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The radio-IR correlation can be written in a general form as:

Equation 1 (1)

where LIR is the IR luminosity of the galaxy, Pnu(nu) is the specific radio luminosity at the radio frequency nu, and kappa and beta are coefficients that depend on the frequency at which the correlation is expressed. The coefficient kappa is dimensional, and its value depends on the units used in the correlation. The presentation above is often referred to as the {Pnu-LIR} presentation of the radio-IR correlation. An alternative is the {Snu-FIR} presentation, in which the radio and IR outputs are expressed in units of W m-2 Hz-1 and W m-2, respectively.

The two presentations are, of course, equivalent for individual galaxies. However, for determining beta, the choice of presentation is important if beta is a priori known to be different from unity (Cox et al. 1988). If all galaxies lie on a line of slope beta = 1 in a log-log Snu vs. FIR plot, then the conversion to a {Pnu-LIR} presentation will move them up or down along a diagonal line, leaving the overall slope of the correlation unchanged. However, if the slope of the correlation is different from unity in a {Snu-FIR} presentation, then fainter galaxies, which tend to be more distant than brighter ones, may move systematically away or toward the beta = 1 slope line in a {Pnu-LIR} presentation. The overal effect will be to flatten the slope if beta < 1, and to steepen it if beta > 1 in a transition from a {Snu-FIR} to a {Pnu-LIR} presentation.

Various investigators examined the radio-IR correlation using a variety of sample selection criteria. All use FIR fluxes measured by the IRAS satellite and construct galaxy FIR luminosities (fluxes) from the 60 and 100 µm detections using the following relation (Sanders & Mirabel 1996):

Equation 2 (2)

where Snu and Lnu are flux and luminosity densities expressed in units of Jy ( = 10-26 W m-2 Hz-1) and Lodot / Hz, respectively.

As we will show in Section 3, the value of beta plays an important role in quantifying the CIB-CRB connection. A value of beta that is significantly different from unity will require detailed knowledge of the evolution of the IR luminosity function with redshift, whereas a value of unity will considerably simplify the relation between the two background emissions. We therefore briefly review previous determinations of beta.

Initial investigations, using a relatively small sample of 91 radio-selected galaxies observed at nu = 4.8 GHz (de Jong et al. 1985) and 38 optically- and radio- selected galaxies observed at 1.4 GHz (Helou, Soifer, & Rowan-Robinson 1985), found the radio-IR correlation to be linear. Wunderlich & Klein (1988) extended the analysis of de Jong et al. (1985) to a wider range of galaxy types, covering about 4 orders of magnitudes in FIR luminosities. They found that the correlation at nu = 4.8 GHz is approximately linear (beta = 0.99 ± 0.38) up to IR luminosities of 9.1 × 109 Lodot, with beta increasing to a value of 1.26 ± 0.33 at higher luminosities. However, we found that their sample of galaxies could also be fitted with a single power law with a slope of beta = 1.05 ± 0.03.

Devereux and Eales (1989) correlated the 1.49 GHz power with the FIR output from an optically selected sample of galaxies with luminosities between approx 108 - 1011 Lodot. They found the slope of the correlation to be significantly larger than unity with a value of beta = 1.28. A similar conclusion was reached by Cox et al. (1988) who studied the correlation for a flux limited sample of 74 radio galaxies at 151 MHz. They found beta = 1.32 ± 0.06 in a {Snu-FIR} presentation, and beta = 1.15 ± 0.04 in a {Pnu-LIR} presentation of the data, and adopted an average slope of beta = 1.23.

Chi & Wolfendale (1990) provided theoretical arguments why the slope of the radio-IR correlation should not be unity. They predicted a break in the slope of the correlation at an IR luminosity above which most of the electrons producing the radio synchrotron emission remained trapped in the galaxy. For these galaxies the slope of the correlation should be unity, but for the smaller and less IR luminous galaxies they argued that beta should be larger than unity. They claimed to have found evidence for this effect in the data and reported values of beta = 1.37 for low luminosity galaxies, a trend exactly opposite to that found by Wunderlich & Klein (1988). Our analysis of the Chi & Wolfendale sample did not confirm either trend. We found that their data could be represented by a single slope with a value of beta = 1.27 ± 0.06.

Condon, Anderson, & Helou (1991; hereafter CAH91) reexamined the radio-IR correlation using the IRAS revised bright galaxies sample (BSG) from which spectroscopically identified AGN were subtracted, and which were detected with the VLA at 1.49 GHz. They found the slope of the correlation to be significantly greater than unity with a smaller value of beta = 1.11 ± 0.02 over the ~ 109 to 1013 Lodot luminosity range.

For the purpose of the present analysis we reexamined the radio-IR correlation starting from the same galaxy sample used by CAH91, the revised IRAS Bright Galaxy Sample, elimating galaxies that were not detected in all 4 IRAS bands, and removing AGN using a more recent catalog of spectroscopically confirmed AGN galaxies (Véron-Cetty & Veron 2001, catalog of Quasars and AGN). The total number of galaxies left in the sample is 222, compared to 258 galaxies used in the analysis of CAH91. We used the additional 12 and 25 µm IRAS bands to derive the total 8-1000 µm IR luminosity from the relation (Sanders & Mirabel 1996):

Equation 3 (3)

Figure 1 depicts the radio-IR correlation for a {Pnu-LIR} presentation of the data, and the best power-law fit to the data. We derive an almost identical slope beta for the same sample of galaxies as CAH91, with a smaller value for kappa, reflecting the fact that the 8 - 40 µm emission was not included in their galactic IR energy budget. The values of the fit for radio-IR correlation are: {kappa, beta} = {2.96 × 1010, 1.083 ± 0.02}. An almost identical slope was obtained for the {Snu-FIR} presentation, suggesting that there were no significant flux related systematic errors in determining the distances of the galaxies in the sample. Also shown in Figure 1 is a forced linear fit to the data with {kappa, beta} = {2.47 × 1011, 1.0}. The linear fit is almost indistinguishable from the best fit for galaxies with luminosities above ~ 1010 Lodot, which produce most of the IR luminosity density in the local and high-z universe. The same linear fit will provide a similarly good representation of the radio-IR correlation even if the AGN were included in the data. The AGN population however shows more dispersion around the fit.

Figure 1

Figure 1. The radio fluxes of galaxies in the IRAS BGS are plotted versus their 8-1000 µm luminosity determined from eq. (3). Star-forming galaxies are represented by crosses, and spectroscopically confirmed AGN by open triangles. The lines are fits to the radio-IR correlation between the star-forming galaxies only. The solid blue line represents the best power-law fit, and the red line the best linear fit to the data. The population of AGN generally follows the same correlation, but with a wider dispersion.

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