  Context: The long-term carbonate-silicate cycle plays an important role in
the evolution of Earth's climate and, therefore, may also be an important
mechanism in the evolution of the climates of Earth-like exoplanets. Aims: We
investigate the effects of radiogenic mantle heating, core size, and planetary
mass on the evolution of the atmospheric partial $CO_2$ pressure, and the
ability of a long-term carbon cycle driven by plate tectonics to control the
atmospheric $CO_2$ pressure. Methods: We developed a box-model which connects
carbon cycling to parametrized mantle convection. The carbon cycle was coupled
to the thermal evolution via the plate speed, which depends on the global
Rayleigh number. Results: We find decreasing atmospheric $CO_2$ pressure with
time, up to an order of magnitude over 10 Gyr. Higher abundances of radioactive
isotopes result in higher $CO_2$ pressures. We find a decreasing Rayleigh
number and plate speed toward planets with larger core mass fractions $f_c$,
which leads to lower atmospheric $CO_2$ pressure. More massive planets may
favor the development of more $CO_2$ rich atmospheres due to hotter interiors.
Conclusions: The dependence of plate tectonics on mantle cooling has a
significant effect on the long-term evolution of the atmospheric $CO_2$
pressure. Carbon cycling mediated by plate tectonics is efficient in regulating
planetary climates for a wide range of mantle radioactive isotope abundances,
planet masses and core sizes. More efficient carbon cycling on planets with a
high mantle abundance of thorium or uranium highlights the importance of
mapping the abundances of these elements in host stars of potentially habitable
exoplanets. Inefficient carbon recycling on planets with a large core mass
fraction ($f_c\gtrsim 0.8$) emphasizes the importance of precise mass-radius
measurements of Earth-sized exoplanets.
