Monday, October 31, 2016

Two Trillion Galaxies

I have a friend who has a very nerdy geeky friend.  In short, he likes numbers.  It seems that whenever a new factoid with a number gets into the news, this fellow likes to play with it.  So, I decided to ask him to write up his reaction to the latest numbers news.  So, here it is, with some slight editing to make it easier to understand.
A few weeks ago, astronomers announced that our (observable) universe contains ten to twenty times as many galaxies as they thought it had (a galaxy is a collection of stars held together by the center of gravity of the galaxy.  Our own galaxy, called the “Milky Way,” contains some 200 billion star systems (a star system is the star at the core, all its planets, asteroids, comets and other junk that is held in place by the star’s gravity)).  And our galaxy is an average galaxy.  Galaxies are HUGE affairs (the Milky Way is some six hundred quadrillion (600,000,000,000,000,000 = 6 * 1017) miles in diameter), and each galaxy is mostly empty interstellar (between stars) space.  Just weeks ago, astronomers believed that the count of galaxies within the observable universe was some 200 billion (200,000,000,000 = 1011) galaxies, roughly the same number as the number of stars in the Milky Way.  Now they believe that the universe contains 2 trillion galaxies, ten times as many.
OK, that’s nice.  So, what?
Did you ever wonder just how far each galaxy is from its closest neighbor, on average of course?  Let’s examine the question.  If we divide the total volume of the universe by the number of galaxies therein, we get the average volume of space that can be said to be inhabited by one average galaxy.  From that single galactic habitation, we can easily derive its radius, and two radii is the distance between the centers of two neighboring galaxies.
Got it?  No?  Go back and re-read, it’s not that difficult.
So, to the math.
  • The observable universe is spherical by definition (as we can see the same distance no matter in which direction we look). 
  • The volume of a sphere is 4/3 * π (pi = 3.1416) * r3.
  • The radius r of our universe is (latest estimate) 13.8 billion light years, and as one light year* is roughly 6 trillion (6,000,000,000,000 = 1012) miles, its radius is about 83 sextillion (83,000,000,000,000,000,000,000 = 8.3 * 1022) miles.
  • So, the volume of the (spherical) universe is 4/3 * 3.1416 * 8.3 * 1022 * 8.3 * 1022 * 8.3 * 1022 cubic miles.  Or, 2.4 * 1069 (I have NO IDEA what this number is called!) cubic miles. 
  • Next, divide the volume of the universe by the (new) number of galaxies to determine the volume of space that each galaxy inhabits, or 2.4 * 1069 cubic miles / 2 trillion (1012) galaxies = 1.2 * 1057 cubic miles / galaxy.
  • Now, to determine that single galactic volume’s radius, divide 1.2 * 1057 cubic miles by 4/3 * 3.1416 to get the r3 (radius cubed).  We get about 6.6 quintillion (6.6 * 1,000,000,000,000,000,000 = 6.6 * 1018) linear miles for the radius.
  • And 2 of those = 1.3 * 1019 miles, which is roughly two million light years between the centers of two averagely spaced galaxies.
There!  Two million light years between galaxies.
The next time you hear a sci-fi tale talk about leap-frogging between galaxies, maybe you will be a bit more skeptical.
One more thing before we leave the math.
Our nearest neighboring STAR is called Proxima (Latin: nearest) Centauri and it is 4 light years away from us (or from our Sun).  Four light years, or 24 trillion miles.  So far, on planet Earth we have sped along as fast as a few times the speed of SOUND, maybe a few thousand miles / hour.  In outer space, we have managed to speed along at some 50,000 miles / hour.  Or about 14 miles / second!  At 14 miles / second – at 50,000 miles / hour  the one-way trip from here to Alpha Centauri will take
Catch your breath …
Fifty-five thousand years.
And the jump between galactic centers would take 500,000 times as long, or some 27.5 billion years, twice as long as the known universe has existed already.
So, lovers of reality, don’t plan on intergalactic – or even interstellar – travel in the lifetime of our species.  We’re stuck here, on planet Earth and the Moon, the asteroids, Mars, and the Moons of our gas giant neighbors.  That’s it.
Worm-holes?  Good luck!
*    A “light-year” is a measure of distance, not of time.  Thus, it is the distance that light can travel in a (Earth) year.  And, as the speed of light is 186,282 miles / second, a light year is that amount times the number of seconds in a year (365.25 days * 86,400 seconds / day).  So, one light-year = 186282 miles / second * 365.25 days * 86400 seconds / day = 5,878,612,843,200 miles or 6 trillion miles for short.

Addendum: Thursday, 11/03/2016
One more arithmetic factoid: how much of the space that contains one average galaxy is galactic (busy) space and how much is intergalactic (empty) space?
The larger space is 1.2 * 1057 cubic miles / galaxy.  The volume of an average galaxy (let’s say, the Milky Way) is what?  As the Milky Way is a spiral galaxy, its raw volume would be a small fraction (maybe less than 1/10) of what it would be if it filled a sphere.  So, the volume of a spherical Milky Way is 4/3 * pi (3.1416) * r3 (as the diameter of the Milky Way is 100,000 light years, its radius r is 50,000 light years) = 4/3 * 3.1416 * (50,000 light years * 6 trillion miles / light year)3 = 4/3 * 3.1416 * (300 quadrillion miles)3 = 4/3 * 3.1416 * (3 * 1017 miles)3 = 4/3 * 3.1416 * 2.7 * 1052 cubic miles = 11.3 * 1052 cubic miles = ca. 1053 cubic miles.  One tenth of that expanse is 1052 cubic miles.  
So, the larger volume divided by the smaller volume is 1.2 * 1057 cubic miles / 1052 cubic miles or about 105.  So, galactic (busy) space is about one part in 100,000 and intergalactic (empty) space is 99.999%.  There.  How’s them apples!