tether mass is like propellent mass in the rocket equation — tether mass goes up exponentially with increasing delta V.

But tossing payloads between tethers breaks the the delta V budget into chunks. And thus greatly reduces the needed tether mass.

How much mass is saved with with ZRVTOs?

The above system tosses payloads up to the moon. For kicks I decided to see what would happen if I made a single LEO tether long enough to throw payloads to the moon.

I placed an anchor mass 1000 kilometers above earth's surface and tweaked tether length until the apoapsis of a flung payload was a lunar distance (384,400 km).

Taper ratio about 35. The tether would need to be about 106 times as massive of the payload it throws given a safety factor of three. It would need to about 1,842 kilometers long.

Let's compare that to my system of tethers in the Tran Cislunar Railroad:

So my system of tethers is actually about 5,000 kilometers longer than LEO tether capable of slinging payloads to the moon. That's a little disappointing. But tether to payload mass is less than 4.

So there is more than a 25 fold savings in tether mass. That is gratifying.

"Would only matter if it was extremely big"

Recall Musk's reply to Carmack was "Would only matter if it was extremely big".

The tether anchor would need to be a lot more massive than the payloads it handles. Or else the act of catching or throwing a payload would destroy the tether's orbit.

That's not an issue for tethers anchored on planetary moons. But it seems like a show stopper for tethers in earth orbits. Or maybe not....

Big Balls of Dead Sats

There is a lot of dead sats that could be harvested for a momentum bank. As of 2015 it was estimated that there were 670 tonnes of dead sats in the graveyard orbit just above geosynchronous orbit. 

Gathering these into a single anchor mass would vastly reduce their surface area and those lessen the likelihood of impacts generating orbital debris.

Many of them still have solar panels that can provide electricity. There are high gain antenna dishes. Some of the harvested momentum mass might even be useful.

What about LEO?

In 2016 I was wondering how we could provide a massive anchor for a LEO tether. But since then Elon Musk and SpaceX have been launching StarLink, a huge constellation of communication satellites.

I expect at the present time the plan is send an aging StarLink satellite down to the upper atmosphere and let it become a shooting star. But couldn't satellites in similar orbits be gathered together to form a momentum bank? If so, Musk has already made huge deposits into an LEO momentum bank.

And the StarLink satellites have a lot of solar panels as well as ion engines that could be salvaged.

Some ZRVTO equations

Some equations to figure lengths of tethers that accomodate Zero Relative Velocity Transfer Orbits. 

Lower tether length would be L1((1+e)^(1/3) - 1).

Upper tether length would be L2(1 - (1-e)^(1/3))

Momentum Exchange

At the outset of this post I mentioned tethers could impart momentum gradually built up by ion engines whose exhaust velocity is a lot higher than chemical rockets.

If there is downward traffic as well as upward that could greatly reduce the argon or xenon propellant used by the ion engines.

Momentum boosting maneuvers

Catching payloads from above or dropping them into lower orbits

Momentum depleted maneuvers

Catching payload from below or tossing them into higher orbits

These could be balanced to achieve most of delta v needed, in my opinion.

Propellent Mass as cost driver?

Reducing propellant mass should be a top priority. In even the best circumstances Gross Lift Off Weight (GLOW) from earth's surface will be dominated by propellant.

Charlie Stross was indignant when I was ridiculing space naysayers. He said he wouldn't dignify my criticisms with a public response. But then proceeded to make several thoughtful public responses. Link. I quote:

"My cost estimate was for near-future transport to LEO.

"Contemporary civil airlines' operating costs are on the order of triple the cost of fuel. (Equal shares: fuel, airframe depreciation and maintenance, and crew/ground support costs.)

"If you want to do it for less than triple the fuel cost, you need to beat the standards of a viciously competitive industry that's been trying to pare costs for around a century.

"SpaceX currently cite the cost of fuel for a Falcon 9 as ~$200,000. So my BOTE would get us to $600,000 for a ~20 ton payload. I gather they're currently quoting about $60M, so there' someroom for improvement."

That's from a comment Charlie made in 2018. And I would agree that spaceflight dominated by propellent cost is optimistic. You would also need very durable, reusable rockets that don't require a great deal of maintenance.

What could we import from above?

For a momentum exchange tether to work you need two way traffic. So what mass from above could provide up momentum?

Lunar propellent might be the first import from above. In 2010 India's lunar orbiter Chandrayaan 1 found evidence of massive ice deposits on the lunar poles Link. The late lunar geologist Paul Spudis would argue an off earth source of propellent could confer a commercial and military advantage to the power that controls it Link. Former NASA administrator Jim Bridenstine also made the same argument Link. 

Jon Goff voiced some objections to lunar propellent. Given the delta V between the lunar surface and and earth orbits, only a small fraction of lunar propellent would arrive to supply propellent depots in earth orbits. If propellent were delivered by conventional rockets. 

However more would arrive if lunar propellent were delivered via momentum exchange tethers. And they would provide up momentum for the momentum exchange tethers and reduce the need for propellent mass. 

A photo of Starship thermal tiles from

About Starships, and the (not what you think) reusability we need.

The Starship upper stages will likely re-enter at a much higher velocity than the booster stages. 35 kilo pascals is typical max Q for ascent. But for for descent 90 kilo pascals is common. Will Starship be able to economically refurbish after re-entry? Unlike the Space Shuttle Starship has a stainless steel hull. It looks like the thermal tiles are mechanically attached rather than glued on.

SpaceX likely has improved on thermal protection since the Space Shuttle Days. But I still expect re-uses after an 8 km/re-entry to be difficult.

If the upper stage could refuel in Low Earth Orbit (LEO), the upper stage could re-enter with an even lower velocity than the booster stage. Re-use would be far less difficult.  Re-usability may even become so advanced that cost of transportation would be triple the fuel costs, as Charlie Stross imagines in his best case scenario.

The Heteroclinic Zone

There are a family of loosely bound lunar orbits where a little delta V and use of earth's tidal influence can make a big change to the orbit. It's possible to move from Earth Moon Lagrange 2 (EML2) to  Earth Moon Lagrange 1 (EML1) with only a tiny burn. These low delta V routes between lunar orbits are called heteroclinic paths. They talk about these paths in chapter 3 of Dynamical Systems, the Three-Body Problem and Space Mission Design by Koon, Lo, Marsden and Ross.

EML1 is around .3 km/s from the top tether I mentioned in ZRVTOs in Other Settings earlier in this post.

From EML it's easy to reach EML2

EML2 is only .9 km/s from Trans Mars Insertion using the Farquhar route.

I am a little obsessed with EML2. I have a post devoted to this Lagrange point.

NHROs

Included in The Heteroclinic Zone are Nearly Rectilinear Halo Orbits NHRO. I like these orbits for a number of reasons.

The perilune of these orbits are near the lunar poles. The lunar poles and cold traps are where I daydream of lunar propellent mines. So orbital insertion from the propellent mines to The Heteroclinic Zone is quite doable. 

I like to imagine advanced propellent mines with a rail guns that launch propellent into NHROs. 

A distant apolune is at the other end of an NRHO. It travels slowly in this region giving it lots of hang time over the lunar poles. It can give long periods line of sight periods to the lunar cold traps. In some ways this is like an earthly Molniya orbit. This would be useful in the early stages of a propellent mine when construction is being done by remotely controlled robots.

Phobos lending a hand with Mars EDL

A 1340 kilometer tether from Phobos could drop payloads into periaerion skimming Mars atmosphere. The payload would enter Mars atmosphere at about 3.6 km/s. Entry from an Earth to Mars Hohmann would be about 5.5 km/s. (3.6/5.5)^2 = ~.43. So less than half the kinetic energy to be shed.

And it may be doable to mine oxygen from Phobos minerals so propellent could also lend a hand in shedding velocity.

A 5680 km/s tether from Phobos would allow entry to Mars atmosphere about about .6 km/s However a Phobos tether going deep in Mars' gravity well is more difficult. Given Zylon and a safety factor of 3, taper ratio would be 84 and tether to payload mass ratio would be 640. See my lower Phobos Tether post.

Park Trans Planetary Vehicles at the edge of gravity wells

.

Vehicles that can keep humans alive for months must be massive. Vehicles for shorter trips can be much smaller.

What is the point of launching a trans Mars vehicle from earth's surface and landing it on Mars' surface? It would be like using a huge Mac truck to deliver a pizza from a restaurant to a customer's front porch.

Far better, in my opinion, to park trans planetary vehicles at EML2 or Phobos. Then there is no need for a thermal protection system to endure re-entry. And there is no need for the ship to be 90% propellent to climb up steepest slopes of planetary gravity wells.

Tran Cislunar Railroad

Three Orbital Tethers

This post revisits Orbital Momentum As A Commodity. But now I will examine these tethers using Wolfe's spreadsheet.

I envision 3 equatorial tethers to move stuff back and forth between LEO and the lunar neighborhood:




The location of these vertical tethers avoids zones of orbital debris:

The orange regions, LEO, MEO and GEO, have high satellite and/or debris density. Thus tethers in those regions would be more vulnerable to damage from impacts.

Dead Sats for tether anchors

Unless elevator mass is lot more than the payloads, the acts of catching or throwing could destroy the tether orbit. At first it looks like the need for a substantial anchor mass is a show stopper. But there are a large number of dead sats in equatorial orbits. By one estimate,  there's 670 tonnes in the graveyard orbit above geosynch.

The dead sats gathered might have functioning solar arrays. According to this stack exchange discussion, solar arrays degrade by 2 to 3% a year due to radiation, debris impacts and thermal degradation. Thus a 20 year old array could still be providing 50% to 66% of the power it delivered at the beginning of its life. The parabolic dishes for high gain antennas might also be salvageable.

Whether functioning or not, solar arrays as well as other paneling might be used as shades to keep propellent cold. If our tethers receive propellent from the moon or from asteroids parked in lunar orbit, shades would help with cryogenic storage.

Consolidating dead equatorial satellites would reduce their cross sectional area and help solve the problem of orbital debris.

Super GEO tether

The circular orbit pictured above is 10,000 km above Geosynchronous Earth Orbit (GEO). The lower part of the tether has a length of 7,000 km and the upper tether is 10,340 km in length.

A Space Stack Exchange answer estimates there are 670 tonnes of dead sats in the geosynch graveyard orbit. Here is a page that tries to estimate total mass in earth orbit.

Delta V to raise the dead sats to this higher orbit is about .28 km/s. This might be accomplished with ion engines. Also the elevator could be used to send some to the sats towards the lower MEO tether. This would help with the .28 km/s delta V budget.

Upper Super GEO Tether, 10,340 km long

Safety Factor	3
Zylon taper ratio:	1.38
Tether to payload mass ratio:	.78
Tether top radius	62,504 km
Tether top speed:	3.3 km/s
Tether top net acceleration:	.07 m/s2 (.007 g)
Payload apogee:	384,400 km
Payload apogee speed:	.53 km/s

The payload apogee is at lunar altitude and the payload's moving .53 km/s. The moon moves at about 1 km/s. So Vinf with regard to the moon is about .47.

Lower Super GEO tether, 7,100 km long

Safety Factor	3
Zylon taper ratio:	1.21
Tether to payload mass ratio:	.47
Tether foot distance from earth	45,000 km
Tether foot speed:	2.4 km/s
Tether foot net acceleration:	.07 m/s2 (.007 g)
Payload perigee:	21,450 km
Payload perigee speed:	5 km/s

The tether foot drops a payload to rendezvous with the MEO tether.

Sub MEO Tether

The circular orbit of the Sub MEO anchor mass is has a radius of 19,425 km. To get satellites from the super synchronous graveyard orbit to this orbit takes about 1.4 km/s. Some of that 1.4 km/s might be accomplished with the super GEO tether. Sending mass downward would help push the remaining GEO sats upward.

Upper Sub MEO Tether, 2,050 km long

Safety Factor	3
Zylon taper ratio:	1.30
Tether to payload mass ratio:	.61
Tether top distance from earth	21,450 km
Tether top speed:	5 km/s
Tether top net acceleration:	.3 m/s2 (.03 g)
Payload apogee:	45,000 km
Payload apogee speed:	2.4 km/s

The payload apogee radius and speed matches the foot of the super  GEO tether's radius and speed.
The top of this tether's radius and speed matches the payload perigee and speed sent from super GEO tether. The Sub MEO and Super GEO tethers can exchange payloads with minimal delta V at tether/payload rendezvous.

Lower Sub MEO tether.

Safety Factor	3
Zylon taper ratio:	1.35
Tether to payload mass ratio:	.78
Tether foot radius	17,375 km
Tether foot speed:	4.1 km/s
Tether foot net acceleration:	.38 m/s2 (.038 g)
Payload perigee:	9,680 km
Payload perigee speed:	7.3 km/s

The Low Sub MEO tether sends and receivse payloads to and from the upper Super LEO tether.

Super LEO Tether

The anchor mass is in a circular orbit of radius 9300 km.

Upper Super LEO Tether, 765 km long

Safety Factor	3
Zylon taper ratio:	1.4
Tether to payload mass ratio:	.84
Tether top radius	10,065 km
Tether top speed:	7.1 km/s
Tether top net acceleration:	.11 m/s2 (.011 g)
Payload apogee:	17375 km
Payload apogee speed:	4.1 km/s

The payload apogee is at lunar altitude and the payload's moving .53 km/s. The moon moves at about 1 km/s. So Vinf with regard to the moon is about .47.

Lower Super LEO tether, 450 km long

Safety Factor	3
Zylon taper ratio:	1.13
Tether to payload mass ratio:	.29
Tether foot distance from earth	8,844 km
Tether foot speed:	6.2 km/s
Tether foot net acceleration:	.7 m/s2 (.07 g)
Payload perigee:	6,778 km
Payload perigee speed:	8.3 km/s

Perigee altitude is about 300 km. Circular orbital speed at this atltitude is about 7.7 km/s. To send a LEO payload on it's way to the Super LEO tether would take about .6 km/s.

Sending a payload from the tether to LEO can take less than .6 km/s as the delta v needed for circularizing can be provided by aerobraking.

Total Tether Mass to Payload Ratio

We've looked at a total of 6 tether lengths, the upper and lower parts of 3 vertical tethers.

Tether Mass to Payload Mass Ratios & Lengths

	T/P	Length (km)
Upper Super GEO	.78	10340
Lower Super GEO	.47	7100
Upper Sub MEO	.61	2050
Lower Sub MEO	.78	2050
Upper Super LEO	.84	765
Lower Super LEO	.29	450
Total:	3.77	22,755

Thus 38 tonnes of Zylon could accommodate 10 tonnes of payload. That's not too bad.

A much larger problem is the anchor mass needed for each tether. There are lots of dead sats just above GEO that could be gathered for the Super GEO tether anchor mass. But anchor masses for the sub MEO and super LEO tethers will be more expensive. This is a possible show stopper.

Facilitating Momentum Exchange

Using Hall Thrusters to restore momentum.

Sending mass from LEO to a lunar height apogee saps our tethers' orbital momentum. The momentum hit is somewhere around payload mass * 4 km/s. Orbital momentum can be restored gradually with ion thrusters. Hall Thrusters can expel xenon with a 30 km/s exhaust velocity.

Plugging these numbers into the rocket equation:

Propellent mass fraction = 1 - e ^-4/30 = ~.125.

About 1/8. So to make up for the momentum lost throwing 7 tonnes of payload, we'd need a tonne of xenon. Better than chemical but still expensive.

Lunar or NEA propellent as a source of up momentum.

Some Near Earth Asteroids (NEAs) can be parked in lunar orbit for as little as .2 km/s. Carbonaceous asteroids can be up to 40% water by mass (in the form of hydrated clays). There may be rich water ice deposits in the lunar cold traps. So far as I know, these are the most accessible potential sources of extra terrestrial propellent.

Catching propellent from higher orbits would boost a tether's momentum. Dropping this payload to a lower tether would also boost momentum.

Thus up momentum can be traded for down momentum. Xenon reaction mass to maintain tether orbits can be cut drastically with two way traffic.

Jon Goff's gear ratios

Jon Goff has pointed out it take some delta V to get propellent from the moon's surface to LEO. Thus only ~10% of propellent mined lunar cold traps would make it LEO. See his blog post The Slings And Arrows of Outrageous Lunar Transportation Schemes Part-1 Gear ratios.

Well, lunar propellent could be a source of down momentum for the Lunar Sky Hook I described recently. And a source of up momentum for the Trans Cislunar Railroad this blog post looks at. NEA propellent could also be a source of up momentum for the Trans Cislunar Railroad.

Using propellent as a source of tether up momentum I believe it's plausible for 40% of the lunar propellent to make it to LEO. In which case it becomes plausible to use reaction mass to mitigate the extreme conditions of re-entry.

Breaking the Genie's Bottle

The human race is a genie in a bottle. Given Tsiolkovsky's rocket equation, it's enormously difficult to cross the boundaries that confine us. But given infrastructure and resources at our disposal, we can build bridges to larger frontiers.

A spiral of tethers

First off let's look at the great granddaddy of vertical tethers, the Clarke tower.

For a vertical tether in circular orbit, there's a point where the net acceleration is zero. Above that point, so called centrifugal force exceeds gravity. Below that point, gravity exceeds so-called centrifugal force. If a payload is released on this point of on the tether, it will follow a circular orbit alongside the tether. This point I call the Tether Center.

In this case, the tether center is at geosynch height, about 42,000 km from earth's center. I set 42,000 km to be 1. What path does a payload follow if released from the tether below the center?

It will be a conic section. Call the conic's eccentricity e. Call the distance from tether point r.

If dropped from below center, r  = (1-e)^1/3.
If released from above center, r  = (1+e)^1/3.

Here's my derivation. Mark Adler also gives a nice demonstration in the comments on that post.

This is true of any vertical tether in a circular orbit.

If there are two prograde, coplanar vertical tethers at different altitudes, there's an elliptical path between them where the perigee velocity matches a point on the lower tether and apogee velocity matches a point on the upper tether.

If a payload is released from the lower tether at the correct time, it will rise to the upper tether which will be moving the same velocity as the payload at apoapsis. Rendezvous can be accomplished with almost no delta V. Cargo can be exchanged between tethers with almost no reaction mass.

Let r for the release point above the tether be (1+e)^1/3 and release point below the tether be (1-e)^1/3. Then both the larger and smaller ellipse will be the same shape.

Center of the above tether is 8000 km. I tried to place it above the dense orbital debris regions of low earth orbit. The tether is 461.6 kilometers long. Dropping from the foot will send a payload to a 150 km attitude perigee. Throwing a payload from the tether top will send a payload to a 9780 km apogee.

From a 150 km altitude orbit, it takes about .33 km/s to send a payload to the tether foot.

Both ellipses have the same eccentricity, about .0864

I repeatedly clone, scale by 126% and rotate 180º:

By ascending and playing catch with a series of tethers, a payload might make it's way from LEO to the vicinity of the moon:

But there's a problem with this scheme. A tether loses orbital momentum each time it catches a payload from below. Ascending and throwing to a higher orbit also saps orbital momentum. How do we keep these tethers from sinking?

Imagine resources parked in lunar orbit. Maybe propellent mined from the lunar poles. Or perhaps platinum from an asteroid parked in a lunar DRO. To send cargo to earth's surface or low earth orbit would entail catching from a higher orbit, descending and dropping to a lower orbit:

If cargo is moved down as well as up, momentum boosting maneuvers can be balanced with momentum sapping maneuvers.

Thus mass in high orbits are sources of up momentum. This itself could be a commodity, a way to preserve orbits of momentum exchange tethers.

This tether spiral scheme cuts tether length, especially in regions of high debris density and the Van Allen Belts.

In this illustration successive ellipses vary by a factor of 2^1/3. Other rates of expansion are possible. Let k be the ratio of one ellipse apogee to the apogee below. k = (1+e)^4/3/(1-e)^4/3. Thus we can wind the spiral tighter or loosen it by choice of ellipse eccentricity.

Golden Tethers

Φ,  also known as the golden ratio, is one of my favorite numbers. It is (sqrt(5) + 1) / 2, approximately 1.618. I've done many paintings and drawings using this number. Here are a couple:

 Two images from my coloring books

The number occurs naturally in designs having a 5 fold symmetry but it also turns up in unexpected places. I was happy to find it when I was playing with orbital tethers.

Vertical Tethers vs Space Elevators


Gravity gradient stabilized vertical tethers are smaller cousins of a full blown space elevator. Jerome Pearson has developed equations giving a space elevator's dimensions and taper ratio.

Some of Pearson's terms:

r₀ planet's radius
g₀ planet's surface gravity
r_s radius of planet's synchronous orbit

For looking at vertical tethers I use P. K. Aravind's equations which I believe are based on Pearson's work. But I substitute the above terms with r_f for r₀,  g_f for g₀, and  r_c for r_s.

Tether Foot
The term r_f refers to distance from planet center to tether foot. Imagine a planet the same mass of earth but with a larger radius, r_f. Then r_f and r₀ become the same. Same with surface gravity, gravity at the tether foot would be the same as surface gravity of a planet with radius r_f.

Tether Center
The term r_c is the distance from planet center to radius at which a natural circular orbit would have the same angular velocity as the tether we're looking. I call this the tether center.  Misnamed since the length above the "center" is greater than the length below, but I can't think of a better word. Again, we can imagine a planet whose angular velocity is the same as our tether's, so r_c would become  r_s.

Tether Top
The term r_t can remain the same. The tether top applies to a vertical tether just as much as it does to a space elevator. The length above the tether center must balance the length below.

Tether Size

I would like to make the tether as small as possible. Smaller size makes for less materials that have to be launched to space. A shorter length makes for greater throughput, less stress allowing less exotic tether materials and smaller taper ratios, and a smaller cross section thus reducing vulnerability to debris impacts.

The tether should be as low as possible. A lower r_c makes for a higher angular velocity and a better Oberth benefit.

How low a tether foot can descend is limited by height of atmosphere. We want the foot above the atmosphere as drag would pull the tether down. So  r_f is one of the first quantities considered in my tether spreadsheet.

How high to make r_t? If releasing a payload from tether top sends the payload on a parabolic trajectory, we can choose any apoapsis by releasing from tether locations between r_t  and r_c.

Releasing a payload from a point (1+e)^1/3 r_c will send the payload on conic section trajectory having eccentricity e. The eccentricity of a parabola is 1. So an r_t  = 2^1/3 r_c would give us a tether able to deliver payloads to any apoapsis.

Adapting P. K. Aravind's equation (5) from his The physics of the space elevator we have

(r_f / 2) * [sqrt(1 + 8(r_c/r_f)³) - 1] = r_t

Recalling we want r_t to send payloads on a parabolic path...

(r_f / 2) * [sqrt(1 + 8(r_c/r_f)³) - 1] = 2^1/3 r_c

Setting our units r_c = 1 ...

(r_f / 2) * [sqrt(1 + 8/(r_f³)) - 1] = 2^1/3

Which comes to the suprising and pleasing result...

r_t  = Φ r_f

Where Φ is the golden mean, the number I was talking about at the beginning of this blog post.

The velocity of the golden tether's foot is about 68.7% the velocity of a normal circular orbit at r_f .

Golden Earth Tether.

The top red orbit is a parabola.

The foot is 300 km above earth's surface. It's moving about 4.8 km/s wrt earth's equator.

Using Kevlar, taper ratio is about 5.1. Tether length is about 4130 km.

Golden Moon Tether.

The top red orbit is a parabola.

The foot is 80 km above moon's surface. It's moving about 1.13 km/s wrt moon's surface.

Using Kevlar, taper ratio is about 1.1. Tether length is about 1125 km.

The tether doesn't have to be golden. Longer tethers would be able to send payloads on hyperbolic orbits (e > 1), useful if interplanetary Hohmann transfers are desired. Shorter tethers would be limited to elliptical orbits (e < 1), but this could still be useful. This spreadsheet allows the user to set eccentricity of exit orbit as well as body's mass and radius. You can also set the altitude of tether foot.

Upper Phobos Tether

This is third in a series of posts that rely on Wolfe's model of tethers from tide locked moons. As with the Lower Phobos Tether post, I will look at possible stages of this tether examining tether to payload mass as well as benefits each stage confers.

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 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.)"

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 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.

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.

The notion of a ZRVTO between Deimos and Phobos tethers is not new. Above is a diagram from an article by JPL engineer Paul Penzo. Page 70 of the 1997 publication Tethers In Space Handbook. Penzo came up with this idea in 1984 (I believe).

Penzo's 940 and 2960 km lengths aren't that far from my 937 and 2942 numbers. It is reassuring that an aerospace engineer's numbers are close to my own.

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 lo-o-o-ng time to climb in and out of planetary gravity wells.

From Mark Adler's explanation on how to model ion rocket trajectories: 

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.