Telescope Reviews: 8" lunar crater resolution...

David (and Pete),

Thanks for clearing this up.

1. Moore's tables in Amateur Astronomy (1968) and The Amateur Astronomer (1990) sound like reprints of Ewen Whitaker's table from Wilkins and Moore's The Moon (1955/1961 - I haven't seen the 1955 edition, so I don't know if it had Whitaker's Appendix). The lines David quotes from the table are identical to those in the 1961 book (although Whitaker originally included at the end of the rows his values for the "Apparent Size of Smallest Cleft" as seen in the eyepiece, which he estimated to be 8 times the "actual" size). In view of where these numbers come from, it would seem to me that the "9 miles/D[inches]" threshold rule for visual observations should be referred to as "Whitaker's formula" (rather than Moore's). As previously mentioned, Whitaker's exact formula (which he presumably used to generate his original table) is "8.7 arc-sec/D[inches]". At an "average" lunar distance of 384,400 km (238,860 mi) that translates to 10.1 mi/D[inches], so Whitaker seems to have assumed a shorter distance -- something closer to 213,000 mi (343,000 km) -- for his table. That is, he seems to be trying to give the size of the smallest craterlet he thinks will be detectable when the Moon is at its closest approach to Earth.

2. The region around Beer was thoroughly photographed by Apollo 15 (1971), but if Moore made his statement about detecting a 500 yard (0.46 km) craterlet there as early as 1968, then I'm guessing he means that there is something on one of his drawings from the 1950's which, when checked against a Lunar Orbiter IV photo (1966) proved to be a 0.46 km crater. The problem with claims of this sort is, as Mardi has pointed out, there are numerous examples of Wilkins and Moore's drawings made with the 33" Meudon refractor in their 1961 book. But unlike the work of their more conservative predecessors, these drawings are invariably peppered with totally imaginary features. Little they show below about 2 km (actual size) has any obvious connection with reality. So Moore may indeed have drawn a hill with a 500 yard pit, but it's very hard to assess the significance of that since (absent the drawing) one has to assume he drew many others of that size that didn't exist. Last week I copied most of the drawings from the library's copy of Wilkins and Moore, but somehow missed Moore's drawing of Beer. As example of Moore's work with the 33" refractor, his drawing of Grove shows only what he records as a 0.5 km pit on the crater's north rim. This turns out to be non-existent, while he completely missed prominent 2, 3 and 4 km craters on the floor, as well as a 3-km diameter central peak. The drawing of J. Herschel, also attributed to him, is all but incomprehensible when compared to photos from space.

3. David, your Plato scale sounds correct to me. Plato looks like it initially had a nearly perfect circular form, now scalloped by slumping. 100 km (59 miles) seems a good estimate for the present inner rim diameter. An exact east-west rim-to-rim diameter, measured from what you've labeled the "west rim bow-in" to the opposite side is 99.5 km (58.9 miles). Features like the "mass of faulted rock" (once known as Plato Zeta) labeled in your illustration look like they've moved the rim-crest out by 5-7 km in places (of course, even the sections that look undisturbed may already have been moved out by collapses whose debris is now hidden under the floor). A sort of least-squares fit through the present rim scallops would probably give a "diameter" of around 104 km.

4. Regarding the accuracy (or lack thereof) of the craterlet diameters I posted on Chuck Wood's the-Moon Wiki, the high resolution USGS scans of the Lunar Orbiter images that have become available in the last few years are a considerable improvement over the LPI's scans from Bowker and Hughes. Also, using the LTVT circle drawing tool removes almost all uncertainty about the scale. By knowing the geometry in which the images were photographed, and by identifying the pixel locations of distant points at known lunar longitude/latitude, the scale is known to a fraction of 1%, and the circle tool permits a sort of visual least-squares-fit to the rim crest. Unfortunately, the definition of the rim-crest position (the transition from light to dark) is not always as sharply defined as one might hope, so there is a certain arbitrariness in how the circle is set that can be affected by magnification, contrast and room lighting. Plus there is my inherent sloppiness and human error. So your guess of +/-0.1 km accuracy is probably not unreasonable, although I feel they are a bit better than that. As a test, the diameters I posted on the Wiki were recorded on LO-IV-127H. Since it takes only a few minutes to do so, I just now re-measured the "Big Four" on LO-IV-127H, and then measured them for a first time on LO-IV-134H and on LO-IV-128H (much less sharp). The scale/geometry of each of these images was independently determined some months ago. Here the results (without editing or fudging -- my memory is too poor to remember the numbers from one photo to the next):

Re-measured Craterlet Diameters [km]

# Wiki 127H 134H 128H Aver SD
- ---- ---- ---- ---- ---- ----
A 2.44 2.47 2.49 2.47 2.47 0.02
B 2.09 2.05 1.99 2.09 2.06 0.05
C 2.22 2.29 2.23 2.23 2.24 0.03
D 1.98 2.05 1.99 1.91 1.98 0.06

The final two columns give the average and the standard deviation (scatter). It looks to me like +/-0.04 km would be a slightly better estimate of the actual experimental scatter in this particular image set. The difference in size between "B" and "C" looks significant to me. That between "B" and "D" may not be.

I didn't think to try to verify these numbers with the Clementine images. In general, their standard resolution is inferior to Lunar Orbiter IV, but if they took some in their Hi-Res mode they would be interesting to evaluate. As best I can measure them on the new (standard resolution) USGS Warped Clementine basemap (where there should be no doubt about the scale), the rim-crest diameters are: A=2.60, B=2.06, C=2.24, D=1.94 km. The diameter for "A" may be an overestimate because the Clementine lighting is from south to north and I'm including what looks like a possible image glitch (it looks like a double image in the mosaic) on the north rim.

It will be interesting to see what diameters Kaguya and Chang'e-1 obtain, but I doubt they'll be much different from these.

5. If one believes the list posted on the Wiki, Maurizio's 25-30 craterlets photographed with a 10" aperture corresponds to a detection threshold of 0.90 km. That can be compared with Bob Pilz' 18 possible craterlets (16 real), or 0.98 km, photographed with an 8" aperture. The rather modest gain (0.98 --> 0.90 km) in going from 8" to 10" seems in keeping with my earlier observation that the craterlet resolution threshold seems to improve only as the square root of the aperture, but Bob may have had some advantage from using a blue filter (shorter wavelengths give less diffraction) and newer image processing software. Even though the individual craterlet diameters given in the list may, as you say, be in error by up to +/-0.1 km, these are, for the most part, random errors; so the general trend expressed there (of numbers of craterlets in various size classes) should, I believe, still be valid. In other words, if, all else being equal and the performance of the telescopes being limited only by diffraction, an 8" scope can detect 16 craterlets ("0.97 km" in the list), we would expect a 10" scope to get to (8/10)*(0.98) = "0.78 km" in the list. "0.98 km" is between numbers 16 and 17 on the list, and "0.78 km" is between numbers 45 and 46. That implies that with "equal" performance, if one can detect 16 craterlets with 8" then it should be possible to detect 45 craterlets with 10", whether the individual craterlets listed were accurately ranked or not.

6. As to how the detectability of Linné compares to Plato craterlet "A", the rim-to-rim diameter of the bowl of Linné appears to me to be 2.23 +/-0.03 km. This was measured on the new medium resolution scan of Apollo AS15-M-0580 from Arizona State University registered to LTO-42A4. On lower resolution scans of AS15-M-0406 and AS15-M-0409 from the LPI, I get 2.28 and 2.24 km; and on LO-IV-097H and on LO-IV-098H I get 2.24 and 2.25 km. These are all independent measures on independently calibrated images. 2.4 km seems too large. My best guess is 2.24 km, which is significantly smaller than the 2.47 km (average) reported above for Plato craterlet "A", and about the same as Plato "C". If this is not convincing, the attachment shows Linné and Plato craterlet "A" in overhead views at the identical scale.

Since Plato "A" is 51.6° north of the Moon's equator versus 27.8° for Linné, there is a significant difference in foreshortening (the apparent north-south dimension of Plato "A" is compressed by a factor of 0.62 vs. 0.88 for Linné), so there would definitely be less total area of shadow to see in Plato "A". However, the east-west foreshortening which affects the separation of the bright-dark components (and therefore, one might think, is more important to the detectability) is not much different. Plato "A" is 9.1° west of the central meridian (on average) and Linné is 11.8° to the east.

I agree with you that the easier visual detectability of the bowl of Linné (if true – photographers may have the opposite experience) would be due to a combination of the larger visible shadow area (by a factor of about 1.3x from foreshortening alone, without even counting the greater hiding of the shadowed floor by the rim in the more northerly craterlet) and the greater contrast of the black shadow with the surrounding bright ejecta blanket.

-- Jim