Symbiotic stars show emission across the electromagnetic spectrum from a wide array of physical processes. At cm-waves both synchrotron and thermal emission is seen, often highly variable and associated with outbursts in the optical and X-rays. Most models of the radio emission include an ionized region within the dense wind of the red giant star, that is kept ionized by activity on the white dwarf companion or its accretion disk. In some cases there is on-going shell burning on the white dwarf due to its high mass accretion rate or a prior nova eruption, in other cases nuclear fusion occurs only occasionally as recurrent nova events. In this study we measure the spectral indices of a sample of symbiotic systems in the Southern Hemisphere using the Australia Telescope Compact Array. Putting our data together with results from other surveys, we derive the optical depths and brightness temperatures of some well-known symbiotic stars. Using parallax distances from Gaia Data Release 3, we determine the sizes and characteristic electron densities in the radio emission regions. The results show a range of a factor of 10^4 in radio luminosity, and a factor of 100 in linear size. These numbers are consistent with a picture where the rate of shell burning on the white dwarf determines the radio luminosity. Therefore, our findings also suggest that radio luminosity can be used to determine whether a symbiotic star is powered by accretion alone or also by shell burning.
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