Refracting telescope and minium angular size

wrote in message news:579a7f2a-ec38-4d6b-a820-

On Apr 29, 2:33 pm, "Robert" wrote:
snip How do I compute the smallest angular size which can
be resolved given the size of the refractor, when no
wavelengths are given?

You should state the assumed wavelength that you applied. Two common
choices include:

1) Dawes original assumption of two equal blue/white binaries at 460nm
wavelength (blue/white light).

2) Rayleigh's criteria that assumes 550nm yellow light.

Depending on the fiducial wavelength used, summary equations for
Dawes' and Rayleigh's criterias will vary slightly between any two
astronomy texts or books.

Another common wavelength that I have seen assumed in astronomy texts
is 500nm.

The common form of equation for these resolution criteria is based on
the angular size of the Airy disk, taken at:

A_radians = 2.44 * lambda / D Eq. 1 where D (aperture) and lambda
(wavelength) are in the same units of measurement.

A_arcsecs = A_radians * 360/2pi() * 60 * 60 Eq. 2

The easiest common length basis for lambda and D is meters. So lambda
- wavelength in nanometers - and aperture D in millimeters are
converted using:

D_meter = D_mm / 1000# Eq. 3
lambda_meter = (lambda_nanometers * (10# ^ -9#)) Eq. 4

Dawes criteria states that two equal blue/white doubles in a 1 inch
aperture can be split when they are separated by 1/2 the Airy disk
size.

Collecting all the above terms and dividing by 1/2 in the aperture
units of millimeters gives:

Dawes criteria = 116 / D_mm at 460nm Eq. 5

Dawes criteria for a one inch aperture is:

4.56 arcsecs (per inch of aperture) = 116 / 25.4 mm

This is the most common statement of Dawes criteria - that the minimum
resolution of a telescope is 4.56 arcsecs per inch of aperture.

Rayleigh's criteria is similar but is based on 550nm light and reduces
to:

Rayleigh's criteria = 138 / D_mm at 550nm Eq. 6

5.45 arcsecs (per inch of aperture) = 138 / 25.4 mm (per inch of
aperture)

This is the most common statement of Rayleigh's criteria - that the
minimum resolution of a telescope is 5.45 arcsecs per inch of
aperture.

The best wavelength value to use, where the wavelength is not
specified, depends on your stated purpose. If you want a stringent
resolution test based on a common easily identified OB class equal
magnitude doubles - 460nm would be the best choice. If you are
talking in generalities regarding the most commonly encountered visual
observing condictions - 550nm or 500nm - might be a better choice. If
you are imaging with a CCD camera through a R or near infrared
bandpass I filter, 656nm might be the best choice.

The take-away point is that there a number of variants of the
simplified equations for Dawes, Rayleigh and/or Sparrow's criterias
floating around in textbooks, astronomy books and on the internet.
Most of the variations can be attributed to slightly different
selections of the fiducial wavelength.

- Canopus56

P.S. - A good practice exercise is to see if you can use Eq.s 1-4 to
derive Eqs. 5 and 6 in a spreadsheet.

The Dawes limit. Put simply, (4.56/d) sec, d in inches. It's an empirical
formula, so the wavelength doesn't matter. Dawes worked it out assuming
yellow stars of magnitude 6.