# I get email: Could we spot the enterprise?

Every now and again, I get a fun question that is totally in appropriate for the column, in that in order to answer it, I need to throw out a few equations. Last week, I got an intensely nerdy email from a reader named Nick:

I’ve always wondered what the chances were we could actually detect an alien spaceship in our own solar system were. Assuming they worked around the the problems with the FTL travel you outlined in your latest column, would any of those give off enough energy for us to detect? Or in general just a ship floating around our solar system, assuming they weren’t worried about being detected? What would it take to detect the ship, if we could not now? If size is a factor, let’s say the ship was the size of the Enterprise D, about 650 metres long, 470 metres wide and 140 metres high.

Unless they were very close, we wouldn’t detect a thing. A spaceship has a couple of advantages over, say, an asteroid, since an asteroid has very low reflectivity while the enterprise, presumably, is both gleaming white and gives off a bit of its own light. For the moment, let’s just assume that the enterprise doesn’t actively want to be seen, but since they have photon torpedoes, isn’t terribly concerned about being seen, either.

That said, let’s consider a perfect mirror sitting in the orbit of Jupiter at opposition:

Since Jupiter is 5.2 times further from the sun than the earth is (5.2 Astronomical Units), then at opposition, it is:
$d=4.2 AU=6.3times 10^{11}m$

from earth.

Now, we’ll suppose that the ship acts as a perfect mirror. How much light will be reflected? The general relation is:

$b=L_{odot}frac{A}{4pi r^2}$

where $L_odot$ is the luminosity of the sun, $A$ is the incident surface area of the ship, and $r$ is the distance of the ship to the sun. Plugging in the numbers, we get:
$b=1.6times 10^7 W$

This total seems like a lot, but objectively, it’s not. From earth, we measure a flux (if it were reflected from only one side):
$f=frac{b}{2pi d^2}=6.41times 10^{-18} W/m^2$

though in detail, we’d have to to worry about the exact shape and orientation of the ship. I’m not going to worry about that. This is a very small number. How small? It’s about $4.5times 10^{-21}$ times dimmer than the sun at noon.

Astronomers use the terribly non-intuitive magnitude system to describe brightness, where a lower number is brighter. The sun has an apparent magnitude of about -26.7. Magnitude =6 is supposed to be the dimmest we can see with the naked eye.

The general relationship to compute the magnitude of an object is:

$m_1=m_2+2.5log_{10}left(frac{f_2}{f_1}right)$

so using the numbers above, we get:
$m_{Enterprise, Jupiter}=24.2$

This is roughly a million times dimmer than anything we could see through binoculars, and just about on par with the very dimmest objects we could observe by taking pictures on ordinary photographic plates. On the other hand, it’s a few hundred times brighter than the dimmest objects seen by Hubble. I should point out, however, that this generally requires a number of orbits.

But remember a few things:

1. While the HST can see down to $m=31.5$, it requires many, many orbits. Even at 24.2, we’d have to be looking at the source for quite a while.
2. The orbit of Jupiter is relatively close, and the flux drops off at $sim 1/d^4$. (Since the ship gets both more distant from us AND more distant from the sun). At 25 AU from the sun (interior to the orbit of Neptune), we’d have no instrument to detect it under pretty much any circumstances.
3. This assumes 100% reflection, but also that the ship has no emission source of its own. However, consider that the reflected light produces nearly 20 million watts on our side alone. This would require something like 100,000 lightbulbs in the windows — or a window every $1.5times 2m$, each with a light on. Unlikely, and even then, it would only double the apparent brightness of the ship at the orbit of Jupiter.

The long and the short of it is that we wouldn’t be able to find the enterprise unless either a) We knew exactly where to look and had large telescopes already pointed in that direction, or b) they beamed direct signals to us, either in radio waves, or with very powerful, highly focused lasers.

In either case, it doesn’t speak well for our long-term alien readiness.

-Dave

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