Here are some formulae that I find useful from time to time. If you have any corrections or others that may be useful please send them to the email address on the contact page.

Eyepiece apparent vs real field of view

CCD arc-sec/pixel and focal ratio

arc-sec/pixel using known object size

f/ratio

focal length using known object size

CCD Critical Sampling

Focal reducers

Focal reducers & refractors/lenses

Focal reducers & SCTs - !NEW!

Eyepiece Projection

CCD Dust Shadows

CCD Filter Reflections

Visual limiting magnitude

Signal to noise ratio

Size of Airy disk

Mount Periodic Error

Star trail lengths

Auto Guider rates

Critical Focus Zone & CCD Focus Zone

Two methods, the first is an approximation but is often the easier to work out due to lack of information about field stop sizes.

The formula for arc-sec per pixel is:

Alternatively you can measure the size of an imaged object in pixels, and divide that objects known size in arc-seconds by the size in pixels.

Solving for focal ratio, this becomes:

You can use this formula to get a reasonably accurate focal ratio from any image where you know the angular size of an object (or angular distance between two stars).

Or if you want the focal length used for an image and know the angular size of the object imaged:

Use this formula to calculate the minimum focal length required to fully sample a high resolution image with any particular CCD.

This is based on the assumption of perfect seeing and the Airy disk being the limit of resolution. For planetary imaging using 'lucky' imaging short exposure techniques this is a reasonable assumption. For long exposures the FWHM value for any particular night would be a better measure of the resolution obtainable.

I have made the assumption that adequate sampling requires the pixel spacing to be three times greater than the scopes resolution (forget Nyquist!)

Use this formula to calculate the resulting focal ratio when using a focal reducer with a refractor or a simple lens:

And this one to calculate the amount of in-focus required by that set-up:

Where

a = Distance of CCD from focal reducer

b = Focal length of FR c = Focal ratio of scope

The Meade/Celestron focal reducers have the following focal lengths

.:

[note: Around 2006-2010 Meade manufactured some f/6.3 focal reducers with a focal length of around half what they should be making them unsuitable for use with SLR cameras or filter wheels, any marked "Japan" are OK, as are later "China" ones.]Note: the Meade and Celestron focal reducers are called f/6.3 reducers, this is because when used as designed they result in a focal ratio of approximately f/6.3 on an f/10 SCT.

They are NOT 0.63x reducers.Meade FR f/3.3 focal length = 85 mm

Celestron/Meade FR f/6.3 focal length = 240 mm

William Optics 0.8x FR focal length = 260 mm

ATIK 0.5x FR focal length = 80 mm* Note that the in-focus figure assumes that the FR-CCD spacing is added to your physical imaging train length (as when using the Meade/Celestron FR's with spacing tubes). If you are using a FR like the ATIK that is fitted internally, then you have to add the FR-CCD spacing to this figure (to make it smaller). If you are using a FR like the Meade/Celestron then you will have to subtract the depth of the FR itself to this figure (to make it a larger negative number).

Use this formula to calculate the resulting focal ratio when using a focal reducer with one of the 'classic' SCTs, note that the calcuations are only valid within 'sensible' ranges of focus. They are based on the FR being placed in its designed position at the back of the SCT:

Moving mirror SCTs require special treatment because their focal length changes (lengthens) as you adjust focus to compensate for the in-focus required from inserting the focal reducer in the optical path.

Where

a = Distance of CCD from focal reducer

b = Focal length of FR

c = Focal ratio of scopeThe Meade/Celestron focal reducers have the following focal lengths

.:

[note: Around 2006-2010 Meade manufactured some f/6.3 focal reducers with a focal length of around half what they should be making them unsuitable for use with SLR cameras or filter wheels, any marked "Japan" are OK, as are later "China" ones.]Note: the Meade and Celestron focal reducers are called f/6.3 reducers, this is because when used as designed they result in a focal ratio of approximately f/6.3 on an f/10 SCT.

They are NOT 0.63x reducers.Meade FR f/3.3 focal length = 85 mm

Celestron/Meade FR f/6.3 focal length = 240 mm

William Optics 0.8x FR focal length = 260 mm

ATIK 0.5x FR focal length = 80 mm* Note that the in-focus figure assumes that the FR-CCD spacing is added to your physical imaging train length (as when using the Meade/Celestron FR's with spacing tubes). If you are using a FR like the ATIK that is fitted internally, then you have to add the FR-CCD spacing to this figure (to make it smaller). If you are using a FR like the Meade/Celestron then you will have to subtract the depth of the FR itself to this figure (to make it a larger negative number).

Use this formula to calculate the resulting focal length and focal ratio when using eyepiece projection:

Where:

epid = Distance of CCD from Eyepiece

epfl = Eyepiece focal length

**Note:** This only gives an approximate value for the resulting focal length and focal ratio. The distance from the eyepiece is hard to measure, and the nodal point of the eyepiece (the point of its effect - which is usually internal to the eyepiece) is normally unknown.

If you are troubled by dust shadows on your CCD images you can calculate the distance that the dust particle is in front of the CCD with the following formula:

Where:

Dist = Distance from CCD surface in mm

p = CCD pixel size in microns

f = Focal ratio of imaging system

d = Diameter of dust shadow in pixels

Want to know where those annoying reflection disks around stars are coming from?...

To calculate this

Where:

Dist = Distance from CCD of reflection surface in mm

D = Diameter of reflection disk in image in pixels

P = CCD pixel size in microns

FR = Focal ratio of imaging system

A rough formula for calculating visual limiting magnitude of a telescope is:

The photographic limiting magnitude is approximately two or more magnitudes fainter than visual limiting magnitude.

A simplified formula for calculating the signal to noise ratio in an image is:

where:

S = total nebula signal

B = total background signal

D = dark current

RN = read noise from bias frame

n = number of sub-exposures

A formula for calculating the size of the Airy disk produced by a telescope is:

and

or

where:

D = Diameter of Airy disk in mm

λ = Wavelength of light (in mm here, normally in nm)

FR = Focal Ratio of system

A = Angular diameter of Airy disk in arcsec

fl = Focal length of telescope in mm

d = Diameter of telescope in mm

Note that this formula gives the diameter of the Airy disk, the resolution limit of the optics is generally taken to be the radius of the first minimum, i.e. half this angular value.

To calculate the periodic error of your mount using a CCD or webcam, you will typically put some numeric deviation data into a spreadsheet and create a graph. The data is normally in the form of a pixel offset and a timestamp. To convert this into an error in arc seconds you need to know how many arc seconds per pixel the images were captured, and the declination of the star used. The formula to plug into your spreadsheet is based on:

where:

D = Deviation of star from base position in pixels

R = Resolution of the camera in arc seconds per pixel

Dec = Declination of the star

Of course you can remove the requirement to use the declination in the calculation by using a star at (or close to) declination 0 degrees - the celestial equator.

You can use the calculator to estimate how long (or short) the star trails will be in a fixed camera image of the sky:

where:

F = Focal length of lens scope (trail length in same units as focal length)

E = Exposure length

T = Length of sidereal day in same units as exposure

D = Declination of the star

Or, for the CCD imagers:

Calculate how many pixels per second your auto-guider will move when the mount is being guided. Note that this assumes your auto-guiders axes are aligned with RA and Dec directions.

where:

str = Sidereal Tracking Rate (15.04 arcsecs/second)

gr = Mount Guide Rate (fraction of sidereal)

aspp = Autoguider arcsecs/pixel

dec = Declination of the star

Calculate the length of the zone in which the focused image of a star is smaller than the size of its Airy disk.

which simplifies to:

For CCD cameras, if we take a 2x sampling ratio:

where:

λ = wavelength of light

Note that because at low f/ratios the size of the Airy disk becomes significantly smaller than typical CCD pixels sizes I have introduced a value for the CCD Focus Zone. The value for the CCD focus zone takes the larger value of the CFZ, or where the Airy disk is half the effective pixel size (2x under sampling ratio) the CCD focus zone value defined above. For small focal ratios the CFZ gives a misleadingly small figure for imagers.