Thursday, February 16, 2012

Murphy's Mangled Math

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


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


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


Earth escape velocity (~11 km/s),

Earth to Mars velocity (~6 km/s)

Mars escape velocity (~5 km/s)


Which totals ~22 km/s.

Murphy's delta V

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


Murphy vs Oberth


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


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


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


But Murphy neglects the use of aerobraking.


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


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


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


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


Tsiolkovy's equation:


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


Where e is Euler's number, about 2.72.


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


Rice

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


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


Refuel In Space?


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


Rice With Depots


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


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


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


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


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


EML1 to Mars delta V map


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


Are there other potential propellant sources?


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


EML1 to Deimos

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



Grab That Asteroid!


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


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


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


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


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


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

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


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


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


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


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






6 comments:

Van Kane said...

Hop David - A nice blog. Would be interested in your take on how much mass has to be launched to put the entire system in place.

Hop David said...

Van Kane, you are the first person to comment on my blog. Thank you.

An interesting pdf:
http://kiss.caltech.edu/study/asteroid/asteroid_final_report.pdf
One of the authors is Chris Lewicki who is also of Planetary Resources. I'm guessing Planetary Resources hopes to use something like this to park an asteroid in high lunar orbit or at EML1.

On page 14 of they describe the Asteroid Capture and Return (ACR): 5.5 tonnes dry mass, 13 tonnes Xenon propellant for a total of 18.5 tonnes. In the illustration it's launched by an Atlas V, but perhaps this won't be the only option by the time they launch.

I am encouraged that one of planetary resources first goals is returning a water rich asteroid:
http://www.planetaryresources.com/asteroids/usage/
A propellant source high on the slopes of earth's gravity well has the potential to make spaceflight much less difficult. And less expensive spaceflight is a prerequisite for achieving ROI on asteroidal platinum and other minerals.

What is the mass of infrastructure needed to make propellant? An open question, but I am anxious about this. Cracking water into hydrogen and oxygen takes 237 kJ per mole. I understand the ISS solar array wings put out about 30 watts/kg. I've heard of solar arrays that have a specific power of 200 watts/kg, but haven't seen yet seen convincing cites. In any case it looks like we'll need a massive power source to crack water at an appreciable rate.

I've seen proposals to bake the water out of hydrated clays using sunlight: placing asteroid material to an airtight kiln at the focus of a parabolic mirror. I believe high specific power is more doable for thermal watts than electric watts, but this is still an ambitious undertaking.

To summarize: ACR mass about 19 tonnes. I don't know what the mining infrastructure mass will be. I know mining the asteroid will be difficult but I haven't yet seen persuasive arguments that ROI is impossible.

kert said...

Funny how he never expanded upon that original post.

Mass drivers ? Solar electric propulsion ( effectively a different type of mass driver ? )
Nuclear propulsion ? ( i.e. something as simple as steam rocket ) etc etc.

Lunar materials as propellants ? ( Aluminium, oxygen )

Hollister David said...

Kert,

The SEP you mention is a big one. The Keck study suggests using SEP with xenon as a propellant with an exhaust velocity of 30 km/s. The study looks at retrieving some objects that can be diverted from the heliocentric orbits to a high lunar orbit for around .2 km/s.

In Murphy's silly asteroid retrieval scenario he looks at fetching a kilometer sized asteroid that would take 5 km/s. He wants to use a propellant with 3 km/s exhaust velocity.

I don't know if Murphy is capable of plugging .2 km/s delta V and 30 km/s exhaust velocity into the rocket equation. If he were, he'd discover his "rough approximation" is off by several orders of magnitude.

eclipsenow said...

This may be some time down the track, but another issue that is ignored is the distant potential for self-replicating droid ships of some sort. Fire a few self replicating ships (or colonies, depending on your scenario for AI technologies) at the asteroid belt with all the fuel you need to brake it there, wait a decade or 2, and eventually it will manufacturer all the ships and fuel necessary to fire resources back to almost any point in the solar system.

Hollister David said...

Eclipse, perhaps Von Neumann machines will come to pass one day. But now they're science fiction. Brin's existence is a nice yarn on such devices.

But I do believe telerobots and robots will be game changers. I talk about that on several of my blog entries, the most recent being Who Needs Humans?

Perhaps as robots become more able they will eventually become able to extract resources from their environments and use them to replicate themselves. That's well beyond present state of the art, though.