Standard stars

Standard stars are required so that different observers are able to compare results with each other. The reason this is true is because every observational setup is likely to have different response functions, so the same stars will not be observed to have the same brightnesses with each separate setup. Differences in response come from many factors: size and condition of the telescope optics, number and type of optics in the system, bandpass and quality of the filter, response function of the CCD, etc. In addition, it is very difficult to measure all of these responses in any absolute sense, so it is almost impossible to go straight from observations to inferences about physical fluxes in absolute units.

To get around these problems, systems of standard stars have been set up so observers can calibrate any new observations against the known brightnesses of the standard stars. A standard system is just a more-or-less arbitrary choice of one particular instrumental setup, but is generally one which is very stable and which someone has spent a lot of observing time establishing. Generally, the standard system is tied to absolute flux observations of some primary standard star so that observations on the standard systems can be converted to physical fluxes.

Let's consider some examples. First consider two identical systems. Clearly these will give the same observed flux as each other. Now let the systems differ only by a multiplicative factor in throughput. Now the observed fluxes will differ by a constant factor, or a constant additive factor in magnitudes. With observations of standard stars, it is straightforward to find the magnitude difference between the observed magntidue and the standard magntidue to calibrate out this constant factor. The additive constant in magnitudes is called the zeropoint.

A slight divergence is called for here to mention that different packages apply some zeropoints to the calculation of instrumental magnitude just to make the numbers (instrumental magnitudes) look ``reasonable'' (i.e., familiar). This is no problem since we determine a zeropoint on top of this to calibrate, but caution is required if talking about instrumental magnitudes measrued with different software systems!

OK, back to differing instrumental systems. Now let's consider the more usual case where the shape of the response curve as well as the absolute throughput differs from the standard system. Let's consider a system which has slightly different filters with a different wavelength cutoff. Now you observe a different number of counts from the standard system, and the difference depends on the spectrum of the object you are looking at. Now spectra can differ for lots of reasons, but the biggest effect is just from spectral slope differences. If your filter has a shorter wavelength cutoff (on the red side), then you'll observe less counts than the standard system, and preferentially less counts for a redder star. To first order, this can be calibrated out by solving for an additive constant which depends on the spectrum of the object being looked at. Since we don't in general know the spectrum, we have to parameterize it by something we can observe, and the best choice here is the stellar color, as inferred from observations taken through more than one filter. Clearly the best bet is also to observe a ``color'' taken near in wavelength to the filter we are trying to calibrate; generally one constructs a color from the desired filter and one nearby in wavelength (but sufficiently different to have real color information!).

Transformation coefficients.

$$m_0 = m - k X$$

$$M = m_0 + t (color) + z$$

where capital letters are the magnitude on the standard system, $z$ is the zeropoint, and $t$ is the transformation coefficent.

The color is generally parameterized by the ratio of the flux at two different wavelengths, or, in magnitudes, the difference between the magnitudes. The two wavelengths should be measured near in wavelength to the wavelength of the filter being corrected; generally, one uses the bandpass being corrected as one of the wavelenghts and an adjacent bandpass as the other. For example, when correcting $V$ magnitudes, people usually use $B-V$, $V-R$, or $V-I$ for the color term.

There are two ways to define the color, either in terms of the observational system or in terms of the standard system. The latter is slightly preferred for using least-sqaures (small errors on the independent variable), and also because it allows observations from different nights to be combined. Note that this formulation does not require you to know the colors of your objects a priori, it's just algebra to figure them out as long as you have observations in both filters.

The use of these first-order transformation coefficients is accurate as long as your filter system does not differ much from the standard system, and additionally, that the spectrum of your program objects does not differ significantly from the spectrum of the standard objects. The more these conditions are not met, the less accurate the results. Some additional accuracy in the case of differing systems can be achieved by using higher order transformation coefficients. However, even in this case, it is always important to remember that if the spectrum of the program object differs significantly from the standards, derived fluxes can be significantly in error.

Certainly, you get to a point when the response of one system is so different than the response of another system that no transformation can be determined. In this case, you have two different photometric systems. In fact, there are several different photometric systems at use in astronomy today, and each has advantages and disadvantages.

It is common practice to combine the equations for extinction and transformation coefficients into a single equation:

$$M = m + k X + t (color) + z$$

where I have ignored second order coefficients. The advantage of combining the equations is that you can use the information about the known magnitudes of the standard stars for the extinction term, so you can combine observations of different standards at different airmasses to derive the extinction coefficient and do not need to observe the same star at multiple airmasses.

In practice, one observes a set of standards of different colors at different airmasses. Then one uses least squares techniques to solve for the values of $k$, $t$, and $z$ which minimize the difference between the observed, transformed, magnitudes and the standard magnitudes. Then one applies these coefficients to measurements of your program objects to derive their magnitudes. Clearly, to do so, you must measure the colors of your program objects by making observations in more than one bandpass since, unlike the standard stars, the colors of your program objects are not known a priori.