Thursday, July 23, 2015

Review -- Elon Musk Quest for a Fantastic Future

by Ashlee Vance.

Each chapter of this book struck a chord with me. As usual this will be a Spinrad style review where I use Vance's book as an excuse to jump on my soapbox.

Winter Is Coming

I often feel like we're trapped in a bleak George R. R. Martin story. Rate of growth may be slowing but the planet's population is still rising. Appetite for consumer goods is climbing as the third world catches up to industrialized nations. In the meantime finite resources are getting harder to come by. Hydrocarbon fossil fuels aren't going to last forever.

ERoEI should be a term on everyone's mind. But policy makers and general populace remain oblivious. What's trending at the time of this writing? Caitlyn Jenner, presidential candidate Trump is saying we need to build a big wall along our southern border to keep out Mexican rapists, Justin Beiber's naked butt on Instagram.

It's like Game of Thrones. Greed, stupidity and cruelty triumph time after time in spite of the hero's best efforts. In slow motion we're watching our derailed train head for a cliff and nobody's putting on the brakes.

A Dream of Spring

But there's a ray of hope. Unlike Martin's gritty realism, Musk seems to be a character from the golden age of Marvel comics. A Tony Stark like character who actually does triumph over insurmountable odds. Vance describes several periods in Musk's life where doom seems imminent. But through sheer tenacity he overcomes one impossible obstacle after another.


Obsessive Compulsive Disorder. Describing Musk's childhood on page 38:
… Soon he owned a Commodore VIC-20, a popular home machine that went on sale in 1980. Elon's computer arrived with … a workbook on the BASIC programming language. "It was supposed to take like six months to get through all the lessons," Elon said. "I just got super OCD on it and stayed up for three days with no sleep and did the entire thing. It seemed like the most super-compelling thing I had ever seen."
(Added emphasis mine). OCD is one of my major faults. How many times have I stayed up til 4 a.m. playing Tetris? How many hours squandered in the labyrinth mazes of Zelda? Etc.

But a crippling flaw can also be an empowering strength. Now I feel less guilty when I get lost doing a drawing or obsessed with a geometry problem.

Musk has a peculiar mental make up. Other factors came together for a perfect storm making Musk a game changing personality. His father was a well to do engineer. Musk had the opportunity and will to master several key skills during a time of dramatic change.

Electric Cars

What happens after peak oil? Nuclear might take care of our electricity needs. But what about transportation? Cars and trucks use gasoline.

There's the Prius -- a misbegotten bastard child of gas and electric with extra mass and complexity.

The fully electric cars I had seen were golf carts that needed to be recharged every 20 miles.

Then Tesla blew my opinions out of the water. An electric car with oomph as well as range. Learning of this car's existence filled me with relief and joy.

I was also delighted to hear of the improved energy storage with lithium ion batteries. This makes solar power more viable, another arena in which Musk is fighting the good fight. I still believe nuclear will be our primary source of power. But improved storage means solar will play a more prominent role.

Tesla and Solar City are traded on the stock exchange. For TSLA and SCTY he regarded going public as a Faustian deal, a necessary evil. He explains that shareholders tend to have short range goals, a profitable quarter while he has a more long range agenda. Well, I've gone on E-Trade and bought one share of Tesla and three shares of Solar City. I am one shareholder that endorses Musk's long range goals. Musk managed to avoid going public on SpaceX

Breaking Free of Cradle Earth

For decades we've been spinning our wheels in low earth orbit. Human Space Flight has been the cash cow of near monopolies that don't care about humanity's future. For our elected officials, HSF is a pork barrel program for buying votes.

New Space pundits like Rand Simberg or Henry Spencer have long been advocating a competitive space market. They rightly point out cost plus contracts encourage graft. That savings can be realized with mass production when R&D expense is amortized over many units.

But these insights weren't sufficient to overcome inertia. That is until Musk came along.

That Musk has built a new aerospace company from the ground up is an amazing accomplishment. I had placed my bets a new player would win the 100 kilometer X-prize. But a new kid on the block achieving orbit and remaining solvent? That took me by surprise.

Many of the possible economies Spencer and Simberg predicted have been realized by Musk. Economies of scale. A supply chain distributed through many congressional districts and policies set by committee guarantees waste.  Much of SpaceX parts are made in house. Central management is lean and mean with little or no duplication of effort. Musk has already brought down the cost of getting to orbit.

One of the most exciting SpaceX goals are reusable space vehicles. How much would a transcontinental trip cost if we threw away a 747 each flight? Musk correctly sees quickly and economically reusable space ships as a prerequisite for breaking free of Cradle Earth.

SpaceX is making good progress towards a reusable booster stage. Since the booster is a lot more massive than an upper stage, reusable boosters could make a huge cut in the cost of getting to space. Presently I'm giving two-to-one odds SpaceX will achieve cost savings with a reusable booster.

How about a SpaceX reusable upper stage? I'm betting against it. An upper stage's big delta V budget makes for a difficult mass fraction. If dry mass is 8% or less, it's hard to have robust structure and thermal protection. Re-entering the atmosphere at 8 km/s subjects the space ship to extreme conditions and I can't see the delicate eggshell of an upper stage surviving this abuse.

But maybe I'm wrong! I already have egg on my face for earlier bets against Musk. If I'm right, I believe Musk will find other ways to deal with upper stage re-entry. Maybe he'll look at solutions like extraterrestrial  propellent and/or momentum exchange tethers.

Musk has beat the odds numerous times in the past. I'm betting he'll continue to surprise people in the future.

Saturday, July 18, 2015

My Geoscapes books are selling!


Sales of Dover Creative Haven Geoscapes recently picked up.

Last week saw sales of about 22 books per day. There was a week in March when 46 books a day were being sold. I have no idea what caused this recent upsurge. Three of my friends and relatives have told me they've seen the book on display at Barnes and Noble stores. My daughter and I walked into a grocery store/delicatessen and saw my book on display along with other Dover Creative Haven books.

I'm proud of this coloring book, it explores geometrical themes: perspective drawings, studies of polyhedra, spirals etc.  I'll post a few screen captures.

A tribute to two of my heroes: Da Vinci and Kepler. Leonardo Da Vinci would make open faced polyhedral models composed of beams along the edges. That way he could study the interior of a polyhedra. Around the edges of this page are Leonardo style Platonic dodecahedra. If you extend the edges of each pentagon to form a five pointed star, you get a Kepler solid, the small stellated dodecahedron (center). Besides revolutionizing our view of the solar system, Johannes Kepler did a lot of wonderful exploration and discoveries in solid geometry.

Study of the dodecahedron-icosohedron symmetry group. Top left: Leonardo style dodecahedron, Top right: Leonardo style icosahedron, Center: Interpenetrating Leonardo style dodecahedron and icosahedron showing the duality between these two solids.

Lower three solids are Archimedean solids that result from truncating (slicing off corners) of either the dodecahedron or icosahedron.


I am fascinated with space filling bricks, a.k.a. honey combs. At the bottom are cubes. Cubes are a special case of a rectangular solid, the sort of bricks we're all familiar with. The structure at the top are alternating octahedra and tetrahedra, a.k.a. an octet structure. The octet faces are equilateral triangles. Height of an equilateral triangle is sqrt(3)/2 length of a side, an irrational number. So at first glance it seems like octet structures would be incompatible with cubic structures. But this gulf is bridged by a third type of space filling brick: the truncated octahedron. That's why I call the truncated octahedron the Bridge brick.

A Lego-like construction toy I invented. But instead of cubic structure, this toy would build octet structures (see image and explanation just above this one). And instead of a single male face and a single female face, every face has a hermaphrodite connector. Bridge truncated octahedral construction units might be a way to make my toy compatible with Legos.

Don't know what to say about these interpenetrating spiral structures. Except that they were fun to draw.

Hoping the people who bought my book enjoy my strolls through strange geometry gardens.

Surreal Visions

Another book I'm proud of is Surreal Visions. But it's still suffering flat sales. I'll post a few images any way:

These rhino monkeys are frolicking about a Klein Bottle. A peculiar object, somewhat like a three dimensional Möbius Strip.

The Riemann Sphere maps the points on the surface of a sphere to a plane (with the exception of the north pole!). Mathematician Chaim Goodman Strauss helped me come up with a related mapping of points from 3-space onto 3-space. Surreal Visions features several images based on this mapping.

And finally one more from Surreal Visions. This image is called Tears.

Wednesday, July 1, 2015

Review: Declaration by James Patrick Kelly

Spoiler alert: I give away events unfolding in Kelly's story. It appeared in the March 2014 edition of Asimov's Science Fiction Magazine.

"Declaration" is a thought provoking extrapolation of existing trends. In the style of Spinrad, I will use this review as an excuse to jump up on a soapbox and deliver my own opinions.

In this tale more and more people are dwelling in softtime. Softime is what today's internet might evolve into, a shared online virtual reality. Depending on sophistication of interface, the virtual reality can be fully immerseive.

A significant part of the population are severely disabled and can't interact with the world using their meat bodies. These disabled people are known as stash. They are more or less stashed in coffin like life support cubicles.

The government mandates that everyone spend an allotted time in hardtime, a.k.a. reality. Stash revolutionaries want to spend all of their existence in softtime. The story title Declaration refers to the revolutionaries' Declaration of Independence. They want to sever their connections with the real world.

But would the revolutionaries become independent of hardtime? In Kelly's story, that's not clear. Are the stashed people dependent parasites or do they provide services and do meaningful work? Robots are ubiquitous in the story but I get the impression machines haven't fully replaced humans. There still seems to be need for meat to interact with the world to maintain infrastructure and take care of business. For example the main character turns her stash brother to prevent bed sores, even though the brother has a carebot.

Primitive versions of these interfaces already exist. For example motion capture sensors control virtual puppets in movies like Shrek or Avatar, as well as virtual avatars in computer games.  Neuroscientist Miguel Nicolelis has implanted a Brain Machine Interface into the cortex of monkeys that they've used to control virtual avatars.

From inner space back to outer space

Virtual avatars aren't the only puppets controllable by motion capture or brain machine interface. Nicolelis' monkeys have also used their cerebral implants to control remote robotic arms. A paralyzed person using a Nicolelis exosuit did the opening kick in the 2014 World Soccer Cup. Surgeons use motion capture sensors to operate surgical telerobots.

Kelly's story has robots as well as a sophisticated brain machine interface. Given telerobots, exosuits, and robotic prosthetics, the boundary between hardtime and softtime blurs. There are hard as well as soft avatars.

Telerobots are becoming major game changers. They are doing work in places too hard to reach or dangerous for humans. British Petroleum uses them to build oil drilling infrastructure on the seafloor. Planetary Resources hopes to use them to mine the asteroids. Paul Spudis and Bill Stone hope to use robots to prospect and establish mining infrastructure on the moon.

Kelly is correct a severely disabled person would want to use a Brain Machine Interface every waking hour whereas a healthy person only a fraction of the time. I believe the severely disabled will be the most practiced users of telerobots. They could be the most intrepid explorers, the most able builders, the heroes of a coming age.

Fashions in Science Fiction

A Brain Machine Interface story from yesteryear was Anne McCaffrey's hopeful and uplifting The Ship That Sang. More recently we have Kelly's bleak dystopia Declaration.

Kelly's story has a lot of currently fashionable themes: overpopulation, terrorism, limited resources, alienation, inevitable decay. More often than not modern SF is gloom and doom exploring catastrophic failure modes of technology. Such storites are worthwhile, we should certainly try to anticipate and avoid possible calamities.

But we also need stories exploring technology's potential for good. I yearn for a return to tales about new frontiers and the triumph of human spirit over adversity.

We need hopeful as well as cautionary tales. Without hope there is no reason to get up in the morning.

Tuesday, June 30, 2015

Making the tether catch

In Phobos - Panama Canal of the Inner Solar System, Doug Plata had asked about tether catches:

How much time would one have to attach to the tether's end? Since it is connecting two different orbits then I'm imagining that it would be fairly brief. If one misses the connection, then what?

I liken a catch at apoapsis to catching a ball at the top of it's bounce. For a brief time, the ball hangs motionless -- and then gravity pulls it back down. The less the acceleration, the longer the ball will hover at the top of a toss.

Regions of the tether that feel a substantial net acceleration will have a greater need for fast reflexes and good timing. The regions of the tether closer to the balance point can catch at a more relaxed space. Catching at the balance point would be like docking with the I.S.S.

For an example I will use the ellipse common to the Phobos and Deimos tether:

The larger red ellipse is the path a payload would follow dropped from the foot of the Deimos tether and/or if thrown from the top of the Phobos tether. At peri and apoapsis, this path matches the speed of the tether. So the moons could exchange payloads while using virtually zero reaction mass.

Note: When I use directional words like top, foot, above, below, up or down, I'm using Mars as the center. Down means Marsward.

Making the catch at the Deimos tether foot

Both the payload and Deimos tether foot are traveling about 1.18 km/s. But it is the relative velocity that counts. After all, I am traveling 30 km/s as earth circles the sun and so is my computer monitor. Do I worry about a catastrophic collision with my computer monitor? Not since I'm moving about zero km/s with regard to my computer.

Catching at the foot of the Deimos tether:

30 minutes before the catch the payload is trailing the foot by a few kilometers and is about 55 kilometers below. It's traveling about 136 miles per hour with regard to the tether, most of that velocity is vertical.

1 minute before the catch, the relative speed is only about 5 mph.

I'll compare this to driving a car. Driving down the road at a leisurely 30 mph, I step on the brakes. I don't stomp on them, mind you. Just decelerating at my usual careful pace, it takes 6 seconds to come to a full stop. Now look at the payload 5 minutes before the catch: 22.6 mph. Compared to my Sunday driving, this payload is moving like a turtle coming out of hibernation.

From Phobos throw to Deimos catch is an ~8 hour trip. During the vehicle can make measurements of it's distance and velocity with regard to the Deimos tether foot and compare it to optimal distance and velocity.

If catching below a tether's balance point, the payload would rendezvous with the tether at the trailing edge. If the tether is in a prograde orbit, the payload would land on the western end of a ramp:

In this cartoon I have a quadpod on wheels entering on the west end of the ramp. The wheels are only partially for comic relief. Wheels would actually be helpful landing on a platform.

The quadpod is a fanciful design not really relevant to tethers. I use it because it can quickly make slight adjustments to speed along any direction. The closing velocity is almost completely vertical. There will be a time when the space ship is quite close to the tether and falling up at a high speed. So it would be good to be able to do a slight tap on the brakes or gas pedal.

Also to give a little error room to the landing on the west ramp, I imagine a folding west end of the ramp that can extend itself after the ship has matched altitudes.

Acceleration, net weight

At this part of the Deimos tether, centrifugal acceleration is -.0681 m/s2 and Mars gravity is .1017 m/s2. Net acceleration is .034 m/22. A Sumo wrestler weighing 400 pounds on earth's surface would weigh 1.4 pounds. A coin falling out of your coat pocket would take ~8 seconds to reach your foot (assuming distance from coat pocket to foot is one meter).

Making the drop to Phobos

To send a payload on it's way to Phobos, simply roll of the east edge of the ramp.

Making the catch at the Phobos tether top

Catching at the Phobos tether:

1 minute out, the tether is moving 18 mph. A faster pace than the Deimos tether catch, but still much more leisurely than me rolling my car into the driveway from a Sunday drive.

Acceleration, net weight

At this part of the Phobos tether, centrifugal acceleration is -.536 m/s2 and Mars gravity is .4027 m/s2. Net acceleration is -.133 m/22. A Sumo wrestler weighing 400 pounds on earth's surface would weigh -5.44 pounds.  A coin falling out of your coat pocket would take 4 seconds to reach your foot (assuming distance from coat pocket to foot is one meter).

From the point of view of someone on Mars, the acceleration would seem upward. Looking through a telescope, they'd see the Sumo wrestler bumping against the ceiling like a helium balloon.

Making the drop to Phobos

To send a payload on it's way to Deimos, simply roll of the west edge of the ramp.

Some general rules for vertical tethers

This is an illustration for any vertical tether in a circular orbit. It also applies to Clarke style beanstalks.

The red orbits below the circular balance point's orbit move faster than the tether except where they cross the tether. At crossing points the orbits move the same speed as the tether. I explain here how tether matching orbits are found.

For points on the tether below the balance point's  circular orbit, entrance/catching ramps are on the trailing edge of the tether (the west end for tethers in prograde orbits). To drop to lower orbits, roll off the leading edge (east end of the ramp in prograde orbits).

For points on the tether above the balance point's circular orbit, entrance/catching ramp are on the leading edge of the tether. To throw to high orbits, roll off the trailing edge.

Making catches in steep acceleration gradients

Things are less relaxed as the tether extends further from the balance point. As soon as I have time, I will look at a Phobos tether whose foot extends into Mars upper atmosphere. The net acceleration at this foot would be about 3 km/s2 or about a third of an earth g.

Wednesday, June 17, 2015

Phobos--Panama Canal of the Inner Solar System

My post Orbital Momentum as a Commodity describes how a tether with a healthy anchor mass can catch and throw payloads. I tried to think of ways a tether might restore orbital momentum lost during a catch or throw. Two way traffic is one way to pay back borrowed momentum.

Well, Mars' moon Phobos masses 1.066e16 kg. With this huge momentum bank, catching and throwing payloads would have less effect than a gnat hitching a ride on a Mack truck. A Phobos anchored tether could catch and throw for millennia with little effect on Phobos' orbit.

The tether illustrated above doesn't suffer the enormous stress of a full blown earth elevator or even a Mars elevator. It could be made from Kevlar with a taper ratio of about 11.

Access to Mars

The tether foot pictured above moves about .6 km/s with regard to Mars surface. This is about 1/10 of the ~6 km/s the typical lander from earth needs to shed. Mars Entry Descent and Landing (EDL) would be vastly less difficult.

Some have suggested Phobos 1.88 g/cm3 density indicates volatile ices. If so, the moon could also be used as a source of propellent. A Phobos propellent source would make EDL even less of a problem. However Phobos' low density might also be due to voids within a rubble pile.

On page 2 of the Acceleration of the Human Exploration of The Solar System with Space Elevators Marshall Eubanks takes a look at how the foot of Phobos-Anchored Martian Space Elevator (PAMSE) might interact with Mars' atmosphere:
The orbital eccentricity of Phobos amounts to 283 km, which is by coincidence comparable to the effective depth of the Martian atmosphere for satellite drag (typically ~ 170 km, but subject to variations due to atmospheric events such as dust storms). The average relative velocity between the lower tip and the surface of Mars is only 534 m/sec, roughly Mach 2 in the cold Martian atmosphere, and slow enough that it should not cause significant heating of the tip. This raises the interesting possibility that the PASME tip could dip down deep into the atmosphere to leave or recover payloads or perform reconnaissance, acting as a supersonic airplane for the period near periapse when it is near the surface.
Eubanks' 534 m/sec is a little slower than the .6 km/s of my tether tip. This might be because I had placed my tether tip 300 km/s above Mars' surface thinking atmospheric friction would destroy a lower tether foot. Eubanks' analysis has changed my view.

In the Facebook Asteroid Mining Group, Eubanks noted:
The orbit of Phobos is equatorial, and there is a big mountain in the way, Pavonis Mons, the middle of the Tharsis volcanoes, straddling the equator and by far the highest obstacle in the path of the elevator tip. Maybe a railroad on top of the volcano could match speeds with the elevator tip, once every 3 days or so (when the orbit and volcano aligned). If so, you would have up to 3 minutes to shift cargo on and off. 
as well as
…the cool thing is that the tip can be something like a tethered airplane (with wings and flaps, etc.) and you should be able to use that to control oscillations. I was hoping to get money to begin actually "testing" this (i. e. in simulation), but, alas, not so far. 
Remember, too, with the PAMSE the counterweight has ~ infinite mass, and so any oscillations have to end there. (of course, anchoring a PAMSE in Phobos is left as an exercise for the reader.)
If Phobos is indeed a loose rubble pile, anchoring the elevator would be difficult. So while Eubanks eased my anxieties on oscillations and atmospheric friction, he calls my attention to a problem I hadn't thought of.

Access to Earth

6155 km above Phobos the tether is moving faster than escape velocity with a Vinf of 2.65 km/s. This is sufficient to toss a payload down to a 1 A.U. perihelion. This could provide most of the delta V for Trans Earth Insertion.

A ship coming from Earth would have a Vinf of 2.65 km/s and so rendezvous with this part of the tether might be accomplished with little propellent.

Access to the Main Belt

7980 km above Phobos the elevator is moving with a Vinf of 3.27 km/s, enough to hurl payloads to a 2.77 A.U. aphelion. This part of the tether might send/receive payloads to/from the Main Belt. There are a lot of asteroids with healthy inclination, though. So there would be substantial plane change expense at times.

Possible Mars exports to the main belt

One thing about the Main Belt, the pace is much more leisurely. Ceres moves about 1º every 5 days. In contrast earth moves about 1º a day and a satellite in low earth orbit moves about 4º a minute.

So a month-long, low-thrust ion burn over there looks a lot more like an impulsive burn than it does in our neck of the woods. I believe high ISP ion engines are well suited for travel about the Main Belt.

The inert gas argon can be used as reaction mass for ion thrusters. Mars' atmosphere is about 2% argon. It is also about 2% nitrogen and 96% carbon dioxide with traces of oxygen and water. Mars also has respectable slabs of water ice at the poles.

Mars would be a good source of propellent for the entire belt as well as CHON for the volatile poor asteroids in the inner main belt.

Ion engines don't have the thrust to weight ratio to soft land on the larger asteroids. But asteroids often have high angular velocity (in other words, they spin fast). High angular velocity combined with shallow gravity wells make asteroids amenable to elevators.

For example the balance point for a Ceres elevator would only be 706 km above Ceres surface, that is the altitude of a Ceres-synchronous orbit. To provide enough tension to remain erect, the elevator would need to extend to an altitude of 2000 km. At 2000 km, the tether tip is moving about .46 km/s, a good fraction of the 2.82 km/s needed fro Trans Mars insertion. If this Ceres elevator is Kevlar, taper ratio would be about 1.02.

If extended to an altitude of 14,500 km, the Ceres elevator top would be moving fast enough for Trans Mars insertion. This would require a taper ratio of around 5 for a Kevlar tether.

Incremental Development

The tether pictured at the top of this post is ~14,000 km long with a taper ratio of 11 for Kevlar. While much smaller than a full blown Mars elevator, this elevator would still be a massive undertaking. But the whole thing doesn't need to be built overnight. Early stages of the elevator would still be useful.

Pictured above a Deimos tether drops a payload to a Phobos tether.

At apoapsis of the large ellipse, payload velocity matches the Deimos tether foot. At periapsis, the velocity matches the speed of the Phobos tether top. Thus payloads can be exchanged between these Martian moons using practically zero reaction mass.

After descending the Phobos tether, the payload can be dropped to a Mars atmosphere grazing orbit.

These tethers are a lot shorter than 14,000 km tether we were talking about and taper ratio is close to 1.

No Moons to Dodge

A full blown Mars elevator capable of throwing payloads to the Main Belt or even earthward would have to dodge Deimos as well as Phobos.

A Phobos elevator for flinging payloads to Ceres ends well below Deimos' orbit. And of course a Phobos anchored tether doesn't need to dodge Phobos.


Tsiolkovsky's rocket equation and big delta V budgets are touted as show stoppers for routine travel to Mars' surface or the Main Belt.

With judicious use of tethers and orbital momentum, rhinoceros sized delta V budgets are shrunk to hamster sized delta V budgets. No bucky tubes needed, ordinary materials like Kevlar can do the job.

Wednesday, June 10, 2015

Mass parameter and ITN

It seems like every other post I'm singing the praises of EML2. I'm also enthusiastic about L1 and L2 necks for big moons orbiting gas giants.

So why do I diss the Sun Earth L2 or the Sun Mars L1? 

Robert Walker put it fairly well:

I've no idea why you think there's some essential difference between e.g. transfers between moons in the Jupiter system and transfer between planets around the sun. Mathematically it's the same situation, multiple masses around a central planet or sun. Obviously the moons of Jupiter are larger compared with Jupiter than planets are compared to the sun, and the orbits are far shorter. But they still have Hohmann transfer orbits, and hill spheres, and lagrange points, and these tubes, which lead out from the lagrange points. 

It's The μ

The big reason is mass parameter. This quantity is often denoted μ in discussion of 3 body mechanics.

It's common to choose units so that mass of central and orbiting body sum to one.

For example if central mass were 90% percent of the system's mass, central mass would be .9 and orbiting body would be .1. In this case μ would be 1/10.

The small the μ, the closer L1 and L2 get to the orbiting body.

What paths do payloads follow when nudged away from the orbiting body at L1 or L2? Well, we can notice a few things about L1 and L2:

The L1 and L2 have the same ω (angular velocity) as the orbiting body.

L1 and L2 are collinear with central and orbiting bodies.

It just so happens I have a diagram of collinear points all having the same ω. It's what I use to model vertical tethers. By scaling this diagram it could also be used to model space elevators. (Space elevators are a special case of vertical tether where the tether foot coincides with planet surface and circular orbit of the balancing point coincides with planet synchronous orbit):

Eccentricity Vertical Tether Conics = |1 - r3|

Release a payload from any point on the tether and the path will be conic section having eccentricity
|1 - r3| where distance from center to balancing point is 1 and a point's distance from center is r.

But this diagram was derived using 2 body mechanics. When nudged away from the orbiting body's Hill Sphere, the payload will quickly enter a regions where the central body gravity dominates and conic sections are fairly accurate.

But while in the neighborhood of the Hill Sphere, there's a short interval when the path should be modeled using both the accelerations of central and orbiting body:

While falling away from the moon near L1,  a payload surges ahead while the moon tugs it backwards and a little up. This has the effect of lowering the apo and periapsis as well as rotating line of apsides in a prograde direction. A payload nudged from L2 away from the moon will lag behind the moon. While in the lavender region, the moon pulls the L2 payload forward boosting peri and apo-apsis as well as rotating line of apsides forward. L2 necks throw higher and L2 drops lower than corresponding points from a vertical tether.

Here are orbital sims for various mass parameters where colored pellets are nudged with slightly different velocities from L1 and L2:

μ = .1

μ = .001

μ = .00001

μ = .000001

Notice as μ shrinks, the orbits get closer and closer to what the tether model would suggest. For μ = .000001, apogee is very close to point of release and eccentricity of ellipses is approaching |1 - r3|. As the Hill Sphere shrinks the lavender two body zone gets thinner and the 2 body model becomes increasingly accurate.

μ for sun-earth is .00000304 and μ for sun-Mars is .000000323. The paths from the planets' L1 and L2 necks don't go far and there's not much variation.

What About Gravity Assists? Just Look At Rosetta

"Well sure, nudging a payload from SEL2 doesn't get us much past a 1.07 A.U. aphelion" an ITN defender replies. "But earth gravity assists can boost that aphelion. Look at Rosetta's March 2005 gravity assist -- it boosted an earth like orbit to an aphelion past Mars."

So let's look at the Rosetta gravity assist.

We can see the March 2005 gravity assist gets Rosetta past Mars orbit. And the orbit from launch to gravity assist looks pretty earth like, right?


The orbit from launch to gravity assist is an ~.9 x 1.1 A.U. ellipse with an eccentricity of around .11. When r = 1 A.U., flight path is about 5 degrees. As it approaches earth, Vinfinity is about 2.6 km/s:

In contrast, Vinf of an orbit departing from SEL1 or 2 will be about .3 km/s. Rosetta's initial orbit couldn't be accomplished via a WSB from an L1 or L2 neck.

Further, a payload from SEL2 remains outside earth's orbit, it does not cross. The closest it comes it .01 A.U. during which time it's flight path is zero. The same is true of an orbit nudged from SEL1:

Synodic Period

Another thing to consider is synodic period. A way to think of synodic period is how often one runner laps another as they race about a circular track. If both runners are going nearly the same speed, it will take a long time.

Synodic period of orbiting bodies is |(T1 * T2)/(T1 - T2)| where T1 and T2 are bodies' orbital periods. Orbital period of a 1.01 x 1.06 ellipse is 1.053 years. Synodic period is 19.88 years. So the payload wouldn't even come close to the earth until almost two decades later!

When the payload finally does lap the earth 19.88 years later, it will be 43º from perihelion.

Instead of being .01 AU from earth, the payload will be more like .02 or .03 A.U. from earth. It won't get close to earth until nearly 8 synodic periods later. That's about 160 years.

The tinier the μ, the bigger the synodic periods of payloads released from L1 and L2 necks.

Synodic Period with a big μ

In closing I'll take a look at synodic period of something dropped with from Earth Moon L1. When it comes to μ's the earth moon's .012 is the 900 pound gorilla of the solar system. It's the biggest I know of except for Pluto Charon's .104.

Dropping from EML1, payloads fall into an approximately 100,000 x 300,000 km orbit:

Period is about 11 days. Synodic period is 20 days. So within a month's time these pellets will fly by the moon when they're near apogee:

EML1 and EML2 can do lots of stuff within a fairly short time. I have seen some crazy stuff running earth moon sims.

But zoom out and it gets boring. I've let sun earth sims run for centuries without seeing any drama. The L1 and L2 necks for the sun/rocky planets are a bunch of duds.

Does the ITN include Gravity Assists?

Given big enough  μ's, some WSBs can snake by another body which can lend a gravity assist.

Does that mean gravity assists are part of the ITN? No. We've been using gravity assists for many decades. They were in common use before Ross' or Belbruno's techniques came on the scene.

Some claim astrogators are now using Belbruno's or Ross' techniques to find opportunities for gravity assists. I haven't seen any evidence of this. So far as I can tell mission planners are still finding such opportunities the old fashioned way: looking for needles in a hay stack. In other words with persistence and hard work.

Monday, June 8, 2015

Orbital Mechanics Coloring Book

“The most sophisticated people I know - inside they are all children. ” 
― Jim Henson

I agree with Henson -- many of the folks I most admire are just big, playful kids. Henson (of the Muppets), Theodor Geisel (aka Dr. Seuss), Carl Barks (Uncle Scrooge, Donald Duck, Huey, Dewey and Louie), Stan Lee (Marvel Comics) are a few of favorite artists/kids.

My coloring book is aimed at kids. It's aimed at adults too. At least I hope there's a lot of adults like myself that like kid stuff.

I try to give info on the conic sections is a visual, accessible manner as well as have some fun. Here's some of the pages:

Click on image to see larger version.

The above are a few of the book's 40 pages. It's my hope it informs and entertains kids as well as adults. Getting folks interested in outer space is also a goal.

It is available at Conic Sections & Celestial Mechanics Coloring Book by Hop David.

My other coloring books can be found on my Amazon author's page.