Author Topic: Some Basic Astronomical Formulas  (Read 2943 times)

buynoski

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Some Basic Astronomical Formulas
« on: September 30, 2014, 10:42:37 AM »
Some Basic Astronomical Formulas

    Caveat:  These formulas are theoretical, and not necessarily perfectly accurate for real world designs, where lenses are not always “thin”, field stops don’t equal barrel diameters, and eyes vary from person to person.  They do, however, serve as a reasonable first-pass guide to some of the optical properties of amateur equipment.

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1. Resolution. The ‘theoretical’ resolution (in arc-seconds) for a circular aperture is

      r = (1.22 * L * 0.206265)/D

                where:   L is the wavelength of the light involved in nanometers
                           D is the diameter of the telescope aperture in millimeters

The result, called the Rayleigh criterion, is roughly equal to the radius of the Airy disk.

Example: For 203mm (8") diameter, and 600nm light, this works out to 0.73 arc-seconds.  There are variations on this formula depending on what one calls “resolution”. Another popular variant is to replace the 1.22 by 1.0. In this case, for the same conditions, the resolution is 0.60 arc-seconds.  What we actually see is the magnified image of the Airy disk (radius: rM where M is the magnification), and what we call resolution depends a lot on the ability of our eyes to discriminate between different objects in the field of view.  The above Rayleigh criterion is for a high-contrast situation (bright stars on a black background) and may well be optimistic in other situations (low-contrast planetary detail).  The best the eye can do is about 1 arc-minute, but that's only in the central small area of the retina called the fovea and for situations that are both well-illuminated and high contrast.  The eye's resolution gets considerably worse for areas of the retina further from the fovea, for lowered contrast, and for lowered illumination. In photography, the value typically used for the eye's resolution (which limits how much an image can be enlarged before "blur" becomes noticeable) is somewhere between 3 to 4 arc-minutes (Kingslake: 3.4, Kodak: 3.7), and this is probably a better guide than rM, the scope's magnification times the Rayleigh criterion, for dimmer, lower contrast situations.  Eyes vary from person to person, leading to a lot of animated discussions on this subject where the participants literally do not see "eye to eye."  (See also under #6 below.)

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2. Magnification: 

      M = F/f =  (tan{A/2})/(tan{T/2})

                where:   F is the focal length of the telescope
                             f is the focal length of the eyepiece
                             A is the apparent field of view of the eyepiece
                             T is the true field of view of the scope/eyepiece combination

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3. Eyepieces and Field of View. The exact field-of-view formula is found by
        solving the equations for magnification above for T: 

        T = 2 * arctan{(f/F) * tan{A/2}}

                   where:    A  is again the apparent field of view
                                      T  is the true field of view
                                 F = focal length of the telescope
                                      f = focal length of the eyepiece

        An approximate formula is T = (f/F) * A. This is a little suspect in amateur astronomy where the A is not necessarily a small angle. 

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4. Exit Pupil. This one is easy. Divide the aperture of your telescope by the magnification, and you have the exit pupil. Because exit pupils tend to be small, it's generally wise to use the scope aperture expressed in millimeters so the exit pupil comes out in millimeters as well.

        P = S * f/F

                where:   P is the exit pupil diameter
                             S is the aperture of the telescope
                             F and f as defined above.

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5. Maximum Apparent Field of View. The barrel diameter of the eyepiece limits the apparent field of view for any given focal length. We take the barrel diameter to be 31mm for 1.25" eyepieces and 50mm for 2" eyepieces.

            tan{Tmax/2} = B/2F   

                             where: B is the barrel diameter (or, if significantly smaller
                                          than the barrel diameter, the field stop diameter)

or, solving for Tmax:

            Tmax = 2 * arctan{B/2F}

If you substitute Tmax as the value for T in the equation of section 3, then you get the maximum apparent field Amax:   

       Amax = 2 * arctan{B/2f}   

plus the focal length of the eyepiece with that Amax:

       f =  B/(2*tan{Amax/2})


That's tedious to use. Solved for some common cases, and put in tabular form, all entries in degrees, we have:

           ----Tmax----                       ----Amax----
F =       B=31   B=50                  f =    B=31    B=50   
----------------------------            ---------------------------
480       3.69    5.96                   2.5    162*     169*
500       3.55    5.72                      5   144*     157*   
750       2.37    3.82                    10   114*     136*
   
1000     1.78    2.86                    15     92       118*   
1250     1.42    2.29                    20     76       103   
1500     1.18    1.91                    25     64        91
   
1750     1.01    1.64                    30     55        80   
2000     0.89    1.43                    35     48        71      
2032     0.87    1.41                    40     42        64 
    
2250     0.79    1.27                    45     38        58    
2500     0.71    1.15                    50     34        53    
2750     0.65    1.04                    55     32        49
      
3000     0.59    0.95                    60     29        45
3250     0.55    0.88                    65       27        42
3500     0.51    0.82                    70     25        39
3750     0.47    0.76       
4000     0.44    0.72       *No current optical designs
                                       achieve these A’s. The
                                       biggest A today is 110 deg.


Eyepieces come in about 5 "flavors" of field of view: 100-110 ultrawides, 82-85 degree Nagler types, 60-70 degree widefield types, 50-55 degree Plossl types, and 40-45 degee ortho types. Compare to the table values to see when a particular type “limits out” for that barrel diameter. For example, an eyepiece focal length greater than about 25mm on 1.25” barrel wide field eyepieces is pointless, ditto for greater than about 55mm on 2” barrel for a Plossl design.  Or, putting it in a table for ‘popular’ values of apparent fields of view:


       AFOV       Max. Focal length ep that can have the AFOV
       of ep                B = 31mm   B = 50mm
     -------------------------------------------------------------------------------
        25degr.       70mm         113mm
        30                      58                93
        35                      49                79
       
        40                      43                69
        45                      37                60
        50                      33                54

        52                      32                51
        55                      30                48
        60                      27                43

        65                      24                39
        68                      23                37
        70                      22                36

        75                      20                33
        80                      18                30
        82                      18                29

        85                      17                27      
       100                     13                21
       110                     11                18     


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6.  Maximum Magnification. The human eye’s limiting resolution is about 60 arc-seconds for color (cones) vision in the central fovea, about 5 times worse than that (300 arc-seconds) in the area of the densest rods around the fovea (used for averted vision), and even worse than that the further away from the fovea the image gets.  This has a bearing on the maximum magnification you can use before reaching “empty” magnification that provides no further detail.  If we take the resolution of the telescope to be the Rayleigh criterion (#1 above), then M times that should be less than 60 arc-seconds for color vision (i.e. for daytime work or viewing stellar spectra at night) and less than 300 arc-seconds for night observing.   That’s for perfect seeing, a case that never exists.  Seeing is usually expressed in arc-seconds, i.e. the angular spread a star appears to have when looking in the telescope.  To account for seeing, add the seeing number to the scope’s resolution (seeing moves the whole Airy disc around) when computing the maximum detail magnification. That is:

           Mmax =  60/(r + s)  for daytime           Mmax = 300/(r + s)  for nighttime       

                       where:  r is the Rayleigh criterion (see #1 above) in arc-seconds
                                   s is the seeing, expressed in arc-seconds
                                                           

Examples.  The Rayleigh criterion for an 8” scope is about 0.7 arc-seconds.  For cone vision, the perfect seeing (s = 0) detail limit for magnification is some 86X  (60/.7 =  85.7).  For averted vision via rods, the perfect seeing limit for detail magnification is 428X (300/.7). If the seeing during the day is 2 arc-seconds (daytime seeing is usually bad), then magnfications beyond 22X [ 60/(2 + 0.7) ] provide no more detail.  If the seeing at night is 1 arc-second, then the maximum detail limit drops to [ 300/(1 +0.7) ] or 176X.

Caveats:  Eyes vary person to person. The spatial frequency of seeing (immediately local blurring of the object, vs. the object’s image moving around on the field of view) can vary as well as its magnitude.  Color vision at night can be problematical; you can still see some color (typically greenish or yellowish) while the eye loses resolution in the blue and red, to which it is less sensitive. What detail a given observer can discern in what scope leads to endless discussion. Lots of people have opinions, some very strongly held, on these matters but hard data is even harder to come by.  Use the numbers generated by these formulae as a general guide, not precise predictions.
« Last Edit: July 04, 2015, 12:05:35 AM by buynoski »