Wednesday, April 25, 2018

Tahoe recreational math conference

I've been invited to talk at a recreational math conference at Lake Tahoe April 28.

I will talk about the family of conics associated with a space elevator or a vertical orbital tether.

The presentation combines stuff from a number of my earlier blog posts. Here is the pdf I will use for this talk.

A screenshot from the cover of the booklet.

Pages 2 — 5 -- includes the visualizations I use to remember expressions for gravity and centrifugal acceleration. Which I use to show how canonical units are a whole lot easier to use than kilograms, kilometers and so on. For example the orbital periods or an orbit whose semi-major axis is k A.U. will have period k3/2 years. An asteroid whose semi-major axis is 4 A.U. will have a period of 8 years. A 9 A.U. semi-major axis gives a period of 27 years.

Pages 13 — 14 -- I believe my general method  for finding a ZRVTO orbit between tethers is new and original. If anyone knows of this method appearing in earlier publications, please give me a heads up.

Page 14 -- Most moons we know of are in nearly circular tide locked orbits. For such moons beanstalks through Planet-Moon L1 and L2 are plausible. I believe this would be a wonderful science device. But today's science fiction writers seems oblivious to any of the Lagrange points other than L4 and L5.

Pages 15 and 16 -- Shameless self promotion. I don't get any royalties on the Dover coloring books, I was paid a straight fee of $100 per page. Regardless, I'd like to see the books sell. A Dover editor placed his bets on me and I hope he is rewarded for this gamble. I do make money on T-shirt sales though. My T-shirts are available at  .

Friday, June 9, 2017

Zylon Mars Elevator

Mars Elevator With Conventional Materials

Mars spins nearly the same rate as earth (about a 24.62 hour day). Mars has about 1/9 earth's mass. At 17,000 kilometers, altitude of Mars synchronous orbit is less than half the altitude of geosynchronous orbit (about 36,000 kilometers).

These considerations have led some Mars enthusiasts to claim a Mars elevator made of conventional materials is possible. No bucky tubes or other science fiction material is needed, Kevlar will do. Is this true? I will take a look using Chris Wolfe's spreadsheet.

Safety Factor

In earlier blog posts using Wolfe's spreadsheet I used a safety factor of 1, a razor thin margin. The slightest scrape or nick will make the tether break. This is like drawing a pentagram to summon the demon Murphy's Law. No sensible entity would risk expensive payloads on such a narrow margin. Much less human lives. I hope to revise my earlier blog posts to include more sensible safety margins.

In later blog posts I looked at scenarios using a safety factor of 3. With this margin a portion of tether can lose up to 2/3 of it's mass without breaking.

In this post I'll use tables looking at a range of safety factors.  With a safety factor of 2, I cut tensile strength in half. A safety factor of 3 cuts tensile strength to a third. Which is a lot like cutting exhaust velocity in the rocket equation. Increasing an exponent can make tether thickness sky rocket.

Mars Equator to Mars Synchronous Orbit

This is the lower part of a Mars elevator. It exerts downward newtons that need to be balanced with upward newtons from elevator mass above Mars synchronous orbit.

Tether to
 Mass Ratio 

Payload is mass of elevator car as well as elevator car's contents. The elevator car will need to include motors and power source.

Mars Synchronous to Sub Deimos Elevator Top

Elevator top is set 50 kilometers below Deimos' periapsis. This is to avoid collision. The counterweight and tether above Mars synchronous orbit must counterbalance the downward force of the lower elevator.

Tether to
 Mass Ratio 
to Payload
 Mass Ratio 

The Whole Shebang

Safety Factor 1

Assuming lifting a 10 tonne elevator car and contents from Mars' surface and given a safety factor of 1, we'd need 10 * (38 + 154) tonnes of tether material. That'd be 1,920 tonnes of Zylon. Perhaps worthwhile if the elevator had a vigorous through put. I think these are the numbers Mars enthusiasts are talking about when they talk about Mars beanstalks made of Kevlar.

Also needed would be a 12,000 tonne counterweight. That's about thirty times the mass of the I.S.S.. This to lift a 10 tonne elevator car from Mars' surface? The need for a stud hoss counterweight sinks the argument for a Mars elevator, in my opinion.

Safety Factor 2

10 * (162 + 955) = 11170. About 11 thousand tonnes of Zylon to lift a 10 tonne elevator car and contents.

We'd need a nearly 150,000 tonne counterweight.

I think it's pretty obvious a Zylon Mars elevator with a safety factor of two isn't worthwhile. I'm not going to bother looking at a safety factor of 3.


The elevator top is moving at about 1.7 km/s. It needs another 1.6 km/s to achieve Trans Earth Insertion (TEI). From the surface of Mars it takes about 6 km/s for TEI. So the elevator cuts saves about 4.4 km/s off of trips to earth.


Given a sensible safety factor, a Zylon tether would need to be much more massive than the payload. The counterweight mass would dwarf the payload mass.

Mars neighbors the main asteroid belt. Some rocks from the belt make their way to Mars neighborhood. Collision with asteroidal debris could cut the tether. Given this elevator's 20,000 km length and healthy taper ratio, there is a large cross sectional area. This increases likelihood of an impact.

Also there is a chunk of Debris named Phobos which crosses the elevator's path every 10 hours or so.

Comparison to Phobos Elevator

A Phobos elevator dropping to Mars' upper atmosphere and extending to Trans Ceres insertion is about 13,700 km. This about 6,000 km shorter than the Mars elevator described above. It also has a smaller taper ratio. This makes for a smaller cross sectional area to intercept debris. Being anchored at Phobos, this elevator won't collide with Phobos. The top is well below Deimos. orbit.

This tether can provide Trans Ceres Insertion as well as Trans Earth Insertion.

It takes about a .6 km/s suborbital hop for a Mars ascent vehicle to rendezvous with this tether foot.

Using a safety factor of 1, the upper Phobos tether has a 3.21 payload to mass ratio. The lower Phobos tether has a tether to payload mass ratio of about 16.1. So from top to bottom, about twenty times the payload mass is needed in Zylon.

The Phobos takes about 1/10 of the Zylon mass for a mars elevator with a safety factor of one.

A sub Deimos Mars elevator can't throw payloads above Mars escape velocity.
But with higher taper ratio, it'd take ten times as much zylon mass than a Phobos elevator.
This is with a safety factor of 1.
A Zylon Mars elevator with better safety factors is impractical.

I hope to revisit the upper Phobos tether and lower Phobos tether pages and include safety factors of 2 and 3. I suspect with a higher safety factor that a Zylon tether from Phobos to Mars upper atmosphere may not be feasible.

Wednesday, February 1, 2017



Rotation Matrix
Proportional Scaling Matrix
Non Proportional Scaling Matrix
Shear Matrix
Reflect Matrix

Determinant of a Matrix

Lorentz Transform Matrix


Vectors are a way to describe point locations with numbers. Vectors can be used to build simple shapes like a cube or just about any shape you can imagine.

We do lots of stuff to these vectors with matrix multiplication.  We can grow, shrink, spin, stretch, squeeze, tilt and flip these guys.

First we'll look at the things you can do to vectors on a plane with computer drawing programs like Adobe Illustrator. Below are some items from the Illustrator tool box that employ matrices.

Rotation Matrix

Rotating a polygon doesn't change it's area. The area remains the same. The determinant of this matrix is 1.

Proportional Scaling Matrix

Doubling size as well as height boosts a polygon's area by a factor of four. This determinant of this matrix is 4.

Non Proportional Scaling Matrix

This matrix stretches the width to twice the original and squeezes the height to half of what it was. Overall the area is unchanged. The determinant of this matrix is 1. However the determinant of a non proportional scaling matrix can be more or less than 1.

Shear or Skew Matrix

When I was using Macromedia Freehand, the graphics program called this transformation "skew". Then Adobe ate Macromedia and I was forced to use Adobe Illustrator. Illustrator calls it "shear".

This transformation transforms a horizontally aligned rectangle to a parallelogram with same base and height. Area remains unchanged. The determinant of this matrix is 1.

Flip Matrix

Making the first term in the main diagonal negative flips polygons about the y axis. Making the lower right term negative would flip polygons about the x axis.

Determinant is -1. Not sure what that means geometrically but absolute value of the area remains the same.

Illustrator Tool Box

Determinant of a Matrix

Below is a general 3x3 matrix multiplied by each of the usual basis vectors in 3-space.
Notice the first basis vector is transformed into the first column of the matrix, the 2nd basis vector is transformed into the second column, and so on.

The 3 basis vectors form edges of a unit cube. Multiplying each of the vertices of this unit cube by our general matrix, we get a parallelepiped with edges (a, b, c), (l, m, n) and (x, y, z).

Of course the volume of the unit cube is one cubic unit. To find the volume of the transformed parallelepiped we take the determinant of the matrix

There are some matrices that don't change the size or shape of the objects they transform. Rotation matrices, for example. These have a determinant of 1. Matrices that don't change the size but flip the chirality of an object (say, change a left shoe into a right shoe), have a determinant of -1.

Lorentz Transformation

In ordinary Euclidean space, a point (x, y, x)'s distance from the origin would be
sqrt(x2 + y2 + z2 ), a metric easily arrived at with the Pythagorean theorem.

But the time space manifold we dwell in is a little strange. The metric is
sqrt(-t2 + x2 + y2 + z2 ). One of these dimensions is not like the other one.

In ordinary Euclidean space, changing Point Of View (POV) entails a translation and/or a rotation. In our space time, changing POV entails a Lorentz Transformation.

Adam Zalcman did a nice job of portraying the Lorentz transformation as a matrix. Here is a screen capture from his physics stack exchange answer:

A Lorentz matrix for a 2 dimensional Minkowski space looks like this:

Above is our two dimensional Minkowski space. As they move through time inhabitants can move either right or left. The ship leaves earth in the present. A year later it has moved half a light year to the right. It is moving .5 c.

Pluggin .5 c into our Lorentz transform matrix we get:

Transforming our Minkowski space with this matrix we get:
The ship's world line has been shoved to the left. From the ship passengers' point of view, they aren't moving. Also they perceive .866 years have elapsed, not a full year. The earth's world line has also been shoved to the left. From the ship's POV the earth is moving .5 c to the left.

Note that the diagonals remain in the same place. Earth folks as well as ship passengers both perceive light photos to be traveling at 1 c (c is the speed of light).

While the transformation stretches along one diagonal, it also squeezes along another diagonal. So the area remains the same. Determinant of this matrix is one.

When I first saw the transformed coordinate system I was thinking "Wait a minute. Earth is now more than half a light year away and only .866 years have passed on the ship. Seems like earth is going more than .5 c.  My mistake was in using the word "now". What were simultaneous events from one frame are no longer simultaneous.

Note that from the ship's P.O.V. Earth's clock is running slower. This is possible because simultaneous events along worldlines change depending on which frame the viewer's in.

Here is an animation showing different transforms where the ship's speed varies from -.9 c to .9 c.

Winchell Chung of Atomic Rockets describes a scene from a Heinlein novel where a student asks:

     “Mr. Ortega, admitting that you can’t pass the speed of light, what would happen if the Star Rover got up close to the speed of light—and then the Captain suddenly stepped the drive up to about six g and held it there?”
     “Why, it would—No, let’s put it this way—” He broke off and grinned; it made him look real young. “See here, kid, don’t ask me questions like that. I’m an engineer with hairy ears, not a mathematical physicist.” He looked thoughtful and added, “Truthfully, I don’t know what would happen, but I would sure give a pretty to find out. Maybe we would find out what the square root of minus one looks like—from the inside.”

Let's take a look at world lines where one rocket is moving .5 c to the left and the other is moving .5 c to the right. At first glance it'd seem like the rocket moving to the left would be moving the speed of light with regard to the other rocket.

Transform the scene on the left to the orange ship's point of view. From the orange ship's P. O. V., the purple ship is moving .8 c to the left. The Lorentz transformation doesn't shift the purple ship all the way to the edge of the light cone. (Click on image to get a larger version).

From the point of view of each world line, his immediate neighbors are moving either .5 c to the left or .5 c to the right. The arrowhead on each line corresponds to the passage of one year from that world line's point of view. The horizontal line indicates simultaneous events from the P.O.V of the central world line after one year. These trace out a hyperbola with the edges of the light cone as asymptotes.

If the above lines were a cone, plane of simultaneous events would cut the cone along a circle and the world lines would pierce that circle in points closer and closer the edge as the world lines approached c. This would be a Poincare disk.

M. C. Escher's Circle Limit prints are based on Poincare disks.

Each angel or demon on this disk perceives themselves to be at the center while their neighbors shrinking towards the boundary of this world as they grow more distant. So it would be with Mr. Ortega's student who would step on the gas when he's moving .999 c. He'd just shift his position to the another part of the disk and he would be no closer to the edge.

Saturday, November 26, 2016

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.


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.

Thursday, September 15, 2016


Xenon The Noble Gas

Xenon is one of heavier Noble Gases

Screen capture from

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.


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.


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.

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.

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.

Saturday, September 10, 2016

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 = 21/3

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 - (ωDeimosPhobos)1/2) / (1 + ωDeimosPhobos)1/2)

Periapsis and apoapsis of the common ellipse:

rperiapsis = (1 + e)1/3 rPhobos
rapoapsis = (1 - e)1/3 rDeimos


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 - 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.


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.

Saturday, August 27, 2016

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.

A tether could be extended from Pluto's near point to Charon's near point. Since the orbit is so nearly circular, 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.