Astronomical Techniques - Photometry

Across the electromagnetic spectrum, many astronomical measurements require precise flux determinations, known as photometry or radiometry depending on the historical background in question. The units of measure are (alas) also dependent on context. Linear flux units (such as W/m2 or ergs/cm2 s Å ) are sensible, but much of the practice in the optical and near-IR is still centered on the system of stellar magnitudes. These are traced all the way back to Hipparchus' catalog (ca. 120 B.C.), and have been formalized (approximately matching the eye's response) to a definition as magnitude = constant - 2.5 log intensity in which the constant is set for each passband to give convenient zero point values. Thus, for example, m1 - m2 = 5 implies I2/I1=100, the intensity ratio corresponding to one magnitude is 5 √(100) = 2.512, and differences between magnitudes at different wavelength correspond to flux ratios. Why keep this archaic system? Quite aside from tradition, the system is internally consistent to greater precision than its external energy calibration. This calibration is based on a few fundamental stars and direct comparison with laboratory blackbodies, extended in wavelength using stellar atmosphere models.

A typical set of energy zero points is given here. Detailed calculations have been given for many passband systems by Fukugita, Shimasaku, & Ichikawa (1995 PASP 107, 945). The usual UBVRI bands have zero points set so that a mean unreddened A0V star has equal magnitudes (zero colors) in all bands. Vega is quite close to this requirement. Other scales exist in which the zero point is tied to constant flux per unit wavelength or per unit frequency. In this table, Fλ is in ergs/(cm2 s Å) and Fν is in watts/(m2 Hz).

 Band λeff, μm Zero point: Fλ Fν U 0.36 4.35(-9) 1.88(-23) B 0.44 7.20(-9) 4.44(-23) V 0.55 3.92(-9) 3.81(-23) R 0.70 1.76(-9) 3.01(-23) I 0.90 8.3(-10) 2.43(-23) J 1.25 3.4(-10) 1.77(-23) K 2.2 3.9(-11) 6.3(-24) L 3.4 8.1(-12) 3.1(-24) M 5.0 2.2(-12) 1.8(-24) N 10.2 1.23(-13) 4.3(-25)

In addition to these passbands, one sometimes sees so-called ABν or ABλ magnitudes, where ABν = -2.5 log Fν -48.594 for Fν measured in ergs/cm2 s Hz. This means that Fν(Jy)= 10 (-0.4 (ABν - 8.906)). Similarly, a widely-used system of UV magnitudes has mag = -2.5 log (flux / Å ) -21.1 for cgs units. And to make this even more confusing, Gunn and the SDSS folks have recently introduced a new sinh magnitude, which rolls over gracefully toward small or negative fluxes, but are useful only for large data sets with very consistent thresholds (such as the SDSS); see Lupton et al. 1999, AJ 118, 1406.

For any detector, the instrumental magnitude (denoted for example by lowercase u,b,v>) is given by a magnitude equation of the form v = const - 2.5 log R for instrumental response R in linear units. We need to correct this for atmospheric and purely instrumental effects before trying to place the data on some standard photometric system. The response is understood to be per unit exposure time; for a photon-counting system, we have R = n / t.

Atmospheric extinction must be allowed for. In precise applications, it must be directly measured for a particular site, night, and passband. It is convenient to parametrize the extinction as a function of air mass X, defined as the ratio of column density along the actualline of sight to that toward the zenith. It is very nearly equal to sec z, where the zenith distance z is the angular distance from the zenith to the line of sight, given by cos z =sin L sin δ +cos λ cos δ cos h where L is the local latitude, δ is the star's declination, and h its hour angle. Departures from the sec z approximation appear at large z, since the atmosphere is not plane-parallel but follows the curvature of the Earth's surface. If we observe the same star at two airmasses X1,X2, the extinction coefficient k will be k = Δ v / (X1 - X2) and similarly for extinction coefficients in color index (which sometimes have significant second-order terms) - see Hardie's article in Astronomical Techniques, for example. Then we may derive an instrumental magnitude "above the atmosphere" v0 = const - 2.5 log (n/t) - kX for further analysis.

To calibrate results to a standard system (as an example, we will take the UBV system), some stars with well-known properties in that system must be observed. If the passband of our system perfectly matched the standard one defining the photometric system, we will have a simple relation of the form V = v0 +zv to be solved as a weighted mean. In the real world, imperfect passband matches introduce color-dependent terms to such a transformation, for example (B-V) = zbv + (b-v)0 + cbv(B-V) Note that in general we do not already know (B-V) for program stars, so iteration using this equation and a first guess generally converges in two trials. Occasionally, for extreme colors, more complex functional relations are found.

Calibration terms are of two kinds: those that depend on properties of a particular observing session (the zero points and extinctions) and those that are instrumentally determined but not rapidly changing (the transformation terms). Thus, transformation terms may be averaged over many nights for greater accuracy; it is not quite right to solve a single night's data for both sets of constants as this introduces unnecessary errors in the transformations. All this is possible only in a cloudless sky, leading to the designation of photometric conditions. Once local standards are available in a field, imaging detectors can allow extension in flux or time coverage without requiring such pristine conditions.

PHOTOMETRIC EQUIPMENT

Photographic data: this and CCD use have the advantage of being all done after the fact. Due to the nonlinear response of emulsions, indirect measures are used. A particularly popular instrument is the iris photometer. Here, a star is centered in a measurement light beam, and a diaphragm (or iris) is adjusted until the light admitted around the edge of the star matches a constant reference signal. There is usually a roughly linear relation between iris reading and magnitude for "suitably set" values of the reference intensity. This is more repeatable than a simple diameter measurement, but can be superceded by densitometry.

CCD data allow sophisticated analysis, fitting point-spread function or analyzing the curve of growth of intensity with radius. A key element is accurate measurement of the sky level for subtraction from the star signal, especially difficult in dense clusters or nebular areas. For stellar images, one may either fit a scaled PSF determined from (an ensemble of) bright but unsaturated images, or use aperture photometry with apertures designed to optimize the signal-to-noise ratio for faint stars, then scale upward to account for light in the PSF falling outside this aperture. Examples include the DAOPHOT, DoPHOT, and ROMAFOT software suites. PSF fitting, though very powerful, may not be robust to changes in the PSF across the field of view. These approaches can also allow one to measure blended star images accurately, given the requirement that both images are of well-known structure.

Photomultipliers were especially powerful for photometry, in either DC-current or pulse-counting modes, and set the standard which later detectors had to meet. The photometer then contains (1) an aperture (possible selectable) to limit the region of sky measured, (2) interchangeable filters, (3) a removeable viewing apparatus to center the star, (4) a Fabry lens, to desensitize the device to slight motion of the star image, and (5) the photomultiplier. In simple photometers, separate sky measurements must be made frequently, using telescope motions, motion of the secondary mirror, or internal chopping with a precisely moveable aperture. For multiple passbands, it is good practice to do multiple obsrevations in a symmetric pattern to remove first-order changes in sky brightness or transparancy, such as UUUBBVVBBUUU.

Pulse counting allows particularly clear tracking of photon statistics and errors, always including dark current and sky counts as noise sources. Photomultipliers are sensitive enough that the faint limit for broad-band photometry with a given telescope may be set by the visual difficulty of locating a faint star. The aperture should be large enough to allow essentially all the starlight in, while minimizing skylight contamination. It may also be important to exclude light from a companion star or surrounding nebulosity.

Since some of the most insidious enemies of accurate photometry are rapid changes in transparency (that is, thin clouds) and sky brightness, multistar photometers allow powerful gains in conditions suitable for variable-star photometry (a very important field). In this case, two or more stars (perhaps blank sky as well) are followed by moveable photomultipliers, so that the variations due to transparency alone will divide out to very high accuracy. If the star are close enough compared to the clouds' motion, they will respond simultaneously. This carries to completion the idea of differential photometry, where alternate observations of a variable and nearby comparison star are made, cancelling out moderately rapid changes in the atmosphere or instrument. A third check star is advisable, since many variable stars (such as Polaris!) have been found while trying to use them as comparison stars. This idea clearly works well with CCD systems of wide field, since you have simultaneous data on many objects.

Photoelectric photometry is a repetitive task, and has proven well suited to automation. A number of telescopes are in operation performing fully automated photometry.

Since variability studies are a particularly important application of photometry, the search for variations and their classification is also important. A variety of methods have been used, including Fourier analysis, phase-dispersion minimization, and special methods for irregularly spaced data. Note that the Earth's orbital motion introduces changes in the timescale compared to an inertial frame, at minimum requiring a heliocentric correction to put the observations in a frame comoving with the Sun and thus avoiding daily or annual variations. High-precision or space-borne observations require great care in timescale, since one may not be able to achieve uniform sampling in an inertial reference frame, and the exact motion of the Earth's barycenter as well as the observer's location are important.

Procedures at other wavelength bands are analogous to those described for optical work (for the IR, they're identical). The atmosphere is of course not an issue for space-based work, and is less important but more complicated for IR work from the ground since some atmospheric absorption bands are saturated and others are not. Space data are especially nice since the calibration changes only very slowly as the instrument ages. For radio work, the gain of an antenna (or interferomater pair) is measured from standard sources whose flux is calibrated against fundamental standards using special antennas with low, but precisely calculable, efficiency. These are bright, generally supernova remnants, and have the interesting property for a standard source of fading slowly with time. For high-energy detectors, it is sometimes possible to calibrate the efficiency directly by measuring the response to known laboratory stimuli.

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