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1 These passbands are the Cousins UBVRI system as characterised by Bessell ([B2], [B3]), the Caltech (CIT) version of the Johnson near-infrared JHKL system and the Gunn ugriz system [F3], The Ks (K-short) passband is truncated at long wavelengths to minimise thermal background radiation. This passband is used in the DENIS and 2MASS near-infrared sky surveys (Chapter 6). Similarly, the L' passband avoids the worst atmospheric absorption in the CIT L-band.

2 References are for flux zero-points in Janskys - units of 10-26 Wattsm-2 Hz-1.

3 m^-p is the sky brightness in magnitudes arcsec-1 at a dark-sky site, such as Mauna Kea. Sky brightness longward of ~2.3 ^m is dominated by thermal radiation from the atmosphere, and is therefore highly variable on short time-scales. The sky brightness measurements for the Gunn filters were computed by A. West, J. Tucker and J. Gunn (priv. commun., 2004).

1 These passbands are the Cousins UBVRI system as characterised by Bessell ([B2], [B3]), the Caltech (CIT) version of the Johnson near-infrared JHKL system and the Gunn ugriz system [F3], The Ks (K-short) passband is truncated at long wavelengths to minimise thermal background radiation. This passband is used in the DENIS and 2MASS near-infrared sky surveys (Chapter 6). Similarly, the L' passband avoids the worst atmospheric absorption in the CIT L-band.

2 References are for flux zero-points in Janskys - units of 10-26 Wattsm-2 Hz-1.

3 m^-p is the sky brightness in magnitudes arcsec-1 at a dark-sky site, such as Mauna Kea. Sky brightness longward of ~2.3 ^m is dominated by thermal radiation from the atmosphere, and is therefore highly variable on short time-scales. The sky brightness measurements for the Gunn filters were computed by A. West, J. Tucker and J. Gunn (priv. commun., 2004).

Broadband photometric systems, with filter-defined passbands of full-width at least 500 A, are used to map the overall spectral energy distribution of celestial objects; narrowband filters are designed generally to examine features in the energy distributions of particular types of star. The most frequently used broadband photometric system is Johnson/Cousins UBVRIJHKLM [J2], [C1], spanning the wavelength range 3,000 A to 5 ^ with each passband having a width of ~ 1,000 A (Table 1.1). Within each passband, the measured flux density corresponds to the stellar flux at the effective wavelength of the filter. The latter quantity is found by convolving the spectral energy distribution of the star (S(A)) with the shape of the filter bandpass (5(A)):

The effective wavelength can vary depending on the spectral energy distribution of the target (the spectral type of the star observed). For example, decreasing temperature moves the peak in the emergent energy distribution towards longer wavelengths, steepening the spectral slope at optical wavelengths and moving the effective wavelength to the red. This effect is particularly important in the broad Cousins R-band, where Aeff changes from 6,380 A for an A0 star to 7,220 A at spectral type M0 [B2]. Other passbands are less affected, with typically 200 A differences in Aeff between spectral types A0 and M0.

Photometric colours are defined as a magnitude difference; for example, (B — V). From the definition of magnitude (equation 1.15), a 'colour' therefore measures the flux ratio in the two passbands:

Traditionally, colours are expressed as the shorter wavelength magnitude minus the longer wavelength, so a negative (blue) colour implies fshonlfiong is high, and a positive (red) colour indicates that fshortlf\ong is low. As discussed further in Chapter 2, colours provide a means of estimating stellar temperatures.

1.5.2 Measuring magnitudes

Observationally, the standard photometric systems were defined using photoelectric photometers and aperture photometry techniques, but most current measurements are made with array detectors. In the former case, a photomultiplier tube was used to measure the brightness of a given source through a circular aperture (usually 20-30 arcsec diameter); in the latter case, software techniques are used to measure the flux within a given radius centred on the star. In either case, since the aim is to measure a large fraction of the stellar flux, the effective aperture size chosen depends on the prevailing atmospheric seeing and guiding accuracy. However, the underlying night sky also contributes to this measurement and this contribution must be subtracted, either by obtaining a separate offset measurement in aperture photometry, or from the 'sky' pixels immediately adjacent to the object in array photometry.

Typical surface brightness values for a dark-sky site are listed in Table 1.1. Airglow (primarily emission from OH and OI; see [R2]) is a strong contributor shortward of 2 and the sky brightness at these wavelengths is well-correlated with the solar cycle, being higher when the Sun is more active [L6]. Volcanic eruptions can also affect msky if dust is introduced into the upper atmosphere, where it gradually diffuses around the Earth [L8]. The eruptions of the Mexican volcano El Chichon (1982) and the Philippine volcano Pinatubo (1990) not only produced spectacular sunsets, visible for several months afterwards in North America and Europe, but also raised the night sky brightness (and the optical extinction) for more than two years. At longer wavelengths, thermal radiation from the atmosphere and telescope dominates the background.

The signal-to-noise of an observation is given (following [L9]) by

AeffNv Avt effNv

where t is the integration time; Av, the bandwidth (Hertz); Aeff, effective area of telescope, in m— ; Nv, the source flux density, in photons, m~ ~2 s—1 Hz—1; Sv, sky brightness, in photons, m—2 s—1 Hz—1 arcsec—1; solid angle of effective aperture (either the physical diameter of the aperture in the photoelectric photometer, or the circle of integration used in analysing the array photometry); D, dark current; and R, readout noise (zero for aperture photometry).

Modern detectors have both low dark-current and low read-noise (3-7 electrons is typical for a CCD working at liquid nitrogen temperatures). Consequently, photon statistics in the sky level constitute the dominant source of noise in photometry of faint objects. In aperture photometry, with a photomultiplier tube, the sky measurements are made separately from the object (+sky) observations. One of the major advantages of array photometry is that the sky level can be determined from the same exposure used to measure the source. Moreover, a 2,048-square optical CCD array covers a typical solid angle of at least 150 square arcmin, encompassing many stellar (and non-stellar) objects in a deep exposure on even a moderate-sized telescope. Since these objects are all observed simultaneously, high accuracy relative photometry is possible even during inclement conditions. A caveat is that individual CCD pixels can have slightly different sensitivities, so the intensity levels in each frame must be normalised using a flat-field exposure - an image made by illuminating the CCD with a diffuse, uniform light-source.

Once CCD images have been normalised, sophisticated profile-fitting techniques can be used to determine the relative flux of each object, minimising the contribution from sky-noise. Clearly, the more concentrated the stellar profile (the better the seeing), the smaller the solid-angle for profile-fitting, the lower the contribution from sky-noise and the fainter the limiting magnitude attained in a given exposure time. The instrumental flux measurements themselves are calibrated through observations of standard stars with well-determined magnitudes on particular photometric systems. Extensive lists are provided by Landolt [L2], [L3] for the frequently-used Johnson/Cousins optical system (Table 1.1), and Persson et al. [P1] provide standard-star lists for the near-infrared.

In general, an instrumental magnitude is measured for each source, defined by (taking the V-band as an example)

where NV is the total number of counts measured for the source in an integration time of t seconds. Observations of standard stars are used to solve for the constants in an equation of the form:

where V is the magnitude on the standard system; v, the instrumental magnitude; z, the angular distance from the zenith; kv, the extinction coefficient; Cv, the colour term; and Zv, the zero-point. The extinction term corrects for absorption through the Earth's atmosphere (sec(z) is known as the 'airmass' of an observation) using a plane-parallel approximation for atmospheric depth along the line of sight. The colour term {which could also be {B-V) or {V-R) in this case) allows for a potential mismatch between the effective wavelength of the reference system and that used in the observations. Defining these terms accurately requires repeated observations of a reasonable number {15-20) of standard stars, well distributed in both colour and airmass. A more thorough discussion of photometric techniques is given by Henden and Kaitchuck [H2].

The observed magnitude of a star is termed the apparent magnitude {m). The absolute magnitude {M) is defined as the apparent magnitude that a source has at a distance of 10 parsecs. Hence, since brightness decreases with the square of the distance:

The quantity {m — M) is known as the distance modulus, a term often used in citing the distances to star clusters.

1.5.3 Bolometric magnitudes and effective temperatures

Summing the total energy emitted at all wavelengths for a star determines its bolometric magnitude, mbol = —2.5log!0 (/totM)) (1.24)

Given a known distance, this can be converted to the absolute bolometric magnitude which, in turn, can be expressed as a luminosity, usually in solar units. Based on absolute measurements of the energy distribution, primarily from satellite data, the Sun has an absolute visual magnitude of MV — 4.79 [L5]. The bolometric correction is approximately —0.12 magnitudes [B7], so Mb0i(©) — 4.67, corresponding to L© — 3.83 x 1026 Watts. Stellar luminosities are then given by

The luminosity of a star is also used to define the quantity known as the effective temperature. An ideal radiator of temperature T produces a pure continuum spectrum, with neither absorption nor emission features, and a spectral energy distribution that is described by the Planck formula. The flux from the black-body distribution can be written as a function of frequency:

where h is the Planck constant, k is the Boltzmann constant, c is the velocity of light, T is the surface temperature and v is the frequency. In this case the usual units for the flux, Fv, are Janskys, where 1 Jansky is 1(T26 Watts m^Hz"1 (MKS units). Alternatively, the Planck curve can be written in wavelength units:

2-Khc2 SA

where the usual units are ergcm~2A_1 or ergcm~2 (c.g.s. units). Integrating this distribution gives Stefan's law, which states that the total energy emitted is proportional to the product of the surface area and the fourth power of the temperature. This leads to the definition of the effective temperature of a star:

4nR2<rT4eff (1.28)

2-K5k4

where a is Stefan's constant, . The effective temperature. Teff, is defined by equation (1.28) as the temperature of a black-body of the same radius as the star which radiates the same total energy.

The black-body distribution peaks at a wavelength, Amax, whose value varies inversely with the temperature. The result is Wien's law, which can be derived by determining when the derivative of equation (1.27) is zero. The solution is

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