old visual and photovisual magnitudes and is thus most relevant for comparison to naked eye estimates. The B band approximates photographic magnitudes. The difference between magnitudes in two different passbands is called a "color index." In the Johnson system, B-V is a widely used color index. The redder the star, the larger the color index; thus, the color index can be considered a "redness index." For a blue star such as Spica, B-V ~ -0.2; for a white star such as Vega, B-V ~ 0.0; for a yellow star such as the Sun, B-V ~ +0.6; and for a very red star such as Antares, B-V == +1.8.

The value of the constant in (3.1) depends on the wavelength of the light under consideration, the width of the passband, and the amount of light received from the stars adopted as calibrating standards. The constant is thus the luminous energy arriving in the vicinity of the earth, per second, per unit area of the receiver, and per unit wavelength interval, corresponding to a magnitude of zero. Note that this quantity is not a direct measure of energy emitted at the source. We use the definitions of Meyer-Arendt (1972/1995), Sterken and Manfroid (1992), and Cohen and Giacomo (1987) to draw the distinctions and provide definitions. First, we note that energy expended per second (say, in joules/second, abbreviated J/s, or in ergs/s) is called power. A common unit of power is the watt,6 abbreviated W (1W = 1 J/s), equivalent to 107erg/s.

The amount of energy radiated per second at the source is called the radiant power or, sometimes, radiant flux (in units of watts), or, considering only the power in a spectral region centered around the wavelength 0.555 mm (micrometer or micron) equivalent to 555 nm (nano-meters) or 5550 A (angstroms: l ~ 10-10m), which is the approximate wavelength of peak sensitivity of the human eye in daylight, luminous power. The unit of luminous power is, naturally enough, the lumen (abbreviated lm), equivalent to about 1/680W (see Meyer-Arendt 1995, p. 351, and remarks below; the sensitivity of the human eye to different wavelengths makes this conversion factor vary with wavelength and bandwidth). Radiant (luminous for the visual region) exi-tance is the power emitted per unit area in units of W/m2 (and in the visual, lm/m2); radiant (again, luminous for the visual region) intensity is the amount of radiant (luminous) power emitted into a solid angle cone of a certain solid angle, W, and has units of W/sr (for the visual, lm/sr). If a source emitting monochromatic radiation at a frequency of 540 x 1012Hz (i.e., at a wavelength of 555 nm) into a given direction, has a radiant intensity of (1/683)W in that direction, the luminous intensity would be 1 lm/sr, a quantity also called a candela (cd). Finally, the amount of radiation

6 Named for James Watt (1736-1819), a Scottish engineer. The joule is named after James Prescott Joule [1818-1889], a British scientist.

7 Angles are measured in degrees or radians (2p radians = 360°). Solid angles are measured in square degrees or steradians (sr). Generally,

W = area/(distance)2. The surface area of a sphere of radius R meters is 4pR2 square meters, so that from the center, W = 4p steradians. A 1 sr solid angle is that subtended by an area of one square meter at a distance of 1m (N.B.: the area can be any shape). Also, 1sr = (180/p)2 = (57.296)2 = 3282.8deg2, and the entire sphere subtends at the center

4p sr = 41,252.88 square degrees.

through a unit area and unit solid angle7 at the source is the radiance; luminance refers to the visual component of the radiance. Radiance and luminance are used to describe the power emitted at different regions of the emitter's surface, and they are sometimes referred to as "brightness" or "surface brightness."They have units of Wm-2sr-1 for radiance, and cd/m2, called nit (for the Latin nitere, "to shine"), for luminance. Alternatively, the luminance is given in units of lamberts (104/p cd/m2).

In contrast to the light produced at a source, the light that we receive, incident on the surface of a telescope or of the human eye, is the irradiance or the illuminance. The irradi-ance is often called flux or flux density in astronomy; it has units of Wm-2. The irradiance in a small spectral region is the spectral irradiance; the illuminance in a unit frequency or wavelength interval is the monochromatic flux, with units of Wm-2Hz-1 or, for example, Wm-2/|mm. In astronomy, magnitudes may be used to describe a logarithmic form of an irradiance; visual magnitudes may describe a logarithmic form of an illuminance. We should emphasize that in the disparate fields of radiometry, photometry, and astronomy (and even among optical, radio, and infrared astronomy), the names, symbols, usages, and units of the technical terms may differ somewhat. For example, in radio astronomy, a common unit of monochromatic flux is the Jansky (Jy) equal to 10-26Wm-2 Hz-1.

For the V band, the constant of (3.1), in Systèm International d'Unités (SI units), is

and for B, it has the value

= -17.857, when € in each case is expressed in the same units. Because the Watt is a unit of power, energy per unit time, we are talking about the amount of energy in the form of light passing through an area of 1 square meter every second. The amount of light is restricted by the passband, indicated by the micron (mm) unit. The zero point of the V magnitude coincides approximately with that of the photovisual magnitude, mpv, which in turn approximates that of mvis. That of B is slightly displaced from that of mpg. Thus,

Equations such as (3.3) are called transformation equations, because they permit us to transform data from one system (here, the photographic) into another (the UBV or Johnson system).

For visual (naked eye) observations, of point (i.e., unresolved) sources, magnitudes are approximated by the simple expression, mv =-13.98 - 2.5log E,

where E is the illuminance, expressed in SI units of lux (lm/m2). In these units, 1 foot-candle (fc) = 1lm/ft2 o 10.76

lux o ~0.01576W/m2), and 1 phot = 1lm/cm2 o 104lux (Allen 1973/1976, p. 26). A source for which mv = 0 (about that of the star Vega) has an illuminance of ~2.54 x 10-6lux = 2.36 x 10-7fcs = 3.72 x 10-9W/m2. From (3.4), and C.W. Allen (1973/1976, pp. 197), a star with an illuminance of 1 lux has a visual magnitude of -13.98. The values of the quantities mentioned here apply outside the atmosphere. The effect of the atmosphere will be considered in the next section, but for now, we mention that it both dims and scatters light, therefore, dimming, making redder, and obscuring astronomical objects. Generally, such data need to be "reduced" to outside the atmosphere. Note that by "visual," we imply that the light has been integrated (collected) over the visual "bandpass," i.e., the range of wavelengths equivalent to the net sensitivity of the eye and centered near the peak wavelength of this range. The range of wavelengths to which the eye is sensitive differs with illumination level, and it differs somewhat from person to person. Thus, this type of formulation, although often helpful, needs to be applied cautiously to ancient observations because it requires that allowance can somehow be made for the effects of the atmosphere, general lighting conditions, and differences among observers. For example, someone studying the brightness of the sky as viewed in the past needs to consider that open fires, lamps, and torches were common near cities and the spectrum of this illumination would have varied from place to place, depending on population density and the predominant fuels—the types of wood/peat/oil— that were available. Except, possibly, for observations in some parts of the third world, these conditions would be different from conditions almost anywhere in the world today (where some degree of artificial lighting exists). The sky background and the extinction due to the soot would be more akin to observing conditions in the vicinity of local fires, which are highly variable, as every photometrist who has tried to observe under such conditions can attest. A redder sky also means a different sensitivity level for the eye as well. The photopic sensitivity level for a wavelength of 600 nm, for instance, is only ~63% that at 550 or 560 nm, for an average individual (Meyer-Arendt 1995, p. 352). See §3.1.4 for differences between day and night vision, which have different color sensitivities. A full formal discussion of the personal equation involved for the conditions under which ancient photometric observations must have been carried out needs to be made.

For extended objects, the situation is somewhat more complicated. In dealing with extended sources, the radiance or luminance must be considered. The perception of the brightness differences from point to point of a nonuniform source such as the Moon, or a limb- or spot-darkened Sun (§5.3.1), depends on the spatial resolution and brightness and contrast sensitivity of the eye, which we take up in §3.1.4. It is interesting, though, that the irradiance at a detector (the eye or a photometric instrument of some kind) of an extended source such as a dense star cluster or galaxy is in fact independent of distance. This is because the flux density falls off with the inverse square of the distance, but the image area at the detector depends on the solid angle of the source, which has the same inverse square dependence on the distance; hence, they cancel out. Distance sources with the same radiant power will appear smaller, and the total radiation received from them will be less, but the surfaces of those sources will look equally bright; i.e., the irradiance of those sources at the detector will be the same. Finally, we note that for uniformly bright, extended sources, the irradiance (in astronomy, the flux or flux density), F, is equal to the product of the radiance (in astronomy, the intensity), I, and the solid angle, W:

Recall that I is in Wm-2sr-1, and F is in Wm-2. At the mean distance of Earth (1 "astronomical unit" = 1.5 x 1011m), the mean intensity of the Sun's disk is 2.000 x 107Wm-2sr-1, and the Sun occupies a solid angle of 6.8000 x 10-5sr, so that F = 1360 Wm-2. This quantity is in fact measured (the "solar constant") and is the flux of total radiation received by the Earth at its mean distance from the Sun. When F is multiplied by the area of a sphere at this distance, the total radiant power (luminosity in astronomy) of the Sun is 4pa2F = 2.8 x 1023m2 x 1360Wm-2 = 3.8 x 1026W. Schae-fer (1993a, p. 319) gives a formula for the illuminance of an extended source. It involves an integration over the solid angle of the source. Rewriting this in terms of the illuminance, F and mean luminance (surface brightness), <b>,

or, in terms of a visual magnitude, mv = -16.57 - 2.5 log F, where the constant applies for Schaefer's units: b is given in nanolamberts,8 W in steradians, and F is in units of foot-candles (lm/ft2) [= 10.76 lux]. Further discussion of extended sources can be found in Schaefer (1993a).

An example of the importance of relating the energy to the observed brightness will be seen in our discussion of the visibility of meteoritic impacts on the Moon (§5.6). We now discuss the correction of observations for extinction and the standardization of photometric observations. Correction for Atmospheric Extinction

As we noted earlier in this chapter, the brightness and color of an object are affected by atmospheric transparency. This section deals with the details of the extinction process.

The atmospheric extinction in magnitudes is usually assumed to be linear with air mass. This is not strictly true, but the approximation in the optical region of the spectrum is not bad. That means that if the light of a star (or other luminous object) traverses twice the thickness of the vertical column of air, its extinction in magnitudes will be twice as great. Equation (3.7) shows the commonly used relation between observed magnitude, m, outside-atmosphere magnitude, m0, and the air mass, X:

8 The lambert, named for the Swiss scientist Johann Heinrich Lambert (1728-1777), is the brightness of a surface emitting (as for the Sun, or reflecting, as for the Moon and planets in visible light) one lumen per square centimeter. In SI units, 1 lambert = 104 lumen/m2 so that one nanolambert (= 10-9 lambert) = 10-5lumen/m2. For reflection cases, the surface is assumed to be be fully diffusing. See modern optics texts such as Meyer-Arendt (1972/1995) or Jenkins and White (1957) for further discussion.

where k' is called the first-order extinction coefficient and k" is the second-order (or, more accurately, the color) coefficient, and c is the observed (and uncorrected) color-index of the object. The quantity k' has a typical value at a sea-level site of about 0.25 for visible light, and it is generally less at higher altitude sites. Its value depends strongly on the wavelength and weakly on the color index of the object observed. The value of k' depends also on atmospheric conditions, so that it varies from site to site, night to night, and often even during a night at the same site. Because of the dependence on the color of the observed object, a color-term is sometimes subtracted from the right side, as shown in (3.7). The quantity k" usually has a value less than 0.01. The color indices can be treated in a way similar to the magnitudes:

Typical values for the B-V color coefficients at sites where astronomical photometry is carried out are k'c ~ 0.15; k" ~ -0.02. One might think that if atmospheric extinction were due solely to molecular scattering effects, then the wavelength-dependence of Rayleigh scattering could be used to predict the extinction in one passband given the extinction in another. This would be true if (1) there were no aerosol (water vapor, dust), or specific absorber content to the atmosphere (e.g., terpenes near forests, discrete chemicals near smelter works, etc.), and (2) all photometry systems were identical (i.e., with the same effective wavelength and bandwidth). For clear air conditions and for one and the same stable system, however, correlations of extinction in the several passbands can be determined (see, for an example in the infrared, Glass and Carter 1989). For the naked eye, sensitivity to color varies widely, and because the atmosphere strongly reddens light, the perception of brightness of a star can be expected to vary from individual to individual (even from eye to eye!). Careful and controlled experimental work to establish or qualify this would be of interest.

The air mass may be precisely computed for most observations. It is related to the zenith distance, z, or the altitude, h, through the expression:

for relatively small values of z. For altitudes down to about 10°, an approximation for the curvature of the atmosphere close to the horizon must be used. One such approximation is given by Bemporad and reproduced by Hardie (1962):

X ~ sec z - a(sec z -1) - b(sec z -1)2 - c(sec z -1)3, (3.10)

where a = 0.0018167, b = 0.002875, and c = 0.000808.

For objects even closer to the horizon, the extinction is much more difficult to determine. Detailed modeling of the scattering and absorption properties of the atmosphere are needed on the night of the observations. For rising stars and planets, an approximation that is in wide use is to assume that the altitude in degrees at which an object can be first observed with the naked eye (i.e., brighter than about visual magnitude 6) is equal to its magnitude: a 4th magnitude star would be first visible at an altitude of about 4°. This affects the measured bearing of the object and hence archeological alignments that depend on it.

Atmospheric extinction in several forms:

(1) Selective absorption by atmospheric gases

(2) Continuous scattering by atmospheric molecules

(3) Scattering and absorption by suspended particles in the air (aerosols)

Selective absorption removes radiation at certain specific wavelengths. A star's spectrum has features that originate in its atmosphere, in the interstellar medium through which the starlight travels, and, finally, in the Earth's atmosphere. The latter include water vapor bands at ~590nm (in the orange region of the spectrum) and 650nm (in the red region), and molecular oxygen bands at 627, 687, and 760 nm. The near infrared contains many features of water vapor and carbon dioxide, among many other molecules. In the ultraviolet, ozone is an important absorber.

The continuous scattering by atmospheric molecules (mainly nitrogen and oxygen that together make up ~98% of the Earth's atmosphere by weight) removes the bluer components of starlight relative to the redder. The radiation is scattered into the night sky. Thus, sunlight is reddened and the sky made blue during daylight hours.

Finally, there is the atmospheric aerosol content, the most variable of the extinction components at low altitude sites. Examples of aerosols are ocean spray, dust from deserts and volcanoes, pollen from trees, and smoke. Aerosol particles, which are generally much larger than the wavelength of light, scatter light more or less equally at all wavelengths.

The very high air mass value near the horizon causes a large uncertainty in the observed magnitude for even small changes in the extinction coefficient. Suppose, for instance, that the extinction coefficient varies by ~ ±0.05 magn. Then, for X > ~20, the uncertainty in the extinction, Dk x X, will vary by ~ ±1 magnitude or more. For assumed values of extinction, sky brightness, humidity, and for the altitude of a site near the Big Horn Medicine Wheel, Wyoming, Schae-fer (1993a, p. 343) calculates an extinction of 0.85 magn. for the star Aldebaran at the extinction angle, which he computes as 0.6°. At the South pole, at an altitude of 3 km, he finds an average kv of 0.14 for both summer and winter, whereas in a site in Athens, Greece (altitude 107 m), he finds 0.25 and 0.31 for summer and winter, respectively. Values for some other sites are 0.22 and 0.28 for Tucson (770 m altitude), 0.18 to 0.28 for Jerusalem (775m), and 0.28 to 0.46 for Los Angeles (100m). The fact that the setting sun may sometimes appear white with only a yellow tinge, a common sight in a clean, dry, western site such as Alberta, for instance, and sometimes a deep crimson, particularly at sea-level coastal sites, shows that color coefficients may vary greatly.

For more detailed work in this area, Schaefer (1989) provides a program to compute the air mass for relevant atmospheric conditions and (Schaefer 1993a, pp. 315-319) provides formulae and tables for X and extinction coefficient k computed from formulae for the Rayleigh scattering, ozone absorption, and aerosol scattering contributors to extinction.

The sensation of color varies from individual to individual, and in the same individual, it varies with light level and other variables. This is so apart from the variation in precision of language used to describe those sensations. The question of the true color of astronomical objects arises in ancient astronomy. For example, the dog star, Sirius, was described by most ancient astronomers (except possibly the Chinese) as some color in the range of yellow to red; yet today it is clearly white. This discrepancy has resulted in much discussion about whether the ancients were describing the true color of the star. Because there is a strong possibility that they were, serious questions about stellar evolutionary time scales then arise. We discuss the Sirius question in §5.8.4. Standardization

Standardization of data can be thought to be a thoroughly modern aspect of astronomy, primarily because we associate such an activity with the scientific method. Scientists try to recreate the conditions of an experiment to test hypotheses by varying one variable at a time. In astronomy, systematic observations of brightness require a further correction for site-induced variation. Observed values of magnitudes and color indices, even after correction for extinction, are still not the same as catalogue values. Each telescope system is different in its sensitivity to the brightness and colors of the stars, and so the data have to be transformed to a standard system. Yet this too is relevant to a discussion of ancient observations because observations made with the human eye may require a "personal equation" to correct for different sensitivies to brightness and color. The transformation equations usually have the form:

where e and m are called transformation coefficients, Z and Zc are zero points, and m0 and c0 are the local system magnitudes and color indices, as above. When mstd is V, the quantity cstd in (3.11) and (3.12) is often B - V. The determination of the extinction and transformation coefficients and zero points is beyond the scope of our discussion, but the reader is referred to one of many articles on the subject (such as Hardie 1962 for the general idea, or Young 1974 for strong qualifications and refinement of methods). As we note below (§3.1.4), the color perception of the human eye depends on three types of cone receptors that act rather like the photographic or photoelectric detectors discussed above and can detect color in sources if they are brighter than ~1500 nanolamberts. The spectral sensitivity of the three types of cones varies with the individual, but normally peaks in the regions 600, 550, and 450 nm, for the red-, green-, and blue-sensitive cones, respectively, and the sensitivity curves overlap. See Schaefer (1993a; 1993b, pp. 87-88) for a terse summary and Cornsweet (1970) for detailed discussion. In principle, one can try to transform data from one individual to a "standard observer." In practice, this is difficult to do, partly because we have too little data about observing conditions from observers in the remote past, so that important complicating effects are not known, and partly because few modern astronomers attempt to perform the time-

consuming visual experiments needed to make such a study. One of the exceptions to the latter is Art Upgren (1991), who studied the effect of increasing light pollution on the visibility of stars labeled "blue" and "red" over a 14-year interval; in that study, however, no strong color-effect was noted, and the detailed dependence of visibility on color index was not investigated. More such studies are needed to isolate the complicating effects of extinction and sky brightness from observer color sensitivity. Modern Star Data

In order to discuss the brightness changes of such objects as the "lost Pleiad," the brightness of Sirius, the pretelescopic detection of variability of the demon star Algol, and so on (cf. §5.6), it is useful to have a list of stars at their current brightnesses. This is provided in Table 3.1, which lists the positions of some of the brighter stars in right ascension and declination coordinates with respect to the equinox of 2000 a.d. [a particular equinox must be specified because of the precession of the equinoxes that changes these coordinates with time (see §3.1.6)]. The brightnesses are expressed in V magnitudes and B - V color indices. All stars of visual magnitude ~3 or brighter, and a few that are fainter (like the seven brightest stars of the Pleiades star cluster), are included. The common names for the stars are given just after their constellation designations; translations and alternatives may be found in C.W. Allen (1963) or in the Bright Star Catalog (Hoffleit 1982). The spectral classification is given in the column marked "Sp." The spectral classification is based on the features in visible spectrum of the star, and these are primarily determined by the star's surface temperature. In order of decreasing surface temperature, the major classes are O, B, A, F, G, K, and M. The lettered class is followed by arabic numbers (0-9) that mark progression within each spectral class. The designation of a Roman numeral following the spectral class indicates the luminosity of the star: V = dwarf (sometimes called a main sequence star); IV = subgiant; III = giant; II = lesser luminosity super-giant; and I = greater luminosity supergiant; gradations a and b are applied to supergiants. Older designations of luminosity include "c" for supergiant, "g" for giant, and "d" for dwarf. Other nuances of spectral classification include "p" for peculiar and "e" for emission features. The star g2 Velorum is classified as "WC8," which means it is a Wolf-Rayet star, a very hot star thought to be the core of a luminous, evolved star that has lost its outer atmospheric envelope. The Sun, a yellow star, has the spectral classification G2V. As in other contexts, a color following an entry such as the spectral class indicates uncertainty. Note the relationship between spectral class and color index in Table 3.1. The bluest stars are O and B stars, and the reddest are K and M stars, but color can be affected by interstellar reddening, even if the atmospheric reddening produced by the extinction coefficient term kBV. has been corrected. Stars can therefore show a color excess, defined as

where (B - V) is the observed color index and (B - V)0 is the intrinsic color index (at the source). If the star has been reddened by the interstellar medium, it has been dimmed as well, by an amount of interstellar extinction, AV ~ 3.1 x EBV. The interstellar extinction is due to the additive contributions of individual dust clouds and averages about 1 magnitude per 1000 parsecs (3260 light years) of distance in the galactic plane. Intrinsically blue stars that are very far away may be reddened by interstellar matter, but their spectral classification will not change. Such a cause is unlikely to effect color changes of stars within millennia, but circum-stellar matter can and does vary over much shorter time scales. In an interacting binary star system, for instance, matter from one star, expanding as it moves toward a red giant phase, may stream around the companion. This may dim the companion, and thus the system as a whole, as well as redden it. No corrections for interstellar extinction or reddening have been applied in Table 3.1. Stars that are unresolved doubles (either a bound binary star system or merely near each other in the apparent plane of the sky), are designated "D" in the "Comment" column; variable stars are designated "Var." Many of the stars marked "D" have only very faint optical companions, which may not be gravita-tionally bound to the naked eye star at all, but located at a much different distance from us; the designation merely serves as a warning that changes in color and brightness recorded in pretelescopic times must be carefully examined to ensure that the effect did not involve a companion star. In those cases of double stars in which the components can be resolved by small telescopes, and for which the component magnitudes and colors are available, the magnitudes and color indices of the combination, VC and (B - V)C, have been computed. The formula is

VC = V2 - 2.5 • log[1 +10-0-4<Vl-V2) ]. (3.14)

A similar expression holds for BC, and from these the difference, (B - V)C can be calculated. The B magnitudes are obtained by adding V and (B - V).

Notice that in computing the combined brightness, magnitudes are not additive: Two stars of magnitude 5 do not have a combined brightness of 10 magnitudes, but of ~4.247.

The stars of Table 3.1 have been incorporated in the star charts of the appendices. Table 3.2 lists the brightness in magnitudes and the color in color indices of solar system objects. In addition to the planets, four minor planets, sometimes called asteroids, are included: Ceres, the first discov-ered,9 and Pallas, Juno, and Vesta. At their brightest, these objects are just visible to the unaided eye. Included also in Table 3.2 are the four brightest moons of the planet Jupiter (Io, Europa, Ganymede, and Callisto); called the Medicean moons by their discoverer, Galileo, they are today called the Galilean satellites. Were they not so close to Jupiter, they would be visible to the unaided eye. The magnitudes shown are for mean opposition for the exterior planets and the moons of Jupiter. Mean opposition refers to an opposition when the object and the earth are at their average distances from the sun. For the inferior planets and minor planets, the brightest magnitudes that these objects can attain as seen

9 The discovery was made on the first night of the 19th century, January

1, 1801, by the astronomer Giuseppe Piazzi, in Palermo, Sicily.

from earth are tabulated. The solar and terrestrial distances of each planet vary, and so its brightness varies, with the inverse square of those distances. Moreover, the reflected light may vary with phase angle (the angle between the direction to Earth and the Sun as viewed from the planet). The magnitudes corresponding to the configuration at which a planet may be seen at any particular time may be calculated by formulae given by Harris (1961, p. 276ff). Greatest brilliancy for the planet Venus occurs about 1.1 months after greatest eastern elongation and before greatest western elongation. At the present time, Pluto is near perihelion and its opposition magnitude is about 13.7. One major mystery is why Uranus was not observed in antiquity, a question first raised by the 18th century astronomer J.E. Bode (1784, p. 217). See Hertzog (1988) for a response.

The B - V color index quantifies the redness of Mars (compare it to the red stars in Table 3.1). The redness of Mars is due to an iron oxide in the surface soil. The redness of Mars, the yellowness of Saturn, and the relative whiteness of Jupiter and Venus provide a kind of scale of color to compare the colors of other objects, such as the stars Antares (rival to Mars) or Sirius, the historical color of which has been the source of much controversy. We will return to the issue of the color of Sirius in §5.8.4. Here, we note only that the present color of Sirius is white; yet, Seneca, writing ~25 a.d., commented that Sirius was redder than Mars (Brecher 1979, p. 97). See Bobrovnikoff (1984) for further discussion of color descriptions of astronomical objects in antiquity. Sky Brightness and Visibility

The usual limiting magnitude for naked eye detection is ~6, but in practice this limit is too optimistic unless the site is exceptionally dark and clear; of course, superior acuity helps. Observations of astronomical objects are limited not only by the atmospheric conditions (cloud, fogs and mists, atmospheric Rayleigh scattering, and absorption), but they are limited also by the brightness of the sky.

The brightness of the sky is the sum of several contributions:

(1) Intrinsic brightness of the sky (the combined direct light of faint stars and distant galaxies and starlight scattered by the atmosphere)

(2) Sunlight (daylight or twilight)

(3) Moonlight (earthshine as well as solar reflection)

(4) Atmospheric emissions

(5) Artificial lighting

(6) Scattering efficiency of the atmosphere

The classic source for the effects of these sources on the visibility of stars is Minnaert (1954), but more recent sources may be more useful. Schaefer (1993a, p. 321) gives formulae for the sky brightness in units of nanoLamberts (see fn 8, § for each of these sources. Most practically, Upgren (1991) provides empirical data of the altitudes at which stars of particular magnitudes are visible. At high latitudes, both north and south, the summer season is marked by increased hours of sunlight; north of the Arctic Circle (south of the Antarctic Circle), (the "land of the midnight Sun"), the Sun

Table 3.1. Positions, brightnesses, and colors of selected stars.



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