# Formulas

By Matthew Buynoski

FORMULAS

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

``                (1.22 * L * 0.20265)/D``

``                where:  L is the wavelength of the light involved``
``                                   in nanometers``
``                        D is the dia. of the aperture in mm.``

For 203mm 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.

Magnification is easy: F/f = (tan&)/(tan&) = M

Eyepieces and Field of View...

The exact field-of-view formula:

``                T = 2 * arctan<(f/F) * tan>``

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

[An approximate formula is T = (f/F) * A where tan & is assumed equal to X]

The eyepieces involved are:

``Manufacturer & Model  Barrel      f         A``
``------------------------------------------------------``
``Vixen Lanthanum       1.25"      2.5mm     45 degr.``
``Orion Ultrascopic     1.25"      3.8       52``
``Pentax XL             1.25"      7         65``
``Televue Plossl        1.25"      8         50``
``Televue Nagler       1.25"/2"    9         82``
``Televue Nagler       1.25"/2"   13         82``
``Pentax XL             1.25"     14         65``
``Televue Plossl        1.25"     15         50``
``Orion Plossl          1.25"     17         50``
``Televue Plossl        1.25"     20         50``
``Televue Panoptic     1.25"/2"   22         68``
``Televue Panoptic      2"        27         68``
``Orion Plossl          1.25"     32         50``
``Pentax XL             2"        40         65``

The magnifications and true fields of view, using the more exact formula, calculate (with F = 2032mm for SCT, 480mm for Ranger) to:

``Eyepiece         Scope  TFOV       Magnif. Exit Pupil``
``------------------------------------------------------``
``2.5mm Vixen     Ranger  0.25 degr.  192X      0.4mm``
``3.8mm Ultrscpc  Ranger  0.44        126X      0.6``
``8mm Plossl      Ranger  0.89         60X      1.2``
``15mm Plossl     Ranger  1.67         32X      2.2``
``20mm Plossl     Ranger  2.22         24X      2.9``
``32mm Plossl     Ranger  3.56         15X      4.7``
``7mm Pentax      SCT     0.25        290X      0.7``
``9mm Nagler      SCT     0.44        226X      0.9``
``13mm Nagler     SCT     0.64        156X      1.3``
``14mm Pentax     SCT     0.50        145X      1.4``
``17mm Plossl     SCT     0.45        119X      1.7``
``22mm Panoptic   SCT     0.84         92X      2.2``
``27mm Panoptic   SCT     1.03         75X      2.7``
``40mm Pentax     SCT     1.44         51X      4.0``
``---------------------------------------------------------``

The max. AFOV possible for any eyepiece focal length is given by putting in the max. TFOV due to barrel diameter (B) limit:

``                tan& = B/2F    (B is about 31mm for 1.25" eyepieces,``
``                                        and about 50mm for 2" eyepieces)``
``or``
``                Tmax = 2 * arctan&``

``so that``

``                B/2F = (f/F) * tan&``
``or              Amax = 2 * arctan&``

``Tabular form, all entries in degrees...``

``       ---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     91.9*  118*``
``1250   1.42    2.29     20     75.6   103*``
``1500   1.18    1.91     25     63.6    90.6*``
``1750   1.01    1.64     30     54.6    79.6``
``2000   0.89    1.43     35     47.8    71.0   *There are no opti-``
``2032   0.87    1.41     40     42.4    64.0    cal designs that can``
``2250   0.79    1.27     45     38.0    58.1    take advantage of``
``2500   0.71    1.15     50     34.4    53.1    these potential large``
``2750   0.65    1.04     55     31.5    48.9    AFOV's.``
``3000   0.59    0.95     60     29.0    45.2``
``3250   0.55    0.88     65     26.8    42.1``
``3500   0.51    0.82     70     25.0    39.3``
``3750   0.47    0.76``
``4000   0.44    0.72       Eyepieces come in about 4 flavors on field``
``4250   0.42    0.67        of view: 82 degr. Nagler type, 65-68 degr.``
``4500   0.39    0.64        widefield types, 50 degr. Plossl types, and``
``                           40-45 degee ortho types. Compare to the table``
``values to see when a particular type "limits out" for that barrel dia. For``
``example, there is no use for f greater than about 25mm on 1.25" barrel wide``
``field eyepieces, or for greater than about 55mm on 2" barrel for a Plossl``
``design.``