Lamentable Lagrange articles

Gravity doesn't cancel at the Lagrange points

"There are places in the Solar System where the forces of gravity balance out perfectly. Places we can use to position satellites, space telescopes and even colonies to establish our exploration of the Solar System. These are the Lagrange Points."

From Fraser Cain's video on Lagrange points. A lot of pop sci Lagrange articles repeat and spread this bad meme. It just ain't so.

The 5 Lagrange points can be found in many two body systems. They can be Sun-Jupiter, Earth-Moon, Jupiter, Europa -- Any pair of dancers has this retinue of 5 Lagrange regions moving along with them. Above are the 5 Pluto-Charon Lagrange points. Also pictured are the gravity vectors these bodies exert. Pluto's gravity is indicated with purple vectors and these point towards Pluto's center. Charon's gravity is indicated with orange vectors and these point towards Charon's center.

For the gravity vectors to cancel each other, they need to be equal and pointing in opposite directions.

L1

The only L-Point where the gravity vectors pull in opposite directions is L1. And here the central body (Pluto) pulls harder than Charon. These two gravities don't balance out.


L3 and L2

Zooming in on the L3 and L2 points, we can see both bodies pull the same direction. These don't balance.


L4 and L5

Zooming in on the L4 and L5 points. Pluto pulls much harder. The angle between these vectors is 60º

The So-Called Centrifugal Force

There is a third player in these Lagrange tug of wars. What we used to call centrifugal force. This is not truly a force but rather inertia in a rotating frame. Here is an XKCD cartoon on this so called force:

Indeed, in a rotating frame, inertia sure feels like a force. The pseudo acceleration can be described as ω²r where ω is angular velocity in radians per time and r is distance from center of rotation. The vector points away from the center of rotation.

Putting Gravity and Centrifugal Force Together

Here's the same diagram but with centrifugal force thrown in (the blue vectors). Also the foot of the Charon gravity vectors are placed on the head of the Pluto gravity vectors -- this is a visual way to carry out vector addition.

For L1, Charon and Centrifugal Force are on the same team and they perfectly balance Pluto's gravity.

For both L2 and L3, Pluto and Charon are on the same team and they neutralize their opponent Centrifugal Force.

But what about L4 and L5? An observant reader may notice that the centrifugal force vector doesn't point away from Pluto's center. Adding Charon's tug to Pluto's tug moves the direction to the side a little bit.

Now the centrifugal force vector points from the barycenter. This is the common point of rotation around which both Pluto and Charon rotate. The same applies to L5.

L4, Charon's center and Pluto's center form an equilateral triangle.

The barycenter lies on the corner of a non-equilateral triangle.

And so it is with all the orbiting systems in our neighborhood. It is a 3 way way tug-of-war between centrifugal force, gravity of the orbiting body and gravity of the central body. Sometimes two players are on the same team, other places they switch. In L4 and L5 everyone pulls in a different direction. But in all 5 Lagrange points, the sum of the three accelerations is zero.

Xenon

Xenon The Noble Gas

Xenon is one of heavier Noble Gases

Screen capture from ChemicalElements.com

The noble gases are the orange column on the right of the periodic table. These are chemically inert. Which means they're not corrosive. This makes them easier to store or use.

Low Ionization Energy

Per this graph is from Wikipedia, Xenon has a lower ionization energy than the lighter noble gases.

Ionization energy for xenon (Xe) is 1170.4 kJ/mol. Ionization for krypton (Kr) is 1350.8 kJ/mol. Looks like about a 15% difference, right?

But a mole of the most common isotope of xenon is 131.3 grams, while a mole of krypton is 82.8 grams. So it takes 181% or nearly twice as much juice to ionize a gram of krypton.

Likewise it takes nearly 4.5 times as much juice to ionize a gram of argon.

The reaction mass must be ionized before it can be pushed by a magnetic field. Xenon takes less juice to ionize. So more of an ion engine's power source can be devoted to imparting exhaust velocity to reaction mass.

Big Atoms, Molar Weight

Low molar weight makes for good ISP but poor thrust. And pathetic thrust is the Achilles heel of Hall Thrusters and other ion engines. The atomic weight of xenon is 131.29 (see  periodic table at the top of the page).

Tiny hydrogen molecules are notorious for leaking past the tightest seals. Big atoms have a harder time squeezing through tight seals. Big whopper atoms like xenon can be stored more easily.

Around 160 K, xenon is a liquid with a density of about 3 grams per cubic centimeter. In contrast, oxygen is liquid below 90 K and a density of 1.1. So xenon is a much milder cryogen than oxygen and more than double (almost triple) the density.

Abundance

Ordinary atmosphere is 1.2 kg/m³ while xenon is about 5.9 kg/m³ at the same pressure. Xenon has about 4.8 times the density of regular air.

By volume earth's atmosphere is .0000087% xenon. 4.8 * .000000087 = 4.2e-7. Earth's atmosphere is estimated to mass 5e18 kg. By my arithmetic there is about 2e12 kg xenon in earth's atmosphere. In other words, about 2 billion tonnes.

Page 29 of the Keck asteroid retrieval proposal calls for 12.9 tonnes of xenon. Naysayers were aghast: "13 tonnes is almost a third of global xenon production for year! It would cause a shortage." Well, production is determined by demand. With 2 billion tonnes in our atmosphere, 13 tonnes is a drop in the bucket. We throw away a lot of xenon when we liquify oxygen and nitrogen from the atmosphere.

In fact ramping up production of xenon would lead to economies of scale and likely cause prices to drop. TildalWave makes such an argument in this Space Stack Exchange answer to the question "How much does it cost to fill an ion thruster with xenon for a spacecraft propulsion system?" TildalWave argues ramped up production could result in a $250,000 per tonne price. That's about a four fold cut in the going market price of $1.2 million per tonne.

Radon

If you examined the periodic table and ionization tables above you might have noticed there's a heavier noble gas that has an even lower ionization energy: Radon a.k.a. Rn.  Radon is radioactive. Radon 222, the most stable isotope, has a half life of less than 4 days. If I count the zeros on the Radon page correctly, our atmosphere is about 1e-19% radon -- what you'd expect for something with such a short half life. Besides being rare, it wouldn't last long in storage.

Scott Manley did a great video on xenon.

Where xenon excels

Great for moving between heliocentric orbits

Ion thrusters can get 10 to 80 km/s exhaust velocity, 30 km/s is a typical exhaust velocity. That's about 7 times as good as hydrogen/oxygen bipropellent which can do 4.4 km/s. But, as mentioned, ion thrust and acceleration are small. It takes a looong burn to get the delta V. To get good acceleration, an ion propelled vehicle needs good alpha. In my opinion, 1 millimeter/second² is doable with near future power sources.

If the vehicle's acceleration is a healthy fraction of local gravity field, the accelerations resemble the impulsive burns to enter or exit an elliptical transfer orbit. But if the acceleration is a tiny fraction of the local gravity field, the path is a slow spiral.

Earth's distance from the sun, the sun's gravity is around 6 millimeters/second². At Mars, sun's gravity is about 2.5 mm/s² and in the asteroid belt 1 mm/s² or less. Ion engines are okay for moving between heliocentric orbits, especially as you get out as far as Mars and The Main Belt.

Sucks for climbing in and out of planetary gravity wells

At 300 km altitude, Earth's local gravity field is about 9000 millimeters/second². About 9 thousand times the 1 mm/s² acceleration a plausible ion vehicle can do. At the altitude of low Mars orbit, gravity is about 3400 millimeters/sec². So slow gradual spirals rather than elliptical transfer orbits. There's also no Oberth benefit.

At 1 mm/sec² acceleration, it would take around 7 million seconds (80 days) to climb in or out of earth's gravity well and about 3 million seconds (35 days) for the Mars well.

Mark Adler's rendition of an ion spiral

where the thruster's acceleration is 1/000 that of local gravity at the start.

From a stack exchange answer on how to model low thrust trajectories. 

The general rule of thumb for calculating the delta V needed for low thrust spirals: subtract speed of destination orbit from speed of departure orbit.

Speed of Low Earth Orbit (LEO) is about 7.7 km/s. But you don't have to go to C3 = 0, getting past earth's Hill Sphere suffices. So about 7 km/s to climb from LEO to the edge of earth's gravity well.

It takes about 5.6 km/s to get from earth's 1 A.U. heliocentric orbit to Mars' 1.52 A.U. heliocentric orbit.

Speed of Low Mars Orbit (LMO) is about 3.4 km/s. About 3 km/s from the edge of Mars' Hill Sphere to LMO.

7 + 5.6 + 3 = 15.6. A total of 15.6 km/s to get from LEO to LMO.

With the Oberth benefit it takes about 5.6 km/s to get from LEO to LMO. The Oberth savings is almost 10 km/s.

10 km/s is nothing to sneeze at, even if exhaust velocity is 30 km/s. Climbing all the way up and down planetary gravity wells wth ion engines costs substantial delta V as well as a lot of time.

Elevators and chemical for planet wells, ion for heliocentric

So in my daydreams I imagine infrastructure at the edge of planetary gravity wells. Ports where ion driven driven vehicles arrive and leave as they move about the solar system. Then transportation from the well's edge down the well would be accomplished by chemical as well as orbital elevators.

Other possible sources of ion propellent.

Another possible propellent for ion engines is argon. Also a noble gas. Ionization energy isn't as good as xenon, but not bad. Mars atmosphere is about 2% argon. Mars is next door to The Main Belt. I like to imagine Mars will supply much of the propellent for moving about the Main Belt.

General template for space elevators

A Family of Conic Sections

Below is a general vertical space elevator. The conic sections are the paths payloads would follow if released from a point on the tether a distance r from body center.

We choose our units so radius of the balance point is 1. Centrifugal acceleration matches gravity at the balance point and net acceleration is zero. For tether locations above the balance point, centrifugal force exceeds gravity and net acceleration is up (away from the planet). For locations below the balance point, gravity is greater than centrifugal acceleration and the net acceleration is down.

This family of conic sections are coplanar, coaxial and confocal. Eccentricity is r³-1, setting r = 1 at the circular orbit of the balance point. (See this stack exchange answer for the math).

In the yellow region are hyperbolic orbits. In the blue region are are elliptical orbits higher than the circular orbit at the balance point. In the orange region, the tether drops payloads into elliptical orbits lower than the circular orbit at the balance point.

A circle of eccentricity zero separates the orange and blue regions, radius of circle = 1.
A parabola of eccentricity 1 separates the blue and gold regions, radius of parabola's periapsis = 2^1/3

Here is the same graphic zoomed in:

Here is the graphic as a Scalable Vector Graphic. I am hoping science fiction writers and illustrators will download this resource and use it.

Scaling this graphic for a variety of scenarios:

The numbers are in kilometers. In the case of earth, the circular orbit is the geosynchronous orbit at an altitude of about 36,000 kilometers.

In general, radius of a synchronous orbit can be described as:

r = (Gm / ω²)^1/3

Where ω is the body's angular velocity in radians, 2 pi radians/sidereal day.

Orbital Elevators

We usually think of an a space elevator anchored at the body's equator. An elevator can also be in a non synchronous orbit. Here the template is scaled to match the orbits of Phobos or Deimos:

Notice Phobos' tether foot  is above Mars surface. The foot is moving about .5 km/s with regard to Mars surface and therefore can't be anchored to Mars. Neither could a Deimos elevator be attached to Mars.

Orbital radius of Phobos is about 40% that of Deimos. So I cloned and shrunk Deimos' tether conics by 40%. I rotated the cloned family of conics by 180º.  The result is an interesting moiré pattern:

It was this pattern that led me to search for a common ellipse.

Eccentricity of the common ellipse:

e = (1 - (ω_Deimos/ω_Phobos)^1/2) / (1 + ω_Deimos/ω_Phobos)^1/2)

Periapsis and apoapsis of the common ellipse:

r_periapsis = (1 + e)^1/3 r_Phobos
r_apoapsis = (1 - e)^1/3 r_Deimos

ZRVTOs

Here's a pic of the ellipse Phobos and Deimos share:

This is an example of a Zero Relative Velocity Transfer Orbits (ZRVTO) - a term coined by Marshall Eubanks. In Marshall's words: "locations (and times, say for a Lunar and Terrestrial space elevator) where you drop things from one space elevator and they approach and hang motionless (for an instant) at a location on the other elevator.  ... what you would want for large scale movement of material."

Eubanks goes on to say "In practice, you might need a little bit of course correction delta-V to make up for radiation pressure, etc."

Also it would be rare for the elevators playing catch to be perfectly coplanar. So a small plane change delta V expense will be the rule rather than the exception. Still the delta V budgets would be a small fraction of what it would take for normal lift off and insertion to Hohmann transfers.

Not just Phobos and Deimos

To be an anchor for a vertical elevator, a moon needs to be in a near circular orbit and tide locked to its planet. This describes most of the moons in our solar solar system. For two moons to share an ellipse, they need to be nearly coplanar. Again, most the moons in our solar system.

Here are the common ellipses between the moons of Saturn:

Judging by the two gas giants and two ice giants in our solar system, families of coplanar, tidelocked moons are common.

Mini Solar Systems

Earlier I had looked at Mini Solar Systems, a notion I stole from Retrorockets. In our solar system Hohmann trip times between planets are on the order of months or years. Launch windows are typically years apart. But for a system of moons around a gas giant, trip times and launch windows are days or weeks. So a Flash Gordon paced story could take place without wildly improbable engineering.

GIELO and ELM

GIELO - Giant In Earth Like Orbit. ELM - Earth Like Moon. I have long been infatuated with this setting. Here is a painting I had done in 2001:

ELM the earth like moon is in the upper right. In the foreground a generation star ship is sending quad pod scout probes to investigate an artifact at the GIELO-ELM L4 region.

James Cameron's Avatar uses such a setting. Pandora is an ELM. I believe this setting could be developed a lot more. If ELM had sister moons and they were all tide locked, it would be a nice mini-solar system setting.

Icey moons with hospitable interiors.

Gas giants in Goldilocks zones aren't the only possibility. Temperature and pressure rise as we burrow deeper into a body. Earth might not be the only location in the solar system that has liquid water at a livable pressure. Thus the icey moons of our own solar system might eventually become "mini solar systems".

Planets of red dwarfs

And recently an approximately earth sized planet was found in the goldilocks zone of Proxima Centauri. Proxima Centauri is a small red dwarf star. The possibly earth like planet has an orbital radius of about 7.5 million kilometers and an orbital period of about 12 days. Planets about small red dwarfs are yet another possible "mini solar system" setting. Planets so close are likely tide locked to the star. Would atmospheric convection mitigate the temperature extremes between the night side and day side? I'm not sure. In any case, I believe there would be a comfortable region hugging the planet's frozen terminator. (By "frozen" I mean stationary).

Delta V and the rocket equation

The Retrorockets guy took a second look at mini solar systems. While trip times are short and launch windows frequent, it still takes a lot of delta V to insert to a Hohmann transfer. I was annoyed he used the incorrect Tom Murphy method of patching conics. But most his math is sound. He is correct that Tsiolkovsky's rocket equation would be a major pain in the mini solar system just as it is in ours. 

This is where a tether system comes into play. Given elevators on tide locked bodies and assuming most the bodies are nearly coplanar, travel between bodies could be done with very little reaction mass. It'd still take a lot of energy to move stuff up and down the elevators. But the difficult mass fractions imposed by the Tsiolkovsky's equation would no longer be a consideration.

Summary

Similar mathematical models and drawings can be used for a wide range of vertical tethers.

A popular misconception is that elevators are only good for getting off the ground. So it's a waste to build an elevator from a small body. But an elevator not only gets the payload off the ground, it can fling a payload towards a destination. The hyperbolic orbits portrayed in this post are especially interesting.

Space elevators would be especially useful in a system of tide locked moons. Or tide locked planets about a small star.

So far the only elevators I see portrayed in science fiction are from major planets. Like Kim Stanley Robinson's elevators in his Mars trilogy. Or Clarke's earth elevator in Fountains of Paradise. There are far more plausible elevators that could be very useful. These doable elevators could also provide many interesting settings.

Pluto Charon Elevator

Double Tidal Locking

Pluto and Charon are mutually tidally locked. That is, they both present the same face to the other planet all the time. They hover motionless in each other's sky. Pluto is in Charon synchronous orbit and Charon is in Pluto synchronous orbit.

What is more, the orbit is very nearly circular.

From Stern et al's 2015 paper:

(Thank you to Dr. Kirby Runyon for pointing me to this paper).

A tether could be extended from Pluto's near point to Charon's near point. Since the orbit is so nearly circular and obliquity is tiny, there would very very little flexing of this tether.

Minimum Tether to Remain Aloft

To remain aloft, a tether anchored to Charon would need to extend past the L1 point more than 10,000 kilometers to within nearly 2,500 kilometers of Pluto's surface.

This tether would be more than 15,000 kilometers long. Using Wolfe's Spreadsheet we find Zylon taper ratio is 1.13. Tether to Payload mass ratio is .88. This is with a safety factor of 3.

All The Way To Pluto

Extending the tether an additional 2,500 kilometers anchors it to Pluto's surface.

Taper ratio is about 1.7 and Tether to Payload mass ratio is 14.36.

Still acceptable but dramatically different from a tether only 2,500 shorter. This is because we dropped the tether foot into a much steeper part of Pluto's gravity well.

Net acceleration is .62 meters/second² at the Pluto end of the elevator. Very close to Pluto's surface gravity. At the Charon anchor net acceleration is -.28 meters/second². Very close to Charon's surface gravity. It is negative to indicate it's in the opposite direction from Pluto's gravity.

At L1 net acceleration is zero.

It's easy to see most of the stress newtons come from the close neighborhoods of Pluto or Charon. It might be worthwhile to build standard compressive towers at the elevator anchor points.

What's The Point?

Pluto's surface escape velocity is 1.2 km/s. Charon's surface escape velocity is .6 km/s. It's not that hard to get off the surface of Pluto or Charon. So what's the point of an elevator?

Space craft with very good ISP have meager thrust. With such space craft soft landings on Pluto or Charon would not be possible. Nor could they leave the surface of these planets.

But a low thrust craft could dock with the elevator at L1.

From L1 a small nudge could send passengers or cargo towards Pluto or Charon. And gravity would pull it the rest of the way down.

I believe Pluto Charon L1 would become  a major metropolis on the corridor between two major city states as well as a port to the rest of the solar system.

Will humans reach Pluto?

The Edge of Sunlight

Sunlight falls with inverse square of distance from sun. Asteroids 3 A.U. from from the sun will receive 1/9 of the insolation we enjoy on earth. Sun Jupiter Trojans at 5 A.U. will get 1/25 the sunlight. We could compensate by constructing large parabolic mirrors to harvest sunlight.

Giant parabolic mirrors could harvest sunlight for spin habs.

But Pluto  has a 30 A.U. by 49 A.U. orbit. And most of the time it dwells in the neighborhood of aphelion. 1/49² = about 1/2400. Mirrors for the KBO nation states would need to be vast. Mike Combs wrote a neat story featuring these sorts of mega mirrors. As much as I enjoy Mike's story, I don't think such monster mirrors are practical.

Fusion Power?

Will our technology achieve practical fusion power plants? Maybe. If so, that would vastly expand our possible frontiers.

The 4th Space Frontier

There's nothing like logistic growth ceilings to motivate opening a new frontier. As we settle and fill up one frontier, we start looking over the horizon. I'm going to make some wildly speculative predictions. This is a science fiction blog, after all.

1st space frontier: NEAs, Luna, Mars, Phobos and Deimos. This would give us one or two millennia of unrestrained growth.

2nd space frontier: The Main Belt. Three millennia of exponential growth. Ceres will be the capital of this United Federation of Main Belt Nation States. This frontier will open within a century or two after we establish a strong foothold on Mars/Phobos/Deimos.

3rd space frontier: The Sun Jupiter Trojans. The Hildas will be our ride from the Main Belt to the Trojans. It will take five hundred years to fill the Trojan petri dish.

4th space frontier: The Kuiper Belt as well as the icey moons of Saturn, Uranus and Neptune. As mentioned earlier, this would require practical power sources other than sunlight. Pluto will be the capital of the United Federation of Kuiper Belt Nation States. This frontier will take 10 millennia to expand into.

5th space frontier: The Oort. The nation states of the Oort will be separated by vast distances. They will be more isolated than even the nation states of the Kuiper. There is a strong incentive to become less reliant on trade and more self sufficient. 20 millennia of unrestrained growth. By the time we reach the outer Oort, nation states will be self sufficient biomes. There would be nothing preventing an outer Oort nation state from achieving solar escape velocity and leaving our sun's sphere of influence.

The Outer Oort Nation States will be natural generation star ships.

Charon Elevator through L2

Enough wild eyed fantasy. Back to mundane stuff like space elevators in the Kuiper Belt.

To maintain tension and remain , an elevator from Charon's far point through the Pluto Charon L2 would need to extend 41,000 kilometers. With a safety factor of 3, Zylon taper ratio would be 1.14. Tether to Payload mass ratio would be  about .3.

Small Problem: Styx

Pluto's moon Styx orbits at a distance of 42,600 kilometers from Pluto. Charon orbits at about 20,000 kilometers from Pluto. So a tether from Charon's far point can only extend about 22,000 kilometers before it runs the risk of an impact with Styx.

A counterweight would need to be placed on the elevator somewhere below the orbit of Styx. If placed just below the orbit of Styx, the tether top could impart a velocity of about .5 km/s. Which would be helpful for injection into heliocentric transfer orbits to other destinations in the solar system.

This elevator could also help with transportation between Charon and the other moons of Pluto: Styx, Nix, Kerboros and Hydra. 

An ion craft could also dock with Pluto Charon L2, so L2 could also serve as a port to the rest of the solar system. There are heteroclinic paths between L1 and L2 so transportation between the two elevators would be easy.

Pluto And I Share An Annivesary

Clyde Tombaugh discovered Pluto on February 18, 1930. February 18 is my birthday! So I guess it's only natural I'm interested in this body, we're practically twins.

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.

Lunar Sky Hook

Kim Holder has been urging me to do this blog post. Her comments in various forums have been helpful in thinking about this.

Vertical Lunar Tether In A Polar Orbit

This sky hook is a gravity gradient stabilized vertical tether. It's in a polar orbit so it will pass over the poles as well as the lower lunar latitudes.

Unlike an equatorial orbit, there are only two occasions during a lunar orbit where a tether's Vinf velocity vector is anti-parallel to the moon's velocity vector. So launch windows to earth would only occur each two weeks. That's still pretty often. These occasions are also good times to rendezvous with the tether.

Playing with earth moon three body simulations, polar orbits seem to remain stable up to a radius of around 20,000 kilometers. That is where I will set the anchor mass at the balance point of this sky hook. I believe this is far enough above the lunar surface that the mascons won't damage this tether's orbit.

Asteroid Anchor Mass Via a Keck vehicle

What to use for the anchor mass? With the asteroid retrieval vehicle proposed in the Keck Report, it is possible for a vehicle of moderate mass to retrieve a much larger mass to the earth moon neighborhood. The Keck authors believe a rock could be placed in high lunar orbit for around .17 km/s. A lunar orbit with a 20,000 km radius has a speed of around .5 km/s. I believe it would take around .7 km/s to park a rock in the orbit we want.

The Keck vehicle includes solar panel arrays and Hall ion thrusters. These would be great to have on a vertical tether. It takes awhile for ion engines to impart momentum, but given time they're about ten times as efficient as the best chemical rockets. A tether can build up momentum over time but release it suddenly. Thus they are a good way to enjoy an ion engine's great ISP and an Oberth benefit.

As well as adjusting the tether's orbit the Keck vehicle's solar arrays might also power elevator cars moving up and down the tether. If water is exported from from the lunar cold traps to the tether, the arrays might also crack water into oxygen and hydrogen bipropellent. There are a number of possible uses for this power source.

Upper Tether

The tether length above the anchor mass can be built in increments. I imagine the tether growing longer and more able with time. Here are three possible stages:

To EML2 or EML1

EML1 and 2 are about 65,000 km from the moon. To reach this apolune, we'd need an upper tether length of about 2700 kilometers. Using Wolfe's spread sheet, this tether length has a taper ratio of 1. With a safety factor to 3, tether mass to payload ratio is about .02.

This is pretty good. I believe this low stress tether length could accommodate copper wires to transmit power to the elevator cars.

Once at apolune, I believe it would take about .3 km/s to park the payload at EML2 or EML1.

EML2 is a good staging location should we want to travel to and from destinations beyond the earth-moon neighborhood.

To a Perigee at Geosynchronous Orbit 

Transfer orbit from GEO to the moon is about a an ~36,000 x 378,000 ellipse. Apogee speed is about .45 km/s. The moon's speed is about 1.02 km/s. So the tether needs to hurl a payload to a Vinf of (1.02-.45) km/s or about .57 km/s.

To achieve this Vinf our tether needs to be 12,200 km. Zylon taper ratio is 1.09. With a safety factor of three, Tether to payload mass ratio is about .167. So a ten tonne tether could accommodate a sixty tonne payload. This is still pretty good. A power cable along this length is also doable.

Perigee velocity of our transfer orbit is ~4.13 km/s. Geosynch orbit velocity is ~3.07 km/s. If the transfer orbit and destination geosynch orbit are coplanar, geosynch circularization would be about 1.06 km/s. But I expect that would be the exception rather than the rule. If the orbit inclinations differ by 20º, 1.6 km/s would be needed to park in geosynch.

To a Perigee at Low Earth Orbit.

A 300 x 378,000 km orbit has apogee velocity of ~.19 km/s. (1.02 - .19) km/s = .83 km/s.

To throw a payload to a trans earth orbit, our tether needs to impart a Vinf of .83 km/s. This takes a tether length of 19,200 kilometers. With a safety factor of three, Zylon taper ratio is 1.2. Tether to payload mass ratio is .38.

If perigee is through earth's upper atmosphere, aerobraking can provide a large part of the 3.1 km/s delta V for circularizing at LEO.

Lower Tether

Again, the tether length below the anchor mass can be built in increments. Incremental growth with time is more doable than trying to do the whole length in fell swoop. Here are some possible steps along the way.

To a Perilune at Low Lunar Orbit.

To drop a payload to a 90 km altitude perilune, length needs to be 7360 km. Given a safety factor of 3, Zylon taper ratio is 1.06. Tether to payload mass ratio is .15.

Velocity of transfer orbit's perilune is about 2.2 km/s. Low lunar orbit is about 1.6 km/s. It'd take about .6 km/s to circularize at low lunar orbit. 

To the Moon's Surface, Impact Velocity 1 km/s.

If the tether is extended to a length of 17890 km, tether foot altitude is about 370 km. Dropping a payload from this tether foot would result in a 1 km/s impact. 

Given a safety factor of two, Zylon taper ratio is 2.88. Tether to payload mass ratio is 26.87.

Note the safety factor is less than in the other scenarios. As we descend further into the moon's gravity well, stress climbs more rapidly. It would be more difficult to include copper wires for power along the lower parts of the tether.

To a Tether Foot Just Above the Moon's Surface.

Dropping the tether foot to an altitude of 10 kilometers gives us a length of 18,252 km. Safety factor of 2 and Zylon taper ratio is 3.72. Tether to payload mass ratio is  about 51.

Dropping from this tether foot, a payload would impact the lunar surface at .184 km/s. 

A .2 km/s payload delta V budget for soft landing seems quite doable. Likewise it would take about .2 km/s to launch a payload from the lunar to rendezvous with the tether foot.

However dropping the tether foot this far is considerably more ambitious than the other scenarios described above.

Travel About The Moon

Kim Holder noted such a tether might serve as transportation between locations on the moon.

Without a tether, going from pole to pole would take about 3.4 km/s: 1.7 km/s to launch and another 1.7 for soft landing. Going from equator to pole would take 1.53 km/s to launch and another 1.53 km/s for a soft landing, totaling 3.06 km/s. 

So a 18,000 km lower lunar tether length would make travel about the moon easier.

A Location to Process Asteroid Ore

It takes about .6 km/s to park ore from some of the more accessible asteroids in 20,000 km lunar orbit. If rendezvous with the tether top is doable, it could take considerably less.

I envision infrastructure accreting about the tether anchor mass 18,262 km above the lunar surface. Water, platinum, gold, rare earth metals, and other materials could be extracted at the anchor. Refined commodities could climb to the top of the tether and then tossed earthward.

A Synergy Between The Moon and Near Earth Asteroids

Moon and asteroid enthusiasts are often at odds with one another. They should be allies. In terms of delta V, it's a lot easier to park asteroids in lunar orbit than lower earth orbits. And given growing infrastructure in lunar orbit, the moon's surface becomes more accessible.

Hildas As Cyclers

Hilda Asteroids - Red,   Sun Jupiter Trojans - Blue,   Main Belt - Green

The above image was made from screen captures of Scott Manley's beautiful animation Asteroids In Resonance With Jupiter.

Jupiter is the dot off to the left, the sun is the yellow dot in the middle. Within the Main Belt can be seen Mercury, Venus, Earth and Mars. I colored the different asteroid populations so we can tell them apart.

The Sun Jupiter Trojans have a 1 to 1 resonance with Jupiter. They co-rotate with Jupiter. The leading Trojans remain in a neighborhood 60 degrees ahead of Jupiter and the trailing Trojans stay in a neighborhood 60 degrees behind.

The Hildas have have 3 to 2 resonance with Jupiter meaning they circle the sun three times for every two Jupiter orbits. Jupiter's orbital period is about 12 years and the Hildas have 8 year periods.

The Hilda orbits only look triangular in Manley's animation because they're being viewed in a rotating frame. You can see Jupiter remains on the left side of the image. In an inertial frame, a Hilda orbit is an ordinary elliptical orbit with aphelion passing through the Trojans and perihelion passing through the main belt.

I envision the Hilda biomes playing a similar role as Marco Polo's caravans shuttling people and goods between east and west. But the Hildas travel between the Trojans and the Main Belt.

The would be a series of regular fly bys for a Hilda Cycler

1) Main Belt to trailing Trojans — 4 years.

2) Trailing Trojans to Main Belt — 4 years.

3) Main Belt to leading Trojans — 4 years

4) Leading Trojans to Main Belt — 4 years

5) Main Belt to Sun Jupiter L3 — 4 years. But there is no asteroid population at SJL3.

6) From SJL3 to Main Belt 4 years

Then back to step 1). The cycle repeats itself.

So not only can a Hilda be a go between between the Main Belt and Trojans, but it can also move stuff between the trailing and leading Trojan populations. Trailing to leading takes 8 years and leading to trailing takes 16 years.

As can be seen from Manley's animation, there is a steady stream of Hildas traveling the circuit. 

Delta V

The Hildas have a variety of eccentricities. I will look at a Hilda orbit having an eccentricity of .31. That would put the aphelion at 5.2 A.U. and the perihelion at 2.74 A.U. (The perihelion is in Ceres' neighborhood, Ceres' semi-major axis is 2.77 A.U.).

Assuming a circular, coplanar orbit at 2.74 A.U.,  it would take 2.6 km/s to leave a Main Belt Asteroid and board a Hilda.

Assuming a circular, coplanar orbit at 5.2 A.U., it would take 2.2 km/s to depart the Hilda and rendezvous with a Trojan.

However, coplanar orbits is a very optimistic assumption. Asteroids have a large variety of inclinations. Making a 10 degree plane change from a Hilda's orbit can cost 2 to 3 km/s.

Ways to mitigate delta V expense

Many asteroids spin about pretty fast. This plus their shallow gravity wells make them amenable to bean stalks, also known as space elevators. 

"Why would an asteroid need a space elevator?" I'm sometimes asked. The questioner will assert "It's very easy to get off an asteroid's surface, and getting off the body's surface is the only reason for an elevator." Which is wrong, of course.

Speed of a body on an elevator is ωr where ω is angular velocity in radians per time unit and r is distance from center of rotation. If r is large, the elevator can fling a payload at high velocity with regard to the asteroid. It is quite plausible for an asteroid's bean stalk to provide .5 to 1 km/s delta V.

Also an asteroid bean stalk allows rendezvous with an ion propelled space craft. Ion ships have great ISP but minute thrust. Soft landings with an ion craft are not possible on larger asteroids like Ceres, or Vesta.

And ion propelled ships are more viable in the outer system. When a ship's acceleration is a large fraction of the local gravity acceleration, an ion burn is more like a chemical impulsive burn. See General Guidelines for Modeling a Low Thrust Ion Spiral. In the outer Main Belt, the sun's gravity is about 1 millimeter/sec². Sun's gravity at the Trojans is about .2 millimeters/sec².

Jupiter's Trojans

Not much is known about Jupiter's Trojans. Japan hopes to launch a mission in the early 2020's. Another proposed mission is Lucy which would also launch in the early 2020's. 

These bodies are on average 5.2 A.U. from the sun and so receive only 1/27 the sunlight earth enjoys. For this reason I am hopeful they are rich in volatile ices. I'd give better than even odds they have lots of water and carbon dioxide ice. Nitrogen compounds like ammonia and cyano compounds are a possibility. Aside from earth, Nitrogen is in short supply throughout the inner solar system and these would be a great export to the Main Belt biomes.

Their numbers are speculation. According to Wikipedia:

Estimates of the total number of Jupiter trojans are based on deep surveys of limited areas of the sky.^[1] The L₄ swarm is believed to hold between 160–240,000 asteroids with diameters larger than 2 km and about 600,000 with diameters larger than 1 km. If the L₅ swarm contains a comparable number of objects, there are more than 1 million Jupiter trojans 1 km in size or larger. For the objects brighter than absolute magnitude 9.0 the population is probably complete. These numbers are similar to that of comparable asteroids in the asteroid belt. The total mass of the Jupiter trojans is estimated at 0.0001 of the mass of Earth or one-fifth of the mass of the asteroid belt.

Two more recent studies indicate, however, that the above numbers may overestimate the number of Jupiter trojans by several-fold. This overestimate is caused by (1) the assumption that all Jupiter trojans have a low albedo of about 0.04, whereas small bodies may actually have an average albedo as high as 0.12;^[16] (2) an incorrect assumption about the distribution of Jupiter trojans in the sky. According to the new estimates, the total number of Jupiter trojans with a diameter larger than 2 km is 6.3 ± 1.0×10⁴ and 3.4 ± 0.5×10⁴ in the L₄ and L₅swarms, respectively. These numbers would be reduced by a factor of 2 if small Jupiter trojans are more reflective than large ones.^[16]

The number of Jupiter trojans observed in the L₄ swarm is slightly larger than that observed in L₅. However, because the brightest Jupiter trojans show little variation in numbers between the two populations, this disparity is probably due to observational bias. However, some models indicate that the L₄ swarm may be slightly more stable than the L₅ swarm. 

The largest Jupiter trojan is 624 Hektor, which has an average diameter of 203 ± 3.6 km. There are few large Jupiter trojans in comparison to the overall population. With decreasing size, the number of Jupiter trojans grows very quickly down to 84 km, much more so than in the asteroid belt. A diameter of 84 km corresponds to an absolute magnitude of 9.5, assuming an albedo of 0.04. Within the 4.4–40 km range the Jupiter trojans' size distribution resembles that of the main-belt asteroids. An absence of data means that nothing is known about the masses of the smaller Jupiter trojans. The size distribution suggests that the smaller Trojans are the products of collisions by larger Jupiter trojans.

I'd love to see science fiction stores set on 624 Hektor.

This article written in memory of Hilda Alvarez May 5, 1929 - July 20, 2016