APOD: Gravitational Tractor (2012 Jul 07)

Comments and questions about the APOD on the main view screen.
User avatar
APOD Robot
Otto Posterman
Posts: 5589
Joined: Fri Dec 04, 2009 3:27 am
Contact:

APOD: Gravitational Tractor (2012 Jul 07)

Post by APOD Robot » Sat Jul 07, 2012 4:06 am

Image Gravitational Tractor

Explanation: How would you change the course of an Earth-threatening asteroid? One idea - a massive spacecraft that uses gravity as a towline - is illustrated in this dramatic artist's view of a gravitational tractor in action. In the hypothetical scenario worked out in 2005 by Edward Lu and Stanley Love at NASA's Johnson Space Center, a 20 ton nuclear-electric spacecraft tows a 200 meter diameter asteroid by simply hovering near the asteroid. The spacecraft's ion drive thrusters are canted away from the surface. The steady thrust would gradually and predictably alter the course of the tug and asteroid, coupled by their mutual gravitational attraction. While it sounds like the stuff of science fiction, ion drives do power existing spacecraft and a gravitational tractor would work regardless of the asteroid's structure or surface properties.

<< Previous APOD This Day in APOD Next APOD >>
[/b]

Flase
Brother of Ture
Posts: 132
Joined: Wed Dec 07, 2011 7:17 am

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by Flase » Sat Jul 07, 2012 5:53 am

Image
I always imagined that this would be the only reasonable purpose of nuclear warheads.

Have them smack into the side of the asteroid at maximum velocity and if you catch it early enough, it wouldn't take much to push it away from a collision course.

If you don't notice the meteoroid until the collision is imminent, it is likely to be small and have a much smaller momentum to change.

Also, it would be fun like a game of asteroids.
It is the year 2464

You are commander Bob, in command of the Terran Empire's greatest warship, The Space Conqueror.

A meteoroid has been discovered on a collision course with Earth and it is now your mission to use your vessel's arsenal of weaponry to save the planet.

You are mankind's last and only hope, Commander Bob, so get your act together and measure up!
Last edited by Flase on Sat Jul 07, 2012 7:47 am, edited 4 times in total.

User avatar
Moonlady
Selenian
Posts: 666
Joined: Fri Apr 13, 2012 9:06 pm

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by Moonlady » Sat Jul 07, 2012 5:55 am

How would you change the course of an Earth-threatening asteroid?

1. Talking: First explain to it that it should take another route, diplomatics and soft skills work best.

2. Send Harry Stemper's drilling team up there, they did it once succesfully and they are experienced, do not forget the russian!

3. Build a giant parabolic sunlight reflector and focus it on the asteroid, then it goes poof!

4. Maybe it will be a second moon around earth?! That would be so cool! A mini moon :D

User avatar
Beyond
500 Gigaderps
Posts: 6889
Joined: Tue Aug 04, 2009 11:09 am
Location: BEYONDER LAND

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by Beyond » Sat Jul 07, 2012 7:16 am

It's going to take a while to get used to a tractor with no wheels, that has enough power to barely hold up a piece of paper on earth(1/50th of a lb of thrust) to expect it to be of any help in changing the course of anything that is headed towards earth. I think I'd rather throw some nuclear bombs at the threats. We've got plenty of those to get rid of anyway. :yes:
To find the Truth, you must go Beyond.

Flase
Brother of Ture
Posts: 132
Joined: Wed Dec 07, 2011 7:17 am

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by Flase » Sat Jul 07, 2012 8:07 am

APOD Robot wrote:The steady thrust would gradually and predictably alter the course of the tug and asteroid, coupled by their mutual gravitational attraction.
What sort of time period would this be?

If you use the continuous acceleration of a thruster, could you not attach a more heavy-duty rocket to the rock's surface, which will give greater power over this same time period? What's more, if you rely upon this spacecraft having a great-enough mass to attract the asteroid gravitationally, it will be very ungainly, difficult to control and the acceleration of the thrusters will be impaired once it's finally in position, will it not?

User avatar
Ann
4725 Å
Posts: 13838
Joined: Sat May 29, 2010 5:33 am

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by Ann » Sat Jul 07, 2012 9:24 am

Click to play embedded YouTube video.
Ann
Color Commentator

henrystar
Ensign
Posts: 97
Joined: Sun Feb 28, 2010 11:40 am

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by henrystar » Sat Jul 07, 2012 10:34 am

Image
Flase wrote:I always imagined that this would be the only reasonable purpose of nuclear warheads.
Have them smack into the side of the asteroid at maximum velocity and if you catch it early enough, it wouldn't take much to push it away from a collision course.


The above is extremely dangerous. The most likely outcome is the asteroid breaking into a zillion fragments which then would hit a much larger area of the earth. That is exactly why the gravitational tractor is such a superb idea. I note another letter suggesting light pressure. That strikes me as an extremely GOOD idea.

biddie67
Science Officer
Posts: 483
Joined: Sun Jan 31, 2010 9:44 am
Location: Possum Hollow, NW Florida

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by biddie67 » Sat Jul 07, 2012 10:58 am

It seems that a spacecraft with the speed and manuverability to catchup and position itself by an asteroid would be better utilized to "nudge" the asteroid from some angle behind it into a path that misses both the Earth and the Moon.

This would avoid problems of transporting the extra weight of rockets or errant rockets coming back to Earth or fragments of an asteroid coming to Earth.

Several of these reuseable ion propulsion spacecraft could be located at L points in case they were quickly needed in an emergency.

smitty
Science Officer
Posts: 145
Joined: Tue Jul 04, 2006 9:57 am

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by smitty » Sat Jul 07, 2012 11:47 am

Realistically (this is supposed to be about reality, right?), how much advance warning would we need to implement such a scheme? Getting the tractor into the vicinity of the asteroid would not happen instantaneously, even assuming we had a supply of such tractors on the shelf and ready to go. Moreover, it sounds as though the tractors would necessarily be sufficiently massive that we probably would need to launch each of them in smaller segments and assemble them in space. And how much would it cost to have this all "in place" and ready to launch at a moment's notice? How much is our world worth?, you might answer. Sounds like an excellent project for a huge international cooperative effort. If we have to finance the whole thing, we're only going to divert asteroids that would have slammed into our part of the world, right? Ooops, but wait; maybe the Chinese and Russians and Brits and French, etc., would all feel the same way? Hmmmm . . . .

User avatar
neufer
Vacationer at Tralfamadore
Posts: 18805
Joined: Mon Jan 21, 2008 1:57 pm
Location: Alexandria, Virginia

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by neufer » Sat Jul 07, 2012 12:06 pm

  • The Burning
-----------------------------------------------------------
Jerry: She said there's some tractor story that she hasn't told me about.

George: Woah, back it up, back it up. Beep, beep, beep. Tractor story?

Jerry: Beep, beep, beep? What are you doing?

George: You know what I think? I bet she stole a tractor.

Jerry: No one's stealing a tractor, it's a five-mile-an-hour getaway.
We're dancing around the obvious, it's gotta be disfigurement.

George: Does she walk around holding a pen she never seems to need?

Jerry: No, she looks completely normal.

George: Oh. Okay, here it is, I got it. She lost her thumbs in a tractor
accident and they grafted her big toes on. They do it every day.

Jerry: You think she's got toes for thumbs?

George: How's her handshake? A little firm, isn't it? Maybe a little too firm?

Jerry: I don't know.

George: Hands a little smelly?

Jerry: Why do I seek your counsel?
-----------------------------------------------------------
Art Neuendorffer

User avatar
neufer
Vacationer at Tralfamadore
Posts: 18805
Joined: Mon Jan 21, 2008 1:57 pm
Location: Alexandria, Virginia

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by neufer » Sat Jul 07, 2012 12:20 pm

Flase wrote:
APOD Robot wrote:
The steady thrust would gradually and predictably alter the course of the tug and asteroid, coupled by their mutual gravitational attraction.
What sort of time period would this be?
  • ~ 10 years.
http://en.wikipedia.org/wiki/Gravity_tractor wrote:
<<Let us suppose that a NEO of size around 100 m, and mass of one million metric tons, threatened to impact Earth. Suppose also that a velocity correction of 1 cm/s would be adequate to place it in a safe and stable orbit, missing Earth that the correction needed to be applied within a period of 10 years.

With these parameters, the required impulse would be: V × M = 0.01 [m/s]×109 [kg] = 107 [N-s], so that the average tractor force on the asteroid for 10 years, = 3.156×108 s, would need to be about 0.032 newtons. An ion-electric spacecraft with a specific impulse of 10,000 N-s per kg, corresponding to an ion beam velocity of 10 km/s (about twenty times that obtained with the best chemical rockets), would require 1,000 kg of reaction mass (Xenon is currently favored) to provide the impulse. The kinetic power of the ion beam would then be approximately 317 W; the input electric power to the power converter and ion drive would of course be substantially higher. The spacecraft would need to have enough mass and remain sufficiently close to the asteroid that the component of the average gravitational force on the asteroid in the desired direction would equal or exceed the required 0.032 N. Assuming the spacecraft is hovering over the asteroid at a distance of 200 m to its centre of mass, that would require it to have a mass of about 20 metric tonnes, because due to the gravitational force we have:

Image

Considering possible hovering positions or orbits of the tractor around the asteroid, note that if two objects are gravitationally bound in a mutual orbit, then if one receives an arbitrary impulse which is less than that needed to free it from orbit around the other, because of the gravitational forces between them, the impulse will alter the momentum of both, together regarded as a composite system. That is, so long as the tractor remains in a bound orbit, any propulsive force applied to it will be effectively transferred to the asteroid it orbits. This permits a wide variety of orbits or hovering strategies for the tractor. One obvious possibility is for the spacecraft to orbit the NEO with the normal to the orbit in the direction of the desired force. The ion beam would then be directed in the opposite direction, also perpendicular to the orbit plane. This would result in the plane of the orbit being shifted somewhat away from the center of the asteroid, "towing" it, while the orbital velocity, normal to the thrust, remains constant. The orbital period would be a few hours, essentially independent of size, but weakly dependent on the density of the target body.>>
  • Note that the mass of the tractor goes with the square of the asteroid's size
    (; e.g., a 1 km asteroid would require a 2,000 ton tractor).
Last edited by neufer on Sat Jul 07, 2012 12:44 pm, edited 1 time in total.
Art Neuendorffer

User avatar
rstevenson
Quis custodiet ipsos custodes?
Posts: 2705
Joined: Fri Mar 28, 2008 1:24 pm
Location: Halifax, NS, Canada

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by rstevenson » Sat Jul 07, 2012 12:26 pm

Flase wrote:If you use the continuous acceleration of a thruster, could you not attach a more heavy-duty rocket to the rock's surface ... ?
What we've learned recently is that many, perhaps all, asteroids are just bunches of rocks and gravel and maybe ice. If you attach anything to pull them, they would come apart. If you go around back and try to push them, they would come apart. I think even pushing them with light pressure would cause them to come apart. Gravity is the only thing that will pull all of the asteroid evenly in one direction. (I suppose you could "bag" them, but we're talking a really, really big bag, and a pretty strong one too!)

The real problem will be getting the large, heavy gravity tug into position in time to do any good. This will only work if we have many years of advance warning.

Rob

Flase
Brother of Ture
Posts: 132
Joined: Wed Dec 07, 2011 7:17 am

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by Flase » Sat Jul 07, 2012 12:48 pm

rstevenson wrote:
Flase wrote:If you use the continuous acceleration of a thruster, could you not attach a more heavy-duty rocket to the rock's surface ... ?
What we've learned recently is that many, perhaps all, asteroids are just bunches of rocks and gravel and maybe ice. If you attach anything to pull them, they would come apart. If you go around back and try to push them, they would come apart. I think even pushing them with light pressure would cause them to come apart...
I suppose you could ascertain the structure of the asteroid before making the decision. Some of them seem to be largely solid iron metal and you could attach a mining operation to build a ship if you have ten years...

Or maybe you could use a large electromagnet instead of gravity to attract it. You can adjust the strength of the magnet on the fly..
Last edited by Flase on Sat Jul 07, 2012 12:58 pm, edited 2 times in total.

User avatar
neufer
Vacationer at Tralfamadore
Posts: 18805
Joined: Mon Jan 21, 2008 1:57 pm
Location: Alexandria, Virginia

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by neufer » Sat Jul 07, 2012 12:51 pm

Flase wrote:
rstevenson wrote:
Flase wrote:
If you use the continuous acceleration of a thruster, could you not attach a more heavy-duty rocket to the rock's surface ... ?
What we've learned recently is that many, perhaps all, asteroids are just bunches of rocks and gravel and maybe ice. If you attach anything to pull them, they would come apart. If you go around back and try to push them, they would come apart. I think even pushing them with light pressure would cause them to come apart. Gravity is the only thing that will pull all of the asteroid evenly in one direction. (I suppose you could "bag" them, but we're talking a really, really big bag, and a pretty strong one too!)

The real problem will be getting the large, heavy gravity tug into position in time to do any good. This will only work if we have many years of advance warning.
I suppose you could ascertain the structure of the asteroid before making the decision. Some of them seem to be largely solid iron metal and you could attach a mining operation to build a ship if you have ten years...
Land on the asteroid first with a ~5 ton spacecraft and robotically collect & bag ~15 tons of "ballast" material.

An extra ion rocket or two would be sufficient to take off.

When the mission is complete use the extra ion power to return the ~15 tons of "ballast" material back to earth for souvenirs(; the asteroid was basically headed in that direction in the first place.)
Last edited by neufer on Sat Jul 07, 2012 1:07 pm, edited 2 times in total.
Art Neuendorffer

User avatar
Chris Peterson
Abominable Snowman
Posts: 18594
Joined: Wed Jan 31, 2007 11:13 pm
Location: Guffey, Colorado, USA
Contact:

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by Chris Peterson » Sat Jul 07, 2012 1:03 pm

Flase wrote:I suppose you could ascertain the structure of the asteroid before making the decision. Some of them seem to be largely solid iron metal and you could attach a mining operation to build a ship if you have ten years...
Iron bodies are extremely rare- the odds of one hitting us are very small. Really big asteroids aren't going to hit us either- we already know where the bodies larger than a few hundred meters are, and that they pose no risk (comets are a different problem... but there's currently no practical technology to deflect those). That leaves mainly asteroids in the ~100-500 m diameter range to worry about, and those are the ones that appear to be rubble piles as opposed to monolithic structures.
Chris

*****************************************
Chris L Peterson
Cloudbait Observatory
https://www.cloudbait.com

User avatar
neufer
Vacationer at Tralfamadore
Posts: 18805
Joined: Mon Jan 21, 2008 1:57 pm
Location: Alexandria, Virginia

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by neufer » Sat Jul 07, 2012 2:12 pm

Chris Peterson wrote:
comets are a different problem... but there's currently no practical technology to deflect those.
Land a surplus Japanese nuclear reactor on the surface and set it for meltdown.
Art Neuendorffer

User avatar
Chris Peterson
Abominable Snowman
Posts: 18594
Joined: Wed Jan 31, 2007 11:13 pm
Location: Guffey, Colorado, USA
Contact:

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by Chris Peterson » Sat Jul 07, 2012 2:30 pm

neufer wrote:Land a surplus Japanese nuclear reactor on the surface and set it for meltdown.
You mean leave it at its default settings?
Chris

*****************************************
Chris L Peterson
Cloudbait Observatory
https://www.cloudbait.com

User avatar
geckzilla
Ocular Digitator
Posts: 9180
Joined: Wed Sep 12, 2007 12:42 pm
Location: Modesto, CA
Contact:

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by geckzilla » Sat Jul 07, 2012 2:32 pm

Oh, snap.
Just call me "geck" because "zilla" is like a last name.

briandonohue

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by briandonohue » Sat Jul 07, 2012 2:34 pm

Would the name of the "dramatic artist" who created this be Joel Hodgson? All that's missing is the show's title carved into the asteroid. "Cambot give me rocket number nine!!!"

User avatar
Psnarf
Science Officer
Posts: 325
Joined: Fri Feb 26, 2010 6:19 pm

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by Psnarf » Sat Jul 07, 2012 3:28 pm

Simply remove all of the asteroid's Higgs bosons. It would then proceed along a vector that is tangent to the asteroid's orbit at the time its mass changed to zero, then Newton's First Law would take it out of the solar system. :b:

Rusty Schweickart

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by Rusty Schweickart » Sat Jul 07, 2012 4:31 pm

Sorry APOD... you are out of date. Many years ago we (B612 Foundation) worked with NASA when their Prometheus mission was in planning. It was a HUGE ion engine propelled spacecraft... which we realized could be easily modified to have gravity tractor (GT) capability. This led detractors to claim that the GT had to be a HUGE spacecraft. NOT! Any spacecraft using electric propulsion can serve as a GT and e.g. deflecting Apophis from the 2029 keyhole would require a 1 metric ton spacecraft (modest size) only a month or so of towing to avoid the keyhole/impact.

I'm including the correct Dan Durda image (https://dl.dropbox.com/u/18255391/gravt ... plumes.jpg) of the GT vs. what you've shown. Dan Durda's original huge version, however, continues to hang on. Please use the correct version showing a reasonable spacecraft... not the one you've used. Thanks, B612 Foundation

smitty
Science Officer
Posts: 145
Joined: Tue Jul 04, 2006 9:57 am

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by smitty » Sat Jul 07, 2012 4:47 pm

And let's hope that if we ever do need to use this technology somebody double checks to see that the correct conversion between metric and English units has been made. Those awkward "oooops" moments are so embarrassing, not to mention potentially lethal.

User avatar
neufer
Vacationer at Tralfamadore
Posts: 18805
Joined: Mon Jan 21, 2008 1:57 pm
Location: Alexandria, Virginia

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by neufer » Sat Jul 07, 2012 6:11 pm

Rusty Schweickart wrote:
Sorry APOD... you are out of date. Many years ago we (B612 Foundation) worked with NASA when their Prometheus mission was in planning. It was a HUGE ion engine propelled spacecraft... which we realized could be easily modified to have gravity tractor (GT) capability. This led detractors to claim that the GT had to be a HUGE spacecraft. NOT! Any spacecraft using electric propulsion can serve as a GT and e.g. deflecting Apophis from the 2029 keyhole would require a 1 metric ton spacecraft (modest size) only a month or so of towing to avoid the keyhole/impact.
You are talking about a very special situation where, presumably, it is only necessary for the asteroid to avoid a 660 meter keyhole (rather than a 12,742,000 meter planet) on its first pass by earth.

If we were 100% certain that Apophis WAS aimed at the keyhole then Apophis' velocity need only to be changed by ~16 cm/h (vs. ~1 cm/s in the APOD) to avoid that keyhole.

However, the size of the 660 meter keyhole is miniscule compared with the current (2006) ~3,200,000 m dynamical uncertainties!

The APOD tractor is, in fact, far too small to guarantee that 400 m Apophis would avoid the keyhole.

If only we could be more certain of Apophis dynamics:
http://en.wikipedia.org/wiki/Rusty_Schweickart wrote:
<<In May 2005 Schweickart told the U.S. Congress that a mission to attach a device such as a radio transponder to asteroid 99942 Apophis (formerly known as 2004 MN4) should be a high priority; it is estimated that this asteroid has a 1 in 6000 probability of striking the earth in the 21st century. The latest data indicates that the chance of Apophis impacting the earth is 1 in 45,000 in 2036.>>
Far cheaper, perhaps, would be to intercept Apophis with an direct impact (or nuclear weapons) at the 660 meter keyhole itself in 2029 if it should come to that.
http://en.wikipedia.org/wiki/Gravitational_keyhole wrote:
<<The asteroid Apophis was once estimated to have a 2.7% (1 in 37) chance of striking the Earth in 2029. Further observations and revisions of the estimated path of the asteroid have ruled out impact in 2029, but identified a 660 m wide keyhole that the asteroid may pass through on its approach in 2029, thereby causing a deflection that may result in impact in 2036; currently, the estimated probability of impact in 2036 is 1 in 250,000 (0.0004%). Apophis is estimated to be as large as 400 m across, and could cause millions of casualties if it were to hit Earth.

Scientists from the B612 Foundation, a private foundation dedicated to protecting the Earth from asteroid strikes, have proposed that Apophis be nudged out of its present orbit into an orbit that takes it further from the keyhole. NASA scientist David Morrison says, "After 2029, the deflection would have to be vigorous enough to miss not just a tiny keyhole but the much larger target of the Earth itself. And such a deflection is far beyond present technology for an asteroid this large."

Additional observations of the trajectory of Apophis revealed the keyhole would probably be missed. As of October 7, 2009, the probability of an April 13, 2036 impact is considered to be 1 in 250,000. An additional impact date in 2037 was also identified; the impact probability for that encounter was calculated as 1 in 12.3 million.

The gravitational keyhole for Apophis is only 660 metres in diameter. Calculations showed that if Apophis' velocity could be changed by only 0.00016 km/h, then in three years its orbit would be deflected by more than a mile, enough to miss the keyhole. The problem is that the keyhole is so small that it becomes extremely difficult to predict precisely if Apophis will pass through it or not. Orbit projections made in 2006 for Apophis in 2029 have a margin of error of about 3,200 km, which as of early 2010 is well within the estimated distance between Apophis (18,600 miles from Earth) and the location of the keyhole (18,893 miles from Earth). As time passes the error ellipse will be reduced but NASA may have to wait until enough data accumulates to reduce the error ellipse to one mile (1.6 km) before it knows if Apophis will hit the keyhole or not.>>
Last edited by neufer on Sat Jul 07, 2012 6:44 pm, edited 5 times in total.
Art Neuendorffer

User avatar
bystander
Apathetic Retiree
Posts: 21592
Joined: Mon Aug 28, 2006 2:06 pm
Location: Oklahoma

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by bystander » Sat Jul 07, 2012 6:21 pm

Rusty Schweickart wrote:Sorry APOD... you are out of date.
This is a repeat. http://apod.nasa.gov/apod/ap051110.html
Know the quiet place within your heart and touch the rainbow of possibility; be
alive to the gentle breeze of communication, and please stop being such a jerk.
— Garrison Keillor

scr33d
Ensign
Posts: 40
Joined: Tue Jun 05, 2012 2:37 pm

Re: APOD: Gravitational Tractor (2012 Jul 07)

Post by scr33d » Sat Jul 07, 2012 7:10 pm

Comments are cheap. Read the cited paper. It's not very long or technical:

"The collision of an asteroid as small as ~200 m with the Earth could cause widespread damage and loss of life1. One way to deflect a threatening asteroid is to dock a spacecraft to the surface and push on it directly2. The total impulse needed for rendezvous and deflection is too large for chemical rockets, but is achievable by 20 ton class nuclear-electric propelled spacecraft proposed by NASA2. Regardless of the propulsion scheme, a docked asteroid tug needs an attachment mechanism since the surface gravity is too weak to hold it in place. Asteroids are likely to be rough and unconsolidated, making stable attachment difficult. Furthermore, most asteroids rotate, so an engine anchored to the surface would thrust in a constantly changing direction. Stopping the asteroid’s rotation, reorienting its spin axis3, or firing the engine only when it rotates through a certain direction adds complexity and wastes time and propellant.
Our suggested alternative is to have the spacecraft simply hover above the surface. The spacecraft will tow the asteroid with no physical attachment using gravity as a towline."

More in a PDF here: http://arxiv.org/pdf/astro-ph/0509595v1
Last edited by owlice on Sat Jul 07, 2012 7:22 pm, edited 1 time in total.
Reason: Removed text; provided link to PDF for rest of paper.

Post Reply