Resolution (and Magnification)
This is a relatively specialised text. If you do not feel like reading it, just “jump to the conclusions” listed at the end.
MAGNIFICATION IN A TELESCOPE. Magnification means how many times larger an object is seen with respect of the naked eye. Given a telescope with focal length H and an eyepiece with a focal length E, the magnification M is given by the simple formula M = H / E and the result is a number followed by an “x” meaning “times”.
For example, the focal length of our SkyMax-102 is 1300mm. If we insert into the diagonal a 10mm eyepiece, the magnification is 1300/10 = 130x. With a 5.5mm eyepiece, the magnification is 1300/5.5 = 236x.
So far it may look as if we can produce from any telescope the magnification we like by just inserting a powerful enough eyepiece. This is indeed true, but any telescope supports a maximum or usable magnification: insert an eyepiece that exceeds it, and you will get a significant loss of sharpness. An old “theoretical” (and as we will see very optimistic) rule is that the usable magnification is twice the main mirror diameter T in mm: this is why SkyWatcher rates this telescope's magnification at 204x. You would expect that the image at this usable magnification is as sharp as the naked eye image. Therefore with the 236x calculated above for a 5.5mm eyepiece we have barely exceeded the 204x usable, and should see very sharp images! However, you can verify with your telescope that this is not the case. To understand why, we have to explore a related but different concept, the resolution.
RESOLUTION OF A VIEWING APPARATUS. We will abridge this complex concept as much as possible. The resolution (of an eye, a camera or a telescope) can be expressed in two different units:
- arc' (arc minutes) or arc" (arc seconds) refer to the minimum arc under which two parallel lines are distinctively perceived.
- distance in mm or mt refers to the maximum distance at which lines 1mm apart are perceived.
We first compute a conversion coefficient g = 60 / TAN(PI()/(60*180). The result is g = 206.3.
Converting mt to arc": if the distance in mt is t, the value r" in is calculated by r"= g /t arc".
Converting arc" to mt: vice versa if the arc" is r", the distance t is calculated by t = g /a mt.
HUMAN EYE. The “theoretically perfect human eye” has a resolution of 1 arc' or 60 arc", therefore its Distance is 206.3/60 = 3.438 mt. This number is found in quite a few webpages, yet it is of limited use: the "average human eye" is much less sharp, and a reasonable average (let us call it c) is 1.8 mt, implying 114.6 arc".
TELESCOPES AND DAWES LIMIT. The “Dawes Limit” is a value w estimating the maximum “expected resolution” of a telescope. It is computed by a formula (derived from scientific measurements) an expressed in arc":
w = 116 / e, where e is the main mirror diameter in millimetres. Therefore, the Dawes Limit for the SkyMax-102 is 116/102 = 1.137 arc". If we compare this with the average human eye value 114.6 arc", their ratio is 100.8x. This means that according to Dawes, for equal-sharpness images, our telescope yields 101x the resolution power of the human eye, not the rated 204x! Quite bad, but a significant loss was to be expected: a reflector telescope with a diagonal and an eyepiece consists of 3 mirrors and 2 groups of lenses, and as it goes through each one of them the light is inevitably distorted, resulting in some loss of sharpness. Luckily for us, the Dawes Limit is not an absolute maximum. Practical measurements have shown that some telescopes are systematically inferior to it, while others (notably Maksutov-Cassegrain such as our SkyMax) can be superior. So how good is the present-author's SkyMax-102?
CHART AND FORMULAE. Our observations will consist of viewing from a distance a common photographic resolution printed chart, with progressively nearer lines and an arbitrary scale in values. Before performing any observation, we need to find (and write down) the place in the printed chart where the lines are 1mm apart: we call this value in the scale k. Then we locate the chart at a distance d in mt from the telescope. Finally we look through the telescope and find that the maximum lines we can distinguish on the chart occur at a value v. Considering the distance of c reported above for the human eye, it is not difficult to reach the following formulae for the resolution r w.r.t. the human eye: r = (d/c) * (v/k) , expressed in “x” or “times”. We will also need a formula for the distance t at which the telescope can resolve lines 1mm apart: t = d * v / k , expressed in mt. Finally the observed resolution r" in arc" (see the conversion formula above) is given by r" = g / t .
ORECCHIELLA TESTS. In the above formulae the parameters k and d are specific to the test setup. Our first observations were performed in August 2018 at sunset in the clear mountain air of the Parco dell'Orecchiella: here we used a test chart with k = 49 and set it at a distance from the telescope d = 140 mt. We used the bundled Skywatcher 25mm and 10mm plus a Gosky 8mm eyepiece.
LUCCA TESTS. The second set of observations was carried out in November 2018 in daylight, using the following test chart (actually we produced a high resolution 120Mpx 13406x8941px and printed it on 10x15cm photo paper):
With this chart obviously we have k = 41. We placed it on the Lucca walls, yielding an unobstructed view from a boulevard. For maximum accuracy we used Google Maps (see the picture below) to determine the position of the test chart and telescope so as to guarantee a distance d = 200mt, with an error smaller than 0.5%. We used the bundled 25mm Skywatcher eyepiece and also our new Explore Scientific Ar 14mm, 9mm and 5.5mm eyepieces.
OBSERVATIONS. The values and results obtained in both test sessions (Orecchiella and Lucca) are shown together in the table below. The Barlow was an Omegon 2X.
| EYEPIECE AND LENS |
EFFECTIVE FOCAL LENGTH |
MAGNI- -FICA- -TION |
VALUE OBSERVED ORECCH. |
VALUE OBSERVED LUCCA |
RES./MAGN. W.R.T. HUMAN EYE |
DISTANCE FOR 1mm LINES |
RESOLUT. IN ARC SECONDS |
|---|---|---|---|---|---|---|---|
| 25mm Skywatcher MA | 25 mm | 52 x | 51 | 81 x | 146 mt | 1.42 | |
| 31 | 84 x | 151 mt | 1.36 | ||||
| 25mm + Barlow 2x | 12.5 mm | 104 x | 58 | 92 x | 166 mt | 1.24 | |
| 35 | 95 x | 161 mt | 1.21 | ||||
| 14mm Explore Sci. Ar | 14 mm | 93 x | 37 | 100 x | 180 mt | 1.14 | |
| 14mm + Barlow 2x | 7 mm | 186 x | 38 | 103 x | 185 mt | 1.11 | |
| 10mm Skywatcher MA | 10 mm | 130 x | 64 | 101 x | 182 mt | 1.14 | |
| 9mm Explore Sci. Ar | 9 mm | 144 x | 41 | 111 x | 200 mt | 1.03 | |
| 9mm + Barlow 2x | 4.5 mm | 289 x | 39 | 106 x | 190 mt | 1.08 | |
| 8mm Gosky Plössl | 8 mm | 163 x | 69 | 110 x | 198 mt | 1.04 | |
| 8mm + Barlow 2x | 4 mm | 325 x | 36 | 57 x | 102 mt | 2.03 | |
| 5.5mm Explore Sci. Ar | 5.5 mm | 236 x | 42 | 114 x | 205 mt | 1.01 | |
| 5.5mm + Barlow 2x | 2.75 mm | 473 x | 39 | 106 x | 190 mt | 1.08 |
- The observed values have an estimated error not greater than about +/-5% for Orecchiella and +/-3% for Lucca.
The 8mm Gosky test was repeated with a top-grade borrowed Vixen eyepiece: the latter showed huge improvements in eye relief and other features, but the minimal improvement in resolution was within the measurement error.
This was later confirmed in Lucca when the 9mm Explore Scientific yielded an almost identical value (111x vs 110x). - The low resolution of the 8mm + Barlow was confirmed by an observation with the 10mm, not included in the table.
- Needless to say, the telescope's resolution is the best one among the observations for the different eyepieces.
As expected, this occurs with the 5.5mm eyepiece: 114x and 1.01 arc ".
CONCLUSIONS:
- Resolutions in the above table appear to improve with increasing eyepiece magnification. This is to be expected: more magnification helps to better observe the telescope's maximum resolution, which is obviously invariant.
- The SkyMax-102 has an excellent resolution, 1.01 arc", 13% better than Dawes Limit. This improvement has been reported for some similar telescopes in field tests.
- The resolution w.r.t. human eye is 114x. This is better than Dawes's 101x, but still nowhere near the declared useful magnification of 204x.
- Replacing the low-cost 10mm and 8mm with a higher-grade eyepiece such as the Explore Scientific 9mm we get significant improvements in image quality and eye relief, but a very small improvement in resolution.
- A value not shown in the table is how much the Magnification of every eyepiece exceeds the maximum resolution of the telescope (114x). With 25mm and 14mm we are well below it. However, with the 9mm the magnification goes up to 144x and therefore we slightly exceed what the telescope can deliver: we should not expect an image as sharp as with the 25mm and 14mm eyepieces. As for the 5.5mm eyepiece, it is good for magnification and can help with astrophotography of a detail, but we will not get any more information than with the 9mm: the higher magnification implies a reduction in visible sharpness.
- The Barlow 2X lens performance was very good: as expected, it slightly increased resolution for wide-angle eyepieces (because the eye can observe the chart lines better) and it decreased it for small-focal eyepieces (because we exceed the telescope's sharpness). There was a remarkable exception: with the 8mm eyepiece and the test chart in a sunset shadow, the luminosity reduction produced by the Barlow caused a significant drop in visibility, explaining the low 57x value.