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    Re: Side error and Lunars
    From: Modris Fersters
    Date: 2025 Jan 26, 09:00 -0800

    Hello everyone!

    Recently Charles McElhill asked Frank: I have a question regarding side error and your favorite subject, Lunars. Is there any negative impact to the lunar distance measurement of the process if there is a small amount of side error?”

    The question was addresed to Frank, but I hope I may dare to make some additional comments.

    From time to time Frank in his posts emphasizes the insignificance of so called “side error”. And he is absolutely right (maybe even more right than he thinks; but this is discussed further in this post). I think there are two aspects of this problem:

    1) resulting numerical error values, caused by side offset;

    2) how large side offset is torelable for sextant sights (ignoring accuracy aspect).

    The first above mentioned aspect is calculable. A good source of sextant error analysis is W.Chauvenet’s “A manual of spherical and practical astronomy” vol.II. 

    Errors produced by inclination of the horizon mirror are discussed in pages 116-117. Chauvenet gives a formula to calculate this error:

    Error=2 k2 sin(1”) cos2ß cotɤ,

    k - inclination angle of the horizon mirror (");

    ß - angle between the sight direction and the perpendicular of the horizon mirror(°);

    ɤ - sextant angle (°).

    I calculated error values for different offsets and put them in a table (see attached file). For clarity I added images of the view through the telescope for different values of the offset. Everyone can take a look at the table, get error values and decide whether the error caused by the specific offset satisfies his needs or not.

    I compared my results with those from the Frank’s post some 3 years ago (27.09.2021) and I got much smaller error values than he. Frank published error values for 5’ offset:

    0,5°….error 1’

    5°….error 0,1’

    ≥25°….error ≤2”

    ≥45°….error ≤0,5”

    My results (using Chauvenet’s formula) for the same input data are:

    0,5°….error 0,36’

    5°….error 0,04’

    ≥25°….error ≤0,007”

    ≥45°….error ≤0,003”

    I don’t know why there is such a discrepancy. Initially I thought that he had used different ß angle (see attached file, where I drew a scheme illustrating this angle), but the effect of this angle is minimal within typical range of this angle for most part of the sextants. I can only speculate: whether Frank used another formula or maybe he made a mistake by entering the formula directly with  offset value. Chauvenet’s formula uses the inclination angle of the horizon mirror (not the offset value). It is obvious that the inclination angle of the mirror is a half of the offset value (for the same reason rotation of the index arm by 60° corresponds to 120° reading on the limb, not 60°). For example, if we want to calculate the error for 20’ offset, we must use mirror inclination angle 10’.     

    The second aspect I had mentioned is:  how large side offset is torelable for sextant sights (ignoring accuracy aspect). When can large offset values  become undesirable? I think this happens when it does not allow to do something that I can do when there is no offset. For example I can comfortably use the Sun for determing the index  corrections when the side offset is within a range of about 0’-10’. To determine the index correction by Star the offset value will depend on the power of scope. For about 6…7x power scope the optimal offset for me is about 1’. Without scope – may be about 5’. Definitley not 15’, 20' or 30’. Of course, these values depend on individual comfort requirements. But anyway I can’t imagine why I should use sextant with very large side offset, without adjusting it. Maybe only if the instrument has damaged adjusting screws. I don’t see why not to once adjust offset to about 1’…2’ and forget about it. OK, maybe there are some plastic or homemade sextants with dramatically changing offset values, but in this case no one will expect high accuracy from them. Besides about 1' offset value allows me to make 4SD sanity check (if I want), practically without effecting results (but, for example, 5' offset will cause measurable negative effects to the results).     

    The main conclusion from all this is: small offsets (1’, 2’, 5’) allows you to measure angles within 0,1’ accuracy limits starting from 2° sextant angles. There is no need to eliminate “side error” to zero! And there is no need to get in other extremity making very large side offsets, thus disabling (or making uncomfortable) to get IC from Sun or Stars. But let's suppose one doesn’t need Sun and Star IC corrections, because he uses sextant only for daylight Sun sights and checks IC, using horizon. In this case (as it can be seen in the attached table) one can get accuracy within 1’ for all angles larger than 10° even with 40’ offset! Thus the demands on the maximal side offset value are dependent on:

    • the tasks, which observer is going to perform;
    • accuracy reguirements;
    • the lowest sextant angles, observer is going to get.   

    Modris Fersters

    PS 

    I am glad that seemingly trivial problems are discussed here in NavList more deeply. Any comments and recheck of my calculations will be highly appreciated. 

    File:
    Errors_from_side_offset_26.01.2025.pdf
       
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