Wednesday, June 16

Welcome To The “Big Empty”

I have written before about the likelihood of intelligent life in our Galaxy, and how the existence or non-existence of life in the Milky Way relates to human religious sensibilities. Now I want to take a more “global” perspective and approach that same question, not from the relatively parochial standpoint of intelligent life “merely” in our Galaxy, but from the standpoint of intelligent life in the entire Universe.  But the questions I pose here are essentially the same in all respects as the questions I posed in the original “Skeptic’s Collection” column.  Given some realistic-seeming, in fact, most likely optimistic, assumptions about the probability that intelligent life will evolve on any given planet orbiting any given star, how widely separated – across the entire Universe – must intelligent races be, given our current understanding of physical law, and therefore how likely or unlikely is such life (a) to exist and (b) to be capable of contacting us? Fair warning:  the answers are not encouraging, especially for Star Trek aficionados … like your faithful Resident Skeptic.

Record of the "Wow!" signal

I should also say that I am skating over some issues that are still far from being settled by the current state of the art, and adopting a consciously naïve stance for want of better assumptions and also to simplify the math.  For example, I assume that the topology of spacetime is basically that of a 4-dimensional hypersphere, whose metric – the formula for measuring distance – is the familiar Pythagorean formula √ ( x12 + x22 +  x32 + x42 ), assuming a 4-dimensional spacetime. This is an important question, but it would seem that the exact topology of spacetime becomes less and less important as the effects of “dark energy” accumulate with time:  eventually everything just … evaporates.  I also freely admit that the proportion of planets with intelligent life – again, Universe-wide – are sheer seat-of-the-pants SWAGs (Scientific Wild-Ass Guesses).  But, if anything, the estimates I use err on the side of optimism:  in the entire history of SETI, there has been one and only one candidate event – the so-called “Wow Signal” – that perhaps ... maybe ... might ... just arguably be a manifestation of extraterrestrial intelligence.  (Problem is, the "Wow Signal" was detected once but never repeated.) Finally, I assume that both galaxies and their constituent stars are more or less evenly distributed across the entire expanse of the Universe -- which is demonstrably, empirically false, but again, let's keep it simple. In any case, almost certainly, the Universe is not a Great Cosmic Tokyo Ginza District or Mall of America teeming with sentience, as depicted on all the Star Trek and Star Wars franchises – which leads us back to Enrico Fermi’s nagging question “Where Is Everybody?” So the particular numbers I derive should only be taken seriously as, at best, order-of-magnitude estimates.

Moving right along – as one piece of Ex-Lax said to the other … we envision the Universe – at least the part of the Universe visible to us, and whose light has had time to catch up with us since the Big Bang – as a sphere 13.8 x 109 light-years (hereafter, ly) in diameter, i.e., 6.9 x 109 ly in radius. (Why 13.8 x 109? Because the best estimate we have for the age of the Universe, i.e., elapsed time since the Big Bang, is 13.8 billion years. So the light of the Big Bang has traveled 13.8 billion ly since then:  13.8 billion ly, i.e., 13.8 x 109 ly.) Remember the formula for the volume of a sphere: 4/3 x π x r3, where r is the radius of the sphere.  Plugging in the above numbers to this formula, we derive: 4/3 x π x (6.9 x 109)3 , which reduces, in scientific notation, to 1.38 x 1030 ly3.  I.e., the total volume, in cubic ly, of the Universe is 1.38 x 1030 ly3. Occupying that immense volume are galaxies, which are themselves composed of stars. How many individual stars altogether? It varies, of course, but the best estimate we have of the number of galaxies is that there are around one hundred billion galaxies, and that within each galaxy there are, typically, around 100 billion stars. So in total there are 1011 x 1011 individual stars in the visible Universe, i.e., 1022 stars. Now, assuming that stars are distributed evenly throughout the available 1.38 x 1030 ly3 of space – again, which we know is not the case … keep reading and don’t nit-pick – that means that each individual star occupies a volume of 1.38 x 1030 / 1022 = 1.38 x 108 ly3 per individual star, i.e., 1.38 hundred million cubic light-years per star, approximately the volume of Bill and Melinda Gates’s garage.

Hubble Telescope "Pillars Of Creation" Image
Hubble Telescope "Deep Field" Image

Now … next question: what would the radius of a sphere be which encloses a volume of 1.38 x 108 ly3? Turns out that is a softball question! Just put that exponential-notation expression on the left side of the above sphere-volume formula, then solve for r by taking the cube root! I.e.,

1.38 x 108 = 4/3 π r3

So r3 = 3  /4π x 1.38 x 108   ly3   = So taking cube roots on both sides to end up with a linear distance, we end up with each star being at the center of a sphere of 320.3 light-years radius. We can call this value the “characteristic radius”, and reference it with the symbol rC. Its neighbor-star is likewise, on average, centered on a sphere with an rC of 320.3 light-years. So the average separation between stars is 320.3 x 2 = 640.6 light-years. Remember:  this is the average across  the entire Universe, not just our Milky Way Galaxy, and for all stars in the Universe, irrespective of chemical composition, mass, age, etc.

But what about stars that have a retinue of planets, with at least one planet hosting intelligent life? No one knows. But let’s suppose, strictly as a seat-of-the-pants SWAG, that one-billionth of one percent of the stars, across the entire Universe, have at least one planet with intelligent life.  Now, given the foregoing, it is possible to ask the following highly relevant question:  What is the average value of rC – the characteristic radius – of all stars hosting intelligent life?

Mathematical model / simulation of the distribution of matter in the Universe

It turns out that it is possible, even easy, to derive a formula for the rC value, as a function of the number of planets with intelligent life (or any other trait).  Trust me:  that formula, a mere generalization of the above steps, is

rC = [ 3  / 4π x (1.38 x 1033 / S) ] 1/3 , where S is the estimate of the number of stars in the entire Universe with the given characteristic (e.g., mass, chemistry, host of a planet with intelligent life, etc.)

Now, if there are, per the earlier estimate, 1022 stars in the Universe, and if one-billionth of one percent have planets with intelligent life, then the number of planets with intelligent life – again, across the Universe as a whole – is 10-9 x 10-2 x 1022 = 1011 stars. Per the above characteristic-radius formula, this yields a characteristic radius for stars hosting intelligent life of

rC = [ 3  / 4π x (1.38 x 1033 / 1011 ) ] 1/3 = ( 3.3 x 1021 ) 1/3 = 1.5 x 107 ly = 15 x 106 ly

In other words, and on the scale of the entire Universe, if one-billionth of one percent of stars host planets with intelligent life, those stars would be separated by an average of 30 million ( 2 x 15 x 106 ) ly. This distance is about 300 times the diameter of the Milky Way Galaxy.

When one pauses to consider the number of sheerly random, fortuitous things that had to go right to evolve intelligence, the more one thinks about it, the more optimistic one-billionth of one percent seems.  So let’s say only one-trillionth of one percent is nearer the proverbial ballpark.  In that case,

rC = [ 3  / 4π x (1.38 x 1033 / 108 ) ] 1/3 = ( 3.3 x 1025 ) 1/3 = ( 33 )1/3 x 108 ly = 3.2 x 108 ly, which gives an average separation of 2 x 3.2 x 108 ly, i.e., 6.4 hundred million light-years.  "Phoning home" across this distance would probably eat even ET's lunch.

We can play these numbers games until Trump says something really nice about Hillary. But regardless of how the numbers fall out for any given iteration, the point will remain:  it’s really lonely ‘way out there in the Big Empty.

James R. Cowles

Image credits

Hubble "deep field" image ... NASA ... Public domain
Distribution of matter in Universe ... Andrew Pontzen and Fabio Governato ... Creative Commons Attribution 3.0 Licence
"Wow" signal ... Big Ear Radio Observatory and North American AstroPhysical Observatory (NAAPO) ... Public domain
"Pillars of Creation" ... NASA ... Public domain





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