April 2014 - mathfun@mailman.xmission.com

[math-fun] Gosper factorial stuff
by Bill Gosper 05 Apr '14

05 Apr '14

WDS>On 2014-04-03 08:18, Warren D Smith wrote: > Well. > First of all, I think whatever Gosper is doing about factorials looks > pretty brilliant, > as fairly usual, but also as usual I do not know how he did it. > Foo, I'm just spazzing with LatticeReduce, collecting empirical relations into a canonicalizing list (called ductions, current length ~57) : In[286]:= ductions[[{1, 2, 3, 4, 5, -2, -1}]] Out[286]= {Gamma[r_Rational] -> (-1 + r)!, r_Rational! :> (\[Pi] r)/((-r)! Sin[\[Pi] r]) /; r < 0, r_Rational! :> Mod[r, 1]! \!\( \*UnderoverscriptBox[\(\[Product]\), \(k\), \(Floor[ r]\)]\((Mod[r, 1] + k)\)\), r_Rational! :> (\[Pi] r (1 - r))/((1 - r)! Sin[\[Pi] r]) /;r > 1/2, (13/30)! -> (26 2^(1/20) 3^(7/20) Sqrt[\[Pi]] (1/3)!)/(5 5^( 5/6) (5 + Sqrt[5])^(5/12) Csc[\[Pi]/30]^(1/60) Csc[\[Pi]/15]^( 1/4) Csc[(7 \[Pi])/30]^( 7/30) (2/5)! ((1 + Sqrt[5]) Sin[(2 \[Pi])/15])^(17/60)), . . . (3/8)! -> (3 Sqrt[\[Pi]] (1/8)! Sin[\[Pi]/8])/(2 2^(1/4) (1/4)!), (1/6)! -> (3 Sqrt[3/\[Pi]] ((1/3)!)^2)/(2 2^(1/3))} You then apply it to your simplificand in a single pass. E.g., In[290]:= Product[((2*k - 1)/24)!, {k, 12}] Out[290]= (1/24)! (1/8)! (5/24)! (7/24)! (3/8)! (11/24)! (13/24)! (5/ 8)! (17/24)! (19/24)! (7/8)! (23/24)! In[291]:= Fold[ReplaceAll, %, ductions] Out[291]= (1301375075 \[Pi]^6 Csc[\[Pi]/24] Csc[\[Pi]/8] Csc[( 5 \[Pi])/24] Sec[\[Pi]/24] Sec[\[Pi]/8] Sec[(5 \[Pi])/24])/150289495621632 In[292]:= %% == FullSimplify[%] Out[292]= (1/24)! (1/8)! (5/24)! (7/24)! (3/8)! (11/24)! (13/24)! (5/ 8)! (17/24)! (19/24)! (7/8)! (23/24)! == (1301375075 \[Pi]^6)/( 2348273369088 Sqrt[2]) In[293]:= N[%] Out[293]= True > Second, basically, there was a line of research started by Chowla & > Selberg 1967. > Here is a recent paper that cites that (I have no idea what the hell > this recent paper is doing, but anyway...) > http://www.math.wisc.edu/~thyang/Colmez.pdf > > C&S found some relations between GAMMA(a/b) values and > elliptic/modular functions. > Putting those together with the usual reflection & n-tupling formulas, > then solving the resulting systems of equations, we find out stuff rather > like the stuff Gosper is seeking. A number of later authors tried to > do exactly > that (in 1967 C&S did not have access to Macsyma, etc, handicapping > them vs later authors), but you'd have to search the literature to > find those (use cite-search on the C&S paper). > > Now about the transcendence and/or irrationality and/or linear > independence (over the algebraic or irrational numbers) results proven > by Nesterenko et al, those could sometimes be used to show that there > are no further magic relations among the GAMMA(a/b) values... > EXCEPT that you'll only be able to show that lack of further relations WITHIN > certain classes of allowed kinds of relations. It isn't going to > tell you a thing about > there being no relations of the form GAMMA(1/5) = exp(sin(GAMMA(3/5)) -- > (which is false), but my point is nonlinear nonalgebraic relations > cannot be ruled out > in this way. > > >> I believe our previous interest in (n/24)! was their rapid computability >> via Dedekind eta. Even assuming it is always possible to convert to etas, >> do their modular properties tell all? And, assuming we can canonicalize >> the eta expression, can we always convert back to Gammas? > > --These questions are just new questions... > If you were to take the attitude you are not even going to permit > elliptic/modular functions > to appear, then it might be the reflection and ntupling is all there > is and Chowla & Selberg might be irrelevant. A tool like "ToElliptic" would be valuable, and might synergize with this duction hack, but the user should keep control. If my memory is right about eta(e^-(pi sqrt(n/d))) always coming out in Gammas, there remains a chance of finding an "inaccessible" identity. --rwg

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[math-fun] bit reversal
by Warren D Smith 05 Apr '14

05 Apr '14

>> Does this, in any sense, beat Knuth's? --rwg >> > > Performance wise it should be identical (unless "rotate by > one bit" is faster than "rotate by k bits" where k is an > immediate operand); so WDS's benchmark looks dubious to me. --hah! Maybe it should be identical with perfect machine language coding (I'll have to take your word for that, and frankly I doubt you are smart enough -- processor pipeline & etc issues are so complicated today nobody can easily tell)... but definitely not on my machine with my compiler. You are correct your assembler bswap version was the fastest, though (ArndtB in my benchmark). Here's another little hacking question for you: I give you a 64-bit word x. Your task is to compute the sum of its 8 component bytes. How fast can you do it? What if I only demand the sum be correct modulo 256?

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[math-fun] Dating the collision that formed the Moon
by Bill Gosper 04 Apr '14

04 Apr '14

WFL> Call for the moderator --- since when has math-fun been a dating site? Besides which, this particular date sounds rather too hot for comfort. Such speculations leave me with an overwhelming sense of how tenuous is humanity's grip on existence, in the context of geological time. WFL And how tenuous is Fred's grip on sobriety, in the context of beer time. Might this event, viewed from Mars say, have outshone the Sun? For how long? --rwg On 4/3/14, Henry Baker <hbaker1(a)pipeline.com> wrote: > FYI -- "The resulting collision would have left the Earth a magma ocean, > blown away its atmosphere..." > > I've been claiming for a long time that the Earth's atmosphere would have > been a lot more like Venus without an event like this one that removed most > of it. For the same reason, we won't be able to turn Earth back into Venus, > no matter how much carbon we burn. > > http://arstechnica.com/science/2014/04/dating-the-collision-that-formed-the… > > Dating the collision that formed the Moon using late-arriving debris > > The magma ocean that resulted acted like a reset button on the age of the > Earth. > > by John Timmer - Apr 2, 2014 5:36 pm UTC > > When did the Earth actually form? There are a number of ways to approach > that question. We can use radioactive dating to look at material that has > fallen to Earth after remaining largely undisturbed since the formation of > our Solar System, or we can obtain a date for the oldest materials we've > found on Earth. But those methods simply provide an upper and lower limit; > the Earth formed some time after the smaller material in the Solar System, > while the earliest materials on Earth would have been produced some time > after its formation. > > Now, researchers have provided a new date for the formation of the Earth, > based on the last time the planet was entirely molten---an event that was > triggered by a collision with a body that ultimately created the Moon. The > data is calculated using what we know about the early Solar System combined > with the debris that fell to Earth after the big collision. > > The early Solar System was a violent place, as small particles aggregated > into bodies that then grew larger by collisions. These collisions > eventually produced planetesimals the size of large asteroids, which merged > to form the current collection of planets. So there was no clear start to > what would ultimately become the Earth, but there was a clear end to the > primary process of its formation. > > That end came when the proto-Earth was smacked by a Mars-sized body. The > resulting collision would have left the Earth a magma ocean, blown away its > atmosphere and any volatile liquids on its surface, and put enough debris in > orbit to form the Moon. In effect, it acted like a reset button for the > timing of the Earth's formation, remixing all its components so that the raw > material for radioactive dating---the stable maintenance of isotope > differences caused by radioactive decay---was eliminated. Everything started > afresh. > > Figuring out the timing of that collision is important if we want to > understand the conditions on the early Earth and in the early Solar System > in general. The new study uses a clever way of providing an estimate, based > on the fact that the last major collision in the Earth's history didn't mean > that the Earth escaped further bombardment from space. > > The logic is pretty clever. In the wake of the Moon-forming collision, the > entire Earth was molten, which allowed the iron to sink to the core. A > number of heavy elements that have an affinity for iron sunk to the core > with it. This would have left the surface of the Earth completely stripped > of these metals. (These elements, which include gold and platinum, are > generically referred to as siderophiles.) But it's possible to mine this > material from the crust. > > The elements are there because the Earth's supply was partly refreshed by > the arrival of new material in collisions with other planetesimals and > smaller asteroids. (The most famous asteroid, the one that killed the > dinosaurs, was first identified through the presence of a layer of another > siderophile, iridium, which it carried to Earth.) If we total how much of > these elements are in the crust and compare that number to the composition > of asteroids in our Solar System, we can get an estimate of how much mass > must have struck the Earth after the Moon-forming collision. > > How do you go from that mass to a date? The authors recognized that over > time, these bodies were lost through collisions with Earth and the other > planets. Thus, early in the Solar System's history, there was an > ever-shrinking stock of planetesimals that could have delivered material to > Earth. To find out the timing of when the stock was depleted, the > researchers relied on models of Solar System formation. > > In fact, they relied on two different types of models: one that keeps the > giant outer planets in their current locations and a second class in which > Jupiter and Saturn engage in what's called a "Grand Tack," moving inward > early in the Solar System's history before receding back out to their > current locations. The Grand Tack simulations were more likely to produce a > set of rocky inner planets that look like our Solar System, but in both > cases, the planetesimals got depleted pretty rapidly. > > Their vanishing act sets limits on when the Moon-forming collision could > have happened and, thus, when the Earth could have formed. Too early, and > the Earth would have seen lots of additional collisions and had its crust > loaded with the elements that are now rare. Too late, and there wouldn't be > enough around to give us any significant quantities. Consequently, the > authors calculate that there's only a 0.1 percent chance that the > Moon-forming collision took place prior to 40 million years after material > started condensing in our Solar System. Instead, they place the likely date > at 95 million years, with a margin of error of 30 million years on either > side. > > The authors suggest that we might want to revisit some of the results that > were generated using isotope data---a few of these put the collision at 30 > million years, and understanding why they got the number wrong may provide > some details of the mechanics of the collision itself. > > They also think that the reloading of the Earth with rare metals tells us > something about the distribution of mass during the formation of the Solar > System, which can constrain our models of planet formation. Right now, > those models don't do a very good job of creating many of the tightly packed > systems that we're currently discovering, so any improvements to them would > be a positive step. > > Nature, 2014. DOI: 10.1038/nature13172 (About DOIs). > > John Timmer / John became Ars Technica's science editor in 2007 after > spending 15 years doing biology research at places like Berkeley and > Cornell. > >

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Re: [math-fun] Dating the collision that formed the Moon
by Henry Baker 04 Apr '14

04 Apr '14

The basis of my claim (which was posted on math-fun a long time ago -- probably before you joined) is that the Earth's atmosphere is very far out of equilibrium with the space surrounding it. I.e., if the Earth's atmosphere basically started in gravitational equilibrium, and lost gas molecules by exceeding escape velocity over 4.5B years, we'd still look a lot more like Venus today. In fact, Venus is the counter-example, since to a first approximation, it was formed at about the same time as Earth. The atmospheric pressure at the surface of Venus is 92x that of the Earth! (That's about the same pressure as 3000' of water on the Earth.) http://en.wikipedia.org/wiki/Atmosphere_of_Venus At 01:26 PM 4/3/2014, Warren D Smith wrote: >> I've been claiming for a long time that the Earth's atmosphere would have >> been a lot more like Venus without an event like this one that removed most >> of it. For the same reason, we won't be able to turn Earth back into Venus, >> no matter how much carbon we burn. > >--interesting and important claim, if true. But it might have worked the opposite way,e.g. >the collision released more gases causing the Earth atmosphere to >become larger, not smaller. I have no idea; I'm just asking if this >"claim" can be backed up somehow.

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Re: [math-fun] Fractional ordinals
by Bill Gosper 04 Apr '14

04 Apr '14

CG> "Sesqui" is 3/2 in Latin, and it was in common use; "Quasqui" for 5/4 apparently existed as well, but I've never seen it in period use. (I have seen it in modern use: "quasquicentennial" for a 125th anniversary.) <Charles Greathouse So last year Jim missed her trisesquith. And don't forget picentennial for the 314th birthday. --rwg Paul Cohen could have told you: Fractional ordinals are irrelevant.

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[math-fun] Fractional ordinals
by James Propp 04 Apr '14

04 Apr '14

My daughter Eliana will be five-and-a-half tomorrow. We are wondering: will tomorrow be her five-and-a-halfth birthday, or her fifth-and-a-half birthday? Jim Propp

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[math-fun] knight distance... how somebody expert with symbolic manipulators can prove it.
by Warren D Smith 04 Apr '14

04 Apr '14

Here is a sketch of how to prove the knight distance formula using MAPLE9. I am not going to prove it, I am merely going to explain how somebody better than I with symbolic manip programs, could prove it. Define f := proc(x,y) local X,Y,p,q,r; X := max(abs(x), abs(y)); Y := min(abs(x), abs(y)); if(X=1 and Y=0) then return(3); fi; if(X=2 and Y=2) then return(4); fi; p := X-Y; q := p-Y; r := (7+sign(2*q+1))/2; return( p - 2*floor(q/r) ); end proc; g := (x,y) -> f(x,y) - min( f(x-2,y-1), f(x-1,y-2), f(x+2,y-1), f(x+1,y-2), f(x+2,y+1), f(x+1,y+2), f(x-2,y+1), f(x-1,y+2) ); Now TO PROVE: g(x,y)=1 unless x=y=0. See? Trivial. It is "merely" an exercise in symbolic simplification. "All" you have to do it to tell MAPLE9 to simplify g(x,y), and then you are done. Absolutely trivial. The trouble is, the MAPLE9 simplifier is way way too stupid to be able to get anywhere near being able to do this "trivial" simplification. It delivers g(x,y) = max(| x |, | y |) - min(| x |, | y |) - 2 floor(1/4 max(| x |, | y |) - 1/2 min(| x |, | y |)) - min(max(| x + 2 |, | y + 1 |) - min(| x + 2 |, | y + 1 |) - 2 floor(1/4 max(| x + 2 |, | y + 1 |) - 1/2 min(| x + 2 |, | y + 1 |)), max(| x - 2 |, | y + 1 |) - min(| x - 2 |, | y + 1 |) - 2 floor(1/4 max(| x - 2 |, | y + 1 |) - 1/2 min(| x - 2 |, | y + 1 |)), max(| x + 1 |, | y + 2 |) - min(| x + 1 |, | y + 2 |) - 2 floor(1/4 max(| x + 1 |, | y + 2 |) - 1/2 min(| x + 1 |, | y + 2 |)), max(| y - 1 |, | x + 2 |) - min(| y - 1 |, | x + 2 |) - 2 floor(1/4 max(| y - 1 |, | x + 2 |) - 1/2 min(| y - 1 |, | x + 2 |)), max(| x - 1 |, | y - 2 |) - min(| x - 1 |, | y - 2 |) - 2 floor(1/4 max(| x - 1 |, | y - 2 |) - 1/2 min(| x - 1 |, | y - 2 |)), max(| x - 1 |, | y + 2 |) - min(| x - 1 |, | y + 2 |) - 2 floor(1/4 max(| x - 1 |, | y + 2 |) - 1/2 min(| x - 1 |, | y + 2 |)), max(| y - 2 |, | x + 1 |) - min(| y - 2 |, | x + 1 |) - 2 floor(1/4 max(| y - 2 |, | x + 1 |) - 1/2 min(| y - 2 |, | x + 1 |)), max(| x - 2 |, | y - 1 |) - min(| x - 2 |, | y - 1 |) - 2 floor(1/4 max(| x - 2 |, | y - 1 |) - 1/2 min(| x - 2 |, | y - 1 |))) ok? Now we can manually help MAPLE9 try to simplify it. We would begin by dividing into cases. The most important case is assume(x>=y+3); assume(y>=3); but there are many more cases that would need to be considered as well. But let us just work on this one case in the present message. We continue with: assume(x>=6); #MAPLE9 too dumb to deduce that by itself! g(x,y); subs(max(y-2,x+1) = x+1, %); subs(max(y,x)=x, %); subs(min(y,x)=y, %); subs(max(y+2,x+1)=x+1, %); subs(min(y+2,x+1)=y+2, %); subs(min(x-2,y+1)=y+1, %); subs(max(x-2,y+1)=x-2, %); subs(max(x-1,y+2)=x-1, %); subs(min(x-1,y+2)=y+2, %); subs(max(y-1,x+2)=x+2, %); subs(min(y-1,x+2)=y-1, %); subs(max(x-1,y-2)=x-1, %); subs(min(x-1,y-2)=y-2, %); subs(min(x-2,y-1)=y-1, %); subs(max(x-2,y-1)=x-2, %); subs(max(x+2,y+1)=x+1, %); subs(min(x+2,y+1)=y+1, %); subs(min(y-2,x+1)=y-2, %); subs(max(y-2,x+1)=x+1, %); subs(max(x-2,y+1)=x-2, %); notice how I have to do almost every goddamn thing by myself because MAPLE9 is too stupid -- and now this case has been simplified down to h := (x,y) -> x-y-2*floor(1/4*x-1/2*y)-min(x+1-y-2*floor(1/3*x+1/3-2/3*y), x-1-y-2*floor(1/4*x-1/2*y),1+x-y-2* floor(3/4+1/4*x-1/2*y),1+x-y-2*floor(1/4+1/4*x-1/2*y),-1+x-y-2*floor(1/3*x-2/3*y)); Notice how MAPLE9's simplifier is so stupid it does not even realize that 1/4*x = x/4 or that -1+x = x-1. It is mind-boggling to me, the incompetence of MAPLE9's designers. But I digress. But anyhow we've got rid of all the maxes and mins except one, and still have a lot of floor functions, namely this list floor(1/4*x-1/2*y), floor(1/3*x+1/3-2/3*y), floor(1/4*x-1/2*y), floor(3/4+1/4*x-1/2*y), floor(1/4+1/4*x-1/2*y), floor(1/3*x-2/3*y), We can now subdivide our case into subcases mod 12. Specifically, there are 144 subcases depending on (x,y) mod 12. We do: for xx from 0 to 11 do for yy from 0 to 11 do print( h(xx,yy) ); od; od; at which point it prints out a long string of 1s, thus proving that h(x,y)=1 mod 12 in our case. EXCEPT actually it does not -- some of the 1s it prints are 3s. Oops. That means somewhere in the manual simplifications above, I made a typo. That is the problem with humans doing the job. Anyhow, ignoring that, at this point we would realize that h(x,y) mod 12=1 if x>=y+3>=6. Furthermore, we would then apply the grey cells to realize via a humanly done proof (or maybe computer could somehow see it, if MAPLE9 got any brains) that |h(x,y)| is upper bounded by a small constant. In fact if we went to mod 24 not mod 12 or something large enough to overwhelm that constant , it'd then become obvious that h(x,y)=1 always. The proves g(x,y), proving our theorem in our one (the main) case. we would then do the other cases such as x=y>=8, or x=y+1>=8 in the same fashion. OK? So I hope I've made it clear (a) exactly how to prove this now (b) Lunnon should see what is inadequate about his "proof", (c) what is inadequate about MAPLE9 simplifier ... this whole proof would have been a triviality if it had a good simplifier. But designing a good simplifier, evidently is not a trivial matter. There are many simplification tasks which are Turing undecidable (diophantine equations). I just find it interesting that the knight distance problem, which on the surface seems so trivial, leads us into utter simplification hell trying to prove it. By the way, if you can do this, then you should also do the (A,B)knight as opposed to the usual (2,1)knight used in chess.

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[math-fun] reversing the order of bits in a word
by Bill Gosper 04 Apr '14

04 Apr '14

Does this, in any sense, beat Knuth's? --rwg * Joerg Arndt <arndt(a)jjj.de> [Apr 03. 2014 17:09]: ulong jj3_revbin(ulong x) { ulong z; z = (x ^ (x >> 2)) & 0x9249249249249249UL; x ^= z ^ (z << 2); // low 3 bits reversed z = (x ^ (x >> 6)) & 0x01c0e070381c0e07UL; x ^= z ^ (z << 6); // low 9 bits reversed z = (x ^ (x >> 18)) & 0x00001ff0000001ffUL; x ^= z ^ (z << 18); // low 27 bits reversed z = (x ^ (x >> 36)) & 0x0000000007ffffffUL; x ^= z ^ (z << 36); // low 63 bits reversed x = (x>>63) | (x<<1); // rotate left by 1 return x; } Can explain if anybody is interested later, got a visitor now. Did run automated testing, but would appreciate independent "OK"s. Yes, tricky. Best, jj Brrrr, version above reverses 63 bits and clears the highest, next reverses all 64 bits, highest bit in first mask dropped now: ulong jj3_revbin(ulong x) { ulong z; z = (x ^ (x >> 2)) & 0x1249249249249249UL; x ^= z ^ (z << 2); // ... rest as before ... }

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[math-fun] Knight distance metric
by Warren D Smith 03 Apr '14

03 Apr '14

Lunnon: It is a pleasure to report the defeat of the horrid spectre of topology, by dint of abandoning attempts to reason about attaching an extra step at the far end [x-2, y-1] in favour of considering (shorter) paths from neighbours [2, 1] etc. of the origin instead. Then straightforwardly the distance function d(x, y) satisfies the same recursion as the explicit expression f(x, y) , whence they are equal. A corrected and slightly expanded screed is posted at https://www.dropbox.com/s/nzmzjswtctju23f/knights_path.txt Reports of attempts to blow further holes in it would be welcome, even if unsuccessful. Particularly if unsuccessful! WDS: I don't agree that your lemma 2 is proven. (Same objection I gave before, I don't see that it was impacted in the slightest.) MInd you, I believe this general approach should work, just a lot more cases will be needed.

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[math-fun] Enceladus has subsurface ocean (The Guardian news story)
by Warren D Smith 03 Apr '14

03 Apr '14

Ocean discovered on Enceladus may be best place to look for alien life Evidence from Cassini spacecraft suggests a large body of liquid water beneath the surface of Saturn's moon Enceladus Ian Sample, science correspondent The Guardian, Thursday 3 April 2014 14.25 EDT Researchers have discovered a deep saltwater ocean on one of the many small moons that orbit Saturn, leading scientists to conclude it is the most likely place in the solar system for extraterrestrial life to be found. Gravitational field measurements taken by Nasa's Cassini space probe revealed that a 10km-deep ocean of water, larger than Lake Superior, lurks beneath the icy surface of Enceladus at the moon's south pole. David Stevenson, a planetary scientist at the California Institute of Technology in Pasadena, said the body of water was so large it "may extend halfway or more towards the equator in every direction. It might even extend all the way to the north." The presence of a saltwater ocean a billion kilometres from Earth more than satisfies Nasa's long-held mantra of "follow the water" to find signs of alien life, but water is not the only factor that makes Enceladus such a promising habitat. The water is in contact with the moon's rocky core, so elements useful for life, such as phosphorus, sulfur and potassium, will leach into the ocean. The first hint that Enceladus might harbour a subterranean ocean came in 2005 when Nasa's Cassini spacecraft photographed extraordinary jets of water vapour blasting out of fissures in the moon's south pole. The source of the water was a mystery, and an ocean of liquid water was only one possibility. The latest discovery, reported in the journal, Science, gives scientists the strongest indication yet that the source of water vapour coming from Enceladus is a large body of water underneath the surface of the icy moon. An international team led by Luciano Iess at the Sapienza University in Rome inferred the existence of the ocean after taking a series of exquisite measurements made during three fly-bys between April 2010 and May 2012, which brought the Cassini spacecraft within 100km of the surface of Enceladus. As Cassini sped past the Saturnian moon, researchers used Nasa's Deep Space Network of giant antennas to monitor signals reaching Earth from the spacecraft's onboard radio. They looked for subtle shifts in the frequency of the radiowaves, which revealed whether the spacecraft was speeding up or slowing down. The measuring technique exploits the Doppler effect, which explains why the siren of a police car has a higher pitch as it approaches, and a lower pitch as it heads away. Cassini, the scientists discovered, sped up and slowed down by a few millimetres per second as it flew past Enceladus. Some of the change in speed was down to variations in the gravitational field of the moon as a result of different densities of material under the surface. After taking account of other factors that could alter the spacecraft's speed, such as drag from the plumes of water vapour, and even the modest pressure produced by sunlight, the researchers created a map of the gravitational field of Enceladus. The shape of the gravitational field pointed to something more dense than ice - but less dense than rock - deep beneath the south pole of the moon. "Given the kinds of materials we know are used to make bodies like this, the natural thing to look for is water, because water is more dense than ice, and because it's a natural thing to have in that environment," said Stevenson. When scientists first speculated that Enceladus might contain an ocean of water, one of the greatest puzzles was how the ice could be heated to make liquid water. Stevenson and his colleagues believe that gravitational forces that act on Enceladus as it orbits Saturn essentially knead the planet, producing enough heat inside to melt the ice. The process is known as tidal heating. Enceladus is not the only moon in the solar system to have a subsurface ocean. Europa, a much larger moon that orbits Jupiter, has a more extensive, global ocean under the surface. But Enceladus has excited scientists because the vapour plumes from the south pole are known to contain organic molecules. These, along with basic elements, a source of heat and liquid water, make Enceladus a prime candidate in the search for alien organisms. "The question is what conditions do you need to form life and, of course, we don't know what temperature the ocean is today, nor do we know what it was back in the geological past. But it's conceivable that it was warm enough, with circulation of water coming from the silicate core as well, to allow life to form even if today that ocean is maintained by antifreeze and is slightly below the freezing point," said Jonathan Lunine, a member of the team at Cornell University in New York. The antifreeze in question is salt, which reduces the temperature at which water freezes. In 2008, Nikolai Brilliantov, a mathematician at Leicester University, worked out how water vapour jets might blast out of Enceladus if it harboured an underground ocean. Confirmation of the ocean made Enceladus the most likely place so far to find extraterrestrial life. "There are definitely regions on Enceladus where the conditions are such that life could exist. You have liquid water, you have chemicals and you have heat. And that is enough for life. To my mind, this is the best place we can expect to find life elsewhere in the solar system," Brilliantov told the Guardian. "On Mars there are some signs of water in the past, but now we have very, very serious indications that liquid water exists in Enceladus," Brilliantov said. "It's technically possible to send a probe that could drill a hole and test the water to see if we are alone in the universe or if something else lives here." Chris McKay, an astrobiologist at Nasa's Ames Research Centre in California, said: "There are now several lines of evidence - the geysers, the plume chemistry, and now gravity - that indicate a substantial body of liquid water. For astrobiology this is confirmation of what we expected and is good news. My one view is that Enceladus should be the priority." Writing in the journal Astrobiology this week, McKay makes the case for an Enceladus flyby mission to return samples of the water vapour plumes and look for biomolecular evidence of life. "With samples of the organic material from the plume, we could search in terrestrial laboratories for organic biomarkers that would be conclusive evidence for life," he writes. --end.

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