  We model the evolution of buoyant magnetic flux tubes in the Sun's convection
zone. A flux tube is assumed to lie initially near the top of the stably
stratified radiative core below the convection zone, but a segment of it is
perturbed into the convection zone by gradual heating and convective overshoot
motions. The ends ("footpoints") of the segment remain anchored at the base of
the convection zone, and if the segment is sufficiently long, it may be
buoyantly unstable, rising through the convection zone in a short time. The
length of the flux tube determines the ratio of buoyancy to magnetic tension:
short loops of flux are arrested before reaching the top of the convection
zone, while longer loops emerge to erupt through the photosphere. Using
Spruit's convection zone model, we compute the minimum footpoint separation
$L_c$ required for erupting flux tubes. We explore the dependence of $L_c$ on
the initial thermal state of the perturbed flux tube segment and on its initial
magnetic field strength. Following an investigation of thermal diffusion time
scales and the dynamic rise times of unstable flux tube segments, we conclude
that the most likely origin for magnetic flux which erupts to the surface is
from short length scale perturbations ($L < L_c$) which are initially stable,
but which are subsequently destabilized either by diffusion of heat into the
tube or by stretching of the anchor points until $L$ just exceeds $L_c$. In
either case, the separation of the anchor points of the emergent tube should
lie between the critical distance for a tube in mechanical equilibrium and one
in thermal equilibrium. Finally, after comparing the dispersion of dynamic rise
times with the much shorter observed active region formation time scales, we
conclude that active regions form from the emergence of a single flux tube
segment.
