Creative Commons License Copyright © Michael Richmond. This work is licensed under a Creative Commons License.

Placing CCD photometry on a standard system

CCDs are very accurate devices: they measure linearly the amount of light which strikes them, as long as the levels remain below the saturation point. However, this can turn out to be a pain: it can reveal very small differences that might otherwise be lost in the noise. If one can make measurements to a precision of better than one percent, several systematic effects start to show themselves. One of these is due to small differences in the effective passband of different observatory systems.

Simple approach to placing differential magnitudes on the standard system

So you have a bunch of measurements of the instrumental magnitude of some variable star, plus the instrumental magnitudes of some comparison stars in the field. 


     CY       A       B       C  
   13.079  12.592  13.670  16.075

How can we turn these instrumental magnitudes into real magnitudes?

Exercise: (and worksheet)
  1. What is the standard magnitude of each star (A, B, C) in passbands B, V, R? Hint: Use Aladin, then overlay the Tycho-2 survey catalog to get B and V; then overlay the USNO B1.0 survey catalog to get R, and B for the faint star
  2. What is the difference between stars B and A in each of these three passbands?
  3. What is the difference between stars C and A in each of these three passbands?
  4. What is the difference between stars B and A, and C and A, in the instrumental system?
  5. What is the best passband for these images of CY Aqr?
  6. In this "best" passband, what is the shift which brings instrumental to standard magnitudes?
  7. What is the standard magnitude of CY Aqr in the "best" passband in the picture above?

This method is simple, quick, and will get you into the ballpark. Its results might be within 10 percent of the truth for "ordinary" objects: stars of intermediate temperature and color. But for very blue or very red stars, or objects with strange, non-blackbody spectra, it can be way off.

The Problem: Small differences in effective passband

Astronomical observations involve a number of components:

As light interacts with each component, photons of different wavelength are reflected or transmitted by different amounts. The overall response to light as a function of wavelength is the convolution of all four individual responses.

Let's consider filters, since they usually place the strongest constraints on the detected photons. Even filters which are supposed to be identical have small differences. Below are the transmission curves of the standard Bessell V passband (which is a model for manufacturers of filters), and the "Harris V" filter at the WIYN Telescope at Kitt Peak National Observatory:

The WIYN filter is slightly broader than the standard passband. How large a difference does that make? Well, there are two ways to look at it.

  1. The broader peak of the WIYN filter will allow a bit more light from a given star to enter the camera. That will decrease exposure times a little bit. But if it affects all stars equally, it won't make any difference in the relative magnitudes of different stars.

  2. The sharper increase in transmission at the blue side of the WIYN filter will allow more light from hot, blue stars to enter the camera than from cool, red stars. This might create a systematic error which makes blue stars seem a bit brighter than red ones when one observes through the WIYN filter.

If one knows the exact shape of a star's spectrum, one can convolve the spectrum with several different passbands to predict its instrumental magnitudes. I have taken the spectra for the hot and cool stars shown on the above graph and convolved them with each of the two "V" filters. The zero point is arbitrary. 

    star             Bessell V       WIYN V     (Bessell V - WIYN V)
---------------------------------------------------------------------
  A0V  (hot)          16.056        16.000           +0.056

 K0III (cool)         16.149        16.109           +0.040

Notice that the change in instrumental magnitude is NOT the same for the two stars. If I use the WIYN filter, my images will show a hot A0 star to be a bit brighter compared to a cool K0 star than it would appear through the standard Bessell filter. The difference isn't large, about 0.016 magnitudes or 1.6 percent, but it can be seen easily in good CCD photometry. There will be a similar effect for all stars of different temperatures.

Diagnosing the problem: systematic errors as a function of color

How can one find out if one's system suffers from a passband mismatch? Easy: just observe a field(s) with many standard stars of known magnitudes and a range of colors (and, hence, temperatures). Then compare the measured magnitudes to the standard magnitudes, as a function of color. If a trend appears in the residuals, there's a problem.

Let's look at an example: Hwankyung Sung and Michael Bessell calibrated the 40-inch telescope and its CCD camera at the Siding Spring Observatory in Australia in the late 1990s. You can find a paper describing their work on the Astrophysics Data System Abstract Service. Searching on their names will reveal the paper Standard stars: CCD photometry, transformations and comparisons, which appeared in Publications of the Astronomical Society of Australia, vol 17, p. 244 (2000). Here's part of their Figure 3, showing the differences between the standard V-band magnitudes and their instrumental system as a function of (B-V) color.

Exercise:
  1. In the Siding Spring data, do blue stars appear brighter or fainter, relative to red stars, than in the standard system?
  2. If you pick 10 stars at random from the Siding Spring data, and compute the difference between the instrumental mag v and standard mag V, what will the typical difference be? In other words, what is the typical error in converting instrumental to standard magnitude?
  3. Describe a possible difference in shape between the Siding Spring V-band filter and the standard V passband which would lead to the observed effect.

Fixing the problem: color corrections

Based on their data, Sung and Bessell came up with an equation which fit the mean trend in the residuals: 


         V  =  a  +  v  +  0.072 * (B-V)
where V is the standard V-band magnitude, a is a zero-point offset in magnitude (i.e. a simple shift of all the magnitudes vertically by a constant amount), v is the measured instrumental magnitude through their "V" filter, and (B-V) is the color of a star in the standard system.

Note that astronomers often use capital letters to denote standard magnitudes, and lower-case letters to denote the corresponding instrumental values.

The coefficient, 0.072, is an example of a first order color term. In the general form of the equation, this is usually written as b, like this: 


         V  =  a  +  v  +      b * (B-V)

Don't confuse this b with the use of "b" to denote instrumental B-band magnitudes.

Exercise:
  1. What are the units of the first-order color term?
  2. Use the data in the graph from Sung and Bessell above to estimate the value of the first-order color term b for the Siding Spring instrument.

Now, the idea is always to correct your instrumental magnitudes so that they match the standard scale. Look carefully at the equation above: in order to calculate the color correction, you need to know the standard (B-V) color of a star; but you don't know the standard V-band magnitude yet! Yikes! It looks like a circular argument. There are three ways around it:

  1. Iterate: make a guess at the standard V and B magnitudes (a decent guess is the measured v, b magnitudes), and use those guesses to determine an initial color term. Then, use the color term to correct the instrumental measurements, yielding better estimates of the standard V and B magnitudes. Use those better estimates to yield a better standard (B-V) color, and so forth. Stop iterating when the standard V and B magnitudes converge.

  2. Invert: one can invert the equation algebraically, turning 
    
         V  =  a  +  v  +      b * (B-V)
    into 
    
         V  =  a' +  v  +      b'* (b-v)
    where the values a' and b' yield the standard magnitude in terms of the instrumental colors.

  3. Use instrumental color from the start: If one starts with instrumental quantities on the right-hand side from the start, one doesn't need to do any extra work at all. Just set it up this way: 
    
         V  =  a  +  v  +      b * (b-v)

   (V - v)  =  a  +            b * (b-v)
    and make a plot of instrumental color on the x-axis versus magnitude error (V-v) on the y-axis. The slope of the trend line will the color term.

After Sung and Bessell make their color corrections, the residuals between their measurements and the standard V-band measurements show no systematic trend with color:

There is still some scatter, but it appears to be random. It's probably due to a combination of photon noise from the star and background.

Exercise:
  1. If you pick 10 stars at random from the CORRECTED Siding Spring data, and compute the difference between the instrumental mag v and standard mag V, what will the typical difference be? In other words, what is the typical error in converting CORRECTED instrumental to standard magnitude?

An example of color terms from RIT Observatory

On July 30, 2013, I used the 12-inch telescope to acquire images of the standard field Landolt PG1633+009. I recommend this good place to find information on Landolt equatorial standards.

I measured the instrumental magnitudes of these stars in both the B and V filters, then compared those to the catalog magnitudes for these stars. The B-band magnitudes showed a significant trend: 



   Q:  Do my measurements show blue stars as fainter than
       the catalog, or brighter?



   Q:  If one writes an equation

            (catalog mag)  =  (instrumental mag) * C   +   K

       what is the first order color term C?

Good fields of standard stars

The best source for magnitudes of stars in the UBVRI system is a series of papers by Arlo Landolt.

The second paper contains finding charts for all the stars. Most appear in groups of three to twenty stars, in "Selected Areas" of the sky. If you observe one of these selected areas, you can acquire data on many standards all at once, which allows you to determine the color coefficients for your system in a short time. Below is a section of the chart for SA101, which is near RA 09:57, Dec 00:00.

Exercise:
  1. What are the exact coordinates of the star marked "363" on the chart?
  2. What is the V-band magnitude of that star?

I have made finding charts for a set of five fields around the celestial equator which have several stars of very different colors with well measured magnitudes.

Creative Commons License Copyright © Michael Richmond. This work is licensed under a Creative Commons License.