Dawn Explores Vesta’s Chemistry

by Paul Gilster on January 20, 2012

The Dawn spacecraft, orbiting Vesta since July of last year, reached its lowest altitude orbit in December, now averaging 210 kilometers from the asteroid’s surface. Ceres is Dawn’s next stop, but that journey won’t begin until the close-in work at Vesta is complete, with the craft in its low altitude mapping orbit for at least ten weeks and then another period at higher altitudes before Dawn leaves Vesta in late July. The spacecraft’s Gamma Ray and Neutron Detector (GRaND) instrument is already telling us much about the giant asteroid’s surface composition.

In fact, the five weeks of mapping at low-altitude have provided the first look at global-scale variations on Vesta. GRaND measures the abundance of elements found in planetary surfaces, and while its investigations are still in the early stages of analysis, it’s clear that Vesta’s surface varies widely as opposed to the mostly uniform composition of smaller asteroids. We know that Vesta developed a core, mantle and crust, making it something closer to a planet than an asteroid. We’ll learn a great deal more as the low-altitude mapping continues and new results arrive.

Meanwhile, have a look at this close-in Dawn image, showing a crater with smaller craters on its edge, with rough textures in the crater wall (far right) that may indicate underlying bedrock. The image shows an area in the Rheasilvia basin in the south polar region, obtained with Dawn’s framing camera on December 18, 2011 — the distance to the surface here is 272 kilometers.

Image: An area in Vesta’s south polar region, centered around 58.3 south latitude and 283.7 degrees east longitude. Acquired during the LAMO (Low Altitude Mapping Orbit) phase of the mission, the image has a resolution of about 25 meters per pixel. Credit: NASA/ JPL-Caltech/ UCLA/ MPS/ DLR/ IDA.

Ion Propulsion’s Slow Push

Let’s keep an eye not just on Dawn’s science results but on the performance of its thrusters as we continue to shake out ion propulsion and tweak its capabilities with missions to the outer system. The spacecraft’s thrusters apply more than 1000 volts to accelerate charged particles (ions) that are expelled out the back at speeds up to 40 kilometers/second. Xenon is used in these thrusters because it is a relatively massive atom, offering higher thrust than other possible propellants. Even so, the acceleration is tiny, underlining the fact that ion propulsion is effective not just in its efficiency but also in its cumulative acceleration over long periods.

As the Dawn team reminds us, the force of an ion thruster on the spacecraft is comparable to the weight of a single sheet of paper. But their enormous fuel efficiency means that the thrusters can continue this tiny push not just for days but for years, allowing the effects to build. Dawn chief engineer Marc Rayman has fittingly called ion thrusting ‘acceleration with patience.’ In a 2006 report on the mission, Rayman summarized the advantages of this kind of propulsion:

All else being equal, for the same amount of propellant, a spacecraft equipped with ion propulsion can achieve 10 times the speed of a craft outfitted with normal propulsion, or a spacecraft with ion propulsion can carry far less propellant to accomplish the same job as a spacecraft using more standard propulsion. This translates into a capability for NASA to undertake extremely ambitious missions such as Dawn.

The rate at which xenon is flowed through the thruster is very low. At the highest throttle level, the system uses only about 3.25 milligrams/second, so 24 hours of continuous thrusting would expend only 10 ounces of xenon. Because the xenon is used so frugally, the corresponding thrust is very gentle. The main engine on some interplanetary spacecraft may provide about 10,000 times greater thrust but, of course, such systems are so fuel-hungry that their ultimate speed is more limited.

Dawn uses a solar array to power up the ion propulsion system. At maximum throttle, Rayman notes, the acceleration is equivalent to about 7 meters/second/day, so a full day of thrusting can change the spacecraft’s speed by something like 24 kilometers per hour. By mission’’s end, the spacecraft will have accumulated over five years of total thrust time, with an effective change in speed of 11 kilometers per second. Slow but steady gets you to your target, and Dawn’s mission planners have noted that using conventional chemical propulsion, the extra mass would have put the mission well beyond budget, yet another plus for frugal ion technologies.

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Nick January 20, 2012 at 18:41

A fascinating future possibility (and essential for any future manned or otherwise non-microscopic starship) is that these chemicals on Vesta or Ceres could themselves provide a propellant vastly cheaper than what could be launched out of earth’s deep gravity well. This however requires radically different (not necessarily difficult, just different) rocket design as well as a very different economic interpretation of the rocket equation than we are accustomed to sitting at the bottom of our gravity well.

Astronist January 20, 2012 at 20:25

So what is Dawn telling us about the surface composition? In particular, what are the proportions of rock and ice?

Stephen
Oxford, UK

Paul Gilster January 20, 2012 at 21:30

I don’t have anything on the relative proportions at this point, but am trying to get a copy of the analysis paper from the GRaND team — we should know more soon.

Tarmen January 21, 2012 at 11:07

Nick is right. Fuel depots can be created at various asteroid belts and low G moons. Further steps should be taken to prospect suitable mining and processing sites. Remotely operated mines can be established, initially solar powered and soon consuming a portion of their own fuel production. The fuel depots will pay for themselves be selling fuel to other robo-miners of metals and to small probes that need to be accelerated to Jupiter or out of Solar System. I envision that much heavy industry will eventually be relocated off of Earth. Also rivel militaries will want to defend off Earth infrastructure. There will be a lot of demand for fuel produced off-Earth .

Doug M. January 21, 2012 at 11:40

@Nick, Vesta is just inside the ice line — meaning, it probably doesn’t have a lot of ice on its surface. It may hold water in other forms, such as hydrated clays; we should know by the time Dawn is pulling out.

That said, if you’ve got the capability of landing hundreds of tons of infrastructure on Vesta — solar panels, bulldozers, a distillery to separate out your ice, a cracking plant to turn it into hydrogen and oxygen, a launch facility to send it somewhere else — well, at that point you pretty much must already have well-developed space industries somewhere else in the Solar System.

Alex Tolley January 21, 2012 at 13:14

@Nick Ceres could themselves provide a propellant vastly cheaper than what could be launched out of earth’s deep gravity well.

I really depends on where you want the propellant. If the idea is to put propellant depots in earth orbit (LEO) then the delta v cost from Ceres (lots of ice) is roughly 12 km/s. Launch to LEO is ~ 7 km/s. So launching dumb water boosters from earth would be cheaper in energy terms.

Where it starts to make sense is when you can use much more efficient, high Isp engines to deliver to water to LEO, e.g. electrothermal or even nuclear engines, solar or beam sails, etc. Then the energy for the orbital velocity change can be delivered with gentle pushing allowing simpler designs.

Joy January 21, 2012 at 17:41

Er, “Lucky Starr and the Pirates of the Asteroids” was a juvenile novel, folks!

Reading the comments on mining ice from Vesta (more likely Ceres) as a rocket propellant for industrial expansion makes me more sympathetic to the philosophers who aver that a total collapse of the current world system is necessary to enable an essential change in human consciousness.

Ice in the asteroid belt is a fossil fuel – deposited when the Sol system was young. Mined and used as propellant, it would be lost forever. With exponential growth, it would not last long. Just to continue the economic and demographic Ponzi scheme a little bit longer? No less a sage than (Lucky Starr author) Asimov wrote and spoke extensively on this issue. Asimov calculated that with a doubling time of 47 years, we would convert all the mass in the universe into human flesh within just 6700 years (superluminal expansion required).

We would like to imagine that we can use unsustainable means to achieve a sustainable end. Admiral Hyman G. Rickover was in the 1950s a proponent of using the short lived bounty of fossil carbon and hydrocarbon to transition to the longer lasting energy stores in (supernova remnant) fossil uranium and thorium (and eventually big bang fossil deuterium). Of course even deuterium cannot sustain exponential growth for long. Depending on one’s assumptions (growth rate, how much deuterium can be mined from giant planets with net energy gain) all of the accessible deuterium in the Sol system would be gone within a time frame shorter than the past history of human civilization.

So even Rickover’s vision was just a more ambitious plan for kicking the Ponzi scheme further down the road into what (in human terms) would seem a distant future. The only vaguely articulated hope embedded in the idea was that after thousands of years of prosperity, humanity would find a loophole in the laws of thermodynamics. Forty years ago Asimov imagined the form such a loophole would take in “The Gods Themselves”, in which humans could extract energy from a high energy parallel universe and use a low energy parallel universe as a heat sink. Delightfully improbable, as Isaac well knew. In the 21st century the Ponzi schemes forever fantasies are focused on the “Singularity”, after which godlike artilects will kindly defeat thermodynamics for us in ways we cannot imagine. Logically and psychologically, this idea is no different from the John Frum cargo cult of Vanuatu.

Large brained cetaceans have likely had a self aware consciousness of a sort for circa 10 megayears. We clever bipedal terrestrial tool users have had a great agricultural/industrial run for the last 10 kiloyears, (just 0.1% of the highly encephalized cetacean time) but are already bumping up against the limits of Ponzi growth. We cannot go any further with the same sort of thinking.

A stable human population using wooden ships with hemp sails for transport could have dwelt on a clean planet Earth for tens of millions of years. If we wish to become an interstellar species, we must think in similar timescales and in similar terms. Larry Niven’s wonderful imagination gave us the concept of the starseeds, great organisms at home in deep space who would travel the galaxy (very slowly) on their organic solar sails (wings). If our descendants have a future abroad in the galaxy, it will be more like that than our ape brained, fossil fueled Ponzi schemes.

Nick January 22, 2012 at 2:02

@Doug M., there are other possible propellants besides water from ice, that is just the most obvious and probably easiest/cheapest one. An interesting possibility proposed by another reader in an earlier thread is ion engines using ionized dust particles. And of course the heavy neutral elements (e.g. xenon) currently used for ion engines, although these may be too dispersed on Vesta and Ceres for efficient production.

@Alex, with cheap propellant peak power output is a far greater cost driver than total energy. Thus Ceres to LEO, low thrust all the way, will be much cheaper than earth surface to LEO. Cheap propellant used outside deep gravity wells and atmospheres really does radically change the economic consequences of the rocket equation. (Low thrust and lack of atmosphere also allows us to lose the heavy tanks, another huge rocket equation win). Besides which, to build a starship we don’t need to ship the propellant to LEO first. Just launch the (orders of magnitude less massive and bulky) engines and payload from earth to Ceres and join them to the propellant there.

@Joy, the interesting thing about a starship is that it both has to radically violate your sustainability principles (by dispersing massive amounts of propellant) and radically follow them (the starship economy will have to be completely sustainable for thousands of years if it is to reach its destination — there are no stops for refueling or trading with the natives on the way).

We on the other hand have barely scratched the surface of our own planet and have a solar system full of matter above, so it will be quite a few centuries before the “essential change in human consciousness” becomes anything like an urgent matter. Until then sustainability will just be something that is sometimes a good idea but is mostly just an interesting intellectual exercise for those with the luxury of wealth (made possible by our squandering ways) to ponder such matters. (Of course, a similar lack of urgency pertains to starships…)

Tarmen January 22, 2012 at 12:01

@Joy
Your dream for sustainable forever human existance sounds bleak and oppresive to me. Anyway you would have to over rule the natural masculine effort to explore the next valley and climb the next hill. We are primates. We can be proud of our past and be cautiously optomisic for then next ten thousand years of discovery and civilizations growth and knowlege. First to the solar system and then to other star systems. Don’t come with us if you don’t want to.

John Q January 22, 2012 at 12:21

> Mined and used as propellant, it would be lost forever. With exponential growth, it would not last long.

That’s not how economics works. It is not a zero-sum game. A market is positive sum/knowledge generating game. Thanks to the pricing mechanism, which follows directly from the nature of markets, we never truly run out of anything. As something becomes more scarce, for whatever reason, the price goes up, the item is conserved, entrepreneurs enter the field coming up with new approaches, companies invest in research to find alternatives, money is loaned to create new firms, the capitalist engine of creation continues. It’s why we don’t worry about the coming whale-oil shortage every time we power up our iPad. Moreover, the laws of economies are truly universal and apply as much to the solar system as they do on earth. Can it be screwed up? Just add an outside source of disruption and interference (at the turn of the last century, government was but 8% of the national economy. Now it is 40%. That is not sustainable growth) and get back to me. But even semi-socialist, barely creeping along economies can get the job done eventually, presuming the pricing mechanism is allowed to function.

>Asimov calculated that with a doubling time of 47 years, we would convert all the mass in the universe into human flesh within just 6700 years

That’s not how demographics works. People do not mindlessly reproduce like replicating robots. As civilizations advance and hit modernity, birth rates plummet. People simply have more options to exercise, less religious imperatives to reproduce, careers, and so forth. But there is a cost. Right now birth rates are declining world-wide too rapidly. Some countries, societies, more than others, but it you look at contemporary examples such as Greece, Russia, Spain, their birth replacement rates are nearing or at 1.3, which historically a very bad number. Societies achieving that rate do not recover. They go right off the demographic cliff. It could be forcibly argued — I won’t make the argument — that the human race is currently heading for extinction. The future belongs to those who show up for it, as one writer has stated. And if no one shows up, there is no future. People are the ultimate resource, given a stable environment of laws, e.g. property rights, that respect individuals and their freedom to produce and create, it’s astonishing what they can do. But the choice is theirs.

>after which godlike artilects will kindly defeat thermodynamics for us in ways we cannot imagine

No one anywhere ever seriously articulated such nonsense. The Singularity can be defined as the convergence of advance molecular assembling technologies and advanced artificial intelligence. If you don’t believe in either, fine. No Singularity, everybody is happy. Now “Mysterians (see Wikipedia),” for example, believe the human brain represents the absolute limit of what is possible in intelligence. Very well, but that belief will at some point become a testable hypothesis. As for molecular-level control, it is already happening. See the nearest living organism. So should scientific knowledge continue to advance, perhaps a big if, the Singularity will happen and it will be interesting.

>A stable human population using wooden ships with hemp sails for transport could have dwelt on a clean planet Earth for tens of millions of years.

Is it really necessary to respond to that? I could not imagine an existence more boring, futile, vacuous, and pointless. My bet is that the universe would wipe them out in short order and good riddance. Let me put it this way: the sun will die someday no matter what we do, but to plan our lives today on the basis of what will happen billions of years in the future makes no sense whatever. Godel’s theorem with its vistas of unlimited knowledge is a far better guide to the future than platitudes about the second law of thermodynamics (didn’t we already go through that in the late 70s, Jeremy Rifkin, etc.? Why the rerun?)
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One final note on entrepreneurs and the economy. The recent book “Steve Jobs,” though I have misgivings about its frequently gossipy aspects, does a good job in showing how a real entrepreneur, flesh and blood and faults galore, gets the job done. If there is to be a solar economy, Jobs is the archetype of the kind of man who will get it done.

Alex Tolley January 22, 2012 at 16:14

@Joy, worrying about continuous exponential growth is not realistic.

If economic growth is related to energy, as it has been in the past, converting all the solar energy impinging on the planet would allow us to grow the global economy 10,000 larger than it is today. That would be a very wealthy world assuming population levels off as expected.

Going further, if humans could use all the energy from the sun, that would mean an economy 100 million times larger again. The population would have to grow immensely to be no better off than today.

If the economy doubled every 47 years (as per your Asimov figure for population growth), humans would have a solar empire well with 2000 years.
So in a period separating us as far as we are from the collapse of the Western Roman empire, humans could have an unimaginably wealthier economy compared to us, which, without growth would be sustainable until the sun died, or we went extinct. Hard to imagine that with such resources we wouldn’t be finding ways to seed life and ourselves throughout the universe.

Sure, we cannot exponentially grow the economy forever, but do we need to? A sustainable economy a million times larger per capita than today, could be wonderful. Certainly better than manning a wood and cloth sailing ship across the oceans forever pursuing some procrustean idea of “sustainability”.

Daniel January 22, 2012 at 17:36

By the way there is a very interesting article that I’d like to share here:

“Alien Life May Depend on Planetary Tilt”

http://www.space.com/14295-alien-life-exoplanets-tilt-red-dwarf-stars.html

Rob Henry January 22, 2012 at 18:27

Joy brings up some very interesting points, though I’m not sure why she goes to such extremes (perhaps an attempt to emphasise their significance through humorous excess?).

1) If we ever use up a vital resource, it would always lead to disaster unless we are sufficiently innovative, and our economy is sufficiently flexible to readjust. In a free, wealthy, and capitalist environment its hard to see how these disaster avoidance requirements would not be met, but being the paranoid sort, I have no qualms about a call for more research into such matters, just to be on the safe side.

2) If reproductive choices are left freely to parents, they should, in the long term, settle to a point that leads to unsustainable growth or decay. Unless there are unheralded natural homeostatic feedbacks in a totally free choice, this indicates that in the very long term governments are likely to be involved in this currently very private decision.

3) If energy use continues to parallel economic growth in an exponentially expanding economy, there must be problems within a handful of centuries from now. I realise that we are already seeing divergence from this simple model, such as the decoupling of growth from energy usage in the very wealthiest nations were the tertiary sector burgeons, but it is at least possible that this problem may reappear in the far future.

The only thing I would really take issue with against Joy is on a philosophical point not an analytic one. If you climb higher, it is true that you have potential to fall further, but it is just subjective as to whether that fall constitutes a greater tragedy. Let us go out and grab that propellant, and fulfil our grand destiny, if for no other reason fewer can stand the claustrophobia of Joy’s envisioned future, than can overcome the vertigo of the high growth path. Thus that path holds the greater good.

Phil January 22, 2012 at 22:45

I for one am very pleased that people like Joy post here. I might not agree with everything, or even most things, in that post, but it is a corrective to the ‘manifest destiny’ mentality that I certainly get stuck in sometimes.

This civilisation, and its ability to turn its spaceflight dreams into things like the Vesta picture above, is a fragile thing that could be snuffed out in an instant. An asteroid could do it, but equally we could do it to ourselves, and a look around might suggest that we are rather blindly stumbling into disaster, all the while trusting in the so-called ‘limitless ingenuity of humanity’.
Is it really ‘different this time’? Ask the ancient Romans. Probably not.
Humility.

P

Eniac January 22, 2012 at 22:59

@Joy

A stable human population using wooden ships with hemp sails for transport could have dwelt on a clean planet Earth for tens of millions of years.

A stable human population using solar cells and hydrogen/oxygen rockets for transportation could dwell on a clean planet Earth for billions of years.

Both of these are futile academic exercises. As others have said much better than I can, there is no such thing as a stable human population. If there was, it would not be an interesting place to live. And wooden ships will never take us to the stars.

jkittle January 22, 2012 at 23:36

The moon is the best source of ice ( and other interesting volatiles for medium to high earth orbit. Ceres is the resource center for transition to the outer solar system .
all this discussion about resource limitations is just Malthus all over again. these discussions never take into account the real ( and unknowable) future needs of a technological society I for one can live with the limits of “fossil deuterium” … if we can fuse deuterium then hydrogen ( proton) fusion is not that far away. there is more water on Ceres than in all of earths oceans, again, I can live with that!

Eniac January 23, 2012 at 9:21

Here is an interesting reference pertaining to the sustainability of a culture based on wood as a resource: http://www.jstor.org/pss/4004530. Looks like Joy’s approach has already been tried, unsuccessfully.

ljk January 23, 2012 at 10:37

John Q said on January 22, 2012 at 12:21:

“Now “Mysterians (see Wikipedia),” for example, believe the human brain represents the absolute limit of what is possible in intelligence.”

Ah, the same kind of thinking several centuries ago (and which still persists in numerous cases across the globe) that humanity was the literal and spiritual Center of the Universe. Then modern astronomy happened and we began to realize, at least intellectually, that we are but a microscopic speck in space and time.

By the way, humpack and sperm whales have brains four times more complex than humans. Maybe it is just for helping them find dinner better and swim through dark and murky waters with more efficiency, but humans do not have the biggest or best brains around, they just think they do because their experience with other minds is so painfully limited.

Neanderthals had bigger brains than us, too, but since they didn’t build New York or televisions or missiles, we Homo sapiens naturally dismiss them outright.

It amazes, saddens, and appalls me that we have so much vastness and resources surrounding our planet, but still most people cannot get out of the crib/nest and think that is all there is. I am reminded of the old adage: “The meek shall inherit the Earth. The rest of us are going to the stars.”

ljk January 23, 2012 at 11:21

Current human society is designed to maintain the status quo on a single planet. This is why if governments and religions (and they are often one and the same) aren’t suppressing those who don’t follow their rules, then society itself tries to enforce the status quo through peer pressure and worse.

This is why science and space are still marginalized and ridiculed by the rest of society, due to a combination of ignorance and pressure from those in power who do not want their subjects to suddenly become smarter and more aware, especially when it comes to the fact that there is much, much more to existence than Earth.

I grew up during a time when it was naturally assumed that humanity would literally conquer the stars, starting with our Sol system. Caution and awareness have their place, but when they become the focus of everything and put a chokehold on our society and species, we might as well just call it quits. This along with human beings being too self-focused and overly cautious (even cowardly) is why we have no colonies on other worlds and sending even a rudimentary mission to Alpha Centauri is still just a dream.

Using Ceres as a resource for further space utilization and exploration is a sensible and farseeing idea; no one should have to feel odd or apologetic about coming up with something our descendants will (hopefully) take as part of their society, just as we hardly even blink about the idea of flying in an aircraft across the oceans these days.

Humility is for monks. Even a casual student of history will know that our progress has come from those who defied conventional thinking and dared to risk their lives for something better and more exciting (to boldly go where no man/one has gone before). Would you rather our species strived to be something better and died trying or have us all turn into couch potatoes wondering who is going to win the next American Idol?

Two quotes from Theodore Roosevelt, the 26th President of the United States of America:

“We are face to face with our destiny and we must meet it with a high and resolute courage. For us is the life of action, of strenuous performance of duty; let us live in the harness, striving mightily; let us rather run the risk of wearing out than rusting out.”

“Far better is it to dare mighty things, to win glorious triumphs, even though checked by failure…than to rank with those poor spirits who neither enjoy much nor suffer much, because they live in a gray twilight that knows not victory nor defeat.”

Eniac January 23, 2012 at 16:37

@LJK: Well said!

andy January 23, 2012 at 18:17

That’s not how economics works. It is not a zero-sum game. A market is positive sum/knowledge generating game. Thanks to the pricing mechanism, which follows directly from the nature of markets, we never truly run out of anything. As something becomes more scarce, for whatever reason, the price goes up, the item is conserved

Tell that to the rhinoceros.

Tarmen January 23, 2012 at 19:05

The brave and bold ae always a minority by definition.
We are going to leave a lot of people behind.
(But someday, ‘we’ will come back)

mike3 January 24, 2012 at 4:08

@ljk: No, “humility” is for everyone. It doesn’t mean be a do-nothing, it means don’t be a smug jerk. It means being able to admit when you’re wrong and being able to work together with others. Both of which are things that are vital for accomplishing great things.

ljk January 24, 2012 at 11:56

mike3 said on January 24, 2012 at 4:08:

“@ljk: No, “humility” is for everyone. It doesn’t mean be a do-nothing, it means don’t be a smug jerk. It means being able to admit when you’re wrong and being able to work together with others. Both of which are things that are vital for accomplishing great things.”

I have noticed that the ones who are often considered to be “smug jerks” are also the ones who have the money, the power, the women/men, etc. Meanwhile the “nice” guys get to go sit back in the economy section of the plane being crushed like sardines and having to eat packets of peanuts while pretending that their humbled (read humiliated ) state is somehow a virtue.

Oh, I think every human being should show humility, too, before a Universe that is so ancient, vast, and powerful that we third-order chimpanzees are hardly a vague thought in comparison, but the fact that we are still little more than smart animals with some fancy toys and a few glimmers of profound thought means most people do not truly appreciate or care about the true state of reality. Earth might as well be the whole Cosmos for most humans.

My sincere hope is that some day enough of our descendants grow up to be truly worthy of more than just being a higher mammal, or at the least make beings who deserve the honor of becoming Citizens of the Galaxy. Why do you think we haven’t found any ETI yet and why our astronauts keep circling Earth rather than spreading out into space.

Eniac January 25, 2012 at 1:06

Why do you think we haven’t found any ETI yet and why our astronauts keep circling Earth rather than spreading out into space.

These are easy: 1) because there aren’t any, and 2) because they’d be marooned and would suffocate ;-)

ljk January 25, 2012 at 12:05

Eniac said on January 25, 2012 at 1:06:

Why do you think we haven’t found any ETI yet and why our astronauts keep circling Earth rather than spreading out into space.

“These are easy: 1) because there aren’t any, and 2) because they’d be marooned and would suffocate ;-)”

If you have ever seen the television series Laugh-In from back in the day, they had a segment where a little old man would invariably sit next to a little old lady on a bench. He would then proceed to say something really annoying to the lady, who would invariably respond by hitting him repeatedly with her purse until he fell on the ground.

Yeah. :^)

Eniac January 25, 2012 at 23:47

Think of me writhing on the ground and screaming in pain… :-))

ljk January 26, 2012 at 10:39

Eniac, I don’t want that. Simply agreeing with me unconditionally will suffice. :^) Just kidding – that would be both unproductive and boring after a while.

Eniac January 26, 2012 at 13:25

You are so right.

Michel Lamontagne February 6, 2012 at 21:18

I’ve always understood that Asimov used the population example to show that no exponential phenomenon, such as the current economical growth mode, was sustainable. Any type of exponential model applied to population eventually results in a pink ball of flesh expanding at the speed of light, quite often within a few thousand years, all depending on the variables. Any phenomenon will eventually outstrip its resources with an exponential equation. For example xt = x0(1 + r)^t, the basic economical/population growth equation, with 5% growth per generation and starting at 1 000 000 people, supplies a Googol of people in 5000 generations, or 100 000 years, and crashes Excel at a googol, googol, googol people (x10^300 ) in 300 000 years. A blink of an eye, in cosmic terms. Magic entrepreneur or no magic entrepreneur, market forces or no.
I favor the inverse logarithmic equation (ln), as a more realistic model, which can give eternal growth, but has a fixed maximum. It just gets verrrrry slow towards the end! The mass fuel ratio of a spaceship is a classic example: vF = v0 ln(M0 /MP). Another interesting aspect of this type of equation is that it can be reconfigured with a new maximum as new resources are discovered.

I think the (ln) model fits the present population growth quite nicely, with a steep start, which looks exponential, followed by a much slower but continuous increase. It gives me hope for the future.

Regards,
Michel Lamontagne

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