Bell9 says that resolution will be noticeably impaired if the off-axis aberrations (which the image may exhibit even at the centre of the field due to imperfect collima-tion) are approximately equal to the empirical resolution limit 4''.56/D (Dawes limit). Despite some statements to the contrary in the literature there is no doubt whatever that this criterion is true, as is fully borne out in my experience by a good deal of very exacting double-star observation at the 0.3-0.4'' level with an f/7 mirror of 12.5 inches diameter. Thus to achieve full resolution we must operate on or near the true optical axis, at Q< 0max where 0max is the angular displacement off-axis at which | + a = the Dawes limit. In view of the comments above regarding the smallness of a, we can approximate this condition closely by the simpler | = Dawes limit (first-order approximation, valid for all normal f ratios) which, with all angles in radians, is

16F2 D where D is the aperture in inches. Hence 0 _ 1.18 x 10-4 F2

max d or in arcminutes:

0.405 F2

This angle is the limitation to the field of critical definition centred on the optical axis and is, therefore, also a measure of the maximum angular error which can be tolerated in collimation of the telescope's optics, specifically, in the squaring-on of the main mirror. The noteworthy point here is the extremely small value of this angle even for unfashionably long Newtonians (which, of course, are far better in this sense since 0max max

^ F2), far smaller in fact than the attainable tolerance of the methods of collimation in general use: for the 12.5-inch at f/7.04, the formula gives 0max = 1.6' - a value again fully borne out by my observational experience. (This implies a maximum field of critical definition of 3.2', compared with an actual field of 2.4' on this instrument at the power used for subarcsecond pairs (x825).) In fact, I would say that for really critical double-star work right at the limit of resolution on a Class I or II (Antoniadi) night, aberrations become quite noticeable even at half this level, so reducing 0max to 0.8', i.e. 48'' - about the size of Jupiter's disk! Furthermore, as this angle varies as the square of the f-ratio, the modern generation of short-focus Newtonians are at a huge disadvantage here and it is probably true that no Newtonian at f/5 or below will ever, in real observing conditions, reach anything approaching its limiting resolution. Even if one can guarantee the hyperfine collimation tolerance demanded (and in my experience these instruments are used most of the time with squaring-on checked only to ± 0.5° or worse, i.e. only the first approximation to collimation is carried out), the objects observed will almost never lie in this minute axial patch of the field of view.

Under what conditions will the first-order approximation above for 0max be valid? We may reasonably say that astigmatism is negligible if, say, o/^ <0.1 and this imposes the condition that F 0max < 3/80, which on substituting the first-order approximation for 0max (in radians) yields 1.18 x 10-4 F3/D < 3/80. Thus the mathematics is self-consistent, and the first-order result for 0max is valid, if and only if F3/D < 318. For the 12--inch telescope this parameter has the value F3/D = 27.9 -well within the "coma-dominated" regime. In fact, there is no focal ratio of Newtonian likely to be encountered in ordinary astronomical use, in which the off-axis limitation to the field of critical definition is due to anything other than the onset of essentially pure coma.

It is worth bearing in mind a few numerical values of this field, 20max, as given by equation (11.3) for some common Newtonian configurations: 8.6' for a 6-inch at f/8; 3.6' for an 8-inch at f/6; 2.0' for a 10-inch at f/5. Equation (11.2) then implies that at the edge of a field n times wider than this, the aberration will be n times larger than the Dawes limit.

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