In last week’s The Universe column, I fielded a reader’s question about galaxy collisions in an expanding universe. The answer deals with vast distances, inscrutable forces and the ultimate fate of the cosmos.
Not all queries are quite so serious. For example, reader David Erickson had this on his mind: “If there were aliens 66 million light-years from Earth, how big a telescope would they need to see dinosaurs?”
Ha! I love this question. I’ve thought of it myself but never worked out the math—except to think, “Probably pretty big,” which turns out to dramatically underestimate the actual answer. But what’s really lovely is that grappling with this admittedly bizarre thought experiment has some real-life implications for the future of the science.
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First off, why does it matter that the aliens are 66 million light-years away? It’s because light travels a distance of one light-year per year through space, and the Chicxulub asteroid impact that wiped out the nonavian dinosaurs occurred about 66 million years ago. The light from that event would just now be reaching a galaxy 66 million light-years away, more or less. At that distance, observers there could still see (the very last of the) dinosaurs, assuming they felt like building a really big telescope.
Now the question needs to be split into two parts: How big is a dinosaur from that distance, and how big must a telescope be to see something that size?
Because the sky looks like a gigantic sphere surrounding us, astronomers use angles to measure apparent size. The basic unit for that is a degree; for example, the angle from the horizon to the point directly above an observer, called the zenith, is 90 degrees. The moon has an apparent size of about 0.5 degree across.
How big an object appears depends on its physical size and its distance from whatever is viewing it. There’s a lovely little formula called the small-angle approximation that relates the two. There are a lot of different ways to represent this equation, depending on the units you use. For degrees, you take the object’s physical size, multiply it by 57.3 and divide by the distance. So an object that is one meter wide, such as a small wide-screen TV, would have an apparent size of one degree at a distance of 57.3 meters.
For our dinosaur, let’s pick everyone’s favorite terrifying carnivore, a Tyrannosaurus rex. T. rexes varied in size, but let’s say the one the aliens wish to observe is 10 meters long.
The distance is 66 million light-years, which is a bit of a hike. We need that in meters, so after converting (“Let’s see, multiply by 10 trillion, carry the 2,” and so on), we get a distance of a staggering 6.6 × 1023 meters.
Plugging that into our formula, we find that a T. rex seen from that distant galaxy would have an apparent size of about 10-21 degree. That’s one sextillionth of a degree, or a zeptodegree, if you like fun math prefixes. That’s incomprehensibly tiny. But to be fair, it’s pretty far away.
Great, that’s one of two key questions answered! Now, how big of a telescope do you need to see something so Lilliputian?
You might think what we need is magnification to spot our beast from so far away, but that’s not exactly the case. In a nutshell, something small and very far away will look like a dimensionless dot. If you magnify that dot in an image, you’re just magnifying pixels. To see it as more than a dot, you need to resolve it. So what we really need to see a T. rex and not a dot is high resolution.
Resolution is an inherent property of all telescopes and depends mostly on the size of the telescope’s mirror. There’s another formula for that, called Dawes’s limit. It too can be expressed in many different ways, but if you use degrees and meters, it becomes: resolution in degrees = 3.2 x 10-5 / D, where D is the diameter of the telescope mirror in meters. We know the size of our object in degrees, so we want to solve for D. When we do so, we find the diameter of our telescope needs to be 3.2 x 1016 meters (32 quadrillion meters).
That’s about 3.4 light-years, which would make for, um, a mighty big telescope. That’s a mirror that would span three-quarters the distance to Alpha Centauri!
Needless to say, we don’t have the tech quite yet to build such a thing. Even if we had the know-how to build this mirror, getting the necessary construction material would be a tall order: given the density of typical telescope mirror glass and assuming a mirror thickness of just one millimeter, our T. rex–resolving mirror would have a mass of about 1030 (one nonillion) metric tons. This turns out to be more than 100 million times the mass of Earth. You’d probably need to raid, destroy and remix a good portion of a big galaxy’s rocky planets to build a mirror like that.
If our peeping aliens are especially clever, they might get around this by building an astronomical interferometer instead. This is an array of smaller telescopes spread out over some area; by using sophisticated mathematical techniques, their observations can be combined to mimic the resolution of a single telescope with a size equal to the separation between the two smaller telescopes that are the farthest apart from each other. But even with the material savings from this godlike feat of engineering, we’d still be talking about a billion trillion metric tons of mirror—a decent fraction of the mass of Earth. I’d love to see the alien contractor’s face when they get that assignment. (Assuming they have a face, that is.)
Just for fun, let’s say our curious alien friends did somehow build a suitable telescope. Other issues would still arise, such as how to point it in the right direction. Just moving it would be a monumental task. Worse, they’d need to keep it locked on our long-dead dinosaur for some time to get a decent exposure. The need to track a target is no small problem because everything is in motion: Earth is spinning and revolving around the sun; the sun is moving through the galaxy; the galaxy is moving through the universe; and the aliens’ galaxy is flying around, too. That apparent motion is incredibly small over such vast distances, but remember just how absurdly small the T. rex appears! From 66 million light-years away, a T. rex is pretty faint; at that distance, even the sun would be too faint to see using something like the Hubble Space Telescope. Myriad celestial motions would smear the image out unless somehow corrected for—and I’ll admit I have no idea how to manage that. Whether as a monolithic mirror or a fancy interferometric array, the telescope would be so big that relativistic effects would come into play.
All this is somewhat whimsical and fun to fiddle with, but it has real-world astronomical ramifications. One goal of astronomy is to build a telescope powerful enough to actually see details such as surface features and cloud patterns on distant exoplanets, those far-off worlds that orbit other stars. Such a telescope would have to be huge, even if it were an interferometer, but it’s technically possible—visually resolving such details on an Earth-sized planet 10 light-years away, for instance, would require a telescope array that stretched a few hundred kilometers across. We aren’t ready to build that now, but in a few decades, perhaps.
How amazing would it be to see continents on a planet in another star system? We just need the will to do it; we already have the brainpower. We’re not dinosaurs, after all.
