Obtaining good resolution in planetary imaging means recording fine detail in the object being photographed. In astronomical images of star fields, it can refer to the ability to distinguish two stars that are close together. In planetary photography it means reproducing small features on a planet or the Sun or Moon. On the Sun, it may be solar granulation and detail in solar prominences on the limb, and fine detail in sunspots. On the Moon it may be small craters or rilles. On planets such as Jupiter, it may be detail in the belts, festoons, barges, or even detail on the Galilean moons. The resolution of detail you can record in an image is determined by a complex set of factors: All of these factors work together in determining the quality of the image that you record. As in most things, the final result will be only as good as the weakest link in this chain. For example, you may have the world's greatest telescope, but if it is not collimated correctly, or if the objective is dewed over completely, or if you have a bird's nest in the tube, you are not going to get the world's greatest pictures. Everything else being equal, larger apertures resolve more detail, but good seeing is absolutely critical. There are two kinds of resolution that are usually discussed in astronomical imaging: angular resolution and linear resolution. Angular Resolution Angular resolution refers to the separation of two points, such as stars, at some distance. The sky and earth are arbitrarily divided up into angular measurements called degrees, minutes and seconds of arc. An arc is part of the curved line of a circle. If you stand facing north and make one complete turn around, you have covered an entire circle, which is 360 degrees of arc. If you face north and put your right arm straight out to the side, it will be pointing east. The angle between due north and due east is 1/4 of a circle, or 90 degrees of arc. If you face north and turn around so you are facing the opposite way, south, you will have turned 1/2 of a circle, or 180 degrees of arc. Each degree of arc can be divided again into finer increments called arcminutes. Each degree equals 60 arcminutes. Each arcminute can again be further divided into finer increments called arcseconds. There are 60 arcseconds in one arcminute. So, we have a total of 360 degrees of arc in a full circle, 60 minutes in a degree, and 60 seconds in a minute. All of measurements of the arc of a circle, not time! The Sun and Moon subtend an apparent size of about one-half of a degree, or 30 arcminutes, in the sky. Jupiter, at its largest apparent size, is about 50 arcseconds. Astronomically, when we get down to the one arc-second range, we are talking about really small angles. If two stars are only one arc-second apart in the sky, they are very close. At this point we are talking about fairly high resolution. The seeing at the best observing locations is usually around 0.5 arcseconds. On rare occasions it can get better than this. The finest detail that has been recorded in amateur planetary photos is around 0.2 arcseconds, but this is exceptional. Linear Resolution Linear Resolution refers to the actual physical size of a star, or smallest planetary detail, at the focal plane of the telescope. We usually need to magnify this small detail with an eyepiece to visually observe it or record it with the pixels in our DSLR cameras.
Stars in the sky are so far away that they are effectively point sources of light. Due to the quantum nature of light, when light passes through an opening, or aperture, such as the objective of our telescope, it is diffracted. This diffraction causes a point source to actually form a small disk instead of a point. This disk is called the Airy disk. It is named after Sir George Biddell Airy, an English mathematician and astronomer who formalized the calculations for the size of the Airy disk in 1835 in his paper On the Diffraction of an Object-glass with Circular Aperture. The Airy disk is comprised of a bright central spot, the maxima, and progressively fainter rings surrounding it, separated by dark interspaces called the minima. Half of the light in the Airy disk is confined to the small central core, a region called the Full Width Half Maximum (FWHM). The FWHM is defined as the width of the star where the star's intensity is 50 percent of its maximum brightness value, measured in arcseconds. In an unobstructed optical system, at a wavelength of 550nm, about 84 percent of the total energy of the star is in the maxima of the central disk. About 7 percent is in the first ring, about 3 percent in the second ring, and about 1.5 percent in the third ring. In an obstructed optical system such as a Newtonian or Schmidt-Cassegrain, with a secondary mirror, more of the energy is transferred into the rings, leading to a loss of contrast on fine planetary detail. While larger telescopes can resolve smaller details, the resolution of a telescope is limited by the diffraction of light. The physical linear diameter of a star formed by a telescope in the Airy disk is usually very small, on the order of one hundredth of a millimeter, or 10 microns. For comparison, a human hair is in the range of 30 to 100 microns in diameter.
All other things being equal, the larger the aperture of the telescope, the finer the detail it will be able to resolve. Unfortunately in the real world, things are more complicated due to factors such as seeing. The Relationship Between Angular and Linear Resolution The angular and linear size of a star at the focal plane of a telescope are related. The angular resolution is determined solely by the aperture. The linear size that angle represents is determined by the focal length at that given aperture.
Lets take the example of a telescope with an aperture of 140 millimeters (5.5 inches). It can form a star with an angular diameter of about 2 arcseconds. If this telescope has a focal ratio of f/5.6, it will have a focal length of 784 millimeters, and this star will have a linear diameter of about 7.5 microns. If we take that same telescope with its aperture of 140 mm, and we use it at f/11, it will have a focal length of 1,568 mm. The angular size of the star it forms will still be the same 2 arcseconds since angular resolution is determined by aperture and the aperture has not changed, but since it is working at a longer focal length, the image is magnified, and the linear diameter will now be about 15 microns. Note that although the scope with the longer focal length produces a larger image, it does not resolve any more detail because the aperture is the same, and therefore the resolving power is the same. The angular resolution and resulting fine detail that can be captured are determined by aperture, and the size of this detail (the linear resolution) is determined by focal ratio (which incorporates focal length). For a telescope of a given aperture, if we double the focal length, we double the linear size of the Airy Disk. The angular size stays the same, and the amount of detail resolved stays the same. The angular resolution of our telescopes determines how fine the detail is that it can image. The linear resolution is also important as we will learn in the next chapter on sampling. Sampling means effectively recording fine detail in the image based on the size of the pixels. Angular and linear resolution are also dependent on the wavelength of the light we are recording. Shorter blue wavelengths form smaller Airy disks compared to longer red wavelengths. On the other hand, seeing affects shorter wavelengths more. All telescopes with the same focal ratio will form Airy disks of the same linear size, regardless of their aperture. But in scopes with larger aperture, these spot sizes will represent smaller angular detail, and will be able to record finer detail. Use the following calculator to determine the linear and angular sizes of the Airy disk. Input the size of your telescope in millimeters, the focal ratio, and the wavelength in nanometers. To convert inches to millimeters, multiply inches by 25.4. Blue light is centered around 475 mm, green around 550nm, and red around 650 mm. Size of the Airy Disk Resolving Double Stars Visually What, exactly, does it mean to be able to "resolve" two stars visually in a telescope?
Our initial, naive, answer might be that to be considered "resolved", we should be able to see two stars as completely separated and not touching. In this case, they certainly would be completely "resolved". But suppose we look at two close stars that are touching and overlapping a bit, but where we can still easily tell there are two stars there? From an information standpoint, if we can tell there are two stars, we could still say they are "resolved". At this point, the question becomes, how close can the two stars be together, and still be able to tell there are two stars?
The images at right illustrate various degrees of separation and resolution. They were created in Cor Berrevoets' Aberrator freeware. In 1879 Lord Rayleigh (John William Strutt), another English astronomer, investigated resolving two stars visually in terms of the Airy disks. The Rayleigh criterion says that two stars are resolved when the center of one star (the brightest part (the maxima) is centered over the first interspace (the minima) between the disk and the first ring of the other star. The Rayleigh criterion gives the angular separation as being the same as the radius of the Airy disk. So an 11 inch telescope can "resolve" two stars that are separated by about 0.49 arcseconds (in green light at 550nm) according to the Rayleigh criterion. Remember, "resolve" here means that the Airy disks are overlapping. Not that there is black space between the two stars and not that they are completely separated without touching.
Real world visual observational experience showed that observers could actually detect two stars at separations smaller than the Rayleigh criteria. William Rutter Dawes had earlier tested observers empirically on double stars of equal brightness and found they could distinguish two second magnitude stars with a 5 percent dip in brightness between the maxima with a 1-inch objective. The dip in brightness for the Rayleigh criterion is about 20 percent. Dawes criterion specified that an observer could tell that a star was double down to about 0.41 arcseconds with an 11-inch scope.
In 1916 C. M. Sparrow proposed the Sparrow criterion that said an observer could detect duplicity when two equally bright stars were elongated. In the same 11-inch scope, they would be separated by just 0.39 arcseconds. Some claim the Sparrow criterion is twice the Rayleigh criterion. But Sparrow himself, in is original 1916 paper On Spectroscopic Resolving Power in the Astrophysical Journal, says "This gives a theoretical resolving power about 26 percent greater than that obtained by the Rayleigh criterion." Criteria for "Resolving" Power Where Extended Detail There is a difference between resolving stars and resolving extended detail. Extended detail is that which is larger than a point source, such as craters on the Moon, or the belts on Jupiter. All planetary detail is extended detail. Stars are high contrast point sources. Extended detail has lower contrast. Extended detail, however, can be considered to be made up of a multitude of Airy disks, since the Airy disk is the smallest fundamental pattern that can be formed by a telescope due to diffraction. It is also possible to "resolve", or more accurately, detect, extended detail much smaller than the Rayleigh or Dawes criteria would suggest. This is because not all Airy disks appear the same size when they are observed or photographed. Some appear smaller than the formulae would indicate. J.B. Sidgwick, in his classic Amateur Astronomer's Handbook, notes that "The image is composed of overlapping diffraction patterns formed by light emanating from every point in the object; if the radii of these diffraction discs were effectively of the full value (of the Rayleigh Airy Disk) there could be no resolution, however high the power used; but the low intensity of the light, compared with that of stars, reduces their effective size, and the Dawes criterion therefore cannot be applied to the resolution of detail in extended images." He goes on to describe different types of extended detail: "a black spot on a white ground; single dark line on light ground; and parallel dark lines on light ground." He notes that a single dark line like Cassini's division was discovered with a 2.5-inch scope, some 3.5x times better than Dawes and that W.H. Pickering glimpsed a dark line 0.03 arcseconds wide with his 10 inch scope, which would be 15x better than Dawes. Sidgwick says "less of the spurious disc is visible with fainter stars, even though the Airy disk is the same size." Sidgwick: concludes "Theory can define the position of the zone of zero intensity in each interspace - the locus of a point maintaining a constant distance r from the center of the spurious disc - but cannot define the point at which the intensity of the disc falls below the threshold of visibility: the visible extent of the disc, like the number of rings visible, arises for a given instrument with the brightness of the source, although the discs are in fact the same size, irrespective of brightness. For this reason r1 cannot be taken as the radius of the spurious disc; a closer approximation would be 1/2 r1." Sidgwick's "r1" here refers to the radius of the Airy disk as calculated by Dawes' formula. The formula for the spot size of the Airy disk does set a fundamental limit on the size of details that can be resolved in a telescope. However, the situation is much more complex than it seems because of the factors mentioned by Sidgwick. The bottom line is that we can detect much finer detail than the classic theoretical formulae would indicate, even though it is not technically "resolved". The Difference Between Resolving and Detecting Detail It should be noted that there is a difference between "resolving" and "detecting" fine detail in extended objects. To completely resolve something, we need to see all of the detail that is present. But in some cases, we can detect, or record, the existence of a feature without resolving it. An example of this is Cassini's famous division in the rings of Saturn. It was discovered in 1675 by Giovanni Domenico Cassini with a 2.5 inch telescope. According to Dawes criterion, this telescope should have only been able to resolve about 3.6 arcseconds. But Cassini's division is only 4,800 km (3,000 miles) wide, corresponding to about 0.75 arcseconds. Clearly, we can detect features much smaller than we can resolve. However, this does not mean that we can see detail in such a detection. The Difference Between Visual Observing and Photography It's interesting to consider the human eye as an imaging device. The average eye has a focal length of about 17 mm and a focal ratio of about f/8 in bright light, and about f/2 in the dark. The eye has a lens with an opening called the pupil. This would correspond to the aperture of the objective of our telescopes. The pupil has an iris that allows it to open up and stop down, so the aperture is variable from about 2 mm to 9 mm. As we get older, our eyes usually can't open as wide as when we were younger. Because the amount of light that is gathered is directly related to the size of the aperture (more light is collected by a larger aperture), when we get older, we usually can't see as faint as when we were young. By age 50, our pupils may only be able to dilate to a maximum size of 5 mm. Light that comes through the pupil eventually falls on the retina. The retina contains the light-sensitive cells that produce vision. There are two kinds of light-sensitive cells in the retina, rods and cones. We use cone cells in the fovea in the central portion of the retina for normal daytime vision. They provide us with color vision and with our sharpest vision for things we are looking at directly. Rod cells are more sensitive to faint light, but are essentially color blind. They are mostly located around the periphery of the fovea. These are the cells we use for observing faint objects with averted vision. The rod cells can detect a single photon of light. However that detection is not passed along to the brain because it is not above a noise level threshold and is filtered out. It takes about 10 photons sensed by a single cell to produce a visual sensation. The retina is about 42 mm in diameter with a total area of about 1,300 square millimeters. The fovea is the central portion of the retina that provides us with our sharpest vision. It is about 1.5 mm in diameter. It contains about 200,000 cones. The central area of the fovea with the highest density of cones is a 50 x 50 micron square with 100,000 to 281,000 cones per square millimeter. This corresponds to about 12 x 12 arcminutes on the sky. In the central area of the fovea, cones have a diameter of about 1.5 microns. The rectangular shaped APS-sized CMOS sensors in our DSLRs are about 15 mm x 22 mm in size. A Canon T2i (550D) sensor has 5184 x 3456 pixels, about 18 million total. Each is about 4.3 microns in size. A modern DSLR sensor can detect about one out of every 3-4 photons that hit an individual pixel, depending on the wavelength of light. Although it would seem, based strictly on the size of the individual detector, that our eyes can out-resolve our cameras, the eye suffers from many more optical aberrations. Still, the eye is a remarkable detector. Before the use of digital cameras with lucky imaging, stacking and post processing, the eye could detect finer features than could be recorded in an image. Today, we can record more in a high-resolution digital planetary image than we can see with the eye. Real World Accomplishments The world's best amateur planetary photographers are recording details as small as 0.2 arcseconds with 14 - 16 inch telescopes. But don't think this is easy. It is not. It is exceptionally difficult. Everything has to come together - great optics in a large scope, perfect technique, and outstanding seeing. We probably won't be able to reach this level of resolution in an image with an unmodified DSLR camera because of the low-pass filter in front of the sensor that camera manufacturers include to reduce the effects of moiré in normal daytime images and because of the Bayer array. However, I have recorded detail smaller than 1 arcsecond on Jupiter's moon Europa on a night of good, but not exceptional, seeing with my C11 and Canon 550D. Planetary Resolution Challenges Everyone likes a challenge, it gives them a sense of accomplishment if they can master it, and it can be fun just trying. This human proclivity probably stretches back to mankind's earliest days when competition was an essential part of survival of the fittest. Today it is easily seen in bars and pubs where boasts are made by drunken astrophotographers that "mine is bigger than yours", undoubtedly talking about the apertures of their telescopes. The Holy Grail of planetary photographers is Encke's Division at the outer edge of Saturn's A ring, which subtends a mere 0.0526 arcseconds. No amateur telescope can really resolve this feature, but it can be detected and recorded in an image under seeing conditions with a large telescope with great optics. Other high-resolution planetary imaging challenges:
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