Long-lived radioactive nuclides, such as $^{40}$K, $^{232}$Th, $^{235}$U and $^{238}$U, contribute to persistent heat production in the mantle of terrestrial-type planets. As refractory elements, the concentrations of Th and U in a terrestrial exoplanet are implicitly reflected in the photospheric abundances in the stellar host. However, a robust determination of these stellar abundances is difficult in practice owing to the general paucity and weakness of the relevant spectral features. We draw attention to the refractory, $r-$process element europium, which may be used as a convenient and practical proxy for the population analysis of radiogenic heating in exoplanetary systems. As a case study, we present a determination of Eu abundances in the photospheres of $alpha$ Cen A and B. We find that europium is depleted with respect to iron by $sim$ 0.1 dex and to silicon by $sim$ 0.15 dex compared to solar in both binary components. To first order, the measured Eu abundances can be converted to the abundances of $^{232}$Th, $^{235}$U and $^{238}$U with observational constraints while the abundance of $^{40}$K is approximated independently with a Galactic chemical evolution model. We find that the radiogenic heat budget in an $alpha$-Cen-Earth is $73.4^{+8.3}_{-6.9}$ TW upon its formation and $8.8^{+1.7}_{-1.3}$ TW at the present day, respectively $23pm5$ % and $54pm5$ % lower than that in the Hadean and modern Earth. As a consequence, mantle convection in an $alpha$-Cen-Earth is expected to be overall weaker than that of the Earth (assuming other conditions are the same) and thus such a planet would be less geologically active, suppressing its long-term potential to recycle its crust and volatiles. With Eu abundances being available for a large sample of Sun-like stars, the proposed approach can extend our ability to make predictions about the nature of other rocky worlds.
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