I was pondering the string model again this morning, and I realized that it was an interesting but not entirely adequate model for the Encke Gap. The Encke Gap isn't a black string at all—it is where two semi-infinite bright planes meet with a small gap between them. This is an important distinction because at a basic level, we're not talking about a long string of Airy disks, but a semi-infinite plane of Airy disks. The edge of a semi-infinite plane won't be a bright line of Airy disks with a sharp set of diffraction rings, but instead a smearing of partial disks nearing the edge, all convolved with their mutual ringing patterns. So in our perfect experiment on just one semi-infinite plane, its edge with an abrupt terminus would not have a sharp edge, but a diffraction smearing across the pixels spanning the edge. And of course, the Gibbs effect would frustrate any attempts at sharpening, given how discrete subito pianissimo transitions are disliked by signal processing.
To test such a hypothesis on the Gap itself, we would need a telescope high above the atmosphere to preserve the Airy pattern and to eliminate the potential artifacts from the sharpening required to try to reconstruct it. The Hubble happens to be such a telescope. And the fact that its diameter is almost—but not quite—large enough to "resolve" the Gap (per the half-angle Rayleigh criterion) ends up making the Gap a useful case study after all.
I found a recent image worthy of study—one from 4 July of this year (at 19:00 UTC, to be precise):
Saturn's parameters at that instant were
- Distance = 1351.494 million km
- Rings diameter = 42.85"
- Gap = 0.0496"
The Hubble parameters are
- Diameter = 2.400m
- Airy disk radius (half-angle) = 0.0562" @ 536 nm (green)
For pixel dimensions, I just used the green channel. The width across the rings was 1055 pixels, which gave a pixel dimension of 24.62 pixels/". This put 1.384 pixels across the Airy disk half-angle—the Rayleigh criterion for resolution. At this pixel pitch, the Encke Gap should be 1.221 pixels—11.8% below the resolution limit.
The actual Gap in the image told a different story. The brightness of the edges of the two semi-infinite planes starts to fall off well before the actual edges of both. I plotted the intensity (i.e., the green level from 0 to 255) across the gap, centered on the dimmest pixel. I did this for both, and for symmetry I plotted them radially from the center of the planet. For a ground truth comparison of quantization levels, I went to the Cassini mosaic of the rings, which is resolved well below the width of the Gap.
The intensity level (0-255) across the gap matched to first order. But the shape of the subito pianissimo Gap was much less sharp and much wider in the Hubble image. The Gap in the Cassini image notably does not fall to 0 levels (apparently it's not an empty gap); the Hubble Gap does even less so. I assume this is because the Gap in the Hubble image is still filled with first-order falloff of Airy disks at the edges of the semi-infinite planes.
So the effect of the Airy disk is to smear the sharp edge of a semi-infinite plane—not just into the adjacent darkness, but also the darkness into the bright plane. You can see this at another semi-infinite plane in the image: the outer edge of the rings also shows a smearing vice a gradual falloff. For the convolved Gap, the dark lane appears to be smeared around 2 radii from the edge. So for our experimental scope above the atmosphere, instead of a bright Gibbs ring from sharpening, we have a filed down edge that widens the high-contrast feature.
Edited by BQ Octantis, 27 August 2020 - 05:21 AM.