One Legged Stools

"It is NOT an issue of either or, Mars, Moon, Asteroids, each is as important to the other as the legs of a three legged stool…"

— Dennis Wingo, author of Moonrush.

I like Wingo's 3-legged stool metaphor. Space advocates have been divided into warring camps, each fighting for their version of a 1 legged stool. And 1-legged stools are doomed to topple. Space advocacy is already a small voice and its influence is watered down even more by this division.

The One Legged Stool to Mars

The chief 1-legged stool guy has been Bob Zubrin. For decades he's been saying we should go straight to Mars. Messing around with the moon and asteroids are a waste of time and money in his book.

In The Case For Mars, Zubrin advocates going to Mars in Heavy Lift Vehicles (HLVs) capable of 140 tonnes to Low Earth Orbit (LEO). Much like the Saturn V of the Apollo era. And, also like the Saturn V, expendable. Saturn V's low flight rate and high development costs resulted in an approximately 10 billion dollars per launch price tag.

HLV apologists say a modern monster rocket would cost less since we now have decades of experience. But the recent Constellation program was over budget and behind schedule. Nor is the current SLS any better. It looks like the SLS HLV will still have high development costs and low flight rate. John Strickland estimates each flight will have a price tag of $14.3 billion.

On page 69 of the The Case For Mars, Zubrin suggests 3 HLV launches per earth-Mars synodic period (about 2.14 years). That could come to more than $52 billion each launch window or $25 billion per year. The launches alone would exceed NASA's entire annual budget (around $18 billion/year as of this writing).

Not that I'm a Mars hater. In various forums I have defended the notion of humans living on Mars and other places in space. Skeptics will say humans aren't evolved to live in such environments. I reply man's extended phenotype (clothes, fire, shelter) has enabled humans to live in northern Europe, Siberia and the Arctic. Places quite different from the environment man evolved in. The naysayers unfailingly state the painfully obvious: Mars is more hostile than northern Europe. I also state the painfully obvious: our modern extended phenotype is much more sophisticated than the animal skins, mud huts and wood fires that enabled our ancestors to settle colder climes.

And I've played the role of naysayer. While our 21st century extended phenotype is more sophisticated, it's also high maintenance. Many would be space colonizers suffer from Home Depot Syndrome. Need a grow light? Go to Home Depot. Thermostat? PVC pipe, electric cord, duct tape, faucet, float valve, WD-40? Home Depot. This stuff is so easy to get that we forget it comes from an extensive planet-wide mining, manufacturing and transportation infrastructure that's taken centuries to build.

Charlie Stross looks at the number of people needed to maintain our modern extended phenotype. It remains an open question how many people and how much infra-structure it would take to make a self-sufficient colony on Mars. But it's safe to say a few decades of Mars Direct style missions wouldn't do it. After 20 or 30 years we would still have an outpost almost wholly dependent on earth.

No matter what era we're in, there's always a budgetary crisis. A high profile, expensive program like Zubrin's Mars Direct would be a lightning rod for politicians looking to trim fat. I would give the program five presidential terms tops. More likely one, just as Apollo bit the dust shortly after Kennedy.

In a best case scenario, Zubrin's Mars Direct would give us abandoned flags and footprints (like Apollo). Maybe a few empty habs gathering dust. But it would cost much more than Apollo. Worst case: the hugely expensive mega rocket program will founder and collapse before even leaving low earth orbit.

The One Legged Stool to the Moon

Paul Spudis is the chief cheerleader for a one-legged moon stool.

There are volatiles in the moon's polar cold traps. But how much? One optimistic estimate says 600 million tonnes of water at the north pole in the form of water ice sheets two or more meters thick. Spudis says "If you convert that to liquid hydrogen and liquid oxygen to launch a rocket ... that is the equivalent of a space shuttle launch every day for 2,200 years." As I.S.S. astronaut Don Pettit notes, lunar propellent has the potential to break the tyranny of the rocket equation. An abundant extra-terrestrial propellent source could make space travel routine in the earth moon neighborhood. Propellent depots at various places would break delta V budgets into manageable chunks making inexpensive, reusable spacecraft possible.

We have extensive and valuable infra-structure in earth orbit: communication satellites, GPS, weather sats, and more. Spudis notes routine cislunar travel would make maintaining and growing orbital infra structure less costly. But does easier access to our orbital assets justify an expensive lunar propellent mine? This seems unlikely.

But there are other possible uses for lunar volatiles.

The moon is close to Earth-Moon Lagrange 1 and 2 (EML1 and EML2). In terms of delta V, these two locations are much closer to Mars as well as the Near Earth Asteroids (NEAs). The moon could ship air, water for radiation shielding and drinking, as well as propellent to a staging platform at EML2. A fully stocked and fueled Mars Transfer Vehicle departing from EML2 has a much better chance of success than the same ship departing from LEO.

Asteroid mining and human Mars exploration could provide the needed market for lunar volatiles.

Planetary Resources or Deep Space Industries are potential customers for a lunar propellent mine. You would think Paul Spudis would support research and development that improves their chances for success. But he attacks the proposed Asteroid Redirect Mission (ARM).

ARM is based on the Keck study for asteroid retrieval. Co-authors include Chris Lewicki of Planetary Resources, John S. Lewis - author of Mining the Sky and Don Yeomans as well as other scientists and engineers with impressive track records. The study shows a moderately sized spacecraft could park a 7 meter rock in high lunar orbit. The delta V for parking a rock like 2008 HU4 would be around .17 km/s. If high ISP Hall thrusters are used, the needed propellent mass isn't prohibitive. The ship could be launched on an Atlas V.

A robust Solar Energy Propulsion (SEP) vehicle would be useful for other things besides bringing rocks to the earth-moon neighborhood. Dennis Wingo has noted such a vehicle would be useful for ferrying payloads from LEO to an orbit closer to the moon.

The proposed SEP vehicle could also deflect hazardous asteroids. This is a point Paul Spudis contests. He claims "Hazardous asteroids are simply too big (many kilometers across) for such an approach."

The Chicxulub rock, thought to wipe out the dinosaurs, is estimated at 10 kilometers. Are dino-killer sized rocks the only hazardous asteroids? Absolutely not.

The Chelyabinsk Meteor was estimated to be 20 meters in size. About 1,500 people had to seek medical attention, mostly from broken glass. The property damage was estimated to be $33 million dollars.

The Tunguska rock was thought be be 60 meters to 90 meters in size. The explosion knocked down 80 million trees over 2,150 square kilometers. This would be a circle 52 kilometers (32 miles) in diameter.
Paris is 105.4 square kilometers.
Chicago is 606.1 square kilometers
New York City is 1,213 square kilometers
Seattle is 369.2 square kilometers.

A Tunguska sized rock could easily wipe out a major city.

Moreover, Tunguska and Chelyabinsk sized rocks are millions of times more common than Chixculub sized rocks. Arguing the Keck SEP vehicle would be useless against Chixculubs is like arguing patching a roof is useless against tornadoes. Never mind that a sound roof is good protection against ordinary rainstorms which are far more common than tornadoes.

Besides Spudis, another well known moon guy is Dennis Wingo. The quote from the top of the page is by Wingo: "It is not an issue of either or, Mars, Moon, Asteroids, each is as important to the other as the legs of a three legged stool".

Surprisingly magnanimous and reasonable coming from hard core moon guy, right? Well, here is Wingo's version of mining asteroids:

John S. Lewis is perhaps the best known asteroid mining advocate. In Wingo's Moonrush J. S. Lewis is quoted a lot. Wingo repeats Lewis' observation that many asteroids are metal rich and that some have very high concentrations of platinum group metals (PGMs).

Then Wingo goes on to say 3 or 4% of the moon craters come from metallic asteroids.

Wingo talks about Barringer Crater on planet earth. He cites models that indicate the Barringer meteorite hit the earth somewhere between 16 and 20 km/s. He also cites models that indicate most of the Barringer meteorite wasn't vaporized.

The Barringer Meteorite may not have been vaporized. But it was blown into many small bits and scattered far and wide. Daniel Barringer was a wealthy mining engineer. He bet his fortune that the Barringer meteorite could be profitably mined. Barringer lost his shirt, as did the unlucky investors who placed their bets on Barringer's scheme.

It is thought that the rich mineral deposits at Sudbury Astrobleme might be meteoritic in origin. On page  94 of Moonrush Wingo has a graphic showing Cu-Ni-Pt-Pd mines along the edges of the Sudbury site. If the minerals are meteoritic in origin, they look like they've been scattered and mixed with the earthly rocks and dirt at the impact site. It is likely the asteroid had richer ore before it became a well smashed meteorite.

Wingo also mentions the Vredefort impact in South Africa. Ores from these mines may contain 5 to 10 ppm PGMs. There are some meteorites with a 100 to 200 ppm PGM contents. Ore becomes less concentrated after impact.

Wingo points out the moon has a shallower gravity well than earth. Therefore meteorites can strike at lower velocities. But the moon doesn't enjoy an atmosphere as earth does. Many earthly meteorites strike the earth at terminal velocity. Mars' atmosphere can also slow a meteorite before impact. The presence of intact iron meteorites sitting on the surface of earth or Mars doesn't demonstrate such objects are sitting on the moon's surface.

On pages 100 and 101 Wingo attempts to estimate impactor velocities on the moon. He cites http://cmex-www.arc.nasa.gov/CMEX/data/SiteCat/sitecat2/crater.htm which evidently is no longer online. He also cites this pdf. The graph on page 101 of  Moonrush gives the average impact velocity as 16.1 km/s. But this average is skewed by a small number of high velocity impacts. Wingo believes a lot of asteroids hit at speeds ranging from about 6.5 to 16 km/s.

On his blog Wingo confidently asserts "The peak of the gaussian curve for lunar impacts is between 14.1 - 16 km/s. Only comets and high inclination rocks hit at over 20 km/sec" But this pdf says the average lunar impact velocity is 20 km/s (fig 3 top of page 7).

It is easy to show Wingo is wrong about only comets and high inclination rocks hitting at over 20 km/s. Consider Asteroid 2011 CP4. With a semi-major axis of .911 a.u., not many would call this rock a comet. Nor does it have a big inclination, it is tilted 9.44 degrees from the ecliptic.

But 2011 CP4 has an eccentricity of .87. This jacks up the flight path angle when it's crossing earth's orbit. When it passes through our neighborhood, 2011 CP4's velocity wrt earth is around 30 km/s.

Not wanting to trust Wingo or his cites, I constructed my own histogram:

I was surprised when my own efforts gave results not that much different from Wingo's guesstimates.

To make the histogram, I went to JPL's NEA orbital elements page and downloaded all the NEAs that were listed as of July, 2013. I eliminated those with a perihelion > 1 a.u. or an aphelion < 1 a.u. -- these don't cross earth's orbit. With the vis viva equation it's easy to get an asteroid's speed when it's 1 a.u. from the sun. This speed plus flight path angle and inclination must be considered to find an asteroid's speed wrt earth. This speed is called Vinfinity wrt earth or V_{Inf_Earth}.

When the asteroid comes within earth's sphere of influence its path can be modeled as a hyperbola wrt earth. The hyperbola's speed is sqrt(V_{Esc_Earth}² + V_{Inf_Earth}²). At lunar altitude, earth's escape velocity is around 1.5 km/s. So when the rock comes near the moon's neighborhood it's moving sqrt((1.5 km/s)² + V_{Inf_Earth}²). This is V_{Inf_Moon} or Vinfinity wrt the moon. An asteroid is traveling a hyperbolic orbit about the moon's center when it impacts. So impact velocity is sqrt(V_{Esc_Moon}² + V_{Inf_Moon}²). On the lunar surface V_{Esc_Moon}  is about 2.4 km/s.

How much is a metallic asteroid scattered when it hits at 3 to 4 km/s? Or at 8 - 9 km/s? To be honest I don't know. More intact than the Barringer meteorite is a safe bet. Wingo is probably right that there are intact platinum rich meteorites in some of the moon's craters.

But which craters? Of the 3 to 4% that come from metallic asteroids, which of those are from low speed impacts?

In this post, Will Baird, a member of the Luna X-Prize Team Phoenicia, argues just the prospecting alone would cost a fortune. Unless space transportation becomes much more economical, looking for needles in the lunar haystack isn't affordable. Baird's argument holds for metal prospecting. However there's some evidence of lunar water ice deposits. And, unlike lunar metal, we know exactly where to look for lunar ice.

Planetary Resources is hoping it will be able to spot the metallic and platinum rich asteroids with Arkyd probes. This would be looking for needles not buried in the lunar haystack.

Wingo talks about revisit times. Opportunities for low delta V trips to an asteroid are rare and trip times are long. This was my major objection against the asteroids. Then I saw the Keck study. The study shows .17 km/s suffices to nudge some asteroids from a heliocentric to a high lunar orbit.

Once an asteroid is parked in lunar orbit, revisit times are no longer an issue. Just like the moon, launch windows would open every two weeks and trips times are less than a week.

Wingo argues even at ~.2 km/s, retrieving a 100,000 tonne asteroid would be more ambitious than multiple trips to the moon. But the Keck report doesn't talk about retrieving asteroids this large, more on the order of 500 tonnes. Not only is it impractical to retrieve larger rocks, but there are also safety issues as described on page 15 of the Keck report.

The One Legged Stool to NEAs (Near Earth Asteroids)

I believe water will be the first and most valuable extra-terrestrial resource. With off-earth propellant we can break the tyranny of the rocket equation. So long as the rocket equation mandates large disposable rockets, transportation expense will prevent profitable space mining. Two possible propellant sources are the lunar cold traps and water rich NEAs. In another post I look at Lunar Ice vs NEO Ice.

Delta V is the major selling point for near earth asteroids. But there are other important metrics: Trip times. Light latency lag. Frequency of launch windows. This last one is a major obstacle for the more accessible NEAs.

Frequency of launch windows.

Above is an illustration of a fictitious comet I have named after Dr. John DeLaughter, who was arguing asteroids would be a better source of water ice than the moon.

This comet has a 1.9 a.u. semi-major axis, 0º inclination and a 1 a.u. perihelion. Although I believe Comet JohnD would be too warm to keep water ice for long, I granted this for the sake of argument.

From LEO, reaching Comet JohnD takes 5 km/s, less than what it takes to reach the moon from LEO (6 km/s).

About 1.3 years before John D is due to pass near the earth, a payload can be nudged into an orbit that will graze earth's atmosphere at perihelion. Courtesy aerobraking, return trips take almost no delta V. Seemingly, this asteroid would have a huge advantage over the moon.

But at JohnD's next perihelion, earth will have moved 223 degrees from the perihelion neighborhood. The right part of the illustration above shows earth's position at each perihelion. The earth doesn't revisit the comet until 22 periods later, about 58 years.

If a mine's commodity can only be delivered to market every 58 years, it's not viable.

And with rare launch windows, it would take longer to establish infra-structure. Let's say it takes three trips to plant the mining infa-structure. With launch opportunities 58 years apart, that'd take 174 years!

Dr. DeLaughter maintains establishing an asteroid mine could be done in a single trip. This is a variation on the Home Depot Syndrome I mentioned above in the Mars section. Broken thermostat? Home Depot! Broken drill bit? Any replacement part? Go to your local Asteroid Home Depot!

Moreover, mankind has millennia of experience mining at 1 g and 1 atmosphere. We have zero experience mining in vacuum and microgravity. It will be a trial and error learning process.  When unexpected problems crop up, new equipment will have to be designed to deal with it. Establishing a mine will take more than one trip.

Of course a lunar mine would have similar issues. We have no experience mining in 1/6 g and vacuum.  Or the deep cryogenic temperatures at the lunar cold traps. A lunar mine would need replacement parts. It's obvious establishing and maintaining a lunar mine would also take multiple trips. But lunar launch windows open every two weeks and trip times are less than a week. Three trips to the moon would not take 174 years.

Trip times and light lag latency

Asteroids in the main belt can have more frequent launch windows. Some, like 24 Themis, seem to have water ice. 24 Themis launch windows open each 1.22 years. Much worse than the moon's windows each two weeks. But not nearly as bad as Comet JohnD's windows each 58 years.

Trip time to 24 Themis is about 1.5 years. We are still a ways off from life support that's trouble free for years at a time. There is also the radiation issue. These are the same problems that make human travel to Mars so difficult. Human miners on 24 Themis won't be plausible for some time to come.

Without humans, the mining infra-structure would be built by telerobots. Light lag latency to 24 Themis would range from 68 to 35 minutes. In contrast, lunar light lag latency is about 3 seconds.

For good dexterity and sensory data, a telerobot needs good bandwidth. Signal strength falls off with inverse square of distance. Given a comparable power source, the lunar signal is anywhere from 1600 to 800 times stronger than a signal from 24 Themis. Given much better bandwidth and only 3 second light lag latency, lunar telerobots would be much more able.

Dr. JohnD maintains autonomous robots can mine the asteroids. The mining entity Rio Tinto uses autonomous robots.  This link shows Rio Tinto hauling trucks transporting dirt from one location to another. The trucks travel well maintained, predictable routes. Most of the mining environment isn't nearly as controlled and predictable as the roads this truck uses. Other Rio Tinto mining equipment such as shovels are still human controlled. And the "autonomous" trucks are closely monitored. And, whether human operated or "autonomous", humans maintain all the equipment. The notion of autonomous robots that don't need humans nearby is almost as silly as the Asteroid Home Depot that provides any and all replacement parts.

Asteroid Delta V not always so small

Compared to main belt asteroids, the moon has better delta V!

Many believe travel to any asteroid takes little delta V due to their shallow gravity wells. This is wrong. Going from one heliocentric orbit to another requires a change in speed. For this, an asteroid's shallow gravity well doesn't help.

The transfer orbit to 24 Themis would have perihelion velocity 36.7 km/s and aphelion velocity 11.7 km/s. Earth travels about 29.9 km/s and Themis about 16.8 km/s. The delta V at earth's end is mitigated by the Oberth effect -- departing from LEO takes 5.2 km/s. But no Oberth benefit at the Themis end. Arriving or departing from 24 Themis takes 5.1 km/s. From LEO to Themis, total Delta V is 10.3 km/s.

Even so, Dr. DeLaughter believes reaching lunar ice would take more delta V than 24 Themis. He asks "how much delta V does it take to land on the Moon's South Pole? ... It is a lot more than a simple equatorial descent..."

For DeLaughter's benefit I described a 6.4 km/s route to the lunar poles that takes 6 days.

The first step in my route is a 3.1 LEO burn. Dr. DeLaughter immediately challenged my number:

De Laughter cites Lance Benner's 6.0 km/s figure for LEO to rendezvous with the moon. DeLaughter was skeptical when I pointed out 6 km/s was from LEO to landing on the moon.

Using the vis-viva equation I calculated the transfer orbit's perigee velocity as well as a low earth orbit velocity. DeLaughter didn't seem to understand the vis-viva equation.

I provided a Wikipedia article saying LEO to lunar surface is 5.93 km/s.

A LEO burn to reach lunar height is known as Trans Lunar Injection or TLI.  I gave DeLaughter this cite saying the Apollos' TLIs ranged from 3.05 to 3.25 km/s.

It's common knowledge that escape velocity near earth's surface is around 11 km/s. I mentioned to Dr. DeLaughter that a 6 km/s burn added to a LEO's 7.7 km/s  sums to 13.7 km/s which is greater than escape velocity. This hyperbolic orbit wouldn't even have an apogee. 13.7 km/s is almost enough for Trans Jupiter Injection.

DeLaughter pointed out vectors aren't scalar quantities. Depending on their directions, a 6 km/s vector added to a 7.7 km/s vector doesn't have to sum to a 13.7 km/s vector. To add two vectors, take the second vector's foot and place it on the first vector's head. Now draw a vector from the first vector's foot to the second vector's head. This third vector is the sum of the first and second vector.

When making a 32.7º plane change, a 6 km/s burn vector added to LEO's 7.7 km/s can indeed give an ellipse with an apogee at lunar height. But plane changes are cheaper at apogee. DeLaughter seems unaware apogee raising LEO burns are done in the same direction as the LEO velocity vector.

Unlike DeLaughter, Benner is skilled in orbital mechanics. It's safe to say Benner's 6.0 km/s figure doesn't refer to a TLI with a 32.7º plane change.

To support his 6 km/s, at one point DeLaughter gives a "grand tour" route to lunar orbit:

DeLaughter seems to forget the 3.1 km/s figure he challenged was for a TLI burn, not getting to lunar orbit. Getting to lunar orbit would require a TLI (Trans Lunar Injection) burn as well as LOI (Lunar Orbit Injection) burn. Generally these two burns total about 4 km/s.

And his "grand tour of cislunar space" is a horribly silly way to get a delta V figure. JohnD is from Texas so I will use his method to get the miles from San Antonio to Houston:

A normal person would say it's about 200 miles from San Antonio to Houston using the Interstate10. But using JohnD's method we can do a crazy path that gives us a grand tour of the Houston area. We can stop at Belville, Conroe and various surrounding communities. If we include enough towns in this crazy grand tour, it demonstrates that one can easily travel 300 miles from San Antonio to Houston.

So far as I know, Dr. DeLaughter still believes he has demonstrated my 3.1 km/s TLI figure is wrong. What he has actually demonstrated is his complete incompetence in orbital mechanics and simple physics.

Dr. DeLaughter has made other silly arguments but right now I don't have the time to address them. Suffice it to say he isn't credible.

Possible Synergies

If Planetary Resources establishes mines in lunar orbit, that would be a substantial infra-structure within spitting distance of the moon -- both in terms of kilometers and kilometers/second. A platform so close to the moon would make the moon more accessible. Infrastructure in luna orbit also makes the propellant and life support volatiles in the lunar cold traps more desirable.

The retrieval vehicle described in the Keck study uses xenon ion drives. At first glance it seems unar water would be little help fetching rocks. But page 29 of the Keck Report says the spiral from LEO to earth escape would take 2.2 years and 3.8 tonnes of xenon. And much of the spiral is a slow trip though earth's harsh Van Allen Belts. An Earth Departure Stage (EDS) driven with lunar hydrogen/lox would take 2.2 years off mission time and increase by almost half the amount of xenon available for moving an asteroid.

The EDS  could attach to a rock fetcher at EML2. At EML2 .4 km/s suffices to drop to a perigee deep in earth's gravity well. Once at perigee, .2 km/s suffices to send the fetcher on its way toward the target asteroid. A .2 km/s burn would not take long, trans asteroid injection could be achieved while still in the perigee neighborhood. After the EDS detaches from the fetcher, it would do another  the .2 km/s burn to shed velocity and return to an elliptical orbit about the earth. Then the Earth Departure Stage could return to EML2 to be refueled and send another fetcher on its way.

With propellant mines in the lunar cold traps and metal mines in lunar orbit, there would be economic incentive to develop Closed Ecological Life Support (CELS). Plants to provide oxygen and food, bacteria to break down wastes. With better CELS, long trips and long stays on other bodies become more plausible. If life support materials come the moon's shallow gravity well, it is also possible to leave with more water, food and air. The life support consumables in a Zubrin style Mars Transfer Vehicle are bleeding edge minimum mass. If a CO2 scrubber or other life support mechanism breaks down, the astronauts are dead. A ship stocked with abundant water, air and food is less vulnerable to these catastrophic failure modes.

Lunar life support consumables make long trips doable. Lunar propellant would shorten trip times and expand launch windows. The revisit issue becomes less of a show stopper. It would open the door to exploiting the larger NEAs and those asteroids a little more more distant in terms of delta V.

Propellant and a staging platform at EML2 takes about 2.7 km/s off an Mars Transfer Vehicle's (MTV's) delta V budget. Or 5.4 km/s off a round trip. This smaller delta V budget allows a larger mass fraction. A larger mass fraction makes reusable MTVs more doable. As I mentioned at the start, single use $10 billion vehicles are a show stopper for establishing a Mars settlement.

A tether anchored on Phobos could be a way station to the Main Belt.

The foot of the Phobos elevator illustrated above is only .6 km/s from Mars' surface. Only .6 km/s from Mars' abundant CO2, argon and water. And from Mars' extensive mineral resources.

Elevator regions above Phobs can help receive or send payloads to or from Earth. Or the Main Belt. Or even Near Earth Asteroids. Dennis Wingo argues Mars infra-structure would open up a whole new set of launch windows to the asteroids. A whole new set of launch windows would dramatically cut revisit times to NEAs as well Main Belt asteroids.

Do we even need a 3-legged stool?

Wingo's Moonrush takes a look at The Club of Rome. As early as 1968 this group foresaw a major challenge: So long as we're trapped on earth, there is a ceiling to our economic growth. Unless we can break free of our cradle, we're doomed to decline or stagnation. Wingo notes space offers a much larger body of resources than our planet's surface.

We can't go very far into earth's surface before pressure and heat prevent us from going deeper. Available resources and real estate can be measured in area. The surface area of the asteroids is hundreds of thousands times that of earth.  Not only is an asteroid's surface accessible but it's entire volume. Access to the asteroids would raise our ceiling many stories and we could enjoy growth for generations to come.

Tom Murphy echoes the Club of Rome warnings. Murphy is a 21st century voice warning us exponential growth isn't sustainable. Like the Club of Rome, Murphy exhorts us to conserve resources, slow population growth and live within our means. I wholeheartedly agree with this sound advice. Even if space resources were available, we should take good care of our cradle Earth.

But Murphy also argues space resources aren't practical. See Why Not Space? and Stranded Resources. Murphy believes the possibility of space resources will lull us into complacency and ignore the urgency of his message. So he sets out from the start to demonstrate his preconceived notion. This approach leads to some embarrassingly bad math.

Neil DeGrasse Tyson notes NASA's funding is small fraction of our budget, less than half a penny on the dollar.  He calls for 1% or a penny for NASA. But what is the purpose of our space program? Is it to find bacteria on Mars? To plant a flag and footprints on an asteroid and return home? If that's the goal, NASA's not worth even .4% of our budget.

But if NASA could help open the gates to an endless frontier, it would be worth much more.  How much is escape from our prison worth? I'd say a dime or even a quarter. Many voters would agree. But more and more voters question whether it's even possible to open the space frontier. Like Murphy, they've resigned themselves to a future where mankind is trapped beneath low earth orbit.

Zubrin, Spudis, and Lewis and like minded people should not be fighting each other.  They should be working together for a common goal. And fighting the common enemy, those who have resigned themselves to stagnation.

Lunar Ice Vs NEO ice

Lunar Poles Aren't So Hard to Reach

"There might be ice in the lunar cold traps," moon haters like to say, "... but if there is, it's hard to reach. It takes way, WAY more delta V to reach the poles than the lower lunar latitudes."

This meme is wrong. Unfortunately it's widespread. The first part of this post will debunk this false notion.

I will describe a route from low earth orbit to the lunar north pole that takes 6.4 km/s and 6 days. It's by no means the only route or even the best route. But I'm using it because it's simple and easy to illustrate.

Start in an equatorial low earth orbit at 300 kilometer altitude. Do a TLI (Trans Lunar Injection) burn to reach an apogee of 1 lunar distance (384,400 kilometers from earth's center). Time the apogee so it is on the moon's line of nodes shortly before the moon crosses the equatorial plane. The ship will enter the moon's sphere of influence south of the moon. I am calling the radius of the moon's sphere of influence 60,000 km.

It pains me to make a cartoon illustration that is nowhere near correct scale. To atone for my sins the above is a scale drawing of what I'm trying to describe.

TLI

The first burn from LEO is TLI or Trans Lunar Injection.

Since the transfer orbit lies on the equatorial plane, the perigee velocity vector is parallel to the low earth orbit velocity vector. Therefore simply subtracting the LEO velocity from the transfer orbit's perigee velocity vector gives the TLI (Trans Lunar Injection) delta V.

But what is the transfer orbit's perigee velocity? And what is LEO velocity at 300 km altitude?

Both these can be found with the vis-viva equation: v = sqrt(GM(2/r - 1/a)

G is the gravitational constant which is about  6.67384e-20 km³ kg^-1 s^-2
M is the mass of the earth, about 5.972e24 kg.
r is distance from earth's center at TLI. At 300 km altitude this is about 6678 km.
a is the semi-major axis of the orbit. For the low earth orbit a is 6678 km, the same as r. For the transfer orbit, a is (6678 + 384,400)/2 or 195539 km.

Plugging these in we get 10.8 km/s for the transfer orbit's perigee velocity and 7.7 km/s for LEO.
10.8-7.7 = 3.1

The TLI burn is 3.1 km/s.

Entering the Moon's Sphere of Influence

When the ship arrives at apogee, r = 384,400 km. For the moon's orbit a = 384,400. Plugging these quantities into the vis-viva equation we get the moon's velocity as 1.018 km/s. The ship's velocity at perigee is .19 km/s.

But these two vectors aren't parallel. The moon's orbit is inclined to the equatorial plane anywhere from 18 degrees to 29 degrees. We'll pick the worst case scenario: 29 degrees.

Given two sides a and b of a triangle, separated by angle alpha, the third side can be found by the law of cosines:

a² + b² - 2ab cos (alpha) = c². For alpha = 90 degrees, cos(alpha) is zero. So the law of cosines is a more general version of the Pythagorean theorem.

For the triangle above, the third side comes to .86 km/s. The ship enters the moon's sphere of influence traveling .86 km/s with regard to the moon.

Inside the Moon's Sphere of Influence, LOI

Our ship enters the moon's sphere of influence at lunar longitude 90º. The lunar 90º longitude line lies in a horizontal plane with regard to the earth. At apogee the ship's vector is also horizontal with regard to the earth so the ship's velocity vector lies in same plane. Inside the moon's sphere of influence, the ship's path can be modeled as a hyperbola with the focus at the moon's center. The moon's center also lies in the 90º longitude plane. Since the hyperbolic orbit is coplanar with a plane that cuts through the moon's poles, this hyperbolic orbit is polar.

Using the Law of Sines we can find the angle between the velocity vector with regard to the moon and the local vertical with regard to the moon. It is 5.5 degrees.

Impact parameter of our hyperbolic orbit is sin(5.5º) * 60,000 km. Or about 6090 kilometers.

Given a .86 km/s Vinfinity vector and a 6090 km impact parameter, the hyperbola's perilune is 633 kilometers above the moon's surface or 2371 kilometers from the moon's center. A hyperbola's velocity is sqrt(V_escape² + V_infinity²). Our V_infinity is .86 km/s. Escape velocity is sqrt(2GM/r). Since we're in the moon's sphere of influence, we will use the moon's mass for M, 7.35e22 kg. At 2371 kilometers from the moon's center, V_escape is about 2 km/s. Sqrt(.86² + 2²) is about 2.2.

Our hyperbola's perilune velocity is 2.2 km/s.

We want to get from a 633 km altitude hyperbola perilune to an elliptical orbit having 633 altitude km apolune and 20 km perilune. This ellipse would have semi axis a = 2064 km. Then we want to move to a low lunar orbit at 20 km altitude. Using the vis-viva equation we can get these numbers:

.9 km/s for LOI (Lunar Orbit Insertion) from a hyperbolic orbit

and

.2 km/s for injection to a low lunar orbit.

We wait until our low lunar orbit takes us to the north or south pole. We need to kill the orbit's 1.6 km/s. As the orbit slows, the ship will lose altitude gaining vertical velocity from gravity. To make a soft landing we need to counteract gravity. I estimate gravity loss will cost .5 km/s.

.5 + 1.6 km/s = 2.1 km/s

2.1 km/s for descent and landing.

The total delta V from LEO to landing is the sum of 4 burns: TLI, LOI, insertion to LLO, and landing.

3.1 + .9 + .2 + 2.1 = 6.4.

6.4 km/s is the total delta V from LEO to landing at the moon's north pole.

Period of an elliptical orbit is 2 pi sqrt(a³ / (GM)). The transfer orbit from LEO to a 384,400 km apogee is about 10 days. We travel half that orbit so
5 days from LEO to 384,000 apogee.

We can find the angular momentum of the hyperbolic orbit about the moon by doing a cross product of it's position vector and velocity vector at perilune. Magnitude of the hyperbola's angular momentum vector is 2371 km * 2.2 km/s which comes to about 5200 square kilometers per second. The magnitude of an orbit's angular momentum is twice the area the orbit sweeps out in a given time.

To measure twice the area the hyperbola swept out from entering SOI to perilune, I did a scale drawing and measured in Photoshop:

310 million square kilometers divided by 5200 square kilometers per second comes to about 60,000 seconds or about .7 days.
.7 days from SOI to hyperbola perilune.

We can use 2 pi sqrt(a³ / (GM)) to get the period of the orbit from a 633 km altitude to a 20 km altitude. This orbit has period of about .1 days. We traverse half this orbit. .5 * .1 = .05
.05 days from 633 km altitude to 20 km altitude.

We can also use 2 pi sqrt(a³ / (GM)) to get the period of a low lunar orbit at a 20 kilometer altitude. The LLO period is .08 days (about 1.8 hours).
Less than .08 days to get to either the north or south pole.

5 + .7 + .05 + .08 = 5.83. I will round up to 6.

6 days is the total time from LEO to landing at the north pole.

Asteroid Ice Isn't So Easy to Reach

Main Belt Asteroids
I will look at water ice in the Main Belt asteroids, short period comets, and what I call accessible NEOs. By accessible NEOs I mean those objects retrievable by a vehicle as described in the Keck Report

There is evidence that some of the Main Belt asteroids have water ice. Three possibilities are Ceres, 24 Themis, and 65 Cybele.

Here is a table showing delta v, launch window frequency (aka synodic period), trip time, and surface gravity:

Asteroid Name	LEO to Transfer (km/s)	Transfer to Rendezvous (km/s)	Synodic Period (Years)	Trip Time (Years)	Surface Gravity (m/s^2)
Ceres	4.9	4.9	1.28	1.29	.28
24 Themis	5.2	5.1	1.22	1.48	.075
65 Cybele	5.4	5.3	1.19	1.65	.07

The above numbers come from a model that assumes circular, coplanar orbits. 24 Themis has less than 1 degree inclination, so coplanar orbits are a good approximation. But 65 Cybele is inclined 3.5º to the ecliptic and Ceres 10.6º. So the above table underestimates the delta V for these two asteroids.

All three take considerably more delta V to reach than the lunar poles.

Lunar launch windows occur every two weeks from a given LEO orbit. Trip time is less than a week. By these metrics, the moon has a huge advantage over the main belt asteroids.

A common myth is that, due to their shallow gravity wells, return to earth from an asteroid takes virtually no delta V. Unless the asteroid has a near earth perihelion or aphelion, the heliocentric transfer orbit have a different velocity at asteroid rendezvous. For example, 24 Themis moves about the sun at about 16.8 km/s. The transfer orbit is moving about 11.7 km/s at rendezvous. Regardless if the transfer orbit is 24 Themis to Earth, or Earth to 24 Themis, the needed delta V is 5.1 km/s.

Some asteroids may be amenable to rendezvous via low thrust but high ISP ion rockets. But not these. The vehicle described in the Keck Report has a thrust of 2 newtons and a dry mass of 5.5 tonnes. That comes to about .0004 newtons per kilogram with no propellant. So to have a thrust/weight ratio greater than 1, surface gravity would need to be less than .0004 meters/second^2. It's hard to imagine an ion rocket getting off the ground on any of these three asteroids.

Short Period Comets
Next I will look at short period comets. A body that has recently outgassed is likely to have volatile ices.

Among the short period comets, Comet Encke has the shortest known semi-major axis, 2.22 A.U. It has a high eccentricity, e = ~.85. This means it has a perihelion quite close to the sun at .34 AU inside the orbit of Mercury. It has an aphelion at about 4.1 AU, out past the main belt.

It gets quite hot at perihelion, a black body temp of 460K (assuming an albedo of .15). And fairly cold at aphelion, 132K. But it spends more time in the aphelion neighborhood, so it's average black body temp is 167 K. If it had a circular orbit of radius 2.22 AU, the average black body temp would be 180K. So more eccentric orbits have the effect of lowering average black body temp throughout the orbit.

While the big eccentricity makes the comet a little cooler, it also makes for healthy delta V. Here are two possible routes from earth to Encke:

 Where rendezvous is at Encke's aphelion....

 ....or perihelion, there is big delta V. This isn't considering Encke's healthy inclination. Most comets have a good inclination that will boost delta V.

But if an asteroid/comet has a 1 A.U. perihelion, that changes the picture quite a lot. For the sake of argument I will imagine there is a comet JohnD with 0º inclination, 1.9 AU semi-major axis and a 1 AU perihelion. The fictitious comet JohnD is named after John DeLaughter, a planetologist I've been arguing with. Much of this blog post comes from that argument.

You may have noticed I've been drawing the transfer orbits sort of purplish. Like Hohmann orbits, the transfer orbits have been tangent to the departure and destination orbits. Thus delta V is only needed for speed change, not direction change. Well, Comet JohnD's orbit is already tangent to earth's orbit. Thus when the comet comes around our neighborhood, the only delta V needed is to leave earth orbit for TJI (Trans JohnD Injection). Rendezvous with the asteroid is virtually no delta delta V. In this case Vinfinity would be about 6.4 km/s. Therefore TJI from LEO is around 5 km/s. It would take less delta V to reach Comet JohnD than the moon.

Likewise for the return trip, wait until asteroid Johnd passes near earth at perihelion. A slight nudge can send a cargo to an earth atmosphere grazing orbit. Let aerobraking take care of most the delta V. Pretty sweet, huh?

Not so fast. The period of Comet JohnD is 2.619 years. When Comet JohnD returns to perihelion, earth will have advanced (360 + 360 + 222.8) degrees. The earth is nowhere near the comet, it's location at the 2nd perihelion is denoted by the number 2 in right part of the graphic above. Each circuit of the comet sees earth advance 222.8 degrees. Earth and the comet won't be in the same neighborhood for 22 periods! 22 * 2.619 = 57.  57.6 years.

If Chris Lewicki of Planetary Resources sets up a water mine on Comet JohnD, he'd have to wait 58 years to send the first shipment back to earth. Or else pay a delta V penalty.

Also microgravity mining in a vacuum is something the human race has zero experience doing. Acquiring the needed experience will be a trial and error process. Thus multiple trips would be needed to establish infra-structure. In the case of Comet JohnD, 3 trips to the comet to establish infra-structure would take 173 years.

The rarity of launch windows more than nullifies the slight delta V advantage this comet has over the moon.

Moreover, it is less likely Comet JohnD would have water ice in it's interior. Recall the average black body temperature of Encke was 167 K. And this is the shortest period comet known. Assuming an albedo of .15, Comet John D would have an average temp of 190 K:

This is 23 K warmer than Encke. A 23 K difference is the same as a 41.4º F difference. The difference between freezing and comfortable room temperature. Water ice in a vacuum starts sublimating at healthy rate at around 150 K. Being surrounded by clay and dust might mitigate sublimation loss for a time. But less so when the average temperature is boosted by 41º F. Encke is a rare comet having the shortest known semi-major axis at 2.22 AU. In my opinion a dead comet with a 1.9 AU axis is less likely to keep volatile ices at its core.

Accessible Asteroids: small asteroids with an earth-like orbit.
While there are many near earth objects quite close in terms of delta V, launch windows to these close objects are rare. Which makes establishing infra-structure more difficult. Also very rare are the opportunities to deliver the asteroid's resources to earth's neighborhood. Due to these considerations, NEOs fell off my radar screen.

Then in April 2012 the Keck Report was published.

Keck Report Authors include Chris Lewicki -- chief engineer of Planetary Resources, John S. Lewis - author of Mining the Sky and Rain of Iron and Ice, Don Yeomans - Manager of NASA's Near-Earth Object Program Office, Rusty Schweickart - chair emeritus of the B612 Foundation. There are many respected engineers and scientists among the authors.

The authors did the numbers demonstrating it's possible to park a small asteroid in high lunar orbit. I've examined the numbers and they are only mildly optimistic. In my opinion the vehicle described is doable.

Parking the rock in lunar orbit completely changes the picture. Now the rock has launch windows each two weeks. Trip time is less than a week.

Moreover, light lag latency from earth's surface is only 3 seconds. Since signal strength scales with inverse square of distance, the rock's proximity makes for good bandwidth. The rock is amenable to being worked by telerobots.

The Keck Report reversed my opinion. I now believe mining a retrieved asteroid is doable.

Retrievable asteroids would be rocks similar to 2008 HU4 -- small and having a semi major axis close to 1 AU. Also small eccentricity and inclination.

Here's a look at 2008 HU4's average temperature (assuming .15 albedo):

This is 117º F hotter than Comet JohnD. What's the life span of a small ice ball at this temperature?

John DeLaughter cites The Stability of Volatiles in the Solar System which says, in part, "a 1 km sphere at 1 a.u. is stable for 3,000 years,"

It's possible that near passages with the earth or other planets could lower a comet's aphelion. I would guess there are some dead comets with orbital elements similar to 2008 HU4. But how many of these were perturbed into their earth like orbits within the past 3,000 years?

And retrievable rocks are much smaller than 1 kilometer. More like 5 to 7 meters.

Equation (6) from Stability of Volatiles:
t_max = r₀/(dr/dt) = r₀ρ/É.

The number of interest here is r₀, initial radius. The ice ball's life scales with initial radius. 7 meters/1000 meters = .007. .007 *3000 = 21 years. A 7 meter ice ball at 1 a.u. would last 21 years.

John DeLaughter believes a blanket of loose soil could reduce water loss by a factor of 10 to 20. He argues the soil's permeability and tortuosity would slow sublimation. But he neglects to mention that it would also lower albedo. The paper he cites assumes a .6 albedo for ice balls. A comet's exterior mantle more typically has albedo of .1. Dark objects absorb more light and get hotter than pale objects. But for the sake of argument, I will give him his factor of 20. That's 21 * 20 years. How many comets have been perturbed into an earth like orbit within the past 420 years?

5 to 7 meter diameter rocks with an ~1 AU semi-major axis are unlikely to have water ice. This is not to say such rocks have no water! I believe there are many accessible rocks with water in the form of hydrated clays. But such rocks are water rich in the same way concrete is water rich. Ice deposits are more easily exploited than hydrated clays. If the lunar cold traps do indeed have large, thick ice deposits, I believe the moon would be a better source of extra-terrestrial propellant.

Study of Escher's Print "Gravity"

In his print Gravity, Escher placed 12 monsters on the star faces of the small stellated dodecahedron. He had thought of using turtles. I wish he had. With all their polygons, turtles are walking polyhedra.

So I did this polyhedra study using polyhedral turtles:

I've tried to follow Escher's patterns. Each turtle is looking at his neighbor's right leg, which forms rings of three. For example the yellow turtle is looking at the green turtle which is looking at the purple which is looking at the yellow. There are 4 sets of triple turtles and these four sets suggest a tetrahedron. There are other polyhedra suggested by other turtle groupings.

Here is a picture of a turtle along with the net used to make his shell:

To make this net I used rotation matrices, vector norms, vector cross products as well as dot products and an equation for finding the intersection of two coplanar lines. It was hard but fun.

Even MIT students can make misteaks.

Finding Delta V between various places in the solar system has been one of my hobbies. I like to learn as well as share what I know about this topic. So I occasionally Google search strings that include the term "Delta V".

Googling: Delta V Mars. The first hit is a Wikipedia article, the second hit is How Much Delta V do you need to get to Mars - Yahoo! Answers.

A fellow who calls himself ronwizfr confidently asserts that the delta V from earth orbit to Mars transfer is 6.6 km/s. Ronwizfr cites an MIT student project for a class called Solving Complex Problems.

The MIT page cited is a well done presentation until the students get to constants:

Then the students made an arithmetic error. The distance from the earth to the sun is 1.49X10¹¹ meters, not 1.49X10¹⁰ meters. Their radius for Mars' orbit is also off by a factor of 10.

The MIT students correctly plug in these wrong quantities to get speeds that are off by sqrt(10):

The velocity of the earth is not 94,384 m/s but closer to 29,846 m/s. Likewise velocity of Mars isn't 74,467 m/s but closer to 23,548 m/s. Their quantities would have been correct if the earth and Mars were .1 A.U. and .152 A.U. from the sun.

Even the very accomplished can make errors. So we should examine every assertion, no matter the source. My dad used to tell me we all put our pants on one leg at a time. His way of saying we're all human and capable of making mistakes.

Surgical Robots

At the Leprecon 39 Science Fiction Convention I attended a talk by Dr. Bruce Davis, a general/trauma surgeon in the Phoenix area.

Dr. Davis has done many laparoscopic surgeries using the da Vinci robot.  Instead of opening up a patient's belly, a small incision is made. A pair of robotic arms as well as a binocular pair of cameras are inserted through a small opening.

Davis reports that the two cameras give good depth perception. The telepresence is so immersive he often forgets he's not physically present inside the patient's body. He finds himself trying to turn his head to look about the cavity. Davis expects the robots will soon have motion capture for the head and neck so the camera motion will mimic the motion of the surgeon's actual eyes.

The arms are operated by motion capture. Robotic wrists mimic the motion of the surgeon's wrists. Instead of a thumb and four fingers, the robotic hand looks more like a crab's pinchers. The surgeon moves these pinchers with his thumb and index finger. Still, a lot can be accomplished with this simple hand.

The cameras give up to 10X magnification. When higher magnification is invoked, the motion of the robotic hands become more minute -- in effect shrinking the surgeon's avatar. We saw a video where a grape seemed like a large watermelon. The surgeon easily cut and peeled back a section of the grape's skin.

The robotic hands lack a sense of touch. Davis says the robot's designers are trying to figure out how to do haptic feedback but it's a challenging problem.

The da Vinci robot Davis uses costs 1.2 million dollars. I noted design and development is a large expense for early technologies. As more robots are made, design cost is amortized over more units and unit price can fall. Davis replied that this is happening to some extent. But da Vinci is the sole vendor in his field of robotic surgery and so has less incentive to drop their price.

I asked Dr. Davis if he foresaw uses of this technology outside of surgery. For example snaking robotic arms and eyes to a problem inside a utility line. This might avoid busting up a busy intersection with jack hammers and backhoes. Davis replied that this sort of application is likely and will probably lead to cheaper robots. Liability is a big expense in surgery. For example, a titanium screw breaking in a hip replacement can result in a major law suit. The same titanium screws that cost air lines $4.00 apiece are sold to surgeons at $400 apiece. Not having a heavy insurance burden, robots for plumbers and electricians can be less expensive and more common place than surgical robots.

I asked Davis if he thought telerobots could be used to build infrastructure on the moon or on an asteroid parked in lunar orbit. He said due to the 3 second light lag, motions would have to be very deliberate. But he thought it was doable. He mentioned there is a.i. being developed that mitigates slow reaction time caused by latency. For example, Big Dog's balance or Google Cars collision avoidance.

In Puppets, Telerobots and James Cameron, I opined that telerobots will be the game changer that enables use of space resources. I wrote that there are many uses right here on earth that are advancing telerobotic technology. So Davis' presentation was encouraging to me. Dr. Davis is also a science fiction writer.

Cartoon Delta V Map

This is the second cartoon delta-v map I've drawn. Clicking on the above can give a larger version.

My first cartoon map gave a lot more space to EML5 and little to L1 or L2. I knew of L4 and L5 through fiction like Gundam which was probably inspired by Gerard O'Neill's The High Frontier. Since then I've become less interested in L4 and L5 and more interested in L1 and L2. This new map reflects that shift in focus.

I had heard of the Interplanetary Transport Network as well as Shane Ross, Martin Lo, and Edward Belbruno. But I knew almost nothing about the low delta V routes achieved with n-body mechanics. I had a vague notion that Lagrange points were involved but that was about it.

Then in 2009 I came across a thread in Nasa Space Flight entitled An Alternative Lunar Architecture. In that thread Kirk Sorensen wrote at length about EML2 and work done by Robert Farquhar. Farquhar's 3 body work was done in the late 1960's and early 70's, decades before Ross, Belbruno and other modern advocates of 3-body mechanics.

Here's one of the Farquhar graphics Sorensen posted to that thread:

This is a 9 day route from LEO to EML2 taking delta V of about 3.5 km/s. It's time reversible so .4 km/s can drop a payload from from EML2 to an atmosphere grazing perigee. There's a 4 day route to EML1 that takes 3.8 days. It was surprising to me that EML2 could be reached with less delta V even though it's on the far side of the moon.

Also on Nasa Spaceflight in 2009, Martijn Mmeijering started a thread looking at an online 3 body text book by Koon, Lo, Marsden and Ross. 

It was in 2009 that I became more interested in L1 and L2.

There are routes between LEO and EML1&2 taking even less delta V, but these are time consuming. These are described by Andreas Stock's Investigation on Low Cost Transfer Options to the Earth-Moon Libration Point Region. 3.1 km/s seems to be the minimum between LEO for EML1 as well as EML2. In the map above I've depicted these routes with darker brown branches.

EML1 moves slower than an ordinary earth orbit at that altitude. An EML1 object nudged a little earthward will fall into an approximately 100,000 by 300,000 km elliptical orbit about the earth. A  .3 km/nudge suffices to send to it to a 36,000 km perigee where a 1 km/s burn can circularize the payload at geosynch orbit. A .7 burn can drop an EML1 payload to a LEO grazing orbit. If the LEO grazing orbit passes through the upper atmosphere, aerobraking can provide the 3.1 km/s needed to circularize at LEO.

EML2 moves faster than an ordinary earth orbit at its altitude. Nudged a bit away from the moon, a payload from EML2 will sail to a 1.8 million km apogee. The Sun Earth Lagrange 1 and 2 are 1.5 million kilometers from earth, so transfer from EML2 to SEL1 or 2 can be done with little delta V. Or an EML2 payload can sail through SEL1 or 2 completely out of earth's sphere of influence.

Nudge either EML1 or 2 a little moonward and they will fall into an approximately 5,000 x 60,000 km lunar orbit. Since the moon's rotating about the earth, a 60,000 km apolune can pass by both EML2 and EML1 over time. Thus it's possible to move between EML1 and 2 with very little delta V.

An object falling to a 300 km earth altitude from either EML1 or 2 will be traveling just a hair under escape when it reaches low altitude. Both EML1 and 2 make a complete circuit each 27.3 days so by timing your drop it's possible to choose longitude of perigee during a launch window. Plane changes are much less expensive at high altitudes so the velocity vector can be pointed in the right direction at perigee. Starting at EML2, injection into Mars or Venus Hohmanns can be done with around .9 km/s delta V (.4 km/sec to drop and a .5 km/s burn at perigee). On arriving at Mars, .7 km/s suffices to exit Hohmann for a 300 x 570,000 km Mars capture orbit.

A Near Earth Asteroid with a Vinfinity of 2 km/s or less can be dropped into an earth capture orbit using a lunar swing by. From there repeated lunar swing bys and little delta V can park the rock in high lunar orbit. Planetary Resources hopes to search for smaller rocks with their Arkyd orbital telescopes. If successful, they will likely find a multitude of rocks within .2 km/s of EML2.

Many NEAs are water rich and the cold traps at the lunar poles may have minable water deposits. So there are a number of potential propellent sources close to the earth-moon L1 and 2. Propellent sources high on the slopes of earth's gravity could break the exponent in Tsiolkovsky's rocket equation. This would give us mass fractions much easier to deal with. The highest delta V budget we'd have to endure is the 9.5 km/s from earth to LEO. Round trips between most other orbits would be in the neighborhood of 4 or 5 km/s.

Some notes on Venus: An earlier version of this map indicated delta V from Venus' surface to a capture orbit was 11.6 km/s. But then I came across an excellent delta V map by a fellow who calls himself Curious Metaphor.  In a discussion of his map, I was convinced Venus' thick dense atmosphere would make for a slower, steeper ascent from the planet surface. I've added 20 km/s gravity loss between Venus' upper atmosphere and surface. In Venus' upper atmosphere I've added a location labeled Landis Land. Named for Geoffrey Landis who noted there is a layer in Venus' atmosphere with earth like temperature and pressure. Moreover, a nitrogen/oxygen mix such as we  breath would be buoyant in Venus' CO2 atmosphere. Landis is a scientist as well as a science fiction writer. Some of his fiction takes place on the cloud cities of Venus.

Catching an Asteroid

In Capturing Near-Earth Asteroids around Earth, Hasnain, Lamb and Ross look at two delta Vs:
1) Delta V to nudge an asteroid's heliocentric orbit so the rock passes through the earth's sphere of influence.
2) Once in Earth's sphere influence, the delta V to make the hyperbolic orbit an elliptical capture orbit about the earth.

They don't try to find minimum delta V to reach an asteroid. Rather they try to determine whether low thrust ion engines can impart the needed delta V within plausible time frames.

I want to find asteroids that take the least delta V.

A good resource is JPL's NEO Close Approach page. To find orbits that already pass close to earth's sphere of influence, choose Nominal Distance less than or equal to 5 Lunar Distances. To find asteroids that don't need to shed too much velocity once in earth's sphere of influence, Sort by V-infinity:

Near the top of the resulting page is 2008 HU4's close encounter in 2015. Here is a picture of 2008 HU4's 2015 fly by:

To draw this picture I used position vectors generated by Horizon's Ephemeris page. I asked for position vectors in 3 day increments ranging from a month before to a month after the fly by:

Both the very helpful pages cited are JPL pages. The folks at JPL are worth their weight in gold, IMHO.

 Nudging 2008 HU4's Heliocentric orbit

Let's take a closer look at the close encounter:

The asteroid is a little ahead of the earth and moon. If it's orbit could be slowed by about a day, it'd be neck and neck with us. Increasing the perihelion speed by .02 kilometers/second would boost aphelion by 500,000 kilometers. This would increase the asteroid's orbital period by about a day -- from 420 days to 421 days.

Now that we have the asteroid running neck and neck with the earth and moon, we need to rotate the asteroid's line of nodes. This is where the asteroid's orbital plane intersects earth's orbital plane. The asteroid is above or below earth's orbit everywhere but at the line of nodes.

I want the asteroid to graze the moon's orbit. At about 6 degrees from the line of nodes, the moon is furthest from the sun, a full moon. Doing a plane change burn 90 degrees ahead of this can rotate the line of nodes where we want the close encounter. 2008 HU4's inclination is about 1.3º. The plane change needed is (1 - cos(6º)) * 1.3º. A .008º plane change would be more than enough to rotate the line of nodes 6º. At the region where we want to do the plane change, the asteroid's moving a little less than 30 kilometers/second. Delta V for plane change  is 2 * sin(.008º/2) * 30 kilometers/second = .005 kilometers/second.

The asteroid's a little further from the sun than the earth and moon. Decreasing the aphelion speed by .015 kilometers/second would shrink the perihelion by 300,000 kilometers. More than enough to get the asteroid to a moon grazing orbit.

So the delta V to nudge 2008 HU4's heliocentric orbit is (.02 + .005 + .015) kilometers/sec. About .04 kilometers/sec.

 Parking The Rock After It Enters Earth's Sphere Of Influence

Now for the second phase of capture. Now that we have the asteroid passing through our sphere of influence, we want it to stay. We need to get the hyperbolic velocity down below earth escape velocity.

2008 HU4's Vinfinity with regard to earth is about 1.4 km/sec. But by the time it reaches the moon's sphere of influence, it will have picked up some speed from earth's gravity.  Speed of a hyperbola is sqrt (V_escape² + V_infinity²). In the moon's neighborhood, earth escape velocity is about 1.4 km/s. So as the asteroid nears the moon's sphere of influence it's moving sqrt(1.4² + 1.4² ) kilometers/second. This is about 2 kilometers/second.

But the moon is moving about 1.02 km/s with regard to the earth. If the asteroid enters the moon's sphere of influence at 5 degrees from horizontal, it's speed will be about .97 km/s with regard to the moon.

When the asteroid enters the moon's sphere of influence, it's path can be modeled as a hyperbola  about the moon with a Vinfinity of about .97 km/s. Let's aim the asteroid so the hyperbola's perilune is about 1800 kilometers from the moon's center, about 70 kilometers above the moon's surface.

The turning angle is about 96º.

When this turned velocity vector is added to the moon's velocity vector, we have a speed under earth escape, about 1.19 km/s.

This would toss the rock into a 113,000 kilometer by 1,110,000 kilometer elliptical orbit about the earth. A .13 km/s apogee burn will boost perigee by about 180,000 kilometers. This path will take about two months so when the asteroid falls back, the moon will there to provide another gravity assist. This new gravity assist will boost the perigee as well as apogee. The new ellipse is a 311,000 x 1,500,000 km ellipse.

Doing a .08 km/s burn a little after apogee would put the rock into a 450,000 km by 1,800,000 km. This ellipse's perigee matches the altitude and speed of EML2.

With the gravity assist from the moon, the second phase takes a total of about .21 km/sec.

Total Delta V

With .04 km/s to nudge the heliocentric orbit and .21 km/s to park the asteroid at EML2, total delta V is around .25 km/sec.

Can Ion Engines Do The Burns? 

I've called for burns at the asteroid's perihelion, aphelion, and a plane change 84º from the node. Also high apogee burns after the rock's been captured to earth's orbit. Do these need high thrust impulsive burns or can ion engines do the job?

Heliocentric and high earth orbits have a much more leisurely angular velocity than low earth orbit.

LEO angular velocity is about 4 degrees per minute. In earth's neighborhood, low eccentricity heliocentric orbits are about 1 degree per day.  LEO orbits have around 6000 times the angular velocity. A 5 minute burn in LEO scaled up to heliocentic proportions would take around 20 days.

Doing a velocity change of 20 meters/sec over 20 days is an acceleration of about .000012 meters/sec² . To accelerate a 500 tonne rock this much, we'd need a thrust of about 5 newtons.

The Keck study for retrieving an asteroid calls for 5 10 kW Hall thrusters, of which 4 would operate at a given time. According to this Nasa page, a 10 kW Hall Thruster gives about half a newton. Four engines would give two newtons.

It's plausible higher thrust ion engines will be developed in the future. It's also possible the nudges to the heliocentric orbit could be done over several orbital periods.

How about the burns needed once in earth's sphere of influence?

At a 1,500,000 km apogee, the rock is moving about .6 degrees a day.

A burn over 20 degrees would take about 34 days. A .13 km/s burn would be an acceleration of .00001 meters a second. Accelerating a 500 tonne rock this much would take about 5 newtons.

If the last ellipse is allowed to return to apogee, it is likely the sun's influence would tear the asteroid loose from earth's capture orbit.  1,500,000 km is the outer edge of the earth's sphere of influence.

If I understand the Keck pdf correctly, they give .17 km/s as the delta V to return a 500 tonne rock, and they say the 5 10 kW Hall thrusters are up to the task. There are some very clever people contributing to that paper and it is likely they've come up with more effective lunar assists then mine. But until I have a better understanding, I'll say 2008 HU4 is .25 km/s from EML2 and retrieving a 500 tonne rock takes 5 newtons of thrust.

How many rocks can be caught? 

2008 HU4 is near the very top of the list of low delta V asteroids. How many other rocks could be grabbed as easily?

In general smaller bodies are more common than larger bodies. If we successfully expanded our asteroid inventory to include most the asteroids 5 meters or greater, we would probably know of 100s or even 1000s of asteroids just as catchable as 2008 HU4.

It is the aim of Planetary Resources to take just such an inventory. They plan to launch a fleet of orbital Arkyd telescopes to accomplish this task.

It's likely rocks fall into temporary earth capture orbits without human intervention. New Scientist says any given time we could have several captured asteroids as tiny moons.

I applaud the efforts of Planetary Resources and Deep Space Industries. Their goals are doable. I'm also pleased NASA has expressed an interest in the Keck Study. Hopefully NASA will award PR and DSI contracts. In the near term I'm hoping NASA will buy time on the Arkyd Telescopes to get a more complete inventory of Chelyabinsk sized rocks.