Figure 10.5 The profile of an isolated star image shows a smooth decline from a peak value and a uniform sky background. From the graph, you can see that the peak value in the star is 1488, well below saturation; the sky background is 388, and the HWHM, at 1.0 pixels, is somewhat smaller than desirable.

the data from the experiment consist of a list of times (starting when the star passed through the zenith) and an instrumental magnitude corresponding to each time.

Look at your data. When you graph the time versus the magnitude, it plots out rather nicely as a curving line. You are pleased because it looks as if there is an underlying law at work; if only you can figure out what the curve is, or better yet, figure out how to make the curve into a straight line.

Since the star is obviously at its brightest when it is straight overhead, you decide to plot the distance from the zenith versus magnitude. A little spherical trigonometry gives you £ , the angle between the zenith and the star:

cos£ = cos (|) cos//cos 8 + sin (|) sin 8 (Equ. 10.19)

where (|> is your latitude, 8 is the declination of the star, and H is the hour angle of the star; i.e., the length of time since the star crossed the meridian. When you graph £ versus magnitude, it is only to experience the disappointment of seeing another wild curve.

You are now forced to think a bit. The attenuation of the star's light clearly depends on how much air it passes through. Each time the light passes through some length of air, it loses a fixed proportion of its light, which means that the star's brightness declines exponentially with the length of the air path (called the air mass). However, you are measuring the star's brightness as a magnitude, which is a logarithmic function. Since the log of an exponential function is linear, you've found the answer: the star loses a constant amount in magnitudes for each unit of air mass it traverses!

Another couple minutes' rummaging around in your math books from college reveals the answer: the length of the air path is proportional to the secant of £, the inverse of cos £, which you computed for your last graph. So at last you can write out the equation for the air mass, X:

X= sec£ = l/(cos(|)cos//cos8 + sin(j)sin8). (Equ. 10.20)

A few more minutes of plotting reveals a beautiful linear relationship between the length of the air path (which equals the secant of the zenith angle) and the dimming that the star has suffered, in magnitudes.

You quickly realize several things. The length of the air path is exactly 1 when a star is straight overhead, at the zenith. However, as a star moves away from that point and the angle from the zenith increases, the air mass increases. The increase comes on slowly at first, and only reaches 2.0 at a zenith distance of 60° (30° up from the horizon). The effect is hardly noticeable to the eye until a star gets within 20 to 25 degrees of the horizon. Lower in the sky, the air mass increases rapidly.

On your plot you see that the slope of the line tells you how many magnitudes dimmer the star becomes for unit of air mass that it traverses—a quick check of the numbers shows an increase of 0.24 magnitude per unit air mass (i.e., the star becomes less bright). Finally the grand revelation: even though you cannot measure a star outside the atmosphere, you can extend your line all the way to zero air mass; in other words, from a series of measurements made within Earth's atmosphere, you can compute how bright the star would appear if measured outside it.

This experiment is well worth doing if you contemplate trying your hand at photometry. Seeing is believing: you can indeed compensate for atmospheric extinction. If you try the same experiment with different filters, you will discover that extinction is greater at short wavelengths and less at long wavelengths, which you already knew because sunsets are red.

On a technical level, extinction is slightly more complicated. The simple equation for the air mass given above must be modified to correct for the curvature of Earth with additional terms:

The secant is valid from the zenith down to 30 degrees elevation; this equation is good down to an air mass of ten, or about 6 degrees altitude.

In your simple experiment, you found that extinction measured through a V-band filter obeys the following simple relationship:

Radius (pixels)

Figure 10.6 The curve of growth graphs raw instrumental magnitude versus the radius of the aperture. Although the curve of growth does not plateau for this star, the difference between radii of 4, 5, and 6 pixels is only about 0.01 magnitude, indicating that radii of 4, 5, and 6 pixels should work equally well.

Radius (pixels)

Figure 10.6 The curve of growth graphs raw instrumental magnitude versus the radius of the aperture. Although the curve of growth does not plateau for this star, the difference between radii of 4, 5, and 6 pixels is only about 0.01 magnitude, indicating that radii of 4, 5, and 6 pixels should work equally well.

where vx is the raw instrumental magnitude measured through the air mass, X, k\ is the extinction coefficient for your V filter, and v0 is the extinction-corrected instrumental magnitude of the star above Earth's atmosphere (with a zero subscript to remind you of zero air masses). At sea level, k'v is typically about 0.24 magnitude per air mass, but it falls to 0.15 magnitude per air mass at dry, high-altitude sites. If this sounds a bit complicated, remember that it's really nothing but a straight-line graph that relates raw instrumental magnitudes to air mass.

However, because extinction is stronger at short wavelengths, as a star sets the blue-end wavelengths in the passband are absorbed more strongly than the red; so the center of the passband shifts toward longer wavelengths. In order to compensate accurately for extinction, you have to measure the star's color also, and then correct for it. The simple equation becomes:

The k"v term is called the second-order extinction coefficient. It is really nothing but a small correction that depends on the (b - v) color of the star. Luckily, this term is so small that astronomers simply ignore it when measuring magnitudes through a standard V filter.

Although it might seem logical to set up analogous equations for each of the Other colors you are measuring (i.e., U, B, R, and /), astronomers measure V and then use the other filters to derive the colors (b - v), (u - b), (v - r), and (v - i). Although photometry seems perverse at times, the internal logic is consistent: color is a differential measurement. For (b - v) and (u - b), the extinction equations look like this:

(¿-v)«, = (b-v)x-k\b_v)X-k"{b_v)X(b-v) {u-b)o = (u - b)x- k\u_b)X-k"(u_b)X{u - b) .

Nothing new happens here: you simply take a raw instrumental color, correct extinction for that color, and make a small second-order correction that depends on the star's color. Second-order corrections are small and hard to measure. In fact, rather than measure k'\b_v), many photometrists simply assume a value of 0.03; and in the UBV system, k'\u_b) is set to zero by definition. Thus, the extinction corrections reduce to:

If these equations didn't look so hideous, their simplicity would be obvious!

There are three basic ways to deal with extinction: the all-sky approach, the shotgun-scatter approach, and the differential photometry approach.

The first approach is used in all-sky photometry; that is, photometry done on stars over the entire sky. In addition to observing program stars, you observe several extinction stars over a range of zenith distances during the night. This enables you to determine extinction coefficients that you can apply to the program stars. To find the best values for the extinction coefficients, you use the method of least squares (or more likely, the software that you are using to reduce the data uses the method of least squares). With any luck, you can piggyback observations of extinction stars with those of standard stars (more about these in the next section).

The shotgun-scatter approach is best for observers in sites with poor skies, because it is fast and uses stars spread over the whole sky. The idea is to measure standard stars low in the east and west, and then several more near the zenith—all in quick succession. These data sample the extinction curve at widely separated air masses, allowing the observer to determine the slope easily. As a follow-up to guard against changing atmospheric conditions, you should observe a few more standards at intermediate elevations during the course of the night. At good sites, the photometrist simply assumes that well-determined standard values are valid, and does not measure extinction at all.

The differential photometry method, which is not valid for the all-sky method, is to ignore extinction and make differential measurements between your program star and a comparison star. The idea behind this method is that extinction is virtually the same for two stars in the same field of view; therefore to an excellent approximation, extinction cancels out. Of course, if the stars have very different colors, extinction will not be the same for the two—so you need to be careful in your choice of comparison stars. Differential photometry is a powerful technique with CCD cameras because you can "sit" on the object (with the comparison star in the same field of view) and take image after image, following the behavior of the program star in time. Differential photometry even works through light clouds, as long as you can still obtain reasonably good images.

10.3.3 Transformation to the UBV(RI) System

To measure the time of minimum light of an eclipsing binary star, or to monitor outbursts from a dwarf nova, it is usually not necessary to convert instrumental magnitudes to the standard system. However, to calibrate the magnitudes of comparison stars on AAVSO charts for visual observers, it is necessary to insure that the magnitudes fit into the standard UBV system. Once you have established a set of reliable transformation coefficients for your telescope, filter, and CCD combination, you can correct your raw instrumental magnitudes for extinction and then transform them to the standard UBV system.

Observationally, you need to take images of a dozen or so standard stars through each of your filters, and you also need images of several extinction stars over a wide range of zenith distance to establish extinction coefficients for the night. After measuring raw instrumental magnitudes for all of the standard stars ind extinction stars from your images, you determine the extinction coefficients and use them to correct the raw instrumental magnitudes of the standard stars.

Consider your data at this point: for each standard star, you have its accepted V magnitude, and your instrumental magnitude, v. If you observed with other filters, you also have the accepted (B - V) color and the "natural" color from your system, (b-v). The transformation looks like this:

where Zv corrects the zero point of your instrumental magnitudes to the standard system, and £ is a color-dependent term that corrects for minor differences in the effective wavelengths of the filters you are using. Since you know V, v, and B-V for a dozen stars, you merely perform a simple least-squares fit to obtain Zv and 8.

You follow a similar procedure for the colors. Since you have the standard colors, (B-V) and (U-B), and the instrumental colors, (b - v) and (u-b), for each of the stars, you look for transforms in the form:

(U-B) = i|f(u-b)-Z{U_B), where |1 and \\f compensate for differences in passbands and effective wavelength, and the two constants Z{B_V) and Z(U_B) adjust the zero points.

Note that because corrections for extinction and transformation in the standard system are linear transforms, they can be reduced to a single linear transform.

In summary, stellar magnitudes must be measured over a well-defined range of wavelength. To define and limit the wavelength range reaching the CCD, images intended for photometry should be taken through a filter that has been de signed to match the passband of a standard photometric color system. The UBV(RI) color system is the de facto standard in astronomy. Magnitudes measured from a properly filtered CCD image are called raw instrumental magnitudes. For all-sky photometry, after the stars are measured, the raw instrumental magnitudes must be corrected for atmospheric extinction and transformed to the standard system. For differential photometry, you simply compare the measured magnitude of the star you are observing with a comparison star in the same image.

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