I have a friend who wants to make some observations to an accuracy of 0.01 mag. I have never worked to such levels of precision, having done most of my stellar photometry at 10 microns wavelength some 40 years ago, and our results were lucky to have 0.1 mag accuracy.

But at such increased levels of precision with today's CCD sensors in the visual region, I'm wondering if anyone has information about the differential extinction posed by Extrinsic (B-V) vs Airmass? Is the atmospheric extinction enough to vary the extinction of observed stars of different colors, differentially, by more than 0.01 mag?

Stated differently, he intends to take area images of his star fields and perform differential photometry. Does he need to account for differential extinction between stars of different extrinsic (B-V) within one image at any reasonable airmass? Or can he assume that all the stars in his image have been extinguished equally in his visual sensor range, to the level of 0.01 mag precision?

His FOV is from a 10-inch RC OTA at F:10. So I'm guessing that his FOV will be on the order of 10-20 arcmin on a side.

- DM

Hi DM,

There are two parts to extinction: first order, which is just an airmass-dependent constant for the particular bandpass in use, and second order, which is the color-dependent correction to first order. The first order correction is ~0.25magnitudes per airmass, so in the small field of view, it won't amount to 0.01mag unless the observer is imaging at low altitudes.

The second order correction could possibly give your friend some problems under rare circumstances, and I think is what you were asking about. Typically, the correction is about 0.03mag per colorindex per airmass. So if you are imaging at 2 airmasses, say with one star near the top of the frame and at airmass 1.99 and the other star at the bottom of the frame at airmass 2.01, and with the target having (B-V) = 1.7 and the comp having (B-V) = 0.7, then the differential corrections are:

first order: 0.25 * (2.01-1.99) = 0.005mag

second order: 0.03 * (2.01-1.99) * (1.7 - 0.7) = 0.0006mag

Not big. However, if you work at higher airmass or with a bigger field of view, so that there is a bigger airmass difference between the top and bottom of your image, or if you are observing a very red target with a very blue comparison star, the correction can be bigger. So whether it is important depends on the situation. In most cases, second-order extinction can be ignored. I always calculate it so that I never have to worry about whether it is important.

Things are very different at 10 microns, of course! I did lots of photometry at JHKL, and found that you could do good differential photometry on one night, but since the atmosphere and water vapor determined the final bandpass, you would get different results on the next night. Optical observers have it easy...

Arne

Hi Arne,

Isn't this a little simplistic? I must admit that my photometry is a little dated but in Auckland the primary extinction would differ from night to night and often widely during a night. Being a city there was also a marked difference between nights with a strong inversion layer and clearer nights. But a 50% spread in values from 0.15 to 0.25 in V was not uncommon. I've just looked up average values for one quarter and on a filter basis we had V = 0.183, B = 0.315, U = 0.452. We always transformed B-V, V, U-B of course. But these are in the southern hemisphere where skies are generally more transparent - the milkyness of the northern skies are clearly apparent when I watch golf on TV. But we're stuck with where we live.

So even there we determined reliable values for sub-standard stars in the area of the target star and used these. One advantage is that most observers now do not measure in U which makes it easier but tends to reduce the value of measures of blue stars. But using average or predetermined values has many disadvantages which the observer needs to understand.

Regards, Stan

Hi Arne, all,

I agree with Stan, I have the same extinction variations here in Burgondy, France. At best e~0.13 and it could reach 0.45 at worst. The mean is about 0.24. I did a survey of B-V during some 500 observations of eps AUR over three years and the result is that the color variations have nearly no correlation with the extinction coefficient and has a significant correlation with AM. But the rest of it is independent of anything. The overall color scatter is about 0.4 mag (B-V) and is not negligible in case of wide field photometry. With a 200 mm lens on APS-C sensor ( 6 deg field ) the impact of the atmospheric chromaticity is critical at elevation below 20 deg. As said Stan we often have issue with inversion layers.

I think there are strong differences between the pro astronomers observation conditions and ours. When I see reports on observatories sky condition, it's clear they are in a different world ! And obviousely this is not well understood, I have seen papers on the subject saying the chromaticity of the atmospher is very stable and just evolve somewhat with the season. Not my case, I could have large variation within a time series and even one image to the next.

Clear Skies !

Roger

Arne,

I think you would agree that first order extinction differs for different filters. From figure G.1 in your book I calculate the following approximate first order extinction coefficients:

0.30 per airmass for v

0.47 per airmass for b

0.67 per airmass for u and the difference between the b and v coefficients gives

0.17 per airmass for b-v

So wold it be correct that the 0.25 figure in your e-mail pertains to v?

As was customary at the time, you book only covered U, B and V filters. Is there a reference that gives rule-of-thumb figures for all the J-C filters and, perhaps, one for the Sloan filters as well?

Thanks,

Brad Walter

Hi Stan and Brad,

Stan, yes, this is simplistic, as the original question was just whether such corrections could be in the 0.01mag range. They can, if you are using widely disparate colors, large field of view, or observing at high airmass. But for most of the time, the corrections will be less than this. While I gave a numeric example to show this, my main point was that my own observing procedure is to include both first and 2nd order extinction for all measures, so that I never have to make a judgement. I also agree that extinction can vary from night to night, but again for small-field CCD work, using means is ok if you don't have a reliable calculation of the current night's extinction.

Brad, in my CCD course I show a figure from Bruce Gary's website:

http://brucegary.net/allsky2011/2011.01.30%20ExtinctionSpectrum%20v1205…

which shows how the first-order extinction varies as a function of wavelength. That is for one site, but gives the general shape and rough size of the correction.

Arne

hi Roger,

Again, you have a 6-degree field, and the questioner asked about 10-20arcmin field. You fall into the "wide-field" exception that I mentioned. To cover all filter variations and all field sizes and viewing altitudes requires a much bigger discussion (which is why there is a full video from the CCD School devoted to the atmosphere).

Not everyone is at a pristine site, though I've found the southwestern U.S. sites to be remarkably stable over a season, even at the lower elevations (like Sonoita).

Arne

Hi All,

Here in Sydney I've been experiencing primary extinction coefficients of effectively infinity for much of the year! It would be nice to have a few nights with "normal" extinction levels of ~0.25 magnitudes per airmass that Arne mentioned. Cheers,

Mark

Arne,

I understood the purpose of your response. You weren't recommending that someone actually use these rules of thumb to do extinction correction. You were ball-parking to understand whether differences in extinction across your field could potentially be a problem.

I just wanted to point out that 0.25 might not apply to all the filters you might be using. and you might want to do the calculation using the rule-of-thumb value for your worst case filter at the highest airmass at which you image as the rough check.

One more thing that it might be worthwhile to point out is that a known systematic of 0.01magnitudes that you can correct should be corrected. It has a much bigger effect than an uncertainty of 0.01 magnitudes. CCD photometry with SNR of 100 or better will have uncertainty in the 0.01 magnitude range or better. In that case, a systematic error of 0.01 magnitudes is probably significant.

Brad Walter

Hi All,

Many thanks for your guidance. I also ran into a paper by Broeg, et al, that seems to directly address the differential extinction with airmass * differential color index. I also liked his idea of taking *all* field stars for comparison. That appears to make automatic extraction more straightforward.

http://bulldog2.redlands.edu/facultyfolder/deweerd/seminars/Broeg_Algor…

- DM

Arne, and any one else with expertise in 2nd order extinction correction. I need a sanity check.

While observing NGC 7790 this month to update my transformation coefficients, I decided to take a look at 2nd order extinction coefficients to see first-hand how much effect they actually have. When I sat down to calculate the coefficients the first thing that jumped out at me is that you have different 2nd order extinction coefficients for a given filter depending on which filter pair you are using analogous to the situation with transformation coefficients. The V filter for example would have v

_{0-bv}, v_{0-vr}and v_{0-vi}for BVRI photometry. Then when I ran the regressions the second thing that popped out, which seemed counter intuitive, is that most of the 2nd order extinction coefficients turned out to be negative. A negative coefficient means that the second order correction increases the extra-atmospheric magnitude larger (dimmer). I realize that first order correction per airmass is much larger than second order in all-sky photometry, but in a reasonably small altitude separation, say, 30 arcminutes or less in differential photometry, the second order extinction correction can easily be the larger correction at airmasses down to 3.0 airmasses if the target and comp stars have significantly different color indexes. For example for the b filter correction with k’b = 0.47 and k”b = -0.0506 and (the rather extreme) b-v color difference between target and comp of 1.0 magnitudes the first order correction is – 0.034 magnitudes but the second order correction is +0.15 magnitudes.The 2

^{nd}order extinction coefficients I derived from 4 nights of data arek" Coefficients

Uncertainties

k"b-v

-0.03842

k"b-v

0.005724

k"b-bv

-0.05055

k"b-bv

0.005126

k"v-bv

-0.01213

k"v-bv

0.003061

k"v-r

-0.02177

k"v-r

0.01093

k"v-vr

-0.02073

k"v-vr

0.006426

k"r-vr

0.001043

k"r-vr

0.006388

k"v-i

-0.01495

k"v-i

0.00473

k"v-vi

-0.01162

k"v-vi

0.003256

k"i-vi

0.003336

k"i-vi

0.003716

k"r-i

-0.00052

k"r-i

0.010398

k"r-ri

0.005335

k"r-ri

0.006908

k"i-ri

0.005856

k"i-ri

0.007803

Attached is the spreadsheet showing the detailed calculations, regressions and plots of the data. Should the coefficients be predominantly positive and I have messed up the process somehow to get negative values for most of them? It seems straight forward since, for example, ∆(b-v) = k

_{b-v}”X∆(b-v) +∆(b-v)_{0}and you plot (or regress) ∆(b-v) against ∆(b-v)X and the slope is the k” coefficient. For single filter colors it becomes, for example, ∆b = k”_{b-bv}X∆(b-v) +∆b_{0}. I Have looked at the calculations several times and don’t see a mistake that would cause the signs of the coefficients to be reversed, but I may be missing the obvious.Brad Walter, WBY

Further to my post above, I checked Appendix G.3 of Astronomical Photometry and It showed a value of -0.042 for k"

_{b-v}(using my notation, k"_{bv}in the book). My value of -0.038 +/- 0.005 (1-sigma) seems reasonable.Thinking about the 2nd order correction in relative terms, it actually does make sense that the extra atmospheric correction is positive for b-v > 0 (b

_{0}dimmed compared b) compared to b_{0}= b for b-v = 0 or negative for b-v < 0 because the first order extinction correction assumes equal source flux at all wavelengths across the filter bandpasses. So if you had two stars with the same instrumental b magn but one has b-v <0 (more b than v) its extra asmospheric b magnitude needs to be corrected negatively (made brighter) above the extra-atmospheric magnitude for something with b-v = 0 since shorter wavelength are attenuated more, and the extra-atmospheric magnitude of a star with positive b-v would have to be increased (made dimmer) because it has more flux toward the v end of the filter bandpass than the b, and, therefore, it experiences lower overall extinction than a star with b-v =0.I think this makes sense and explains why most of the 2nd order extinction coefficients are negative, even though it seems counter intuitive at first.They make adjustments required by the first order assumption that the flux is constant across the filter bandpasses. As a result most of the 2nd order correctionss are in the opposite direction (or at least the opposite sign) of the first order extinction corrections we are more accustomed to working with.

Brad Walter, WBY

Hi Brad,

I cannot download your spreadsheet using the link shown - I ended up with about 50 1980s IBVS instead. Could you email it to me directly at astroman@paradise.net.nz

Regards,

Stan

There's a nice paper which looks in detail at the effects of extinction as a function of small differences in airmass across an image, including color effects. The paper describes both the observed systematic errors and a set of calculations which agrees pretty well with the observations. The instrument is part of the Dark Energy Survey (DES), but one can make some decent generalizations to other systems without _too_ much extra work.

See the preprint at

http://arxiv.org/abs/1601.00117

TG calculates a transformation coefficient Tr-vi. It seems to me that the correct second oder extinction correction to use for the TG input for this transformation would be r values corrected using k"

_{r-vi}applied to (v-i)X where K"r-vi is defined in the equation∆r = k”

_{r-vi}∆(v-i)X + ∆r_{0}, where the ∆ magnitudes are the differences between magnitudes of two stars in the same image at essentially identical airmassesCan anyone expert in applying second order extinction correction confirm whether my conclusion is correct?

Thanks,

Brad Walter, WBY