Ask a Mathematician / Ask a Physicist

Physicist: About why you’d expect: they’re just too damn small and too damn far away.  Nothing fancy.  That’s not to say that we can never get images, just that you need to be a lot closer.  The lunar landers are each about 4 meters across and about 384,400,000 meters away, which makes them about as hard to see as a single coin from a thousand miles away.  You gotta squint.

A picture of the Apollo 17 landing site taken by the Lunar Reconnaissance Orbiter which, as the name implies, was in orbit around the Moon when it took this presumably reconnaissance-related picture.  Those meandering lines are tracks left by a lunar rover.  Click to enlarge.

In fact, a big part of why we (humans) bother to go to the Moon, other planets, and space in general is that photographs from Earth leave a lot to be desired.  In addition to being far from everything else, here on the surface of Earth we’re stuck at the bottom of an ever-moving sea of air.  In exactly the same way that the surface of water scatters light, air makes it difficult for astronomers to practice their dread craft.

Also, not for nothing, telescopes are terrible at retrieving material samples.

The Apollo 17 landing site from even closer.

You and every telescope on Earth (and the Hubble Telescope in low Earth orbit) are all about a quarter million miles from the Moon and the landing sites thereon.  If we ever get around to building something bigger on the Moon, like mines or cities or president’s heads, then we shouldn’t have nearly as much trouble seeing it from Earth.

Answer Gravy: It turns out that the best/biggest telescopes we use today on Earth can’t detect things the size and distance of the lunar landers using visible light.  This isn’t due to poor design; the devices we’re using now are, in a word, perfect.  They literally cannot be made appreciably better (at detecting visible light).  The roadblock is more fundamental.

The “resolving power” of a telescope, is described in terms of whether or not you can tell the difference between a pair of adjacent points.  If the two points are too close together, then you’ll see them blurred together as one point and they are “not resolved”.  If they’re far enough apart, then you see both points independently.

Whether it uses mirrors or lenses, the resolving power of every telescope is limited by some fundamental constraints determined by the wavelength of the light that’s being observed and by the size of the aperture.

Every point in every image is surrounded by a rapidly diminishing “Airy disk” which are a symptom of light being wave-like.  This is only a problem really close to the diffraction limit.  You don’t see these when you take a picture on a regular camera because these rings are smaller than the individual pixels in the camera’s CCD (by design).

Because light is a wave it experiences “diffraction” which makes it “ooze around corners” and generally end up going in the wrong directions.  But the larger a telescope’s opening, the more the light waves have a chance to interfere in such a way that they propagate in straight lines, which makes for cleaner images where the light ends up more-or-less where it’s supposed to be when it gets to the film or CCD or your retina or whatever.

It turns out that the relationship between the smallest resolvable angle, θ, the wavelength, λ, and the diameter, D, of the aperture is remarkably simple:

Visible light has a wavelength of around 0.5 micrometers (about 2,000,000 per meter) and the largest visible-spectrum telescopes on Earth are about 10 meters across (Hubble is a more humble 2.4m across).  That means that the absolute best resolution that any of our telescopes can hope to achieve, under absolutely ideal circumstances, is about .  Or, for the angle buffs out there, about 0.01 arcseconds.  This doesn’t take into account the scattering due to the atmosphere; we can do a little to combat that from the ground, but our techniques aren’t perfect.

By carefully looking at how the atmosphere distorts a laser beam shot upwards from a telescope on the ground, we can take into account how the atmosphere affects light coming into the telescope from space.

The lunar landers are a little over 4 meters across (seen from above) and are about 384,403,000 meters away.  That means that the landers subtend an angle of about 0.002 arcseconds.  In order to see this from Earth, we’d need a telescope that is, at absolute minimum, about 200 meters across.  If we wanted the image to be a more than a single pixel, then we’d need a mirror that’s a few miles across.

So, don’t expect that anytime soon.

Physicist: Not too bad!  Any black hole that humanity might ever create is very unlikely to harm anyone who doesn’t try to eat it.

Black holes do two things that make them (potentially) dangerous: they eat and they pop.  For the black holes we might reasonably create on Earth, neither of these is a problem.

Home-grown black holes: not a serious concern.

The recipe for black holes is literally the simplest recipe possible; it’s “get a bunch of stuff and put it somewhere”.  In practice, you need at least 3.8 Sun’s worth of stuff and the somewhere is anywhere smaller than a few dozen km across.  That last bit is important: the defining characteristic of black holes isn’t their mass, it’s their density.

For a given amount of mass the same amount of gravity “flows” through every containing surface.  In this picture, the same total gravity points through both outer surfaces if they contain the same total mass.  But if all of the mass is concentrated in a tiny place (as it is on the right), then the gravity through the smaller surface must be stronger in order to equal the weaker gravity through the larger surface.  Fun fact: this can be used to derive the inverse square law of gravitation and/or is a consequence of it.

If you’re any given distance away from a conglomeration of matter, it doesn’t make much difference how that matter is arranged.  For example, if the Sun were to collapse into a black hole (it won’t), all of the planets would continue to orbit around it in exactly the same way (just colder).  The gravitational pull doesn’t start getting “black-hole-ish” until you’re well beyond where the surface of the Sun used to be.  Conversely, if the Sun were to swell up and become huge (it probably will), then all of the planets will continue to orbit it in exactly the same way (just hotter).

To create a new black hole here on Earth, we’d probably use a particle accelerator to slam particles together and (fingers crossed) get the density of energy and matter in one extremely small region high enough to collapse.  This is wildly unreasonable.  But even if we managed to pull it off, the resulting black hole wouldn’t suddenly start pulling things in any more than the original matter and energy did.

For comparison, if you were to collapse Mt. Everest into a black hole it would be no more than a few atoms across.  It’s gravity would be as strong as the gravity on Earth’s surface within around 10 meters.  If you stood right next to it you’d be in trouble, but you wouldn’t fall in if you gave it a wide berth.  In fact, that’s why mountain climbers aren’t particularly bothered by Everest’s mass; even if you’re literally standing on it, you can’t get within more than a few km of most of its mass (fundamentally, Mt. Everest is a big, spread out, pile of stuff).

But the amount of material used in particle accelerators (or any laboratory for that matter) is substantially less than the mass of Everest.  They’re “particle accelerators” after all, not “really-big-piles-of-stuff accelerators”.  The proton beams at the LHC have a mass of about 0.5 nanograms and when moving at full speed have a “relativistic mass” of about 4 micrograms (because they carry about 7500 times as much kinetic energy as mass).  4 micrograms doesn’t have a scary amount of gravity, and if you turn that into a black hole, it still doesn’t.  A black hole that small probably wouldn’t even be able to eat individual atoms.  “Probably” because we’ve never seen a black hole anywhere near that small.

The other thing that black holes do is “pop”.  Black holes emit Hawking radiation.  We haven’t measured it directly, but there a some good theoretical reasons to think that it’s a thing.  Paradoxically, the smaller a black hole is, the more it radiates.  “Natural” black holes in space (that are as massive as stars) radiate so little that they’re completely undetectable (hence the name: black hole).  The itty-bitty black holes we might create would radiate so fast that they’d be exploding (explosion = energy released fast).  The absolute worst case scenario at CERN (where all of the 115 billion protons in each of the at-most 2,808 groups moving at full speed are all piled up in the same tiny black hole) would be a “pop” with the energy of a few hundred sticks of dynamite.

That’s a good sized boom, but not world ending.  More to the point; this is exactly the same amount of energy that was put into the beams in the first place.  This boom isn’t the worst case scenario for black holes, it’s the worst case scenario for the LHC in general (cave-ins and eldritch horrors notwithstanding).  It is this “pop” that would make a tiny black hole a hazard.  The gravitational pull of a few micrograms of matter, regardless of how it is arranged, is never dangerous; you wouldn’t get pulled inside out if you ate it.  However, you wouldn’t get the chance, since any black hole that we could reasonably create would already be mid-explosion.

A black hole with a mass of a few million tons would blaze with Hawking radiation so brightly that you wouldn’t want it on the ground or even in low orbit.  It would be “stable” in that it wouldn’t just explode and disappear.  This is one method that science fiction authors use for powering their amazing fictional scientific devices.

The kind of black holes that we might imagine, that are cold (colder than the Sun at least), stable, and happily absorbing material, have a mass comparable to a continent at minimum.  Even then, it would be no more than a couple millimeters across.  These wouldn’t be popping or burning things with Hawking radiation.  The real danger of a black hole of this size isn’t the black hole itself, so much as the process of creating them (listen, I’m making a black hole, so I need to crush all of Australia into a singularity real quick).

We have no way, even in theory, to compress a mountain of material into a volume the size of a virus.  Nature compresses matter into black holes by parking a star on it.  That seems to be far and away the best option, so if we want to create black holes the “easiest” way may be to collect some stars and throw them in a pile.  But by the time you’re running around grabbing stars, you may as well just find an unclaimed black hole in space and take credit for it.

Physicist: The problem with magnetism and the electric force is that they tend to cancel themselves out.  For example, if you have a positive charge the first thing it does is repel all the other positive charges around it and attract all the negative charges.  In short order you end up with a positive and negative charge right next to each other, pulling and pushing on every other charge with the same force (however much the positive charge pulls, the negative charge next to it pushes and vice versa).

(Top) Like charges repel and unlike charges attract. (Middle) A pair of opposite charges will tend to grab onto each other, but this pair pulls as much as it pushes on other nearby charges. (Bottom) The result is that the effect of the charges cancels out and we’re left with “electrically neutral” matter.

Same thing with magnets, if you have two bar magnets floating around, they’ll try to line up with their north side next to the other magnet’s south side.

As a result these “positive/negative” forces tend to balance out really fast.  There are “dipole forces” (one charge might be a little closer, so it pulls just a skosh harder), but dipole forces are tiny and decrease much faster with distance (technically, all magnets are dipole).  In your body right now you have somewhere in the neighborhood of 10²⁸ or 10²⁹ (between ten and a hundred thousand trillion trillion) charged particles in the form of protons and electrons.  The number of extra, unbalanced charges on a good Van de Graff generator that’s dangerous to approach is less than a billionth of a billionth of that.

A slight imbalance of charge.  The ratio of positive to negative charges here is on the order of 1 to 1.0000000000000000001.

Point is; with magnets and charges you always have a problem with things canceling themselves out almost perfectly.  The strength of the electric force between (for example) two protons is just a hell of a lot stronger than the gravitational force (about 1,000,000,000,000,000,000,000,000,000,000,000,000 times bigger), but you’d never know it since those huge forces are all balanced and cancelled out by all of the negative charges around.

Gravity, on the other hand, has only one kind of “charge”: matter.  All matter attracts all matter, so despite being far and away the weakest force, gravity is basically the last man standing on large scales.  You might imagine that if gravity acted like magnetism there would be planets and stars pushing and pulling each other every which way, but in all likelihood we just wouldn’t have large structures in the universe like planets in the first place.