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Stars are intrinsically broadband emitters. As a consequence, the integrated light of a stellar population at any wavelength contains contributions from many parts of the Hertzsprung-Russell diagram and necessarily from all ages present in the population. However, there is a significant wavelength dependence in sensitivity to age, which can be used to extract information on the history of a population.

A useful way of quantifying the dependence of the integrated light on the star formation history is in the form of a "history weighting function." This describes how much the stars of a given generation affect the resultant luminosity at each wavelength. Here we give a simple analytic illustration of weighting functions in three important wavelength regimes discussed in the main text: the ultraviolet ionizing continuum (lambda < 912 Å), the far-UV continuum (~ 1500 Å), and the optical band (here, the V band).

Detailed spectral synthesis models (e.g., Larson & Tinsley 1978; Bruzual & Charlot 1993) show that the integrated light of a given generation of stars decays roughly as a power law in the age t: L(lambda,t) ~ a(lambda)t-beta(lambda). The function a(lambda) describes the initial spectral energy distribution per unit mass of the generation. The power-law time dependence is a good approximation over most of the interesting range of ages (at least after ~ 2 million years) and for most wavelengths. Although some treatments in the literature assume the decay is exponential, this is clearly unphysical because it implies a constant decay time whereas the actual timescale for stellar evolution increases as the turnoff mass decreases.

A power-law dependence implies that photometric measures for the integrated light, stated in conventional magnitude units, respond in proportion to log t. This means that we should imagine the history of a galaxy as a set of bins of constant size in delta log t stretching from the present to a lookback time of ~ 15 Gyr. The appropriate size for the bins is governed by the number of independent data points available, the wavelength range, and the photometric precision. The log t binning implies that much less information is available on the details of the early history than on more recent times. This is true for both integrated light and resolved color-magnitude diagrams.

From a spectral synthesis routine written by W. Landsman (see Hill et al. 1994b), which uses the Geneva stellar evolutionary tracks, we find for the Lyman continuum, the 1500 Å continuum, and the V band that beta ~ 4.3, 1.5, and 0.8, respectively. (At 1500 Å there is an abrupt increase in beta for times after 500 Myr.) These values are for solar abundance. Other model sets (e.g., Bruzual & Charlot 1993) yield similar results, but the exact figures are not important for our discussion.

The integrated present-day luminosity of a galaxy is then given by

L(lambda) ~ a(lambda) integ SFR(t0 - t) t-beta(lambda) dt ,

where SFR is the star formation rate (Msun yr-1) at a given cosmic time, t0 is the present cosmic epoch, and t is the lookback time to the formation of each generation. Here, we are not so much interested in the actual spectrum of the galaxy as in visualizing how activity in log t bins over four decades in age (1-15,000 Myr) affects observations at each wavelength. So, we can rewrite this expression as

L(lambda) ~ L1(lambda) integ SFR(t0 - 10zeta) W(lambda, zeta) dzeta ,

where zeta = log10 t and L1 is the luminosity produced for a continuous star formation rate of 1 Msun yr-1. The normalized weighting function is W(lambda, zeta) = b(lambda) exp{2.30 [1 - beta(lambda)] zeta}. It measures the relative contribution of each generation of stars in log t space to the present-day luminosity at each wavelength. The integral of W over log t is unity in this formulation. For our three wavelength regimes, b(lambda) = 74.0, 1.74, and 0.08, respectively, where we have truncated the 1500 Å integral at 500 Myr.

The history weighting functions are plotted in Figure 24. The three wavelengths offer very different age sensitivities and therefore serve to "dissect" the star formation history. The weighting functions other than that for the Lyman continuum are broad and smooth. This is typical of most wavelengths. The absence of structure in the weighting functions makes analysis of star formation histories difficult and is an important contributor to a number of well-known ambiguities and controversies in integrated light analysis. Our ability to extract information on the history of star formation from integrated light depends strongly on the range of wavelengths observed.

Figure 24

Figure 24. Normalized history weighting functions showing the contribution of star formation activity at various times in the past to the integrated light in three wavelength regimes. The vertical scale is linear. The curves for 1500 and 5500 Å are plotted at 3 times their actual values.

The sensitivity of Lyman continuum radiation, and hence strong emission lines or radio free-free continuum, to star formation declines rapidly with age and is essentially 0 for times over 10 Myr in the past. These indicators are very useful but also of very limited scope. The 1500 Å continuum is also most sensitive to the youngest stars, but its usefulness extends another factor of 50 in age beyond 10 Myr. The integrated V-band light of a single generation decays slowly, so the present-day V-band light of a galaxy can be strongly composite. It is most sensitive to populations over 1 Gyr in age. The 1500 Å and V-band continua are about equally sensitive to stars of age ~ 100 Myr.

An alternative way of quantifying the sensitivity of different wavelengths to the star formation history is by determining light-weighted "mean ages" at each wavelength (e.g., O'Connell 1990; Hill et al. 1994b). Defined as in the latter reference, mean ages for the Lyman continuum, the far-UV continuum, and the V band are 2 Myr, 35 Myr, and 2 Gyr, respectively.

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