Basic All Sky Photometry

My goal was simple: Obtain four values nightly: the first and second order extinction coefficients, the first order transformation coefficient to a standard system, and the zero point. Since I am also working with high school students to introduce them to astronomy in general and photometry in particular, I needed a solution that could be implemented with simple aperture photometry and a spreadsheet program like Excel.

There are nearly as many references to photometric methodology as there are stars, but none were just what I wanted. Sometimes the methods were too simplistic - ignoring coefficients that I deem important. Other methods were overly theoretical with little practical applicability, or they depended on some software package to use, or some combination of all of these things.

I decided to simply start from the basic equation of photometry and work a method out for myself. I like what I got because one need only measure a standard field twice. From that, you get a full solution for the first and second order extinction coefficients, the zero point and the first order transformation coefficient.

I have ignored the U band here because:

- most amateur CCD’s are pretty much dead in the UV,

- transformation of the U band is tricky and non-linear due to the Balmer discontinuity

- B-V is mostly what we are after anyway.

Equation of Photometry

In the following equation the subscript ‘B’ refers to the B photometric band, but is really just a placeholder for whatever photometric band is in use. Apply the equation to other passbands (with their corresponding unique coefficients and zero points) by simply replacing the ‘B’ subscript with some other passband (say, ‘V’). I have ignored the second order transformation coefficient as it is insignificant for the level of work I am doing (or just about anybody else for that matter).

B_i = b_ij - k_BX_j - k’_B(B_i - V_i)X_j + g_B(B_i - V_i) + P_B (1)

where:

b_ij = instrumental magnitude for the i^th star and the j^th airmass.

B_i , V_i = standard magnitude for the i^th star.

X_j = j^th airmass.

k_B = first order extinction coefficient.

k’_B = second order extinction coefficient.

g_B = first order transformation coefficient.

P_B = magnitude zero point.

Note that it is possible to reference the both the second order extinction coefficient (k’) and the transformation coefficient (g) to either the usual B-V color index, or a color index that includes the passband under consideration. Using a color index that includes the passband being observed reduces the need for additional filtered observations if one is not otherwise observing in B or V.

Measuring a Single Landolt Field

Our goal is to find values for four items so we can do our all sky photometry:

k_B, k’_B, g, and P_B

Consider equation (1) in the case of a single observation of a standard field at some airmass X_j = X₀. After some re-arranging, we get:

B_i – b_i0 = (g_B - k’_BX₀)(B_i - V_i) - k_BX₀ + P_B (2)

Equation (2) is a linear equation in (B_i – V_i) with the following slopes and intercepts:

slope

S_A= g_B - k’_BX₀ (3)

intercept

I_A = P_B - k_BX₀ (4)

To find these values, go measure a standard Landolt field in say, B and V. Plot the standard magnitude minus the instrumental magnitude (B_i – b_i0) for each standard star against its color index (B_i - V_i). Do a least squares linear regression to recover the slope and intercept.

At this point, the problem remains unsolved. Both the slope and the intercept equations have two unknowns.

The slope equation can be solved for the transformation coefficient without further data if you are willing to assume a vanishing second order extinction. This is not a terribly bad assumption in V, and a pretty good assumption in R and I. The B band however can have a somewhat significant second order extinction coefficient.

The intercept equation can be solved for the first order extinction coefficient if you have faith that your zero point has not moved (and you have measured it previously!). This may work if you system configuration has remained absolutely stable including the CCD parameters such as cooling, but is not often a good assumption.

Measuring a Single Landolt Field at Two Airmasses

By repeating the observation of our standard field at a differing airmass, we can solve for all four of the variables (k_B, k’_B, g, P_B) that we need to do our all sky photometry. We do this by subtracting equation (1) in the case of one airmass from equation (1) with the other airmass:

[B_i - b_i0 = - k_BX₀ - k’_B(B_i - V_i)X₀ + g_B(B_i - V_i) + P_B ]

-[B_i - b_i1 = - k_BX₁ - k’_B(B_i - V_i)X₁ + g_B(B_i - V_i) + P_B ]

------------------------------------------------------------------

b_i0 - b_i1 = k’_B(X₀ - X₁)(B_i - V_i) + k_B(X₀ - X₁) (5)

Equation (5) is also linear equation in (B_i – V_i) with the following slopes and intercepts:

slope

S_B= k’_B(X₀ - X₁) (6)

intercept

I_B = k_B(X₀ - X₁) (7)

Because we know the airmasses at which we took the data (X₀ - X₁), both these equations are readily solved to find k_B and k’_B directly.

To get the slope and intercept, measure the same standard Landolt field at a new airmass, and make a plot of the difference in instrumental magnitudes (b_i0 - b_i1) for each standard star against it’s color index (B_i - V_i). Again do a least squares linear regression to recover the slope and intercept.

Armed with values for k_B and k’_B , we can plug them into equations (3) and (4) to solve for g_B and P_B.

We now have all of k_B, k’_B, g_B, and P_B

My values

My observatory is almost at sea level, so I expect and find that my first order extinction coefficients will be higher than for high mountain observatories, and they are. Second order extinction coefficients are expected to be small, especially for V, R, and I. They are assuredly not zero (since first order extinction is in fact higher in V than R than I) but I presently can't resolve the difference with a small data set. I do however get a small non-zero value for B band color dependent extinction.

Finally, my transformation coefficients are almost certainly peculiar to my system (STL-11000m, Bessel filters, C-14). In any case, below are my values for a recent run. First order extinction is given per airmass, second order extinction is per airmass per magnitude of (B-V) color index, and gamma is per magnitude of (B-V) color index.

Johnson-Cousins Band	k	k’	g
B	0.330	-0.05	0.003
V	0.163	~0	-0.063
R	0.135	~0	-0.154
I	0.107	~0	-0.003

Amazingly, for (B-V), my transformation coefficients cancel out to the few millimagnitude level per magnitude of color index. That means that I will generally get a pretty good CI straight off the instrument! Life throws you a bone once in a while I suppose.

Standard ‘Landolt’ Fields

We are all in Arlo U. Landolt’s debt for having created a marvelous set of standard fields distributed in RA near the celestial equator. Refer to:

A.U. Landolt “UBVRI Photometric Standard Stars in the Magnitude Range 11.5 < V < 16.0 Around the Celestial Equator”

Astronomical Journal, July 1992, volume 104, number 1, page 340.