Wednesday, September 19, 2012

Beanstalks, Elevators, Clarke Towers

Planetary Beanstalks

Arthur C. Clarke well described a space elevator in his novel The Fountains of Paradise:
In the very decade that the first satellite was launched ... one daring Russian engineer conceived a system that would make the rocket obsolete. It was years before anyone took Yuri Artsutanov seriously. ... 
Go out of doors any clear night and you will see that commonplace wonder of our age — the stars that never rise or set, but are fixed motionless in the sky. We ... have long taken for granted the synchronous satellites ... which move about the equator at the same speed as the turning earth, and so hang foerever above the same spot. 
The question Artsutanov asked himself had the childlike brilliance of true genius. A merely clever man could never have thought of it — or would have dismissed it instantly as absurd. 
If the laws of celestial mechanics make it possible for an object to stay fixed in the sky, might it not be possible to lower a cable down to the surface, and so to establish an elevator system linking earth to space? 
When you build a bridge, you start from the two ends and meet in the middle. With the Orbital tower, it would be the exact opposite. You have to build upward and downward simultaneously from the synchronous satellite, according to a careful program. The trick is to keep the structure's center of gravity always balanced at the stationary point. If you don't, it will move into the wrong orbit, and start drifting slowly around the earth.

Besides popularizing the notion of Artsutanov elevators based on geostationary orbit, Clarke also invented geostationary communication satellites.

What is the altitude of a stationary orbit? Speed  of a circular orbit is (Gm/r)1/2. Speed is also ωr, where ω is angular velocity in radians. For a geostationary orbit, ω would be 2 pi radians/sidereal day. Using these two equations we can find radius of a body's stationary orbit:

ωr = (Gm/r)1/2
ω2r2 = Gm/r
r3 =  Gm/ω2
r = (Gm/ω2)1/3

Altitude of the stationary orbit is the orbit's radius minus the body's radius.

Stationary orbit altitudes of a few inner system bodies:

Vesta 265 km
Ceres 706 km
Mars  17030 km
Earth 35784 km

Vesta has a low stationary orbit because of it's low mass and high ω.

There are two accelerations at play: gravity and inertia in a rotating frame (the so-called centrifugal force).

Centrifugal acceleration is ω2r and gravity is Gm/r2. We can choose our units so that Gm as well as ω are 1. Then the acceleration gradient can be graphed like this:

The slope is steeper below the geostationary orbit. For the above and below portions to balance, the pink area must equal the blue area. Assuming uniform thickness, the blue lengths above need to be longer than the red lengths below geostationary orbit.

A friend pointed out "But why assume a uniform strand? The part near synch has the most tension, and so is thickest in most designs,"  Thickened portions can be modeled as several strands. Each strand would need to be asymmetrical to balance.

A 108,000 km strand above geostationary would counterbalance the 36,000 km strand from geostationary to earth's surface.

Ratio of tether thickness at stationary altitude to thickness at planet surface is called the taper ratio. Taper ratio varies depending on Gm, ω as well as tensile strength and density of tether material. This Wikipedia chart gives tensile strength and density of various materials. Equations from this The physics of the space elevator by P. K. Aravind can be used to find altitude of the elevator top as well as taper ratios.


Tide-locked Moons

It is also possible to build bean stalks from tide-locked moons. For tide-locked moons, the stationary starting points would be L1 and L2. Here are some tide-locked moons sorted by altitudes to L1 and L2:

Body L1 (km) L2 (km)
Phobos 3.25 3.27
Deimos 16.63 16.64
Io 8655.52 8831.64
Europa  11987.17  12172.28
Ganymede 28737.39 29362.77
Callisto 49749.25 48241.32
Luna 56292.23 62789.73

These numbers come from equations on pages 133 to 138 of Szebehely's "Theory of Orbits - The Restricted 3 Body Problem".

With tide-locked moons, there are three accelerations: 1) gravity of central body, 2) inertia in a rotating frame (aka centrifugal force, 3) gravity of moon.

The angular velocity ω is 2 pi radians/moon's orbital period. We can set time unit as orbital period/(2*pi), length unit as moon's orbit radius, and mass unit as mass of central body. Then ω and Gm are 1 and the accelerations can be graphed like this:

The constant k is ratio of moon's mass to central body mass. On the left side of moon orbit, moon pulls away from earth so moon acceleration is shown as positive. On the other side, the moon pulls stuff towards the earth, so moon acceleration  is negative.

To counterbalance, the blue strands extending away from the moon must be longer than red strands dangling towards the moon. the asymmetry is even more pronounced on the EML2 beanstalk.

A length extending 234,000 kilometers from EML1 earthward would balance a 57,000 kilometer length from EML1 to the moon's surface. Liftport proposes a Lunar elevator somewhat like this. An 11 tonne Zylon tether would extend 264,000 km from the moon's surface earthward. That's a little shy of the length needed but their diagrams indicate a counterweight at the earthward tether end. 

Here's a picture of the Liftport proposal:

Speed at apogees of red ellipses match ω * r. So virtually no delta V is needed for rendezvous with tether at apogee. The tether is within 3 km/s of Low Earth Orbit (LEO) and 1 km/s of Geosynchronous Earth Orbit (GEO).

Jerome Pearson et al have talked about lunar elevators. They point out a counterweight near EML1 has few newtons per kilograms, so a weight in that neighborhood would need to be quite massive. The chart on upper right of page 7 of this pdf indicates the counterweight mass at 60,000 km would be between 100 and 1000 times the tether mass. Here is a more detailed lunar elevator pdf by Pearson and friends.

Phobos Elevator

At 1.08e16 kilograms, Phobos is a large momentum bank. A Phobos tether could catch or fling many payloads with little effect on its orbit.

Mars fans like to point out that Mars' shallow gravity well allows a beanstalk made of conventional materials like Kevlar. They suggest a Mars elevator could be a gateway to the resource rich Main Asteroid Belt. To sling payloads to Ceres, a Mars elevator would need to be at least 46,350 kilometers tall. Taper ratio for Kevlar would be 45.

In contrast a Phobos tether less than 14,000 km can fling stuff to Ceres.

Kevlar taper for a Phobos tether is about 8, less than 1/5 of the Mars tether's taper.

Here is a graphic comparing taper and length of Phobos and Mars elevators capable of slinging payloads to Ceres:

A Phobos elevator accomplishes many of the same goals for a small fraction of the materials. It doesn't descend to Mars' surface, however. The Phobos tether foot is moving about .6 km/s wrt Mars' surface. So a small suborbital hop would be needed for a Mars ascent vehicle to rendezvous with the Phobos tether foot.  A Mars lander departing from the tether foot would need to shed .6 km/s, much less difficult than the typical 6 km/s.

A Mars elevator would need to avoid Phobos as well as Deimos. Not a problem with a Phobos elevator. The top of the Phobos elevator is below Deimos' orbit. And of course a Phobos elevator doesn't have to worry about collisions with Phobos.

Given a Phobos tether and a Deimos tether, it is possible to travel between the two moons with virtually no delta V. If a payload is released 937 kilometers above Phobos, it will follow an ellipse whose apo-aerion is 2942 kilometers below Deimos. At this apo-aerion, the payload is traveling the same speed as the Deimos tether at that altitude.

The eccentricity of this ellipse is (1 - (ωDeimos/ωPhobos)1/2) / (1 + (ωDeimos/ωPhobos)1/2).

Ellipse Peri-aerion is (1 + e)1/3 * Phobos orbital radius.

Ellipse Apo-aerion is (1 - e)1/3 * Deimos orbital radius.

Given two co-planar tide-locked moons orbiting a planet, there can be similar transfer ellipses between tethers. I like to imagine a system of tide-locked moons about a gas giant using such tethers. The tethers would need to lie outside of the gas giant's rings, though. Else the debris flux from the ring would likely cut the beanstalk.

A little bit of nay-saying (added 11-18-2012)

I'm not embracing elevators as the panacea that will open the cosmos. There are problems. Problems should be examined. 


A Spaceward article The Space Elevator Feasibility Condition looks at throughput. Elevator cars and their cargo add to elevator mass but not tensile strength. So unless the cars are a tiny fraction of elevator mass, they'll boost the taper ratio. How fast can the elevator cars move? If their horse power comes from solar arrays on the car, they may move fairly slowly. The distances are huge, it could easily take a car months to climb to its destination.

Initially the space elevator material must be delivered with rockets. If the mass delivered by rockets is hundreds of times the mass an elevator can deliver in a year,

The Space Elevator Feasibility Condition notes that throughput might not be enough to even maintain an elevator.

The longer the elevator, the more serious the throughput problem. It'd be much less of an issue in the shorter elevators like the Phobos or Ceres elevator.


Orbital debris could sever an elevator. This is a big problem for an earth surface to GEO elevator. This elevator passes through LEO which has a high debris density and this debris is moving about 8 km/s with regard to the elevator.

Tether Experiments is a page listing various tether missions. One of the missions was SEDS-2, a 20 kilometer tether deployed "to see how long it would remain intact in the face of collisions with space dust and other orbital debris. ... it was cut after only four days"

The other elevators I've looked at occupy volumes with a lower debris density. And the orbital velocities are more leisurely so the debris flux is more tolerable. But even if an impact is a long shot, it's a concern if a very large investment is at risk.

Balancing act during construction

This is mostly directed at the Lunar elevator. The Liftport elevator starts at EML1 and sends tether ends simultaneously moonward and earthward. A slight nudge from EML1 can send a mass along a chaotic orbit, sometimes wildly chaotic. Station keeping is important. During construction, this balancing act must be maintained while one end is traveling approximately 200,000 kilometers and the other end 60,000 kilometers. After the elevator is anchored to the lunar surface, this station keeping isn't necessary but it's unclear how long it will take for the anchor to reach the moon's surface. If the anchor impacts the lunar surface at near lunar escape velocity, it would likely vaporize. If the lunar anchor is lowered gently, the duration of the balancing act would be prolonged.


I'm not attacking these notions. Quite the contrary, I believe criticisms from a thoughtful Devil's Advocate can help a worthwhile idea more than cheer leading.

Tuesday, August 21, 2012

Mf is a Mofo

The Tyranny of the Rocket Equation is an excellent article by astronaut Don Pettit written while he was aboard the I.S.S..

The foundation of the article is this version of Tsiolkovsky’s rocket equation:

Mf = 1 - e-delta v/vexhaust

Mf is the fraction of the spaceship’s mass that is propellant.

Vexhaust for chemical propellant ranges from 3 to 4.5 km/s. Delta V to get into orbit is around 9 km/s.

Plugging these in we can see a spaceship must be more than 80 percent propellant. Pettit notes The 3-stage Saturn V rocket was 85 percent propellant and the Soyuz rocket 91 percent.

Pettit explains this sort of mass fraction is very challenging and drives up engineering expense:

If a vehicle is 10 percent propellant, it is typically made from billets of steel. Changes to its structure are possible without detailed engineering anyalysis: you simply weld on another hunk of steel to reinforce the frame according to what your intuition might indicate. I can easily overload my three quarter ton pick-up truck by a factor of two... 
Once vehicles become airborne, engineering structures become more serious. Lightweight structures made of aluminum, magnesium, titanium, and composites of epoxy-graphite are the norm. To alter a structure requires significant analysis: one does not simply weld on another chunk to your airframe or drill a hole though some convenient section if you want to live. ... Overloading an airplane by a factor of of two results in disaster. Even though these vehicles are 30 to 40 percent propellant (and thus, 60 to 70 percent structure and payload), there is eough ‘wiggle rooom’ to comfortably operate aircraft, which is how we have a robust, safe, and cost effective aviation industry.  
Rockets at 85 percent propellant and 15 percent structure and payload are on the extreme edge of our ability to fabricate, not to mention pay for. They require constant work to keep flying. The seemingly smallest modifications require monumental analysis and testing of prototypes in vacuum chanbers, shaker tables and test launches... 
The common soda can... is 94 percent soda and 6 percent can by mass. Compare that to the external tank (ET) for the space shuttle at 96 percent propellant and thus, 4 percent structure. The ET is big enough to hold a barn dance inside and contains cryogenic fluids at 20 degrees above absolute zero pressurized to 60 pounds per square inch, four times atmospheric pressure...”

One very expensive aspect of Mf Pettit didn’t talk about: Delta V budgets for reaching low earth orbit and beyond strongly encourage multi-stage expendable rockets. Expendable is another word for disposable. After transporting it’s payload, much or all of the engineering marvel is typically thrown away. Imagine how expensive a transcontinental plane ticket would be if we threw away a 747 each trip.

Pettit suggests a way to break the tyranny of the rocket equation:

A rudimentary and basic skill to master is to learn how to use raw materials of space to create new capabilites there. Our nearest planetary neighbor, the moon, is close and useful – it contains the material and energy resources we need to build a permanent space transportation system. Extracting and producing useful materials useful products from the raw materials of the Moon (particularly water, useful for life suport, rocket propellant and many other space applications) would relieve us from the need to drag everything we need in space from the bottom of Earth’s deep gravity well. This eventuality would significantly alter the consequences of the rocket equation in our favor.
I’d like elaborate on this. Here's a delta V map of our neighborhood, cislunar space:

A propellant tanker from the Moon to EML1 would have a round trip delta V budget of 5 km/s. Mf is 67 percent (assuming hydrogen/oxygen propellant).

A propellant tanker from EML1 to LEO would have a round trip delta V budget of 4.5 km/s. Mf is 63 percent.

A tanker from EML1 to GEO would have a round trip delta V budget of 2.6 km/s and a 43 percent Mf.

With propellant available at these locations, vehicles for moving about cislunar space would have mass fractions ranging from 25 percent to 58 percent

No herculean feats of engineering needed to meet these mass fractions. And they allow single stage reusable vehicles.

Lunar water isn’t the only way to cut Mf. Planetary Resources wants to move a water rich asteroid into high lunar orbit or EML1. Whether propellant comes from the Moon or a Near Earth Asteroid, it would revolutionize space transportation in our own neighborhood as well as deep space destinations such as Mars or asteroids.

Pettit's article appeared in the Fall 2012 issue of Ad Astra published by the National Space Society.

Wednesday, June 27, 2012

Inflated Delta Vs

"What's delta V from Earth orbit to Mars orbit?" -- a common question in science fiction or space exploration forums. The usual answer given is around 6 km/s, the delta V needed to go from a low, circular Earth orbit to a low, circular Mars orbit. A misleading answer, in my opinion.

There are a multitude of possible orbits and low circular orbits take more delta V to enter and exit. A science fiction writer using 6 km/s for Earth orbit to Mars orbit has a needlessly high delta V budget.

There are capture orbits that take much less delta V to enter and exit. By capture orbit I mean a periapsis as low as possible and apoapsis as high as possible. A capture orbit's apoapsis should be within a planet's Sphere Of Influence (SOI).

On page 124 of Prussing and Conway's Orbital Mechanics, radius of Sphere Of Influence is given by:

rsoi = ( mp / ms ) 2/5 rsp


rsoi is radius of Sphere Of Influence
mp is mass of planet
ms is mass of sun
rsp is distance between sun and planet.

The table below is modeled after a mission table at Atomic Rockets, a popular resource for science fiction writers and space enthusiasts.

• Departure and destination planets are along the left side and across the top of the table.
• Numbers are kilometers/second
• Numbers below the diagonal in blue are delta V's needed to go from departure planet's low circular orbit to destination planet's low circular orbit. These are about the same as the blue quantities listed at Atomic rockets.
• Numbers above the diagonal in red are delta V's needed to go from departure planet's capture orbit to desitnation planet's capture orbit.

Venus Earth Mars Jupiter Saturn Uranus Neptune
.7 3.6 5.6 6.7 7.5 7.5
Earth 6.8
1.1 3.5 4.6 5.3 5.4
Mars 7.9 5.7
3.0 4.5 5.6 5.8
Jupiter 25.8 24.0 21.8
.1 .3 .3
Saturn 20.0 18.1 16.2 27.8
.1 .2
Uranus 16.6 14.7 13.2 23.8 16.6
Neptune 17.3 15.4 14.1 24.5 17.3 13.1

It's easy to see the red numbers are a lot less than the blue numbers. I used this spreadsheet to get these numbers. The spreadsheet assumes circular, coplanar orbits.

A  graphic comparing delta Vs from earth to various destination planets:

If a low circular orbit at the destination is needed, it's common to do a burn to capture orbit with the capture orbit's periapsis passing through the upper atmosphere. Each periapsis pass through the upper atmosphere sheds velocity, lowering the apoapsis. Thus over time the orbit is circularized without the need for reaction mass. The planets in the table above have atmospheres, so the drag pass technique can be used for all of them.

A delta V budget is from propellant source to destination. If propellant depots are in high orbit, the needed delta V is closer to departing from a capture orbit than departing from a low circular orbit.

Thus it would save a lot of delta V to depart from Earth-Moon-Lagrange 1 or 2 (EML1 or EML2) regions. The poles of Luna have cold traps that may have rich volatile deposits. This potential propellant is only 2.5 km/s from EML1 and EML2. Entities like Planetary Resources have talked about parking a water rich asteroid at EML1 or 2. Whether EML propellant depots are supplied by lunar or asteroidal volatiles, they would greatly reduce the delta V for interplanetary trips.

Mars' two moons, Phobos and Deimos, have low densities. Whether that is from volatile ices or voids in a rubble pile is still unknown. If they do have volatile ices, these moons could be a propellant source. It would take much less delta V departing from Deimos than low Mars orbit.

All the gas giants have icey bodies high on the slopes of their gravity wells. However the axis of Uranus and her moons are tilted 97 degrees from the ecliptic. The plane change would be very expensive in terms of delta V. So the moons of Uranus wouldn't be helpful as propellant sources.

Venus has no moon. So of all the planets listed above, only Uranus and Venus lack potential high orbit propellant sources.

Anyway you look at it, the blue numbers from conventional wisdom are inflated.

Wednesday, April 25, 2012

The Next Continent

Hard science fiction set in our solar system nearly died in the 1960s.

The Tigers of Barsoom were slain by Mariner Probes to Mars. The Jungles of Venus were defoliated by probes to Venus. H. G. Wells’ Selenites were exterminated by Apollo. The Mariner Probes as well as Apollo told us the neighboring islands are barren places inhospitable to life.

Science fiction moved from neighboring planets to neighboring stars. Stories told over time spans shorter than decades or centuries were forced to resort to faster than light travel. The Golden Age of hard science fiction passed away and so called science fiction became more about fantasy than science.

In the meantime space exploration has moved on.

We’ve learned water is abundant in our solar system. A multitude of icey bodies dwell in the Kuiper Belt in the outer system. The Sun-Jupiter L4 and L5 have healthy populations of small bodies thought to be icey. There’s evidence Main Belt asteroid Ceres has a liquid water ocean within. Four main belt asteroids have been seen outgassing, an indication of volatiles. A thin layer of volatile ices was detected on the surface of Main Belt asteroid 24 Themis.

We have learned Europa, Enceladus and other icey moons of gas giants may have liquid water oceans beneath their frozen crusts. Tidal flexing creates an internal heat source that could sustain ecosystems just as deep ocean ecosystems on earth are sustained by chemicals and heat from volcanic vents.

There are regions on the moon’s surface that never see sunlight. Temperatures in these lunar cold traps can be as low as 40 degrees Kelvin. Colder than Pluto. There are indications of large bodies of ice in these crater basins as well as an abundance of other volatiles including various compounds of carbon, hydrogen, oxygen and nitrogen. Neighboring some of these polar craters are plateaus that enjoy nearly constant sunlight.

It turns out our solar system is much more interesting and mysterious than we had imagined. I had hoped these revelations would result in science fiction re-embracing our local neighborhood. But the path of main stream science fiction remains dominated by inertia, little affected by the perturbations of new discoveries and ideas.

There are exceptions, of course. A lot of optimistic, hard science fiction is coming out of Japan. Haikasoru is a publishing house that translates Japanese science fiction for the English speaking market.

The Next Continent by Issui Ogawa is one of the Haikasoru books.

“The Next Continent” is earth’s moon.

Ogawa has done his homework. He has invested some time and effort learning the nuts and bolts of aerospace, life support and other engineering aspects of his story. I have reassessed that. See postscript at bottom of this post (spoiler). While scientifically plausible, the story is still entertaining, it doesn’t get bogged down in technical details.

The book revealed to me a chauvinism I didn’t know I had. Many stories by U.S. writers feature American heroes who are more tenacious and clever than characters from other nations. And I never notice. But it was jarring to see Ogawa’s Japanese heroes show up their U.S. counterparts. But it’s only natural a Japanese writer would put Japanese characters center stage. Which isn’t to say Ogawa is disrespectful of the United States. He portrays a mixture of international cooperation and competition that will propel humanity to space. But in this story the U.S.A. isn’t the first to establish a beachhead on an extraterrestial body.

At the rate we’re going, I wouldn’t be suprised if China, Japan or other nations establish a lunar base before the United States. If that comes to pass, I would be delighted. Humanity must break the boundaries that confine us to a single planet. Which nation leads the way isn’t important just so long as we do it.

Postscript (spoiler alert):

October of 2010 I e-mailed Haikasoru's Nick Mamatas, letting him know of an error. In the third part of Chapter 8, Sohya and Tae face almost certain death. But they seem to have found a way out! They can escape the sun's searing heat by making a break for it during an eclipse:

Except the moon doesn't orbit the earth at 1.68 km/sec. That's about the figure for low lunar orbit. The moon's average orbital speed about the earth is more like 1.022 km/s. Tae and Sohya would definitely have been cooked!

Sadly the above screen capture is from a Kindle book downloaded in July of 2015.

It is extremely disappointing that Haikasoru doesn't give a damn about scientific accuracy or getting the math right. I guess their science fiction is a lot more typical than I had thought.

Thursday, February 16, 2012

Puppets, Telerobots and James Cameron

Cameron’s movie Avatar looks at telepresence and remote interaction. The biological telerobots portrayed are well beyond our present state of the art. However telepresence and telerobots made of metal, silicon and plastic aren't science fiction, they are being used today.

Avatar also portrays more plausible puppets made of metal, silicon and plastic. The mercenaries will don exosuits for heavy work or hand to hand combat. An exosuit user will slip inside a motion capture suit within the exosuit. The user’s movements are mimiced by the exosuit’s movements. If a robot puppet can be operated by a motion capture suit from within, it could also be operated by remote motion capture. The notion of exosuits is related to the notion of telerobots.

Cameron’s movie The Abyss featured Remotely Operated Vehicles (ROVs). James Cameron and his brother Mike developed ROVs for underwater exploration and filming. Their ROV dubbed “Snoop Dog” was used to explore the Titanic in preparation for making of the movie. Later ROVs named “Jake” and “Elwood” were used for further exploration of the Titanic as well as the sunken battle ship Bismarck. Cameron and Vince Pace developed 3-D cameras to film the sunken ships. 3-D cameras bring us a little closer to the goal of a fully immersive telepresence.

The Cameron brothers aren’t the only players developing telerobots and telepresence.

Existing markets are pushing advances in the state of art. As easy to reach resource bodies are exhausted, industry is looking to ore bodies in wastelands and under the ocean. Any hard to reach and/or dangerous workplace could benefit from telerobots. Rio Tinto Mining company is developing teleoperated devices (see page 10 of this pdf). British Petroleum uses submersible ROVs with their underwater oil platforms. There are also military applications. It is becoming more common to use drones for reconnaissance or telerobots to disarm bombs.

The movie industry uses motion capture suits. Actor Mike Meyers operates the virtual puppet Shrek with motion capture. The blue beings in Avatar are virtual puppets operated in a similar fashion. Motion capture is starting to move into the video game market, Wii and Kinect being early platforms.

Given various market forces, it’s inevitable telerobots will climb in ability as they drop in price.

Improved telerobots could be a huge game changer in efforts to settle and exploit space.

Remotely operated rovers Spirit and Opportunity were a spectacular success in gathering science on Mars. Cameron had offered to put his 3-D camera on the Mars Science Laboratory rover, but there wasn’t enough time to redesign the rover before the November, 2011 launch window.

For several reasons, a lunar telerobot could be far more able than a Martian telerobot. Light lag to a Mars rover ranges from ten to fifty minutes. Lunar light lag is about three seconds. Another factor is bandwidth. An able telerobot needs to send lots of sensory data as well as receive complex instructions. Signal strength falls with inverse square of distance and a weak signal is more easily lost to noise. So the moon’s proximity makes high bandwidth less difficult. The Lunar Reconnaissance Orbiter achieved 100 megabytes per second.

James Cameron sits on the board of the Google Lunar X Prize. This competition will award 20 million dollars to the first team that lands a rover on the moon. The rover must travel 100 meters while sending video images back to earth. It is my hope that Cameron will eventually work with these teams to land lunar rovers with his 3-D cameras on board.

Telepresence may become an early form of space tourism. A tourist could move about the lunar surface, picking up rocks and interacting with the lunar environment in other ways. All while his flesh and blood body moves about in a motion capture suit safe and comfortable on earth’s surface.

The Cameron brothers took great satisfaction in capturing light passing though the Titanic’s lead windows. Windows that hadn’t felt light since 1912. I like to imagine the Cameron/Pace 3D cameras filming the terrain of a crater floor at a lunar pole. An environment that hasn’t felt sunlight for billions of years. Inky black pits that fall to 40 degrees above absolute zero, even colder than Pluto. These craters are bound to contain some of the most bizarre and surreal landscapes in the solar system.

Besides strangeness and mystery, the lunar cold traps are thought to have abundant water and CHON (Carbon, Hydrogen, Oxygen and Nitrogen). Lunar water ice is exciting. Ice can be used for drinking water, radiation shielding, and water can be split into hydrogen and oxygen for rocket propellant. The nitrogen and oxygen compouds could provide air to breathe. Besides giving us vicarious experience of alien landscapes, we might use lunar telerobots to prepare an enduring human home on the moon.

The Futurist: The Life and Films of James Cameron
Big Dog. A robot with balance, a technology that could mitigate the slow reaction time from a 3 second light lag.
Google Cars. A robot with collision avoidance, another technology that could mitigate a 3 second reaction time.
Shackleton Energy Company. A TED video by Bill Stone. Stone hopes to mine lunar propellant with the aid of robots. The video features Stone's semi-autonomous robot, DepthX, a device for exploring subterranean caves.
Spudis and Lavoie's Lunar architecture. Dr. Paul Spudis and Tony Lavoie also propose to utilize lunar propellant with the aid of telerobots.

Murphy's Mangled Math

In his blog Stranded Resources Tom Murphy argues that space resources will likely remain beyond our reach. He concludes humanity should learn to live within its means and conserve our resources. This sound advice is the theme for most of his Do The Math blogs.

But the math on which he builds his argument is wrong.

To calculate delta V from earth to Mars he adds 3 quantities:

Earth escape velocity (~11 km/s),

Earth to Mars velocity (~6 km/s)

Mars escape velocity (~5 km/s)

Which totals ~22 km/s.

Murphy's delta V

But you don’t simply add these three quantities. Break the Earth to Mars velocity into two parts. These parts form legs of two right triangles. The other legs being Earth escape velocity and Mars escape velocity. Add each hypotenuse for the actual delta V.

Murphy vs Oberth

So the total delta V is around 17 km/s, not 22 km/s.

But wait. Murphy did generously round his 22 km/s to 20 km/s.

And there is also a ~2 km/s gravity loss incurred during vertical ascent. Add this 2 km/s to 17 km/s and you get 19 km/s. Murphy isn't shy about mentioning gravity loss. But he doesn't include it in his calculations, giving the impression that he's being quite generous to the addled space cadets. Including gravity loss takes the actual delta V to about 19 km/s. This isn't too far off from Murphy's 20 km/s.

But Murphy neglects the use of aerobraking.

For the Mars orbiter missions, a small burn is done to park the probe in a capture orbit rather than a low circular orbit. This can be done with as little as .7 km/s. The lowest point in these capture orbits pass through Mars upper atmosphere. Each time the probe passes through Mars' upper atmosphere a little velocity is shed by atmospheric friction. Using aerobraking, a capture orbit can be reduced to a low circular orbit using virtually zero propellant.

For the Mars landers, aerobraking sheds around 6 km/s.

Including gravity loss and using aerobraking the delta V budget for Earth surface to Mars surface is more like 14 km/s, about the same for delivering a comsat to geosynchronous orbit. So even Murphy's apparently generous 20 km/s is 6 km/s too much.

Given that the exponent of Tsiolkovsky's rocket equation scales with delta V, 6 km/s is a serious error.

Tsiolkovy's equation:

(start mass) / (final mass) = e(delta V/exhaust velocity)

Where e is Euler's number, about 2.72.

The dramatic power of exponential growth is illustrated by The Legend of Paal Pasam. An east Indian king enjoyed challenging his guests to a game of chess along with a friendly wager. Unknown to the king, one of his guests was Krishna. Krishna offered this wager: 1 grain of rice on the first square, 2 on the second, 4 on the third, doubling the grains each square of the chess board. The king agreed. Only after losing to Krishna did the king realize the enormity of his bet. Krishna revealed his true identity and told the king he could pay his debt over time. To this day the king’s estate gives rice to Krishna’s followers during their pilgrimages through that land.


Exhaust velocity of hydrogen and oxygen is about 4.4 km/s. 3 / 4.4 = ~ln(2). Each 3 km/s added to the delta V budget is a square on the above chess board. That is, each 3 km/s doubles the starting mass.

Murphy's 6 km/s error quadruples the starting mass.

Refuel In Space?

If you can get propellant along the way, it changes the picture:

Rice With Depots

At each square with a propellant depot, you get to start over at 1 grain of rice.

Murphy takes a look at refueling in space. A good propellant source would be close to earth in terms of delta V. So what does Murphy suggest? Jupiter or Titan! If he is looking for the most absurd propellant sources to debunk, he would do better to look at sources from Alpha Centauri. Or better yet, the Andromeda galaxy.

What are potential propellant sources that are close in terms of delta V? Earth’s moon is one.

At the lunar poles are craters floors which never see sunlight. Temperatures in these basins are as low as 40 degrees Kelvin, colder than Pluto. After a comet impact, volatile gases that don’t escape spread over the lunar surface. Gases reaching the cold traps will freeze and stay there. India’s Chandrayaan-1 lunar orbiter found evidence of thick, relatively pure ice sheets in many of these cold traps. It is estimated the anomalous north pole craters have at least 600 millions tonnes of ice.

These lunar volatiles are potential propellant only 2.5 km/s from Earth Moon Lagrange 1 (EML1) and Earth Moon Lagrange 2 (EML2). Using 3 body mechanics, there are paths that enjoy delta V savings over Hohmann orbits. And EML1 and EML2 are hubs for this Interplanetary Transport Network.

EML1 to Mars delta V map

Lunar volatiles can also provide water for radiation shielding, water to drink, as well as nitrogen and oxygen to breath. All 2.5 km/s from EML1. This is a huge mass that doesn’t have to be lifted from the bottom of earth’s gravity well.

Are there other potential propellant sources?

The low density of Mars’ moons Phobos and Deimos could indicate volatile ices. The low density could also be caused by voids within the moons, so the jury’s still out. If these do have ice, they are potential propellant sources quite close in terms of delta V. It is about 3 km/s from EML1 to Deimos. Possibly a little less if aerobraking is used.

EML1 to Deimos

Murphy looks at delta V from one low planet orbit to another. This is common, Atomic Rockets does the same, for example. But there are a multitude of possible parking orbits. Parking in a low circular orbit takes the maximum delta V. A high apogee capture orbit can take much less. Given the possibility of departing from propellant sources high on the slopes of a gravity well and shedding velocity using aerobraking, he would do better to look at delta V between elliptical capture orbits.

Grab That Asteroid!

Murphy suggests 5 km/s to capture an asteroid in earth orbit. There are near earth asteroids that could be captured with much less. The comet Oterma suggests a possible capture method using 3 body mechanics. Oterma will sometimes fall through the Sun-Jupiter L1 (SJL1) neck into Jupiter’s realm. It spends some time in Jupiter’s realm and then exits through the Sun Jupiter L2 (SJL2) neck. Then later it will fall back into the SJL2 gate, dwell in Jupiter’s realm, then exit trhough the SJL1 gate. This is described in the online textbook Dynamical Systems, The Three-Body Problem and Space Mission Design, a 17 Mb pdf.

An asteroid slowly drifting by the Sun-Earth L1 (SEL1) or Sun-Earth L2 (SEL2) could be parked in these regions with a minute nudge. From SEL1 or 2, a tiny amount of delta V suffices for delivery to EML1 or 2. For some asteroids .3 km/s can suffice for capture.

Only a small number of asteroids are amenable to capture this way though. A much larger number of Near Earth Asteroids pass within 1 km/s of EML1.

Murphy’s hypothetical asteroid is a cubic kilometer. The Tunguska object is thought to have been about 50 meters in diameter. Murphy’s asteroid is about 10,000 times larger than a meteorite big enough to wipe out a major city. So his absurd asteroid is a nonstarter due to safety considerations as well as the difficulty of moving such an enormous mass.

If we find a 20 meter asteroid of value, this could more safely be parked in earth’s orbit. This is small enough to burn up in earth’s upper atmosphere.

If we find a large ore body, it makes no sense to park the entire asteroid in earth orbit. Rather import the resources in small enough loads that it’s safe and doable. This also avoids flooding the market and thus devaluing the commodity.

Given a 20 meter object and 1 km/s delta V, the energy required differs by a factor of about two and half million from Murphy’s scenario -- somewhat less difficult.

Murphy ignores a number of things: 1) The Oberth Effect. 2) Aerobraking. 3) Moving between capture orbits rather than low circular orbits. 4) Nearby propellant sources. 5) Exploiting 3 body mechanics for delta V savings. 6) Small asteroids close to EML1 or EML2 in terms of delta V.

Tom Murphy does use weasle words like "simplified, approximate terms" or "crudely speaking". But his errors are truly enormous, too big to be salvaged by these disclaimers.

So I have to give Stranded Resources a grade of F.

Which is a shame. Murphy is correct to urge less consumption. But he doesn't have to resort to wrong arguments to support his view. That only subtracts from his credibility.