Resolved disk of Ceres
Posted 17 January 2013 - 02:45 AM
I don't have the reference before me but I recall reading that W. Herschel measured Ceres and Pallas. His estimates of diameters were off the mark, but its clear he was resolving the disks, recognizing they were more than simple points of light.
Posted 17 January 2013 - 10:14 AM
Actually, anything about 1/4th the Airy disc diameter will begin to produce a PSF larger than a point source.
Classic MTF. According to MTF theory, a white line on a black background appears wider than it really is. This is diffraction at work. By the same logic, small disk on a black sky appears larger than it really is.
Every point on the circumference of the disk will act as the center of a "Airy disk" so that what you wind up with is esentially an almost but not quite stellar object.
For example, if a telescope produced an Airy Disk that was 1 arc second in size and you looked at a disk that was .75 arc seconds in size, the image would be a point that was bigger than the .75 arc second target. It would in fact appear to be about 1.75 arc seconds across (75% larger) than the 1 arc second that a similar brightness star would produce.
It would not look stellar.
I think when the source gets to be about 30% of the size of an Airy disk size for the instrument, it starts to show more of a classic diffraction effect with something that looks like a fat, soft first diffraction ring, but not quite like a fat, soft diffraction ring.
Bottom line.. Diffraction makes it possible for an aperture to "Resolve" a disk smaller than the diffraction pattern the instrument creates.
Another example is a Sparrow split. Two point sources so close to one another that they produce an elongated Airy Disk.
In fact, this is a great case of how an apture "Resolves" targets that are less than half the radius of the Airy Disk. As long as the resulting image is bigger in diameter or in any direction than an Airy Disk for a star of the same magnitude, we can say that we have detected (Norme would say "Resolved") some detail. Now the human eye starts to struggle (Sparrow splits were almost all photographic) because at some point the eccentric image is to spherical for the eye to really see as elongated, but precision instruments can measure the elongation. We haven't really "Resolved" the split because we cannot see a dip in the valley, but we can infer that it is there if other evidence shows the source to be to distant to be non-stellar. So while we can't see the Sparrow split as a true binary star, we can infer that it is one because of the elongation of the Airy Disk.
The exact same thing is happening when a disk is smaller than the Airy Pattern of a star in an instrument. The disk "Expands" the size of the "Airy Disk" by the diameter of the source. Agian, the theory for image formation of extended targets is that it is a composite of an infinite number of overlapping Airy Disks being formed by every point on the surface of the source.
Diffraction explains all of this easily.
The difference is that in a much larger aperture, the disk will have progressivly harder, more distinct limbs so that at some point, it is decidely not stellar, while in the smaller instrument, it will be expanded angularly by the diffraction, but the center of the spot will be broader and flatter than it would for an Airy disk, but the limb itself could not be resolved because it falls inside of the diamater of an Airy Disk size for the instrument. In the smaller aperture, we see no distinct limb, but a soft sided ball of light. The important point though is that it will not appear stellar because the circumference will be too large and the central "Spike" will be very broad.
And this is what MTF is all about. It describes the contrast transfer. The contrast of the limb is lost in the smaller aperture. But careful measurment (or a skilled observer) would easily see that in the case above, the target is not stellar. And if it appears to be slightly larger than a star with no defined first diffraction ring, then the most logical explination is that it is not a point source, but rather a disk that is the source of the light. It will indeed look more like a disk than a star.
Posted 17 January 2013 - 11:41 AM
Now, say there is a less bright area on Ganymede's limb. The collection of point sources that comprise it will be dimmer (contributing less intensity to the enlarged PSF) than the surrounding brighter collection of point sources. On Ganymede, at about the size of a 6" aperture Airy disc, one can easily fit one line pair - which is resolution at the Raleigh limit (which really does not apply, I think, in the pure sense of two Airy discs.) In fact, one can include more than one line pair at the Dawes limit. So, depending on the contrast level, Ganymede is plenty large to separate features of sufficient contrast. The same should be true for Ceres in an 18" scope, provided sufficient contrast and great seeing, of course.
"...according to the optical theory, a point-source image has to be less than 1/4 of the Airy disc in diameter; larger image enlarges the central disc, and alters energy distribution in the area of rings (at the image size of ~0.25 Airy disc diameter, the FWHM is enlarged ~2%, at twice that size it is about 8% larger, and with the image equaling the Airy disc in diameter the FWHM is nearly doubled, and the ring structure greatly suppressed)."
"Diffraction image of a point source on the surface of most extended objects could be detected only if separated from the rest of surface, not because it is small and relatively faint, but because it is typically of much lower intensity than that of the surface. "
"Consequently, diffraction image of an extended surface can be evaluated as a product of surface dots not larger than 1/4 of the Airy disc diameter..."
"Surface of an extended object can be decomposed on point-sources, that overlap and grow into a larger diffraction image of it. Any distinctive area on such surface also can be decomposed on its effective point-sources. Whether such an area - a surface detail - will be visible in the telescope image depends on the multiple factors: its size, brightness and contrast and, if colors are present, hue specificity and saturation."
Again, I don't yet fully understand how object resolution transitions from point source to extended object. But, this is how I read it.