Wilmslow Astro

Astronomy from a Cheshire suburb

Useful Formulae

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
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


Eyepiece Apparent vs Real Field of View

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

Method 1

Magnification = Focal Length Scope / Focal Length Eyepiece

Real FoV = Apparent FoV / Magnification

Focal length of scope
Focal length of eyepiece
Eyepiece apparent FOV
 
Calculated Real Field of View

Method 2

RealFoV = (Eyepiece Field Stop / Focal Length Scope) * 57.3

Focal length of scope
Eyepiece field stop diameter
 
Calculated Real Field of View

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CCD arc-sec/pixel & Focal Ratio

The formula for arc-sec per pixel is:

arcsec/pix = (pix size/focal length) * 206.3

Focal length of scope
CCD pixel size (microns)
Pixel binning
 
Calculated arcsec/pixel

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.

arcsec/pix = objectsize / imagesize

Object size in arc secs
Image size of object in pixels
 
Calculated arcsec/pixel

Solving for focal ratio, this becomes:

f ratio = (pix size * 206.3) / (aperture * arcsec per pix)

CCD pixel size (microns)
Pixel binning
Image arcsec / pixel
Scope aperture (mm)
 
Calculated focal ratio

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:

focal length = (object[pix] * pix_size * 206.3) / object[arcsec]

CCD pixel size (microns)
Pixel binning
Object size in pixels
Object size in arc seconds
 
Calculated focal length

 

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CCD Planetary Critical Sampling

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!)

f length >= (aperture * pixel)/(wavelength * 1.22) * 3

Light Colour
Wavelength (nm)
Scope aperture (mm)
CCD pixel size (microns)
 
Calculated focal ratio
Calculated focal length

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Focal Reducers

Use this formula to calculate the resulting focal ratio when using a focal reducer:

f ratio = (1 - (a / b)) * c

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

in focus = a - (b * a) / (b - a)

Focal Reducer
CCD-FR distance (mm)
FR focal length (mm)
Focal ratio of scope
 
Calculated focal ratio
Calculated in-focus required*

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 Meade manufactured some 0.63 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.]
:

FR 0.33x focal length = 85mm
FR 0.63x focal length = 285mm

William Optics 0.8x FR focal length = 260mm

ATIK 0.5x FR focal length = 80mm

* 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).

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Eyepiece Projection

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

magnification = (epid - epfl) / epfl

Resulting FL = scope FL * Magnification

Resulting fRatio = scope fRatio * magnification

CCD-EP distance (mm)
Scope focal length (mm)
Scope focal ratio
Eyepiece focal length (mm)
 
Calculated focal ratio
Calculated focal length (mm)
Magnification factor

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.

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CCD Dust Shadows

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:

Distance = (p * f * d) / 1000

CCD pixel size (microns)
Pixel binning
Imaging system focal ratio
Diameter of dust shadow (pixels)
 
Calculated distance from CCD (mm)

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

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CCD Filter Reflections

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

To calculate this

Distance = (D * P * FR) / 1000

CCD pixel size (microns)
Pixel binning
Imaging system focal ratio
Diameter of reflection (pixels)
 
Calculated distance from CCD (mm)

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

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Visual Limiting Magnitude

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

VLM = 7.5 + (5 * Log(Aperture [cm]))

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

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Signal to Noise Ratio

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

SNR = S / SQRT(S + B + D + nRN^2)

where:

S = total nebula signal
B = total background signal
D = dark current
RN = read noise from bias frame
n = number of sub-exposures

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Size of the Airy Disk

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

D = 2.43932 * l * FR

and

A = 2 * ArcTan(D / (2 * fl))

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

Light Colour
Wavelength (nm)
Telescope Diameter (mm)
Imaging system focal ratio
 
Calculated size of Airy disk (mm)
Calculated angular size of Airy disk

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Calculating Mount Periodic Error

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:

Error = (D * R) / cos(Dec)

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.

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Star Trail Lengths

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

Star trail length

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:

Star trail length in pixels

Focal length (mm)
Exposure time (secs)
Declination
Pixel size (microns)
Pixel binning
 
Star trail length

Auto Guider Rates

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.

pixels/sec = (str * gr * cos(dec)) / aspp

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

Guide scope focal length (mm)
Mount guide rate (fraction of sidereal)
Declination of guide star
Auto Guider pixel size (microns)
Pixel binning
 
Guide rate

Critical Focus Zone

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

cfz = 2 x f-ratio x AiryDisk

which simplifies to:

cfz = 4.88 x f-ratio^2 x wavelength

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

ccd fz = f-ratio x pixel size

where:

λ = wavelength of light

Light Colour
Wavelength of light
Focal Ratio
Camera pixel size (microns)
Pixel binning
 
Critical Focus Zone
CCD Focus Zone

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.

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