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This APOD image shows two nested rings in the X-ray spectrum surrounding a gamma ray burst. Or maybe I should consider it to be a single ring around a disc. The APOD description says,

...the X-rays also bounced off regions high in dust right here in our Milky Way Galaxy, creating the unusual reflections. The greater the angle between reflecting Milky Way dust and the GRB, the greater the radius of the X-ray rings, and, typically, the longer it takes for these light-echoes to arrive.

This seems reminiscent of how sun halos form, where there is a preferential angle for sunlight to reflect off of ice crystals in the atmosphere, leading to a ring at a certain angular radius around the sun. But the second sentence quoted above seems to suggest that the angle could be different, although it says nothing about what could make it different. Is the analogy with sun halos valid? Is it known what determines the deflection angle?

Bright spot in disk with gray ring surrounding

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    $\begingroup$ It can't be a perfect analogy, since halos are formed by internal reflections in crystals Perhaps a corona would be a better analogue, those are formed by diffraction rather than reflection. $\endgroup$
    – James K
    Commented Oct 18, 2022 at 5:25

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Imagine the X-ray pulse from the GRB travelling outwards like a spherical shell. If it then encounters a slab of dust on its way into our galaxy, then the intersection of the X-ray shell and the slab will be a ring drawn on the slab.

Some of the X-rays will then be scattered and change their original direction and head towards the Earth. We then see those scattered X-rays as a ring, coming from the slab, surrounding the GRB. Because the scattered X-rays have travelled further than the direct photons from the GRB, then the X-ray ring will appear after the GRB.

GRB dust ring geometry

Although the scattering is not equal in all directions there is no preferred angle - it is a form of Rayleigh scattering (see Smith & Dwek 1997). In that respect the Physics is not the same as visible light scattering from ice crystals in the Earth's atmosphere. What determines the deflection angle is how close the dust slab is.

As a formula $$ \theta (t) = \bigg [ \frac{2c}{d} \frac{(1-x)}{x} (t-{T}_0) \bigg ]^{0.5}, $$ where $T_0$ is the time of the GRB, $c$ is the speed of light, $t$ is the time the ring is observed $d$ is the distance to the GRB and $x$ is the ratio of the distance to the dust slab and $d$. Usually $x\ll 1$ and this simplifies to (Pintore et al. 2017) $$\theta (t) = \bigg [ \frac{2c (t-{\rm T}_0)}{d_{\rm dust}} \bigg ]^{0.5}. $$

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    $\begingroup$ Usually I wait longer than this to accept an answer but this seems unlikely to be surpassed. At least one thing I read mentioned "slabs" of dust, and I was familiar with the idea that a pulse of radiation creates a spherical shell. But I didn't have that thought in my head because I was so entranced by the spurious comparison to sun halos. Good answer! $\endgroup$ Commented Oct 18, 2022 at 15:33

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