Atmospheric extinction

We've already desribed the absorption/scattering effects of earth's atmosphere:

$$F = F_0 \exp (-\tau(z=0) X)$$

or

$$m = m_0 + 1.086 \tau(z=0) X$$

$$m = m_0 + k X$$

$$m_0 = m - k X$$

where $k$ is the extinction coefficient. As we've discussed, this can vary from night to night, so if you're interested in accurate photometry (better than a couple percent), you need to measure it on your night. Also remember that the extinction coefficient is wavelenghth dependent, so you need a separate number for each filter. It's also critical to measure this to check for photometric weather: even if you don't require a few percent, if there are clouds you might get a lot worse! So you need to check.

The extinction coefficient can be determined by making multiple observations of a star at different airmasses. Then you can solve for $k$ and $m_0$ using least squares. Note that you need to sample a good range of airmasses to get good leverage on the fit, and you must bracket the airmasses of all of your program objects.

When doing broad-band work, however, there's an additional subtlety because of the wavelength dependence. Two different stars might have different extinction coefficients in the same filter because the stars might have very different colors, which has an effect if the filter bandpass is broad. This problem can be solved by using second order extinction coefficients, where the extinction is a function not only of the airmass, but also of the stellar color:

$$m = m_0 + k X + k_2 (color) X$$

where $color$ can be either an observed or a known color for the star.

Using the above formalism, you have to solve for extinction coefficients plus a magnitude for every star you observe. This requires a fair bit of observing to constrain all of the parameters. Clearly, if you can observe stars of known brightnesses, you will have better constraints on the extinction coefficients. These leads us into the discussion of standard stars.