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Converting surface brightness in stellar magnitudes to photon flux?

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#1 Jon Rista

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Posted 11 May 2016 - 04:26 PM

I like to plan out my imaging sessions a bit before I dive into them, when I can. I have been trying to figure out what kind of SNR gives me the noise levels I want for fainter structures, and I am not quite sure what the math is to do that. I ultimately need photon flux (photons per second per unit area, say per arcsecond), however for objects that actually have the data available, brightness is usually specified in stellar magnitudes. Is there a formula that can convert surface brightness in stellar magnitudes to photon flux? Along the same lines, I use an SQM-L meter to measure my skyfog brightness, and that reports in SQMs which as far as I know is also stellar magnitudes...so I assume that once I have an SQM reading, I could convert to photon flux the same way? 

 

Anyway, if anyone knows how to do this, I would be much obliged!


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#2 jhayes_tucson

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Posted 11 May 2016 - 04:54 PM

Jon,

Look in:  "Photonics Rules of Thumb," Ed Friedman, John Lester Miller, SPIE Press/McGraw-Hill, 2003, 2nd ed., pp 40-41.  It looks like they have exactly what you are looking for.  Of course the TOS prevent me for simply scanning and posting the relevant equations for you so you'll either have to buy the book or PM me for another, perhaps quicker way to get at it.

John


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#3 korborh

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Posted 11 May 2016 - 05:18 PM

It will depend on what Survey/catalog/filter the magnitude corresponds to.

Some catalog uses Pogson. SDSS uses asinh  (see here for example : http://classic.sdss....ms/fluxcal.html ).

 

You will have to study the catalog literature to see what is exact conversion, but then the filters used are photometric. If you are using LRGB filters with arbitrary bandpass, then I am not sure how the conversion will work.

 

I recently moved to using Sloan g', r', i' filters and so can directly calibrate with reference SDSS star magnitudes for the flux scalars.


Edited by korborh, 11 May 2016 - 05:18 PM.


#4 freestar8n

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Posted 11 May 2016 - 05:32 PM

Every factor of 100 is 5 magnitudes. If you have a 5 mag object and it is spread over 100 square arc-seconds, it would be 10 mag/arc-sec^2 (5+5). If it is over 10,000 square arc-seconds, it is 15 mag/arc-sec^2. Etc.

So if you know the size and the total magnitude you can get a good estimate - assuming fairly uniform brightness.

If you want to find the actual photon flux you would need to know spectral properties and how exactly they defined magnitude for something like an emission nebula.

If you have an image of a nebula and you know its magnitude and you know the total electron count you obtained in your exposure for the whole nebula - you can find how many "magnitudes per electron" you have and figure out the mags per arc-second or mags per pixel based on that - I guess. Just keep track of the log conversion.

Frank

#5 mikefulb

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Posted 11 May 2016 - 05:58 PM

If you have an image of a nebula and you know its magnitude and you know the total electron count you obtained in your exposure for the whole nebula - you can find how many "magnitudes per electron" you have and figure out the mags per arc-second or mags per pixel based on that - I guess. Just keep track of the log conversion.

Frank

 

I have wanted to be able to do this with DSS images but have never found a good photometric calibration between DSS counts and flux - anyone ever look into this?

 

It would make planning observations of extending objects a bit more quantitative.

 

Michael



#6 Jon Rista

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Posted 11 May 2016 - 06:21 PM

It will depend on what Survey/catalog/filter the magnitude corresponds to.

Some catalog uses Pogson. SDSS uses asinh  (see here for example : http://classic.sdss....ms/fluxcal.html ).

 

You will have to study the catalog literature to see what is exact conversion, but then the filters used are photometric. If you are using LRGB filters with arbitrary bandpass, then I am not sure how the conversion will work.

 

I recently moved to using Sloan g', r', i' filters and so can directly calibrate with reference SDSS star magnitudes for the flux scalars.

 

This would only apply if I was measuring the photon flux rate using data I acquired myself, right? If I want to refer to existing information about object surface brightness, say an object that is listed as being 26mag/sq", and simply convert that magnitude into photons/second. Then filters, and their photometry, wouldn't actually come into play, right?

 

I think Frank pointed out an important point, that I would need to account for the object's total area. That should be easy enough to determine to within a good enough precision to fit my needs, which is mostly just planning. I'm not actually looking to do photometry myself, I just want to figure out what minimum SNR gives me the kind of quality I am looking for, then figure out how long I would need to integrate in order to achieve that SNR for faint details. 



#7 freestar8n

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Posted 11 May 2016 - 06:49 PM

Yes - doing it accurately would require filters and calibrating with field stars and so forth - to correct for atmospheric extinction, star color, etc. But for just getting ballpark values you can use a reference star and its visual magnitude. I think Maxim is set up to do this - so you can calibrate the star aperture to a known star magnitude - and then read off the mags of other stars.

If a mag 5 star has 10,000 electrons in it - then a mag 10 would have 100. And a sky of mag 20/arc-sec^2 would have 1/100 electrons per square arc-sec - or 1 electron in every 10x10 arc-sec square. If I did the math right. In most deep sky images a mag 5 star would have a ton of electrons in it - and it would be saturated so you couldn't use it. I'm just giving the idea.

Frank
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#8 catalogman

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Posted 11 May 2016 - 07:23 PM

SNR catalogues generally do not list any magnitude (the flux listed is a radio flux).

 

For other uniform objects:

 

-first convert the total catalog magnitude to surface brightness (in mag/arcsecond^2):

 

http://www.users.on....brightness.html

 

-then convert the surface brightness to candelas/m^2:

 

http://www.unihedron...sky/magconv.php

 

                                                                                                -- catalogman



#9 freestar8n

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Posted 11 May 2016 - 07:36 PM

SNR catalogues generally do not list any magnitude (the flux listed is a radio flux).


I think Jon is talking about a different SNR.

Frank
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#10 mikefulb

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Posted 11 May 2016 - 07:45 PM

Thank you for the suggestions - I'd tried that route many years ago planning H-alpha runs but never felt I had a consistent solution.  I think I'll give it another try.

 



#11 catalogman

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Posted 11 May 2016 - 08:03 PM

See if you can work out consistent solutions given by the links:

 

- Look up the total magnitude T

 

- Calculate the area A in arcseconds

 

- Compute surface brightness in mags/arcsecond2:

 

       S = T + 2.5 * log(A)   [cf. first link]

 

- Compute flux in candelas/m2:

 

        F = 108000 * 10-0.4 * S   [cf. second link]

 

                                                                                      -- catalogman



#12 catalogman

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Posted 11 May 2016 - 08:04 PM

 

SNR catalogues generally do not list any magnitude (the flux listed is a radio flux).


I think Jon is talking about a different SNR.

Frank

 

 

Yes, SNR = signal-to-noise ratio   :foreheadslap:



#13 korborh

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Posted 11 May 2016 - 09:20 PM

Yes - doing it accurately would require filters and calibrating with field stars and so forth - to correct for atmospheric extinction, star color, etc. But for just getting ballpark values you can use a reference star and its visual magnitude. I think Maxim is set up to do this - so you can calibrate the star aperture to a known star magnitude - and then read off the mags of other stars.

If a mag 5 star has 10,000 electrons in it - then a mag 10 would have 100. And a sky of mag 20/arc-sec^2 would have 1/100 electrons per square arc-sec - or 1 electron in every 10x10 arc-sec square. If I did the math right. In most deep sky images a mag 5 star would have a ton of electrons in it - and it would be saturated so you couldn't use it. I'm just giving the idea.

Frank

 

Yes your calculation looks right. 

 

Skytools 3 Pro has a exposure calculator that tries to help with SNR calculation and it takes into account filter bandpass, object magnitude/color etc.

 

It is mostly useful as a guidance for SNR for different imaging windows. Here is a gui snapshot:

 

http://www.skyhound....es/ECalcM13.jpg



#14 Jon Rista

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Posted 11 May 2016 - 09:38 PM

See if you can work out consistent solutions given by the links:

 

- Look up the total magnitude T

 

- Calculate the area A in arcseconds

 

- Compute surface brightness in mags/arcsecond2:

 

       S = T + 2.5 * log(A)   [cf. first link]

 

- Compute flux in candelas/m2:

 

        F = 108000 * 10-0.4 * S   [cf. second link]

 

                                                                                      -- catalogman

 

Does anyone know the relationship between candela and photon flux? I guess that would probably be dependent on a specific wavelength at a time, say Ha or OIII, since you would need to know the energy of the photons, correct?



#15 catalogman

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Posted 11 May 2016 - 10:08 PM

See if you can work out consistent solutions given by the links:

 

- Look up the total magnitude T

 

- Calculate the area A in arcseconds

 

- Compute surface brightness in mags/arcsecond2:

 

       S = T + 2.5 * log(A)   [cf. first link]

 

- Compute flux in candelas/m2:

 

        F = 108000 * 10-0.4 * S   [cf. second link]

 

                                                                                      -- catalogman

 

Hmmm...

 

Well, I just noticed that the first link in Post #8 gives S in mags/arcmin2  but the second link (F) assumes

S is mags/arcsec2.

 

So when you find the area A of the object, be sure to use arcseconds for the dimensions.

 

As a partial check: a dark sky is about 10-4  foot-candles.

 

                                                                                                                                   -- catalogman


Edited by catalogman, 11 May 2016 - 10:11 PM.


#16 catalogman

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Posted 11 May 2016 - 10:40 PM

As a rough check of the surface brightness calculation, here's a paper
which does the computation for several galaxies:

 

http://articles.adsa...000321.000.html

 

For M81 (NGC 3031), the B3 surface magnitude ranges between 20.82 (center) and
24.53 (edge of disk).

 

Assuming dimensions of 8' X 3' (480" X 180") and a total B mag of 8.5, a rough
calculation of S gives

 

A = pi * 240" * 90" = 67,900 arcsec2

S = 8.5 + 2.5 * log (67900) = 20.58

 

which is close to the paper's value of 20.82 mags/arcsec at the center.

 

Assuming the disk to be of this intensity everywhere (which it isn't), the flux is

 

F = 108000 * 10-0.4 * 20.58 = 6.33 * 10-4 candelas

 

                                             -- catalogman



#17 Jon Rista

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Posted 11 May 2016 - 11:16 PM

Thanks for the insights, catalogman. 

 

If I run the calculation for both my dark site (21.3mag/sq") and for an ideal dark site (22mag/sq"), I get 3.26 * 10-5 for 21.3mag/sq" dark site, and 1.71 * 10-5 for 22mag/sq" dark site. 


Edited by Jon Rista, 11 May 2016 - 11:24 PM.


#18 catalogman

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Posted 12 May 2016 - 03:43 PM

According to Figure 13.1 in the Explanatory Supplement to the Ephemeris (USNO, 1977), at
astronomical twilight the indirect illumination of the Sun is about 6 X 10-5 ft-candles, which is

 

6.5 X 10-4 lux = 6.5 X 10-4 lm/m2 = 6.5 X 10-4 cd-sr/m2

 

although the actual brightness depends on local conditions.

 

Here's a document you might find useful:

 

http://members.ziggo...my/Formules.pdf

 

                                                               -- catalogman



#19 Jon Rista

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Posted 12 May 2016 - 03:50 PM

So...what is the relationship between candela and photon flux? I really need a photon count per second, or per some time period. Aren't candelas an energy measurement, rather than a count measurement?



#20 spokeshave

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Posted 12 May 2016 - 05:57 PM

A candela is about 1.5E-3 watts per steradian.

 

Jon Rista, on 12 May 2016 - 4:50 PM, said:

So...what is the relationship between candela and photon flux? I really need a photon count per second, or per some time period. Aren't candelas an energy measurement, rather than a count measurement?

A candela is a unit of power per solid angle. You want power per unit area. I'm sure there is a way to get there mathematically, and I may work on that later since it is an interesting question. In the meantime, according to http://www.astro.umd...TR620/mags.html the photon flux at the top of the atmosphere in the "B" bandpass (covering most of the visual spectrum)  is around 4000 Jy for magnitude 0, which converts to about 6E10 photons per second per square meter. From there, you can adjust for magnitude and area.

 

Tim


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#21 Jon Rista

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Posted 12 May 2016 - 07:28 PM

Thanks, Tim. I think we are getting closer. John H. sent me a formula that I think will get me the answer directly, however there are some details I don't quite have, so I haven't been able to apply it yet. (Namely, transmittance, of the atmosphere and of the scope...not sure how to get or measure either of those, and it seems they could be quite variable, particularly the transmittance of the atmosphere.) It seems somewhat surprising how difficult the answer to what seems to be a simple and benign question actually is. ;)

 

Gotta love solid angles, not the most intuitive concept. I've worked with them before, again in a photographic context. But they are just...odd. No one understands them, and I don't remember if I was ever able to find a way to convert to power per unit area in the past. 



#22 catalogman

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Posted 12 May 2016 - 08:20 PM

So...what is the relationship between candela and photon flux? I really need a photon count per second, or per some time period. Aren't candelas an energy measurement, rather than a count measurement?

 

Yes, the units in Post #16 s/b candela/m2 (not candela). Good catch!

 

If the light is monochromatic, then

 

  E (of each photon) = h * c / wavelength (of each photon)

 

The candela is power per solid angle, so multiply candelas by the object's solid angle
in steradians to find the luminous power in lumens.

 

The problem is to convert lumens to power P in watts. At 555 nm, 683 lumens = 1 W = 1 J/s.
But at other monochromatic wavelengths, the conversion is different.

 

Once the power P is found, P/E has the units

 

  J s-1/J photon-1 = photons/s (photons per second)

 

                                                                                                              -- catalogman


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#23 spokeshave

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Posted 12 May 2016 - 08:47 PM

Jon Rista, on 12 May 2016 - 8:28 PM, said:

Thanks, Tim. I think we are getting closer. John H. sent me a formula that I think will get me the answer directly, however there are some details I don't quite have, so I haven't been able to apply it yet. (Namely, transmittance, of the atmosphere and of the scope...not sure how to get or measure either of those, and it seems they could be quite variable, particularly the transmittance of the atmosphere.) It seems somewhat surprising how difficult the answer to what seems to be a simple and benign question actually is. ;)

 

Gotta love solid angles, not the most intuitive concept. I've worked with them before, again in a photographic context. But they are just...odd. No one understands them, and I don't remember if I was ever able to find a way to convert to power per unit area in the past. 

Most of the measurements for light intensity are intended to describe a point source, so solid angles make sense, since the intensity is the same, regardless of distance from the source. While stars are point sources, they are so distant that it makes no sense to use solid angles, so flux is a better concept.

 

I suspect that there will be enough uncertainty in the calculation that atmospheric attenuation and losses in the scope will be lost in the noise. I think that, at best, you'll be able to estimate within an order of magnitude.

 

Tim



#24 Jon Rista

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Posted 13 May 2016 - 11:44 AM

Ok, so I used the formula John sent me. I stuffed it into Excel and played around with the numbers a bit. I am wondering if someone could check my results. 

 

For a 21.5mag star, although I assume that would work for any mathematical point in space including a point on an extended object that has that magnitude, with a bandpass of 300nm (basically the full visible spectrum, Luminance filter), with 80% transmittance from the atmosphere and my scope combined (honestly not sure what the actual transmittance of the atmosphere may be at the zenith on a clear night, so I am just guessing here), with a 150mm aperture, I get a photon flux of 1.065 γ/s. For a magnitude 24 star, I get a flux of 0.1065 γ/s. For a narrow bandpass of 3nm (wavelength of 0.555 for now, as I have not yet figured out the irradiance of a mag 0 star at other wavelengths), that flux drops to 0.001065 γ/s. If I drop the aperture down to 80mm, the flux drop down to 0.000303 γ/s, and if I increase the aperture to 203mm the flux jumps to 0.002 γ/s. 

 

Do those results sound reasonable? 



#25 jhayes_tucson

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Posted 13 May 2016 - 01:44 PM

 

So...what is the relationship between candela and photon flux? I really need a photon count per second, or per some time period. Aren't candelas an energy measurement, rather than a count measurement?

 

Yes, the units in Post #16 s/b candela/m2 (not candela). Good catch!

 

If the light is monochromatic, then

 

  E (of each photon) = h * c / wavelength (of each photon)

 

The candela is power per solid angle, so multiply candelas by the object's solid angle
in steradians to find the luminous power in lumens.

 

The problem is to convert lumens to power P in watts. At 555 nm, 683 lumens = 1 W = 1 J/s.
But at other monochromatic wavelengths, the conversion is different.

 

Once the power P is found, P/E has the units

 

  J s-1/J photon-1 = photons/s (photons per second)

 

                                                                                                              -- catalogman

 

 

Just be aware that candelas and lumens are part of the photometric system, which is weighted to the response of the human eye and ultimately relates back to the brightness of a candle. I have an easier time dealing with simple radiometry, which uses radiance (W/m^2-str), irradiance (W/m^2), Power (W), and Energy (J).  You can certainly make the conversion between the two systems but I personally find that folding in the responsivity of the eye makes things confusing--particularly when computing a detector response.

 

John


Edited by jhayes_tucson, 13 May 2016 - 04:53 PM.

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