Well, after the saga of the disappearing edits and the successful outcome of my “Mars occultation” project, plus all the other diversions life throws at one, where were we? Ah yes – the problem of different diameters for Sun and Moon.
On which topic, I can but say “Yes, you are quite right”. I have to confess that I committed the cardinal error of looking for the answer before fully analysing the problem! The analyses of the general problem on the Internet all seem to take the centre-to-centre distance as the variable parameter rather than the chord distance, which does indeed result in some rather involved calculations – sometimes involving integration between limits! Conversely, using the chord distance and the w-sin(w) formula makes the extension to unequal diameters almost trivial, and I agree with your answer.
Using this principle in my spreadsheet, I also agree that the result for f=0.99253 is 18.08%. Interestingly, if I use the values 1201 and 845 for the Sun diameter and chord length respectively (instead of 1200 and 846) together with f=0.99253, the result is 17.955%, or a magnitude of 0.29093, compared to the online answer of 17.960% and 0.29098, which is exceedingly close and tends to confirm my estimate of “accurate to 1 pixel”. I am thus still happy with my result as it agrees with the online answer to within a variation of 1 pixel whether the online calculator used the f factor or not.
As to the operative limit of the formula, do you not mean “until w or W is GREATER THAN pi”? Fortunately, if f=1 this can never happen as when the angle is pi the event is an exact total eclipse and the formula gives the correct answer of 100% and 1.00 for the obscuration and magnitude. If f<1 the formula can indeed fail as we are then heading for an annular eclipse, for which the chord length actually decreases at some point and then becomes undefined during the annular phase of the eclipse. If f>1 the formula seems to give sensible answers until the chord length is equal to the Sun’s diameter i.e. the case which would be an exact total eclipse if f=1, after which it will give wrong answers becuase, as in the case of an annular eclipse, the chord length will decrease as the Moon moves further over the Sun’s disc. The formula will thus give reducing values of obscuration whereas it is of course increasing. Further analysis required, I think!