In a PEP thread (New PEP Observers - #15 by bskiff) Brian Skiff raises points about the constancy and measurability of second-order extinction coefficients in B band. I’d like to explore the topic further but not in that particular thread. Though second-order coefficients exist for any band or color index, in my comments below, k" (k double-prime) is to be understood as the second-order extinction in B band unless otherwise stated.
The reference works with which I am familiar are (A) Photoelectric Photometry of Variable Stars 2nd ed, Hall and Genet, 1988; (B) Astronomical Photometry, Henden and Kaitchuck, 1990; and (C) Astronomical Techniques, Hiltner (ed.) chapter 8 by Hardie, 1962.
H&G (page 156) says that k", “proves to be between about -0.02 and -0.04, depending primarily on the width of the filter.”
H&K (pages 90&91) says that second-order coefficients for the B-V color index, “have been found relatively constant and probably do not need to be determined any more often than are color transformation coefficients.” Given that k" in V band is considered to be little different from zero, that would seem to imply that k" of B is relatively constant. Admittedly, I think that professionals pushing the limits of accuracy establish their transformations every night, so it is not totally clear what H&K mean by their statement.
In AT (page 187) Hardie states that k" is generally between -0.02 and -0.04 depending upon filter bandwidth and, “appears relatively constant [compared to primary extinction],” with some provisos. Below I attach a PDF excerpt of Hardie’s chapter. Hardie_2nd.pdf (266.4 KB)
These are the origins of my claim to the stability of k".
In his post, Brian references a paper by Arlo Landolt and a paper by Stetson, et al. In the Landolt paper the author lays out extinction information for his 13-year project in Table 1, that I partially reproduce below.
I assume that “k1” is primary extinction and “k2” is second-order extinction, both for B-V. Again, given the near-zeroness of k" for V I think we can assume the variation in k" for B-V is due primarily to B band.
Landolt reports a significant range of variation. He offers an “average” value that I take to be the mean. However, he does not report either a standard deviation or the median, which makes it hard to judge the scatter in his numbers. In a different context, Arlo once told me that when he performed a multi-night run of observations he would measure k" every night, but that when he reduced the whole run’s data, he would first compute an average k" from the nightly values, and use that average as his k" for every night of data. He said that this gave better results than distinct nightly k".
It is unclear to me whether Landolt explicitly measured k" in the red/blue star pair manner described by H&K (and the PEP Guide), or if it was extracted as part of a multi-parameter “fit” applied to assorted standard stars observed during the night. Someone better versed than I in statistics would need to weigh in on whether one method might be more reliable than another. How I would love to see Arlo’s notebooks!
The Stetson paper describes photometry performed (largely) on archival CCD images of globular clusters. It states that second order extinction, “…is too subtle to be measured with any precision in our actual photometric data.” He adopts a fixed 2nd order coefficient of -0.016 for B-V based upon modeling (see Appendix B1).
If Landolt could measure k", why couldn’t Stetson? These two papers seem contradictory.
Hi Tom
Good to see your notes.
I have not read the papers, but your notes seem to indicate that they are largely in agreement rather than contradictory. One says “too subtle to be measured with any precision” and the other says that it varies so much from night to night that, in practice, he uses an average of many nights.
IMHO it is good to go after the best measurements possible.
I should probably chase k’ at least once just for the exercise.
In practice, having comps typically 1’ to 6’ from the variables, I use “standard” values for k’ , skip K’’ and call it good enough.
I find many other factors that produce larger errors over periods of years or even months. Especially when monitoring difficult FOVs. Probably doing something wrong here, but I put more efforts into beating down the larger general errors in hundreds of FOVs, each having their own peculiarities. I am learning that at the levels of 5 to 25 mmag most stars and many comps seem to be variable.
I’ve found a few papers which present measurements of the second-order extinction coefficient k" for (B-V) taken on different nights. Perhaps reading these papers and looking at the figures and tables might help to answer some of your questions.
Unless I missed them, I’m pretty sure these references show only the direct extinction coefficients, not second-order ones. The ones for La Silla in Chile do cover enough seasons to get the annual variation. They seem not to have used second-order extinction in the seven-color Geneva system, perhaps from having the pairs of filters closely spaced in wavelength. Figure 10 in the Palomar PTF paper again shows the run of the color terms and extinction as a function of calendar month. They shouldn’t have second-order effects in the red region.
…specifically the column headed A6 in Table II, then the column called k2 in Table III. Some of the older data were taken with the original tube and filters that Harold Johnson used to define the UBV system (and the same telescope for that matter). But do notice the night-to-night jitter in the k" determinations. As Tom C has noted, the means over many nights are pretty good (about -0.03 or so), but the values from any single night are sketchy.
I might also mention the results in Table V of this paper, which includes bright standard stars measured far more times than the original UBV standards and over many seasons. These have typical mean errors of about 0.003 mag in V and B-V.
The author of the Lowell Bulletin, Mike Jerzykiewicz, I regard as “world’s best photometrist that nobody’s heard of”. He made several long-term visits to Lowell, partly for observing outer planets, but mainly to get data on multi-mode B stars with many pulsational frequencies having tiny amplitudes. See SIMBAD for papers about nu. Eridani that have him as a co-author. Those are now almost commonplace with space-based instruments, but they were getting this stuff with simple photoelectric photometers from the ground and very careful observing procedures. He and his wife are now happily retired somewhere near Warsaw.
Brian is right – I did mistake figures showing results for first-order color terms in those papers for second-order (color and airmass) terms. Sorry about that. The Palomar Transient Factory paper does describe finding the second-order terms … but does not show any specific values as a function of time.
Thanks to Brian for catching my mistake and then providing a proper example!
SInce the width of the filter is a key factor in k" it stands to reason that the actual transmission curve shape matters, too, at least at the blue end of the B passband. That being the case, then the response curve of the detector also matters.
I raise this point because H&K, H&G, Landolt, and Jerzykiewicz are all working with photomultipliers, while AAVSO PEP observers are largely working with photodiodes. In B band, the diode response curve has positive slope, while the PMT has primarily negative slope (for 1P21).
In our admittedly limited collection of k" measurements with photodiodes, we seem to be getting values more like -0.04 to -0.06.
Tom
PS: Brian, thanks for the Lowell Observator paper - I need to examine Appendix I and II in detail.
The 1966 Jerzykiewicz paper appendices worry over making the color corrections in the extinction before or after transformation (i.e. does one use instrumental colors or standard colors for the extinction determinaton?). Mike’s answer was that it does not make much difference except down at the few-millimag level.
The ‘modern’ approach advocated in the 1981 Harris et al paper:
…is to include the extinction as one of the coefficients in the transformation rather than treating it separately. This fundamental paper includes some heavy going mathematically, but the early parts and the examples at the end should be relatively easy to follow. It deserves multiple re-readings.
Back in 2005, I studied the 2nd order extinction at my backyard observatory (coastal southern California). The results were published in the SAS Proceedings; see https://socastrosci.org/wp-content/uploads/2021/12/2005_Proceedings_final.pdf and go to p. 111.
Bottom line: k"B was pretty small (~ -0.03) and seemingly pretty constant. The paper also includes some references that I found showing measured first- and second-order extinction at some professional observatory sites.
\Bob Buchheim
