gamma-ray burst, a supernova, a magnetar flare, or whatever — will go off, lighting up the sky more or less spectacularly and then fading. Normally that's the end of that event, and we're left to wait for the next one. (In particular, this means that if you study this sort of thing, you need to be prepared for one of these things to happen at any time, so that you suddenly need to write target-of-opportunity proposals, analyze data, and release preliminary results in a tearing hurry. This typically happens while you're supposed to be on vacation.) Once in a while, though, we see a peculiar phenomenon called "light echoes".
A "light echo" arises a little like an ordinary (sound) echo: some event happens producing a bright flash (loud noise) and in addition to the light (sound) making its way to you directly, some of the light (sound) goes in a different direction, bounces off something, and makes its way from the other object to you, arriving a little later. From the place I usually stand to watch the summer fireworks competition, you hear the big skyrockets go off, then a second or two later you hear the echo from a nearby building. Of course, on human time and distance scales, the light from the fireworks reaches us instantaneously, so it's obvious that both the original sound and its echo are delayed. In an astronomical setting, we only receive light, and it takes very much longer. But it's still possible to receive a delayed echo, and studying these echoes can be very informative.
(Photo, courtesy of ESA, to the right is X-ray dust echoes around the magnetar 1E 1547.0-5408, one of the objects people in the group here at McGill study. This interesting dust-echo work is from another group, though. The echoes are from a massive X-ray outburst, which we think was caused when the extremely strong internal magnetic field stresses cracked and twisted a piece of its crust; this twisted the external magnetic field, and the twisted magnetic field produced and accelerated massive numbers of electrons and positrons, which blasted out a torrent of X-rays. At least we think that's how it happened; we saw the torrent of X-rays.)
Light echoes in astronomy usually come from light scattered off dust clouds. We usually analyze the problem in what is called the "thin-screen" approximation. What this says is, assume the dust is all concentrated in a thin, two-dimensional screen perpendicular to our line of sight to the original object, and assume that this screen scatters some small fraction of the light striking it in all directions. In particular, this means that some of the light striking the screen is scattered towards our telescope, so that we can observe it. Since this light had to travel further than the light that came to us directly, we see it later than the original event. Since it is coming to us from some position in the dust screen but not on our line-of-sight to the original event, we see it coming to us at some angular separation from the original object. If we saw the original event (so that we know when and where it happened), we can use the delay T and the angular separation a to work out the distances D to the object and d to the dust screen:
The exact expression looks something like cT=d(sec(a)-1)+((D-d)^2+d^2 sin(a)^2)^1/2-(D-d), which is not at all instructive. But if we assume a is small (in practice it's usually arcseconds) we can approximate this by the much simpler cT=da^2/(2(1-d/D)).
This formula is fairly easy to work with. For example, if D is much larger than d, as happens when a dust cloud in the Milky Way produces a light echo of a gamma-ray burst from a distant galaxy, unfortunately this tells us almost nothing about the distance D to the gamma-ray burst, but on the other hand it locates the dust cloud tidily; and in fact light echoes of gamma-ray bursts have been used to try to map out dust clouds in our Galaxy.
If d is not too much smaller than D, then we have one equation with two unknowns, so we'll need more information to pin things down. The people studying 1547's dust echoes, for example, used a dust scattering model to fit the radial and energy distribution of X-rays scattered as part of the echoes. The uncertainties in this model, unfortunately, limit how well they can measure the distance to 1547.
We can also see that the angular size of the light echo grows as the square root of time since the event was observed.
What happens if there are multiple layers of dust? Well, fortunately, we can distinguish them easily: at any given time T, for fixed distance to the source D, there is a unique angular separation a for each dust cloud distance d. So multiple dust clouds just mean we see multiple rings. (Be careful, though: if the source had multiple events, that can produce multiple rings at different radii from a single cloud.)
What about this "thin screen" approximation? It seems weird and improbable — after all, how likely is it that all the dust would lie in a thin plane? But let's look at just how thin our thin screen needs to be to make the approximation work. If we've got a thick cloud, it'll behave roughly like several thin sheets. (Don't worry about multiple scatterings; for the most part if the probability that any given photon will be scattered is small enough for any to reach us, the probability of any being scattered twice is the square of that, and minuscule. Even if it happens, it'll just result in a smeary low-level background.) So how far do we need to move a thin sheet to change the ring appreciably? Well, the fractional change in a is about half the fractional change in d, so to appreciably change the ring, the screens would have to be a sizable fraction of D away from each other. On the other hand, the perpendicular size of the dust cloud only needs to be about ad, and remember a is small. So even a could that is as thick as it is wide is probably a "thin screen".
That said, it seems to me that there's another way you could produce light echoes that are rings: if all of space is filled with uniform dust, then the model above predicts you should see echoes at all angular separations (given the approximations we made). But the model above talks only about the geometry of the light echo, not about the intensity, which depends on the scattering process itself, and the nature and density of the dust. For the moment let me deal with the density of the dust by simply assuming it's uniform everywhere. This means that a time T, you could potentially be receiving light echoes from anywhere on an ellipsoid with you at one focus and the source at the other.
Obtaining a ring-like light echo will depend on the details of the scattering process. The details are no doubt complicated, but many scattering processes scatter most strongly at small angles: that is, even if a photon is deflected, it is most likely to be deflected by a very small amount. (Not all scattering processes are like this, for example scattering ball bearings off posts sends them in all different directions if there's any interaction at all. It was this large-angle scattering that Rutherford's famous experiment observed in gold foil, overthrowing the then-current "plum pudding" model of the atom.) If your photons are likely to be scattered only a little, then you might find that of the light being scattered off the dust and into your telescope, most of it comes from that point on the ellipsoid that requires the minimum possible scattering angle. This, it turns out, happens at the point equidistant to you and the source. So you'll see a "ring" where the emission is at its brightest corresponding to d=D/2 in the thin screen model. That said, the radial dependence of the ring will look something like the angular dependence of the scattering model, so unless the scattering model is very strongly biased to small angles, the ring will be quite fuzzy. This is not observed. So this is a purely imaginative model on my part.
Thin-screen light echoes, on the other hand, are good science, and are regularly observed following powerful transient events.