Thursday, May 23, 2013

Prime Roots

What's in a number?  Well, prime roots, for one thing.  I'm sure there are many other "significant" ways to understand a given integer, but prime roots seem like they get at some kind of fundamental essence.  Take 12, for instance.  2, 2 and 3.

Just for fun, let's pretend that quarks represent prime numbers.  If up quarks were "2", and down quarks were "3", that would give you a proton, so a proton would be 12.  A neutron would be 18, since it has 2 down quarks and 1 up quark (2, 3 and 3).  We might have to create some imaginary quarks to go much further, but this obliquely conveys my excitement about prime roots being "fundamental building blocks of numbers".

Not seeing a particularly simple prime root calculator, I created this:
  prime root finder


How strong is the gravitational pull of Jupiter from Earth?  On average, it turns out to be 0.2 micrometers per second per second (between 0.147 and 0.32, depending on how close we happen to be).  That doesn't seem like much, but after an hour you would have moved about 4 feet, and after a day you would have gone 780 meters and would be moving at a more noticeable 2cm/second.

That's only 1/160th the strength of the Moon's gravity, which causes tides.  And the Moon's pull is 1/180th that of the Sun's, which is 1/1700th of the familiar gravity on Earth's surface.

But it's still strong enough to be noteworthy.  Even at about 1/50 millionth of what we consider normal "falling", it would be quite noticeable under certain conditions.  For instance, if your keys suddenly lost their responsiveness to all gravity save Jupiter's, they would be on the ceiling (or wall, or floor, or under the bookshelf) well before morning.

Before considering this attraction to be small, we should consider that the gravity on Earth's surface is really quite huge.  If you were to free fall in Earth's gravity for only 47 minutes you would have gone far enough to circle the globe (not including atmospheric drag, of course).  So a 50 millionth of that still seems significant.

Enough to lose one's keys overnight, anyway.

The Sun is a reference point between those extremes that we can use to help conceptualize this amount of pull.  If you were to fall off the edge of the Earth, toward the Sun, it would take 18 seconds to fall your first meter but within an hour and a half you would have cleared the atmosphere.

For another reference point, the acceleration toward the galactic center is about 8 femtometers per second per second, 24 million times weaker than toward Jupiter.  At that rate, you would only fall 4 meters in your first year, and only 30km in your entire lifetime.  But it manages to swing the entire solar system around in a circle every 220 million years.  So you can't say it isn't strong.  You would just have to be very patient to notice it.

Unchecked attraction, more often referred to as "falling", can quickly get quite out of control.  If you were to fall at the familiar, Earthly rate for just a minute (ignoring friction), you would have gone over 10 miles and would be going 1300 mph.  Fortunately, there is a stable way to relate to gravity: by orbiting.  This arrangement lets you continuously fall, but with a nice, smooth mix of angular momentums that prevent you from picking up too much speed.

And speaking of falling in circles, we are already doing that, and at impressive speeds.  The rotation of the earth causes us to move a little over 1000 mph (at the equator).  But the Earth is orbiting around the Sun much faster than that: at about 67,000 mph.  And our solar system's orbit around the galactic center is even faster: over 500,000 mph.  And our whole galaxy is moving even faster than that: somewhere in the neighborhood of 900,000 mph, depending on what you measure against.

All of these various fallings make up the fabric of our local space-time continuum, this familiar, invisible grid that we assume to be Euclidean but is in reality quite lumpy.  Jupiter, being the strongest gravitational influence after the Moon, represents a frontier of unacknowledged lumpiness.

(It's really more of a slowly varying incline.  But if you look at the shape of space-time across the whole solar system, "lumpy" seems a better term.)

In any case, even if one's keys are not likely to be sucked into outer space by Jupiter's gravity, it is fascinating to imagine the effects of that force, subtly mixed into the the more potent gravities in our lives.

One last calculation.  Two humans, placed in outer space with about a foot between them, would take about 3 hours to fall into each other.  This force of attraction is about 1/18th that of Jupiter's.

Lumpy indeed.