Upper Phobos Tether

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

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

I used Wolfe's spreadsheet to find location of tether top where tether length Phobos side of L2 balances the length extending beyond L2. This occurs 6.6 kilometers from the tether anchor. Having the tether extend 7 kilometers is sufficient to maintain tension.

Safety  Factor	Zylon Taper Ratio	Tether to Payload  Mass Ratio
1	1.01	.04
2	1.03	.06
3	1.04	.09

Benefits
Docking with a facility at the L1 or L2 regions is easier than landing on Phobos. In the words of Paul451: "Instead of a tricky rocket landing at miniscule gravity on a loosely consolidated dusty surface, you just dock with the L1-hub of the ribbon (same as docking with ISS), transfer the payload to the elevator car and gently lower it to the surface. Reverse trip to bring fuel from Phobos to your ship (Assuming ISRU fuel is available on Phobos.)"

Also this small tether can serve as scaffolding on which to add longer tether lengths.

937 kilometer upper Phobos tether - transfer to Deimos tether

Given tethers from two coplanar moons tidelocked to the same central body, it is possible to travel between the two moons using nearly zero reaction mass.

Above I attempt to show how peri-aerion and apo-aerion of elliptical transfer orbit matches velocity of the tether points this ellipse connects. Tether Vs are red, transfer ellipse'sVs are blue.

Above I try to explain the math for finding the tether lengths from Deimos and Phobos.

Trip time between the two tethers is about 8 hours.

Safety  Factor	Zylon Taper Ratio	Tether to Payload  Mass Ratio
1	1.02	.035
2	1.04	.070
3	1.05	.107

With a safety factor of three, one tonne of Zylon could accommodate about 9 tonnes of payload.

I look at the Deimos tether here.

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

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

Benefits

Easy travel between Deimos and Phobos is a benefit in itself. 

But this would be a huge help to ion driven Mars Transfer Vehicles.

I like the notion of reusable ion driven MTVs. Ion engines have have great ISP thus allowing a more substantial payload mass ratio. However they have pathetic thrust. Andy Weir's fictional Hermes spacecraft can accelerate at 2 millimeters/sec^2. Which actually is very robust ion thrust. However ithis is only medium implausible. Low thrust means little or no planetary Oberth benefit. Plus a lo-o-o-ng time to climb in and out of planetary gravity wells.

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

300 km above Mars surface in low Mars orbit, gravitational acceleration is about 3 meters/sec^2. For a 300 km altitude low earth orbit, gravitational acceleration is about 9 meters/sec^2. 2 mm/s^2 acceleration is less than 10^-3 of the gravitational acceleration at initial orbit velocity in both these case. However I will be kind and go with Adler's .856 * initial orbit velocity.

At 2 millimeters/s^2 it would take Hermes 38 days to spiral out of earth's gravity well from low earth orbit and 17 days to spiral out of Mars gravity well. Most of the slow spiral out of earth's gravity would be through the intense radiation of the Van Allen belts.

I was very disappointed when Neil deGrasse Tyson's trailer had Hermes departing from low earth orbit and arriving in Mars' orbit 124 days later.

Besides adding 10 km/s to the delta V budget, climbing in and out of gravity wells would add about two months to Hermes' trip time. Tyson's video describes an impossible trajectory.  I wish he'd fact check himself with the same enthusiasm he applies to others.

It would be much better for Hermes to travel between the edges of each gravity well. At least as close as practical to the edge. In earth's neighborhood, Hermes could park at EML2 between trips. In Mars' neighborhood, parking at Deimos would save a lot of time and delta V. From Deimos, astronauts and payloads can transfer to Phobos and then to Mars surface. In this scenario, Hermes' 124 day trip from earth to Mars is plausible.

2345 kilometer upper Phobos tether - Mars escape

If anchor in a circular orbit, escape velocity can be achieved if tether top is at a distance 2^(1/3) anchor's orbital radius. I try to demonstrate that here. Phobos is in a nearly circular orbit. To achieve escape, the tether would need to be 2435 kilometers long.

Safety  Factor	Zylon Taper Ratio	Tether to Payload  Mass Ratio
1	1.11	.204
2	1.22	.436
3	1.35	.700

A 7 tonne Zylon tether could deal with a 10 tonne payload, even with a safety factor of three.

Benefits:

Achieve mars escape.

6155 km kilometer upper Phobos tether - To a 1 A.U. heliocentric orbit

A tether this long can fling payloads to a 1 A.U. heliocentric orbit, in other words an earth transfer orbit.

Safety  Factor	Zylon Taper Ratio	Tether to Payload  Mass Ratio
1	1.80	1.57
2	3.24	4.77
3	5.82	11.16

With a safety factor of three, an 11.2 tonne elevator could lift a one tonne payload. Not great, but it'd be worthwhile if we were tossing lots of payloads earthward.

Benefits

Catch/throw payloads to/from earth. Phobos is about 24º from Mars orbital plane. Mars orbit is about 1.5º from the ecliptic. So there may be some plane change expense.

7980 kilometer upper Phobos tether - to a 2.77 A.U. heliocentric orbit.

Safety  Factor	Zylon Taper Ratio	Tether to Payload  Mass Ratio
1	2.5	3.1
2	6.4	12.5
3	16.2	39.5

With a safety factor of three, it would take a 40 tonne Zylon tether to handle a 1 tonne payload. We would need to be tossing many payloads for this to be worthwhile.

Benefits:

2.77 A.U. is the semi major axis of Ceres. A tether this long could catch/throw payload to/from Ceres. But this doesn't take into account plane change because of Ceres inclination.

Even with plane change expense, this tether could be very helpful for traveling to and from The Main Belt.

This could also throw payloads into a faster than Hohmann transfer orbit towards earth.

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.

Safety  Factor	Zylon Taper Ratio	Tether to Payload  Mass Ratio
1	1	.000003
2	1	.000006
3	1	.000009

Even with a safety factor of three, a tenth of a kilogram tether (about 3 ounces) can handle a 10 tonne payload.

Benefits

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.

Benefits

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

Safety  Factor	Zylon Taper Ratio	Tether to Payload  Mass Ratio
1	1.05	.1
2	1.1	.21
3	1.15	.33

Even with a safety factor of three, a 1 tonne zylon elevator could handle 3 tonnes of payload.

Benefits

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.

Safety  Factor	Zylon Taper Ratio	Tether to Payload  Mass Ratio
1	1.9	2.6
2	3.4	8.3
3	6.3	20.4

Benefits

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.

Safety  Factor	Zylon Taper Ratio	Tether to Payload  Mass Ratio
1	4.4	16.1
2	19.2	114
3	83.8	638

Given a reasonable safety factor of three, it would take a nearly 640 tonne elevator to lift a one tonne payload. I don't think a Zylon elevator from Phobos to Mars' upper atmosphere is practical.

Benefits

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.

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:

Lower Phobos Tether

Upper Phobos Tether

Deimos Tether

Mars Elevator

Liftport Lunar Elevator through EML1

Lunar Sky Hook

Trans Cislunar Railroad

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 Acceleration = ω² * 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.

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:

Earth
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.

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

Mars
Phobos anchored vertical tether.
Deimos anchored vertical tether.

Ceres
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.

Vesta
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.

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)

9/11

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.

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.

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 (r_s).

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

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

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

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