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Figure 3.1. The geometry of the plane-parallel atmosphere approximation and the definition of air mass. Drawing by E.F. Milone.

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Figure 3.1. The geometry of the plane-parallel atmosphere approximation and the definition of air mass. Drawing by E.F. Milone.

objects are similarly reddened, and the closer the object is to the horizon, the longer the path length through the atmosphere and, thus, the greater the scattering and the redder the object appears.

Dust and smoke also scatter light, but the ash created by forest fires may sometimes cause different color effects (there are reports of distant forest fires causing a literal "blue moon" effect2—causing more scatter in the red region of the spectrum, due to the large size of the scattering particles). Under normal circumstances, at sites near sea level, it is not uncommon to find even a totally clear sky blocking more than half the near-ultraviolet light from an object at the zenith. At larger zenith distances, still more light is lost. Astronomers use the term extinction to describe the diminution of light caused by its passage through the atmosphere and discuss the extinction of starlight in terms of magnitudes of extinction per air mass (magnitudes are defined below). Air mass refers to the thickness of the column of atmosphere through which the light passes compared with that at zenith. Very roughly for low altitudes, but a good approximation for higher altitudes (greater than ~45°), the extinction is proportional to the inverse cosine (or the secant) of the zenith distance (or 90-h). Figure 3.1 illustrates the basic geometry, ignoring a correction for the curvature of the Earth's atmosphere, which is important for objects near the horizon. The dimming and reddening of starlight (and sunlight and moonlight) has important consequences for the visibility and the ancient descriptions of these objects. Schaefer (1993a, b) summarizes and discusses the visibility of objects due to various effects, and he may be the best source of information about the class of problems that he calls "celestial visibility."

In dry, high-altitude sites, the total visual extinction in an otherwise clear sky may amount to little more than 10%, but at low, moist sites, it may exceed 30% to 40%. As a consequence, Schaefer has challenged the capability of establishing precision alignment at such sites. In order to be quantitative about these matters, and to be able to compare

2 Perhaps because this term connotes rarity, it has also been applied recently to the second of two full moons within a civil calendar month. Because there are either 30 or 31 days in all months but February, the average 29d.53 length of the Moon guarantees that it will occur whenever the full moon occurs on the first day of the month.

observations from different observers and different times, it is helpful to understand a few observational concepts. The next few sections introduce these concepts. An excellent general source for this material, and one that relates general usage to astronomical photometry, is Sterken and Manfroid (1992).

3.1.2.1. Magnitudes and Color Indices

Here, we define the brightness and color of a star. The traditional way to describe brightness is by magnitudes. Astronomers in antiquity, and even much later, did not hesitate to refer to the brighter and fainter stars as "bigger" or "smaller," respectively. However, the word magnitude in modern contexts has nothing to do with size, despite its etymology3 and common usage, because the actual surface area of a star other than the Sun cannot be resolved by the human eye. Magnitude is an index of the faintness of light from an astronomical source: The fainter the star, the bigger the magnitude. The visual magnitude of the total collected light of the Solar disk is —27, that of the star Sirius, -1.6, that of Vega, ~0.0, and that of Barnard's Star (invisible to the naked eye), ~+10.

Systematic estimates of star brightness were first recorded according to current knowledge by Hipparchos4 [~146—127 b.c.], who assigned a magnitude of 6 to the faintest stars visible to the eye and 1 to some of the brightest. The response of the eye can be modeled by a logarithmic function to changes in light; so the ratio of the brightest to the faintest star represented by Hipparchos's five magnitude difference is about 100. The present magnitude scale now in use was suggested by Norman Pogson in 1856. It sets a difference of 5 magnitudes exactly equivalent to a ratio of 100: 1 in brightness. A difference of 2.5 magnitudes is then equivalent to a ratio of 10:1, a 7.5 magnitude difference equivalent to 1000:1, and so on.

The magnitude, m, is related to the detected light, € (its units will be discussed below) by the following expression in terms of base 10 logarithms:

The relationship between the brightness of two objects is expressible in the following equation:

The quantities € and €2 represent energy in the form of light received per second per unit area and in some passband, a particular region of the color spectrum, from two astronomical point-like sources.

Before the development of photography, the implied passband was the entire range visible to the eye. These "visual magnitudes" are commonly designated mvis. Amateur organizations such as the American Association of Variable Star

3 From the Latin magnus from the Greek megas, size;Ptolemy used the related word megathos for magnitude.

4 "Hipparchus" in its Latin form. See §7.2 for a discussion of

Hipparchos's many other contributions to astronomy.

Observers encourage observations of variable stars by providing observers with star charts with calibrated (standard or constant) stars to assist in gauging the relative brightness of variable stars, although amateur astronomers today increasingly use detectors other than the eye to make such observations. The measurement of the brightness of naked eye variable stars such as Algol or b Lyrae (see §5.8.1) is sometimes assigned by astronomy instructors as an exercise in basic astronomical observing. Because the eye's color and sensitivity varies from person to person, a "personal equation" needs to be applied in combining observations from different observers. Such an equation contains a "color term" and a zero point, typically. We discuss this "transformation" problem more generally below.

The use of photographic plates, beginning in the latter part of the 19th century, permitted more restricted spectral regions to be studied. The first one was essentially created by the natural, somewhat bluer sensitivity of the emulsion compared with the eye. Magnitudes in this system are designated mpg. By means of special dyes added to the emulsion, other regions could be defined. One of these was the photovisual, replicating that of the eye: mpv.

The modern photoelectric photometer and, more recently, the "charge coupled device" (CCD), provide higher precision in the measurement of the energy in starlight. A problem in comparing data from different observers is that the data are typically obtained with different equipment and under different observing conditions. To overcome the uncertainties in interpretation of the results, modern observers need to "standardize" their data. It is helpful to bear this in mind when dealing with ancient observations too. So, how do we do this?

Standarization requires measurements to be made in precisely defined passbands, which are defined by the spectral sensitivity of the detection system (these days): the telescope, detector, and color filters; or (throughout most of human history) the eye, alone. One of the most widely used modern systems is the UBV system of Johnson and Morgan (1953). In the 1960s, the Johnson system was extended to five passbands in the optical part of the spectrum: the ultraviolet (U), blue (B), visual (V), red (R), and near infrared (I) (Johnson 1966; Landolt 1983/1992). Johnson extended his system to include infrared (>1 mm) passbands also, but these were badly placed with respect to the transparency windows of the atmosphere,5 and they are difficult to standardize, especially at sites where the water vapor content varies strongly with time. The passbands most important for our purposes here are the B (centered at ~0.440 mm or 4400 A wavelength) and V (~0.550 mm or 5500 A), because of the large numbers of published observations in these passbands, and in the closely related visual and photographic systems. The V band is calibrated to approximate the

5 Water vapor, carbon dioxide, ozone, and other atmospheric constituents absorb light in the infrared, creating regions of high opacity broken by regions of relative transparency—the atmospheric "windows" in the infrared spectrum. See Milone (1989) for a discussion of the problems of standardization in the infrared and Young, Milone, and Stagg (1994) for solutions to some of them.

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