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.

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 √ ( x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2 }), 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 10^{9} light-years (hereafter, ly) in *diameter*, i.e., 6.9 x 10^{9} ly in *radius*. (Why 13.8 x 10^{9}? 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 10^{9} ly.) Remember the formula for the volume of a sphere: 4/3 x π x r^{3}, 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 10^{9})^{3} , which reduces, in scientific notation, to 1.38 x 10^{30 }ly^{3}. I.e., the total volume, in cubic ly, of the Universe is 1.38 x 10^{30} ly^{3}. 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 10^{11} x 10^{11} individual stars in the visible Universe, i.e., 10^{22} stars. Now, assuming that stars are distributed evenly throughout the available 1.38 x 10^{30} ly^{3} 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 10^{30} / 10^{22} = 1.38 x 10^{8} ly^{3} per individual star, i.e., 1.38 hundred million cubic light-years per star, approximately the volume of Bill and Melinda Gates’s **garage**.

Now … next question: what would the *radius* of a sphere be which encloses a volume of 1.38 x 10^{8} ly^{3}? 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 10^{8} = 4/3 π r^{3}

So r^{3} = 3 /4π x 1.38 x 10^{8 }ly^{3 } = 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 r_{C}. Its neighbor-star is likewise, on average, centered on a sphere with an r_{C} 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 r _{C} – the characteristic radius – of all stars hosting intelligent life?*

It turns out that it is possible, even easy, to derive a formula for the r_{C} 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

r_{C} = [ 3 / 4π x (1.38 x 10^{33} / 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, 10^{22} 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 10^{22} = 10^{11 }stars. Per the above characteristic-radius formula, this yields a characteristic radius for stars hosting intelligent life of

r_{C} = [ 3 / 4π x (1.38 x 10^{33} / 10^{11} ) ] ^{1/3 }= ( 3.3 x 10^{21} ) ^{1/3 }= 1.5 x 10^{7} ly = 15 x 10^{6} 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 10^{6} ) 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,

r_{C} = [ 3 / 4π x (1.38 x 10^{33} / 10^{8} ) ] ^{1/3 }= ( 3.3 x 10^{25} ) ^{1/3 }= ( 33 )^{1/3} x 10^{8} ly = 3.2 x 10^{8} ly, which gives an average separation of 2 x 3.2 x 10^{8} 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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