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tokens: ( 2015 ), Lestrade Thilliez ( 2015 ), Booth et al. ( 2017 ), Chavez-Dagostino et al. ( 2016 ), and this paper (see Table 2 ). In the case of the observed millimeter emission, the rapid rise of the flux density with frequency (a spectral index of 4.9 ± 1.8) favors emission from warm dust (e.g., Ricci et al. 2010 ). In contrast, the centimeter emission is remarkably flat (a spectral index of −0.07 ± 0.25), suggesting an optically thin free–free nature. Below we examine the expected properties of the free–free emission. Figure 2. Spectrum of the emission associated with Eridani at centimeter and millimeter wavelengths from the references given in the text. The long-dashed lines are separate power-law fits to the centimeter and to the millimeter emission. The data point at 3 GHz (0.48 in the logarithm) is a 3 σ upper limit and was not taken into account in the fit. 4.1. Optically Thin Free–Free Emission The free–free power per unit volume per Hz is given bywhere Z is the ion charge, n e and n i are the electron and ion number density, respectively, T is the plasma temperature, and is the velocity-averaged Gaunt factor. The total power per unit volume is obtained integrating over the frequency where is the frequency average of the velocity-average Gaunt factor, which is in the range 1.1–1.5. Choosing gives a 20% accuracy (Rybicki Lightman 1986 ). The power per unit volume over a frequency range is given bywhere the normalized energy is u = hν / kT . For an optically thin plasma at a constant temperature, the luminosity in a normalized energy range ( u 1 , u 2 ) is obtained by integrating the emissivity over the volume V : The X-ray luminosity of Eridani has been measured with XMM-Newton in the energy range [ E 1 , E 2 ] = [0.2, 2] kev, with values in the range (1.5–1.68) × 10 28 erg s −1 (Poppenhaeger et al. 2010 ; Loyd et al. 2016 ). Let L R be the luminosity in the frequency range [ ν 1 , ν 2 ] = [6, 43] GHz where the spectrum is flat. Then, from Equation ( 5 ), the ratio of the free–free luminosity in the X-ray range L X and in the radio range L R is given bywhere the lower value corresponds to T = 2 × 10 6 K and the upper value corresponds to T = 3 × 10 6 K. These temperatures are similar to those measured by Schmitt et al. ( 1996 ) for the peak of the temperature distribution of the coronal material of Eridani. The observed radio luminosity in the range 6–43 GHz is given bywhere D = 3.2 pc is the distance to Eridani. Then, this radio luminosity implies a free–free emission in X-rays offor a temperature in the range (2–3) × 10 6 K, in agreement with the observed X-ray luminosity observed by XMM-Newton . Therefore, the radio emission in the flat-spectrum region is consistent with free–free emission for this range of plasma temperature. Note that line X-ray emission is also important in hot plasmas (e.g., Mewe et al. 1985 ; see their Figure 1). Thus, the total X-ray emissivity should be larger than the continuum free–free emission calculated in