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a Bessell, M. S. in Astronomy & Astrophysics Encyclopedia, ed. S. Maran, Van Nostrand Rienhold, 1992, p. 406. Glass, I. S., Handbook of Infrared Astronomy, Cambridge University Press, 1999.

b 1.0 nm (nanometer) = 1.0 x 10-9 m. The effective wavelength is the average wavelength in the band for the spectrum of a star of type A0V. c 1.0 THz (terahertz) = 1.0 x 1012 Hz. The effective frequencies veff and passbands Av were obtained from columns 2 and 3 by the present author, using veff = cAeff and Av = c(AX)/X2. The former could give slightly different results than (18) depending on the spectral shape.

d 1.0 Jy (jansky) = 1.0 x 10-26 W m-2 Hz-1. Approximate spectral flux density given for magnitude zero at the "effective frequency". At other magnitudes, m, the spectral flux density is: S(v) = S<)(v) 10-0 4m (Jy).

wavelengths (see below) Aeff and passbands AX are given in Table 2 along with the corresponding frequencies v and passbands Av. The bands in the table are one of several such systems of standard filters.

Conversion from magnitudes to SI units

Since magnitudes are not physical (SI) units, it is often necessary to convert magnitudes to (spectral) flux densities S (W m-2Hz-1). The flux densities S0 that correspond to the magnitude zero for any color are given in the rightmost column of Table 2. Each entry S0(v) is the spectral flux density at m = 0, with units 10-26 W m-2 Hz-1 (i.e., the jansky) at the effective frequency v = veff, which is roughly the central frequency of the band.

The flux densities S corresponding to magnitudes other than m = 0 can be calculated easily from S0. The (energy) spectral flux density S decreases with increasing magnitude in the same manner as the photon flux density. Comparison to (14), for a reference magnitude m1 = 0, yields

where S0(veff) is obtained from Table 2. This enables one to convert any magnitude in a given color band to a spectral flux density in SI units.

A value of S0( veff) in Table 2 is nominally the spectral flux density at the indicated effective frequency of the filter passband for a Vega-type spectrum. The effective frequency is the average frequency of the recorded photons, n I C ^

veff = vSp(v) e(v) dv / / Sp(v) e(v) dv (Effective (8.18)

0 0 frequency)

where e is the efficiency function of the filter in question and Sp is the spectral distribution of the source being observed. For a Vega-like spectrum, the result should yield the frequencies quoted in Table 2, but see the caveat in footnote c to the Table.

One can multiply both sides of (18) by the Planck constant h to obtain the photon energy hveff at the effective frequency on the left. The integral in the numerator is then the photon energy hv times the detected photon flux summed over all frequencies, i.e., the detected energy flux F(W m-2). Since the denominator is the measured photon flux density Fp (s-1 m-2), the energy flux F is simply the product of h veff and the measured photon flux

Unfortunately the effective frequencies depend on the spectral shape of the incident photon flux, i.e., on the relative numbers of low and high frequency photons arriving within the passband of the filter. Consider the extreme case where all the photons in the passband are concentrated in a narrow spectral line at the low-frequency end of the band. Clearly, the average frequency will occur at the low end of the band. In contrast, the average value will be near the midpoint of the band for a source with a flat, continuous spectrum, or one that linearly increases or decreases with frequency, if it dominates the spectral lines.

The effective wavelengths in Table 2, as noted, were calculated for the spectrum of an A0V star, like Vega, with an expression similar to (18) where X replaces v and the weighting factor is SXp (photon flux per unit wavelength) replaces Sp. We calculated the values of veff in Table 2 from the Aeff rather than from (18), but the differences should be minor.

The widths Av of the transmission function of the filters (Fig. 2) are the full width at half maximum, FWHM. The approximate photon spectral flux density Sp at the effective frequency may be found by dividing the width into the measured flux density F p

Sp(veff) ~-= — / Sp(v) e(v) dv (Incident photons; (8.19)

Av Av 0

The magnitude system is very much an instrument-driven system. As noted above, the efficiency functions of different types of color (filter) systems will generally be different. Thus, two different color systems will yield different ratios of blue responses FB^/Fb 1 for two stars that have different spectral shapes within the blue passband. That is, one filter system might produce a ratio of 1.5 indicating that star #2 is 50% brighter than #1, while the other system might yield 1.7.

The calibration of different broadband detection systems against one another is thus dependent on the distribution in frequency of the incoming radiation. Measurement of the distribution of energy from a star thus requires knowledge of that distribution! How does one do this? One either uses exactly the same detection system as other colleagues, or, if one uses a non-standard system, the measured response can sometimes be corrected iteratively to a standard color system. The latter is feasible if the spectrum is continuous and well behaved, but not so if unknown spectral lines are involved.

Measurements of color magnitudes of stars, known as photometry, is always accompanied by measures of standard stars with well determined magnitudes and spectral shapes. These allow one to correct for instrument and atmospheric effects. These days, most photometry is carried out with CCD detectors.

Color indices

The ratio of fluxes in two color bands for the same star, e.g. the ratio B/ Fpy for the B and V bands, yields through (12) the difference of magnitudes, mB — mV. This quantity is called the color index B — V. This is a color as we would perceive it. We see an object as blue if it emits more blue flux than yellow flux. The blue magnitude alone does not indicate a color.

Colors can be defined for any two adjacent spectral bands, with the redder magnitude subtracted from the bluer, e.g., U — B, R — I, etc. Since these are given in magnitudes, larger values of a color indicate a redder color. The B — V color is often used as an indicator of temperature for the quasi blackbody radiation from stars (Section 9.4).

Absolute magnitudes - luminosity

The luminosity of a celestial body can be cast into the system of magnitudes. The absolute magnitude M is a direct measure of luminosity, the power (W) emitted by the star in the specified spectral band. It is defined as the magnitude of a star if it were at the standard distance of 10 parsecs. (1 pc ~ 3.26 LY = 3.09 x 1016 m; see (4.9) and associated discussion.) The use of a standard 10-pc distance removes the distance dependence and thus yields a quantity equivalent to a luminosity. The upper-case "M" is invariably adopted for absolute magnitude. Again, a subscript can specify the frequency band, e.g., MV for the absolute visual magnitude. To distinguish this quantity from the ordinary magnitude mV, the latter is sometimes called apparent magnitude because it is based on the distance-dependent appearance or faintness of a star.

For example, what is the absolute magnitude of a star at a distance of 10 000 pc with an apparent visual magnitude of mV = 20? The star would be 1000 times closer if it were at 10 pc and would thus appear to be 106 times brighter (F a r—2). The magnitude difference for this flux change, from (12), is -2.5 x 6 = —15. The absolute magnitude therefore is MV = 20 —15 = 5. A shortcut method makes use of the fact that a change of flux by a factor of 100 corresponds to 5 magnitudes (13). The change in flux by a factor of 106 thus corresponds to three factors of 100 brighter, or 3 x 5 = 15 magnitude brighter.

The absolute (visual) magnitude of the sun is M© ,V = 4.82, about the same value as in our example. Thus the sun would appear to be about V = 20 at the distance of 10 000 pc (32 600 LY) which is somewhat farther than the galactic center at 25 000 LY distance. The sun lies much closer than the standard 10 pc (= 32.6 LY) distance; it is only 499 light seconds (= 1.58 x 10—5 LY) distant (semimajor axis). The apparent magnitude of the sun turns out to be mV = —26.75; the sun is "apparently" very bright indeed! Thus stars similar to the sun could in principle have apparent magnitudes m ranging from -26.75 to more than +20 due solely to their distances from the earth.

The expression that relates the apparent and absolute magnitudes is

modulus)

where r is the distance to the star in parsecs. This first equality in (20) follows from the definition (12) where the two stars being compared are identical but at different distances, in this case the actual distance r and the hypothetical distance of 10 pc. The inverse square relation between the flux and distance F~p a r-2 gives / ^p, 10 = (r/10 pc)-2. Substitution of this into the second term yields the third. The fraction in the latter term may also be expressed in terms of light years, i.e., [r (LY)/32.6 LY]. The apparent magnitude of the sun given just above was derived from (20).

Equation (20) is an important relation that one encounters often. It shows that the quantity m - M is a direct measure of the distance of the star from the earth; this quantity is thus called the distance modulus. If the luminosity of a star is known from spectroscopic studies (which can provide a stellar type of known luminosity), the value of M is known. Measurement of the apparent magnitude m then yields the distance r to the star according to (20), r = 10°-2(m-M) 10 pc (8.21)

Tables of stars may well give both their apparent and absolute magnitudes and leave it to you to find thedistance from (21). Note that a distance modulus m — M = 5 magnitudes leads to a distance 101 x 10 pc = 100 pc, that 10 magnitudes gives 1000 pc, and that 15 gives 10 000 pc.

Bolometric magnitude

The bolometric magnitude mbol was defined above (see Table 1). It is based upon the total energy flux in and near the optical band (IR, optical, UV). The distribution of radiation from a star approximates a blackbody spectrum with superimposed absorption lines; it peaks in the visible range for many stars and falls toward zero at shorter and longer wavelengths (See Chapter 11). The bolometric magnitude is a measure of the energy spectral flux density (W m-2 Hz-1) integrated over the entire appropriate frequency band; that is, it is a measure of the bolometric flux density F"(W/m2).

The bolometric magnitude is often not measured directly because photometric detectors usually count photons in wide energy bands; they do not keep track of the exact energy of every photon. Furthermore, a large part of the flux may not be measurable because it is in a region of the spectrum that does not penetrate the atmosphere, for example, in the ultraviolet beyond 1.0 x 015 Hz (X = 300 nm) or in the near-infrared below 4 1014 Hz (X = 750 nm) (Fig. 2.1). However, the missing flux can often be estimated from the observable part of the spectrum.

The bolometric magnitude can be used in both the apparent and absolute senses. The absolute bolometric magnitude Mbol is the bolometric magnitude that would be measured if the emitting body were at a distance of 10 pc. Radiation at extreme wavelengths (radio or x-ray) is small for most stars, as noted above, and is traditionally not included in the bolometric magnitude.

Bolometric correction The bolometric correction (BC) is the amount in magnitudes by which the visual magnitude, apparent or absolute, must be corrected to obtain the bolometric magnitude,

The BC is a useful quantity because one can not directly measure the entire IR-optical-UV spectrum of a star (except from space). The BC is the same value in (22) and (23) because an additive correction in magnitudes represents a fractional change in flux which is the same for the apparent and absolute cases.

How does one obtain the bolometric correction? For many stars, the distribution of light with frequency will, as noted, approximate a blackbody spectrum. The temperature of a blackbody completely determines the distribution of light emitted by it. A hot body is blue-white and a cool body is reddish. The amount of light that falls outside the optical band is well known for a perfect blackbody. The spectra of actual stars deviate from the blackbody curve in ways that are now quite well known from direct measurements and theoretical studies, even for stars that have a large portion of their spectrum outside the optical range.

These studies provide a bolometric correction for each temperature and/or type of star so that one can look up a BC for any star whose type is known from spectroscopic studies. The BC is set to zero for a yellowish star (type "F0V") which is a bit hotter than the sun and a bit cooler than Vega. For much cooler or hotter stars, the visual filter misses large portions of the light; the correction is increasingly negative in both directions (Table 3). The table also lists sample stellar types.

Absolute bolometric magnitude and luminosity The absolute bolometric magnitude is a direct measure of the bolometric luminosity of a star, i.e., the energy output integrated over the IR, visual, and UV, as appropriate for the temperature of the star. The bolometric magnitude Mbol of an arbitrary star is related, by definition, to its luminosity L as

which has similarities to (12) and uses solar quantities, Lq = 3.845 x 1026 W and Mbol Q = 4.74, as a reference. Solve for the luminosity L to obtain the numerical relation between Mbol and L ,

Table 8.3. Star types: absolute magnitudes and bolometric correctionsa
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