Thursday, December 24, 2015

Lower Phobos Tether

A Phobos tether can be built in increments, it is useful in the early stages. So there's no pressing need to build a huge structure overnight. I will look at various stages of a Phobos tether, examining mass requirements and benefits each length confers. To model the tether I am using Wolfe's spreadsheet. I will use Zylon with a tensile strength at 5,800 megapascals and density of 1560 kilograms per cubic meter. Here is the version of the spreadsheet with Phobos data entered.

7 kilometer lower Phobos tether - tether doesn't collapse but remains extended

At a minimum, the lower Phobos tether must extend far enough past Mars-Phobos L1 that the Mars-ward newtons exceed the Phobos-ward newtons. This will maintain tension and keep the elevator from falling back to Phobos.

I used Wolfe's spreadsheet to find location of tether foot where tether length Mars side of L1 balances tether length from Phobos to L1. That occurs when tether foot is about 6.6 kilometers from tether anchor:

So going past that a ways will give a net Marsward force.

At this stage tether to payload mass ratio is about .01. The tether length exerts negligible newtons compared to payload force. Therefore a payload descending the tether to Phobos' surface would exert enough force to collapse the tether, especially as it nears Phobos' surface. So a counterbalancing mass would be needed at the tether foot.


Escape velocity of Phobos is about 11 meters/sec or about 25 miles per hour. A small rocket burn would be needed for a soft landing. This burn could kick up dust and grains of sand, some of which could achieve orbit. This would create an annoying debris cloud.

However a spacecraft could dock with a station at Mars Phobos L1 much the same way we dock with the I.S.S.  Payloads could then descend the tether and arrive at Phobos without kicking up debris.

It would also allow low thrust ion engines to rendezvous with Phobos.

It would also serve as a foundation which can be added to.

It would take a Mars Ascent Vehicle about 5 km/s to leave mars and rendezvous with this tether. Trip time would be about two hours, so the MAV could be small.

From this Phobos tether, a .55 km/s burn can send drop a lander to an atmosphere grazing periapsis. Aerobraking can circularize to a low Mars orbit moving about 3.4 km/s. If Phobos is capable of providing propellent, much of that 3.4 km/s could be shed with reaction mass.

In contrast, a lander coming from earth will enter Mars atmosphere at about 6 km/s. Since it takes about 14 km/s to reach this point, the lander will not have reaction mass to shed the 6 km/s. For more massive payloads like habs or power plants, shedding 6 km/s in Mars atmosphere is a difficult Entry Descent Landing (EDL) problem.

87 kilometer lower Phobos tether - copper pulls it's own weight

It would be nice to have power to the elevator cars. However copper only has a tensile strength of 7e7 pascals and density of 8920 kilograms per cubic meter. Have copper wire along the length of the Zylon tether would boost taper ratio. Using the spreadsheet, I set tensile strength and density to that of copper and lowered the tether foot until I got a taper ratio of 1.1. That gives a length of about 87 kilometers.


Along this length of the tether, copper pulls it's own weight, as well as supports the payload. A massive power source can be placed at L1 -- at L1 there are no newtons either Phobos-ward or Mars-ward. A copper only tether of this length would be about .2 times that of payload mass.

Elevator cars can ascend this length without having to carry their own solar panels and battery.

If descending from L1 Mars-ward, Mars' gravity can provide the acceleration and no power source is needed.

Of course copper wires can be extended further but this would boost taper ratio as well as tether mass to payload mass ratio.

From this tether foot, it takes .54 km/s to drop to an atmosphere grazing orbit. Trip time is about two hours.

1,400 kilometer lower Phobos tether - release to an atmosphere grazing orbit

With Zylon, tether to payload mass ratio is .11. The tether mass is still a small fraction of payload mass.


Releasing from the foot of this tether will send a payload to within a 100 kilometers of Mars' surface. Skimming through Mars upper atmosphere each periapsis will shed velocity and lower apoapsis.

Low Mars orbit velocity is about 3.5 km/s. The payload arrives at 4.1 km/s.

4,300 kilometer lower Phobos tether - payload enters atmosphere at 3 km/s.

With Zylon, tether to payload mass ratio is 2.55. Tether mass is almost triple payload mass.


At 4,300 kilometers from Phobos, dropping a payload will have an atmospheric entry of 3 km/s, about .5 km/s less than low Mars orbit.

5800 kilometer lower Phobos tether - maximum length

Phobos orbit has an eccentricity of .0151. It bobs up and down a little. Mars' tallest mountain is about 25 kilometers tall. Given these considerations, tether can't be more than 5800 kilometers. Else the foot might crash into the top of Olympus mons.

With Zylon, tether to payload mass ratio is about 16.10.


The tether foot will be moving about .57 km/s with regard to Mars. Mars Entry, Descent and Landing (EDL) is far simpler with .57 km/s. If Phobos is a source of propellent, much of that .57 km/s can be taken care of with reaction mass.

For an ascent vehicle, only a small suborbital hop is needed to rendezvous with the tether foot.

Wednesday, December 16, 2015

How Wolfe's tether spreadsheet works

I plan to do a series of posts examining elevators and tethers. I will link to them as posts are completed:

LEO Rotovator
Pluto Charon elevator

They will be based on Chris Wolfe's spreadsheet for modeling tethers.

I'll try to explain how Wolfe's spreadsheet works.

Tensile strength

Density and tensile strength are important quantities for tether material. Tensile strength is measured in pascals.

A pascal is a newton per square meter, newton/(meter2). A newton is a unit of force, mass times acceleration.

Zylon has a tensile strength of 580 megapascals or 580 meganewtons per square meter. On earth's surface with it's 9.8 meter/sec2 acceleration, it would take a 591,836,735 kilogram mass to exert that much force. It would take a zylon cord with a cross section of one square meter to support this force. But that's more than half a million tonnes!

10 tonnes is more plausible payload for space cargo. A much thinner cord could support this. Cross section of a Zylon cord need only be 1.72e-9 square meters. If a circular cross section, cord would be about 47 micrometers thick. Strands of hair can be anywhere from 17 to 181 micrometers thick.

So number of newtons determines tether cross sectional area.

How many newtons?

How to figure number of newtons at the tether foot? First we set maximum payload mass as well as foot station mass. The default in Wolfe's spreadsheet is a ten tonne payload mass and a foot station massing 100 kilograms. But how many newtons does this 1,100 kilogram mass exert?

The net acceleration on this foot mass is acceleration from planet's gravity minus centrifugal acceleration minus moon's gravity.

(Click on illustration to embiggen)

This spreadsheet sets the origin at the planet center.
Tether foot radius is the foot's distance from planet center.
Barycenter radius is Orbital Radius * mass planet / (mass moon/(mass planet + mass moon)
Tether anchor radius is Orbital Radius - Moon Radius. The tether anchor is assumed to be at the near point of a tide locked moon.
Distance from Barycenter to Tether Foot is Tether Food Radius - Barycenter Radius.

The three force equations:
Gravity Planet = G * Mplanet / Tether Foot Radius2
Centrifugal Accelerationω2 * Distance from Barycenter to Tether Foot. ω is constant, it is the angular velocity of the orbit.
Gravity Moon = G * Mmoon / (Orbital Radius - Tether Foot Radius)2

Net acceleration is the sum of these three.

An illustration of the accelerations with net acceleration in red. Moon gravity is negative because it is pulling away from the planet. Centrifugal acceleration is also pulling away from the planet except left of the barycenter it is towards the planet. 

When a curve crosses the axis the value is zero. Centrifugal crosses the axis at the barycenter. In most cases barycenter will be beneath planet surface. The illustration above has an exceptionally large moon. 

Net acceleration crosses the axis at L1, at this point the three accelerations sum to zero. to the right of L1, net acceleration is towards the moon.

To approximate the tether we chop it into many small lengths:

To find tether volume in step 1, we multiply the cross section by length of step 1. (Recall cross sectional area is set by number of newtons coming from tether foot.) Multiplying this volume by tether density gives step 1 tether mass. Multiplying this mass by net acceleration gives us the newtons this length exerts.

Adding the newtons from step 1 to payload newtons means the next step has a thicker cross section. We multiply this new cross section by tether length * tether density * net acceleration to get newtons from the tether length along step two.

And so on.

Summing all the masses from each step gives us total tether mass.

This is an approximation. The finer we chop the tether, the closer the approximation. The spread sheets we'll be using cut the tether length into 1,000 parts.

Our sheet can be found here. It is a 1.7 megabyte file.

For an upper moon tether, anchor will be on the far side. Moon's gravity will be added instead of subtracted from planet's gravity. I'll label tether end "Tether Top" instead of "Tether Foot".  Otherwise, the spread sheet will be the same as the lower moon tether spreadsheet.

Friday, October 30, 2015

Hope to resume space blogs soon.

I'm not dead. I've been up to my ears in alligators lately -- with paying projects (thank God!).

Hope to resume blogging soon. Some things I want to do:

Tethers and elevators

I'm eager to adapt Chris Wolfe's spreadsheet and examine a variety of tethers and elevators.

A few scenarios I want to look at:

There's a large population of dead sats in a graveyard orbit just above geosynch. These could act as a momentum bank for a vertical tether above geosynch.
I also want to look at a low earth orbit rotovator. It will be tricky adding tidal stress to the rotovator's stress from centrifugal force, but I think I can tweak Wolfe's spreadsheet to do the job.

Lunar elevator going from Mösting Crater through a balance point at EML1
Lunar vertical tether from an anchor mass at 30,000 km altitude

Phobos anchored vertical tether.
Deimos anchored vertical tether.

Clarke style elevators from Ceres. Using Chris' spreadsheet I will be able to look at elevators of various lengths. Down the road a Ceres beanstalk might even throw stuff to trans-earth orbits.

I want to examine Clarke style elevators from Vesta.

Boundaries of our bodies/extended phenotype

Awhile ago I reviewed a James Patrick Kelly story where a large fraction of the populace dwells in cyber-space. A trend in science fiction was been to explore artificial digital worlds rather than outer space. I opined that as telepresence improves, robotic avatars will become common place. The line between digital existence and meat space will blur.

Well, recently Kelly wrote an essay on prosthetics. It was a rich source of information, full of great web links (the norm for Kelly's Asimov articles). I want to talk about Kelly's essay. Also prosthetics and boundaries of our body. Many already regard dentures or lens implants as part of ourselves. I believe the same will become true of robotic arms and other body parts. And if we come to regard a prosthetic arm as an extension of our body, what is the difference between an arm attached to our shoulder or a robotic tele-arm thousands of kilometers away?

Robert Reed has written science fiction stories of god like beings whose minds and bodies extend throughout multiple star systems. That's not going to happen any time soon, but I do hope to see our "bodies" extend to the moon and near earth asteroids. In my lifetime.

I believe it will be advances in robotics that enable us to move beyond Cradle Earth.

Monday, August 24, 2015

Neil Tyson -- Incompetent Ass

A little more fact and a little less fiction, please

At one time I was a Neil DeGrasse Tyson fan. Our culture needs more charismatic pop figures promoting science and rational thought.

But Tyson has a propensity for just making stuff up. A true devotee of science makes it a top priority to disseminate accurate info.

I'll give a few examples.

Tyson likes to promote space exploration -- that's a good thing. He also notes our space programs have generated spin off technologies that help the economy -- also a good thing. But he exaggerates and embellishes. Here Tyson credits the space program for miniaturizing electronics:

The urge to miniaturize electronics did not exist before the space program. I mean our grandparents had radios that was furniture in the living room. Nobody at the time was saying, "Gee, I want to carry that in my pocket" Which is a non-thought

Ummm….   No.

Making electronics more compact, less massive and less expensive was an extremely obvious thought. Tyson's statement is utterly ridiculous.

Tyson would do well to read Wikipedia's history of the transistor. There were efforts to replace vacuum tubes as early as the 1920's. TR-1, the first transistor radio, hit the market in November of 1954.

NASA was formed in 1958.

The Regency TR-1 transistor radio hit the market in 1954,
4 years before NASA was formed.

NASA and military aerospace have made substantial contributions to the development of electronics. For example, the funding of integrated circuits R & D. Let's crow about these real contributions. But please don't give our space program credit for the notion of miniaturizing electronics.

It damages our cause when someone like Tyson spouts complete B.S.. The tendency to exaggerate and embellish is one of the reasons space advocates suffer from a lack of credibility.

"He's an entertainer," Tyson defenders might say. "Embellishing the truth is standard practice to boost ratings. If he can recruit more supporters, who cares if he doesn't cross his t's and dot his i's?"  I didn't buy that but had to acknowledge the man has gathered a following. So I didn't grumble too loudly.

Then Tyson stepped over the line.

A malicious fabrication
(Just to be clear -- by "fabrication" I mean a made up story. Not necessarily with intent to deceive)


Like everyone else, I was horrified by the events of September 11, 2001. I braced myself for President Bush's reaction. I thought Bush would use the tragedy to demonize Muslims. Exploiting xenophobia is an all too common political device.

But Bush's response was a pleasant surprise:
The face of terror is not the true faith of Islam.  That's not what Islam is all about.  Islam is peace.  These terrorists don't represent peace.  They represent evil and war. 
When we think of Islam we think of a faith that brings comfort to a billion people around the world. Billions of people find comfort and solace and peace. And that's made brothers and sisters out of every race -- out of every race. 
America counts millions of Muslims amongst our citizens, and Muslims make an incredibly valuable contribution to our country. Muslims are doctors, lawyers, law professors, members of the military, entrepreneurs, shopkeepers, moms and dads. And they need to be treated with respect. In our anger and emotion, our fellow Americans must treat each other with respect.
What Speech did Tyson recall hearing after 9-11? He remembers Bush loosely quoting Genesis "Our God is the God who named the Stars". Tyson says he was "attempting to distinguish we from they".

Watching the Tyson's video I'm scratching my head. Bush wasn't trying to stir up hatred against Arabs, just the opposite. And where did this stuff about star names come from? Below is Tyson's rant. Tyson starts talking about Bush at around 1:30.

Bush did give a speech where he quotes Genesis saying God named the stars. But it wasn't a post 9-11 speech slamming Arabs. It was a eulogy for the astronauts killed in the Space Shuttle Columbia disaster.

From Bush's Columbia disaster speech:
In the skies today we saw destruction and tragedy. Yet farther than we can see, there is comfort and hope. In the words of the prophet Isaiah, "Lift your eyes and look to the heavens. Who created all these? He who brings out the starry hosts one by one and calls them each by name. Because of His great power, and mighty strength, not one of them is missing. 
The same Creator who names the stars also knows the names of the seven souls we mourn today. The crew of the shuttle Columbia did not return safely to Earth; yet we can pray that all are safely home. May God bless the grieving families. 
And may -- may God continue to bless America.
Somehow Tyson conflated Bush's Columbia disaster speech with post 9-11.

Chemically enhanced perception?

How on earth did Tyson manage to conflate 9-11 with the Space Shuttle Columbia disaster? I have a theory. Carl Sagan, Tyson's hero and mentor, was thought to indulge in a little weed now and then.

Perhaps Tyson has discovered psychotropic drugs can be used as a vehicle to explore other worlds and alternate realities. I'm not the only one speculating Tyson's a stoner.

Tyson's "Apology"

In a Facebook post Tyson admits conflating the two events. And he reposted this admission on July 1st, 2015. The more recent admission is still open to comments as of this writing.

Well, he makes the admission buried in the 10th paragraph. The first six paragraphs are devoted to glowing descriptions of Tyson's favorite subject: himself. The seventh paragraph he slams those petty and small minded "lawyers" who doubt Tyson's word. How dare they question his credibility just because they catch him in the act of making stuff up?

Here's Tyson's admission:
What followed fascinated me greatly.  As others had uncovered, the President indeed utter the following sentences: 
In the words of the prophet Isaiah, "Lift your eyes and look to the heavens. Who created all these? He who brings out the starry hosts one by one and calls them each by name. Because of his great power and mighty strength, not one of them is missing."  The same creator who names the stars also knows the names of the seven souls we mourn today. 
But I was wrong about when he said it.  It appears in his speech after the Columbia Shuttle disaster, eighteen months after September 11th 2001.  My bad.  And I here publicly apologize to the President for casting his quote in the context of contrasting religions rather than as a poetic reference to the lost souls of Columbia. I have no excuse for this, other than both events-- so close to one another -- upset me greatly.  In retrospect, I’m surprised I remembered any details from either of them.
This is all Tyson needed to say. Had his apology just consisted of these few paragraphs, he might have salvaged a little credibility.

But Tyson goes on to write:
Of course very little changes in that particular talk.
Utterly and completely wrong. As usual. That talk was about President Bush's idiocy. The whole Arabic star name thing was to "confound Bush's point". Sadly for Tyson, the point being confounded comes from an imaginary Bush that lives in Tyson's crack pipe.

And still more B.S. from Tyson:

I will still mention Islamic Extremists flying planes into building in the 21sth century. I will still contrast it with the Golden Age of Islam a millennium earlier. And I will still mention the President's quote. But instead, I will be the one contrasting what actually happened in the world with what the Bible says: The Arabs named the stars, not Yahweh.
Of course the Arabs named the stars. Does that falsify the passage from Isaiah? No. How does Tyson know God didn't name the Stars? God's existence or nonexistence isn't something that can be demonstrated with experimental evidence. His confident assertion isn't a testable hypothesis. It is beyond the purview of science.

But then again, Tyson's cult of personality has never been about science.

Saturday, August 8, 2015

Lunar pogo hopper

Paul Spudis recently offered some thoughts about Drones on the Moon. He notes conventional drones would not work on an airless world.

Spudis writes:
Sub-orbital “hops” (ballistic flights from point-to-point) are possible, but come at fairly high cost—it takes nearly as much energy to fly hundreds of kilometers on the Moon in a ballistic hop as it does to go into orbit and then descend elsewhere.
This is incorrect. Here I look at suborbital hops on airless worlds. A minimum energy ellipse going from point A to B would have a focus on the midpoint of the chord connecting A & B:

The other focus would be at the moon's center, of course.

The vis viva equation tells us
v=sqrt(GM(2/r - 1/a)

In this case GM is the moon's gravitational parameter, r is the moon's radius and a is the semi major axis of the ellipse.

Let's say A is 300 km from B. That'd be  about 9.9 degrees separation. Here's a pic:

.67 km/s to hop and another .67 km/s for a soft landing. For low lunar orbit that would be 1.68 km/s to take off and another 1.68 km/s to soft land. Energy goes with square of speed. (.67/1.68)2=.16. The energies differ by more than a factor of 6! How on earth did Spudis conclude these are nearly the same?

Here is my Lunar Hopper spreadsheet. There's a tinted cell user can input distance between point A and B. This is the first document I've uploaded to Google docs, hope it works.

Spudis suggests spherical pit bots for lunar drones. These bots use micro thrusters to hop and hover. Whether the hop is 5 meters or 500 kilometers, the most efficient hop is the minimum energy ellipse described above. On the moon a ten minute hover costs about one km/s delta V. Spudis justifiably grouses about the tyranny of the rocket equation. But these pit bots rely on reaction mass to move. They don't circumvent said tyranny.

Pogo Hoppers

Various folks talk about lunar drones at Spudis' forum. Someone who goes by the name finkh mentioned pogo sticks. An interesting notion, in my opinion.

When I was a kid, my pogo stick used a spring. Solar cells might provide energy over time to compress a spring, thus avoiding the use of reaction mass. No more nasty rocket equation! On landing the spring absorbs the impact. The compression on impact might be a way to recover some energy.

I moved over a variety of terrains with my pogo stick. I could move forward, backward, left or right. It seems feasible to develop a robot with similar abilities.

But as I recall, getting from point A to B on a pogo stick was more strenuous that walking. So I'm not sure compressing a spring on impact is a great way to regain energy. Looking at existing robots like Big Dog, it looks like powerful engines are needed to power the device. Once again, the need for a better Alpha rears its ugly head. Elon Musk seems to be working on improved solar panels and energy storage. Hopefully Tesla Motor's R&D will have applications in space exploration.

How much impact can a pogo stick take? The 300 km hop pictured above hits the ground at .67 km/s or about 1500 miles per hour. No, I wouldn't want to be on that pogo stick.

This list of pogo stick records says Biff Hutchison jumped nearly 3 meters high. By my arithmetic he hit the ground at about 7.5 meters/sec or about 17 miles per hour.

Assuming 17 miles per hour is maximum jumping and landing velocity, a lunar pogo could jump about 36 meters (assuming the hop was a minimum energy ellipse from point A to B). This hop would be about 9 meters high.

A jump 36 meters long and 9 meters high isn't spectacular but such a device might have uses. And I like the image of a pogo stick on the moon.

Friday, August 7, 2015

A new tether spreadsheet

 A few notes on my old spreadsheets

Up to now, most of my spreadsheets for vertical tethers and elevators have been based on Jerome Pearson's work. The taper ratio I've used is based on equation (10) from Pearson's The orbital tower: a spacecraft launcher using the Earth's rotational energy. A screen capture from Pearson's PDF:

Pearson is looking at a very specific vertical tether here, a space elevator whose foot is at earth's surface (r0) and whose balance point is at geosynchronous orbit (rs).

But I make some substitutions to make Pearson's equations more general:

Substitute r0 with rfoot, the distance of the tether's foot from body center. 

Substitute rs with rbalance. By rbalance I mean the point on a vertical tether where centrifugal acceleration exactly balances gravity.

Substitute g0 with gfoot. By gfoot I mean gravity at tether foot. Pearson sets g0 at 9.8 meters/sec2, earth surface gravity. I set gfoot as G*(mass of central body) /  rfoot 2.

My resulting spreadsheets were cumbersome and complicated. I don't like complicated -- more opportunities for error. And indeed my early versions had many errors that gave obviously wrong results. Through careful proofreading and many bottles of aspirin I started to get numbers that matched Pearson's. But I remain uncomfortable with these sheets.

A new tether spreadsheet

Then Chris Wolfe sent me his Phobos tether spreadsheet. Chris' approach is simple and straightforward.

First he figures payload force at the tether foot: payload mass*net acceleration. In the linked spreadsheet, net acceleration is Mars gravity - centrifugal acceleration - Phobos gravity. Given the tensile strength of the tether material, this sets tether cross sectional area at the foot. Setting a safety factor of two will double this cross sectional area.

Then to the payload force he adds the force exerted by the length of tether just above the payload: density * cross sectional area * dr * net acceleration. This sets the cross sectional area of the next short length of tether. Again, the safety factor number multiplies this cell.

And so on up to the balance point.

Summing the mass of these cylinders gives a very good approximation of tether mass.

I had used a similar approach for calculating tether mass:

Tether mass is tether density times an integral giving tether volume from foot to balance point. For cross sectional area I used Pearson's equation 9 (see top of page). Being unable to solve the integral analytically, I chopped the tether into many small lengths and did a Riemann sum. In other words my numeric method for calculating mass is nearly identical to Wolfe's. Except Wolfe has a simple and straightforward method of getting the cross sectional area. 

The numbers from my spreadsheets closely match Wolfe's which also seem to match numbers from credible folks like Pearson or Aravind.

Chris' method is more versatile. A few advantages:

We can look at mass of decoupled upper tether length

If you have a huge anchor mass (like Phobos), the upper length becomes decoupled from the lower. The need to balance newtons from above with newtons from below is no longer an issue when anchor mass is 1.07e16 kg. 

If the upper length is independent of the lower, gravity at the tether foot isn't relevant. But Pearson's methods make heavy use of h, the tether material's characteristic length at the tether foot. This characteristic length has  gfoot in the denominator.

With Wolfe's method we can ditch the irrelevant h. Just as with the foot, we can start with net newtons at the tether top and then work our way down to the balance point.

We can look at moon tethers balanced from L1 or L2

Pearson's elevator model assumes two accelerations, earth's gravity and centrifugal acceleration. Wolfe's model can easily include a moon's gravity in the net acceleration. I am looking forward to tweaking Wolfe's spreadsheet to look at lunar elevators from EML1 and EML2.

More to come

Chris Wolfe's tether model enables me to scrutinize many of my favorite scenarios in more detail.

Besides this wonderful spreadsheet, Chris sent me a lot of other neat stuff. As time and energy allow, I'll use his ideas as the basis of drawings and discussions.

August 8 edit: Chris Wolfe has started a new blog Bootstrapping Space. I am predicting it will be an increasingly valuable resource as time goes by.

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.

A few mass parameters

Here are a few mass parameters for central and orbiting bodies:

Pluto/Charon 1.043E-01
Earth/Moon 1.216E-02
Sun/Jupiter 9.545E-04
Sun/Saturn 2.856E-04
Saturn/Titan 2.374E-04
Jupiter/Ganymede 7.789E-05
Jupiter/Callisto 5.684E-05
Sun/Neptune 5.153E-05
Jupiter/Io 4.700E-05
Sun/Uranus 4.366E-05
Jupiter/Europa 2.526E-05
Saturn/Rhea 4.046E-06
Sun/Earth 3.039E-06
Sun/Venus 2.448E-06
Saturn/Dione 1.935E-06
Saturn/Tethys 1.091E-06
Sun/Mars 3.229E-07
Saturn/Enceladus 1.935E-07
Sun/Mercury 1.659E-07
Saturn/Mimas 7.037E-08
Mars/Phobos 1.682E-08
Sun/Pluto& Charon 7.149E-09
Mars/Deimos 2.803E-09
Sun/Ceres 4.741E-10

I hope I'll some time to play with the Pluto/Charon 3 body system. I believe there are some wonderful possibilities in this setting.

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.