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Astronomy Calculations


Sky background magnitude

One big problem when doing astro photographing is the light polluted sky. The light pollution can be expressed in magnitude per arc second area, arcsec2 .

With the usual Sky Brightness meters you can see what light pollution values you have for the moment, the goal of this calculation is to see what light pollution you had on your old photos.

To find a value of the Light pollution we need a reference star, it will be better results if the reference star is high up in the sky and free from surrounding stars, close to Zenith. On the other hand, when examine old photos you take a ref star in the center of the photo whatever the camera pointing to.

The wavelength where the magnitude is related to shall correspond to the filter you have on the camera. The visual wavelength band is the most common, it's not the full visual spectra, it's more narrow, the mv (magnitude visual) is centered at 550 nm and 160 nm wide. The human eye is most sensitive in the green / yellow spectra during low light condition then this will be wrong to use the blue spectra.

In books, at least earlier ones the magnitude is only denoted mag, I thought that must be mv (visual). After I got answer on my question to my earlier teacher prof Gösta at the University, he told me that's normally a blue spectra because of the earlier photo technique. The color system in astronomy is complicated, I can read that more then 200 different system exists.

Better to use a star catalog where the magnitude data are more in detail which spectra it belongs to. I normally use CdC (Sky Chart), a star chart software to find the stars magnitudes, just right click on the star and an info box comes up.

If you have a color camera you can use the green channel as a rough estimation of mv.

We also need to know something about the telescope and camera to get this correct, we must know how many arcsec2 one pixel cover, we get that from the pixel scale.

  • If you don't already know your systems pixel scale you can calculate it here:
    Pixel scale

The image you take your data from must be dark and bias calibrated. Use the center of the sensor, then less problem with vignetting.

Because the star normally covers many pixels you must add up all of them that the stars covers. But first you must subtract the background, normally done by automatic in your astro tool you use.

Say that your ref star covers 5 pixels and the background mean ADU is 100 per pixel in the dark areas.

Pixel 1: 150refADU - 100bgADU = 50,
Pixel 2: 200refADU - 100bgADU = 100,
Pixel 3: 250refADU - 100bgADU = 150,
Pixel 4: 200refADU - 100bgADU = 100,
Pixel 5: 150refADU - 100bgADU = 50.

These add up to 450 ADU star flux.

It's very important that the reference star is not oversaturated!

If you have a color camera it's little bit more complicated. I think it's best to use the CFA format, calibrate it with dark and bias and then do a demosaic and keep the green channel, not debayer.

Some software give two green channels when demosaicing which is the correct way to do it, then just average them together.

If you find the math behind this interesting:

msky = 5 / 2 * log10 ( Iref / Isky ) + mref

Where Isky = IskyADU / pixelscale2


Type in your data

It start to calculate as soon you change or write new figures in the white or dark red boxes. Do not exceed the maximum number of characters, delete characters if necessary.

You use the information on your own risk! There can always be a mistake in my equations behind the calculations, check that the result is correct. Let me know if you find something wrong and I try to correct it. Some calculations are very simple done and not correct in the small details.

From a reference star we calculate the background brightness or light pollution:

Reference star magnitude
1.00 to 25.00 m  

Pixel scale of your optic system:

Pixel scale  
    = Calculated results
0.10 to 100.00 arc seconds / pixel       arcsec2 / pixel

ADU sum from all reference star pixels:

Star flux ADU counts  
1.0 to 1000000.0 ADU counts  

Mean background flux ADU around the reference star:

ADU background  
    = Calculated results
1.0 to 10000.0 ADU / pixel       ADU / arcsec2

Sky background magnitude

  = Calculated results
  m / arcsec2

Sky backgrounds from magnitude of 19 or above are good.

My own result:

Test no 1:
From the balcony 10 km west from city center, reference star close to M97 at altitude 40 degree, ref star's magnitude comes from CdC (Sky Chart):
Data Value Comments
Reference star magnitude 9.33 mv Visual
Pixel scale 1.5 arc second / pixel Canon 6D / f 910 mm
Star flux  10095 ADU  
Sky flux background 839 ADU / pixel mean value
Sky brightness magnitude 12.9 mgreen / arcsec2 very light polluted !
From that if everything was correct calculated I know now that if I move from my balcony 10 km from city center to a really dark place the background signal will decrease about 6 magnitudes or 250 times.
Test no 2:
From the summerhouse 30 km east from the city center, reference star close to M31 at altitude 50 degree:
Data Value Comments
Reference star magnitude 8.46 mv Visual
Pixel scale 2.48 arc second / pixel Canon 5D / f 684 mm
Star flux 37323 ADU  
Sky flux background 50.07 ADU / pixel mean value
Sky brightness magnitude 17.6 mgreen / arcsec2 much better then balcony
This was much better, 1.4 magnitude (4 times) higher sky background compare to a really good place (19 magsky).
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