  Earth and other rocky objects in the inner Solar System are depleted in
carbon compared to objects in the outer Solar System, the Sun, or the ISM. It
is believed that this is a result of the selective removal of refractory carbon
from primordial circumstellar material. In this work, we study the irreversible
release of carbon into the gaseous environment via photolysis and pyrolysis of
refractory carbonaceous material during the disk phase of the early Solar
System. We analytically solve the one-dimensional advection equation and derive
an explicit expression that describes the depletion of carbonaceous material in
solids under the influence of radial and vertical transport. We find both
depletion mechanisms individually fail to reproduce Solar System abundances
under typical conditions. While radial transport only marginally restricts
photodecomposition, it is the inefficient vertical transport that limits carbon
depletion under these conditions. We show explicitly that an increase in the
vertical mixing efficiency, and/or an increase in the directly irradiated disk
volume, favors carbon depletion. Thermal decomposition requires a hot inner
disk (> 500 K) beyond 3 AU to deplete the formation region of Earth and
chondrites. We find FU Ori-type outbursts to produce these conditions such that
moderately refractory compounds are depleted. However, such outbursts likely do
not deplete the most refractory carbonaceous compounds beyond the innermost
disk region. Hence, the refractory carbon abundance at 1 AU typically does not
reach terrestrial levels. Nevertheless, under specific conditions, we find
photolysis and pyrolysis combined to reproduce Solar System abundances.
