Small Moves

Hey, remember this post? In it, we estimated a few quantities that are going to be useful here--the size of the Earth and Moon, along with the distance between the two. We were off a bit, so here are the actual figures:

Diameter of the Earth = $D_+ = 1.3 \times 10^7$ meters

Diameter of the Moon = $D_m = 3.5 \times 10^6$ meters

Distance from the Earth to the Moon = $R_{+M} = 4 \times 10^8$ meters

As with all big numbers where the zeroes have started to pile up, that can be hard to visualize. Most people imagine the Earth and Moon looking something like this:

(Source)

But actually, a scale diagram of the two is a little more extreme-looking:

Earth is the blue one.

The distance between the two is about 31 times the diameter of the earth--so on this scale, where Earth is about a centimeter wide, the moon is nearly 1/3 of a meter away. That is the farthest from home that a human being has ever traveled.

But what about what's past there? The nearest another planet ever comes to Earth is Venus, when it's at its closest approach (the term for this is "opposition"). In the case of Venus, this is about $3.8*10^{10}$ meters.

That's 96 times the distance to the Moon. I was going to add Venus to the scale drawing above, but apparently Google Docs drawings don't handle 32-meter long lines very well. WolframAlpha has helpfully suggested that this is approximately the length of a blue whale, so if you want to imagine the distance between the two, you could try envisioning two dots the width of your pinky at opposite ends of this guy.

The guy on the right, specifically (Source)

Or you could stick to the more formulaic analogy of one dot in a football field's end zone and another on the 35-yard line. Mars is a little trickier, since its orbital eccentricity--at 1%--is fairly high. In theory, it's possible for Mars to come as close as $5.4\times 10^{10}$ meters (which would put Mars 1.4 blue whales away, or almost exactly on the 50-yard line), but this hasn't actually happened in human memory. The 2004 opposition between the planets, where they reached $5.6 \times 10^{10}$ meters (1.5 blue whales, or halfway between the 51- and 52-yard lines), was the closest we've been to the red planet in 50,000 years.

But this is still just a third of the way from the Earth to the Sun, clocking in at $1.5 \times 10^{11}$ meters. Rather than measuring that in blue whales, it might make a bit more sense to apply that to something a little more familiar. Let's put the Earth, still a centimeter across, right here:

It's a better sport than football anyway.

If the centimeter-Earth is one one end of the rubber, the moon is sitting at the other. Venus, at opposition, is sitting in shallow center, just past the edge of the infield dirt. Mars at its 2004 opposition would be just at the bottom of the "B" in center field. The sun, meanwhile, makes it over the wall, landing in the fifth row of bleachers in straightaway center. At a meter across, it's also taking up two seats.

No human being has been this far, but our robotic probes have made it substantially further. Saturn is currently $1.5 \times 10^{12}$ meters away, with Cassini circling it. On our scale, that works out to about $1.1 \times 10^3$ meters, putting Saturn right on the Boston side of the BU bridge:

Interestingly, Saturn also could go near the mound of MIT's baseball field (top). But that's a bad analogy because no one knows where that is. (Source: Google Earth)

Let's go a bit further, to Pioneer 10--the first spacecraft to achieve escape velocity from the solar system. It was launched in 1972 and performed a flyby of Jupiter 21 months later. Contact with Pioneer 10 was lost just eleven years ago, when it was a healthy 12 billion kilometers (sorry, $1.2 \times 10^{13}$ meters) from Earth. Judging from its trajectory at the time, that should put it about $1.7 \times 10^{13}$ meters away today. On our scale, that would mean Pioneer 10 would be stuck in traffic at the intersection of the Pike (I-90 for non-natives) and 128 (I-95) in Waltham, MA.

Pioneer 10 should've taken Route 16 and skipped the traffic. (Source: Google Earth)

Pioneer 11, launched the following year, is a comparable distance away, but headed in the opposite direciton. So now, finally, let's indulge the Voyager fanbase out there (and yes, it is a fanbase--you can tell that much just from the fact that, while I had to do a bit of math to figure out just how far away the Pioneer probes are, the Voyager 1 Wikipedia page was updated to reflect its distance from Earth as of this morning). Courtesy of a very avid Wikipedia updater, I know that Voyager 1 is $1.9 \times 10^{13}$ meters away right now. While that would, finally, get us out of Greater Boston on our scale, we'd have to be going due west. If not, well, it's almost exactly the distance from Fenway to the intersection of 128 and route 2 a little farther north...

(Source: Google Earth)

So that's the farthest manmade object from Earth. It's been traveling for 36 years, 5 months, and 12 days as of today (thanks, Wikipedians!), and on the centimeter-Earth scale, it doesn't even make it into another area code.

Of course, I could scale this however I wanted to make that distance seem insignificant ("If the Earth were a grain of sand on the mound, Voyager would barely have made it to the warning track! Take that, NASA!"). So let's zoom out a bit more. This is Proxima Centauri:

Hello. (Source)

It's a pretty interesting star. A seventh the diameter of the sun, it's so faint it wasn't discovered until 1915--all despite the fact that,  4.2 light years distant, it's our nearest stellar neighbor. Traded in for slightly more useful units, those 4.2 light years come out to just about $4.0 \times 10^{16}$ meters. Plug it into the super-complicated math I've been doing to these distances (AKA multiplying them by $1 \: cm / D_+$), that works out to $3.1 \times 10^7$ meters. Where does this land Proxima Centauri? Well, if you were to travel along the 42nd parallel (just south of Boston) this would leave you out in the Atlantic, halfway between London and Boston--if you were to go west. At 31,000 kilometers, Proxima Centauri is a long way away. In fact, the Earth's semicircumference is 20,000 kilometers, which means that no two locations on the planet are farther apart than that (if you're planning on traveling 31,000 kilometers to get anywhere, you might want to consider turning around--it'd be a whole lot faster).

I personally see the Apollo Program as one of the greatest achievements in the history of our species, and I think rightly so. The idea that fellow humans have set foot on another world, however small, should quietly blow the mind of anyone looking up at the night sky. But it's worth keeping in mind that, even by the standards of our stellar neighborhood, we've just barely started to move.

Baby steps. (Source)