Abstract
Stellar convection is customarily described by Mixing-Length Theory, which
makes use of the mixing-length scale to express the convective flux, velocity,
and temperature gradients of the convective elements and stellar medium. The
mixing-length scale is taken to be proportional to the local pressure scale
height, and the proportionality factor (the mixing-length parameter) must be
determined by comparing the stellar models to some calibrator, usually the Sun.
No strong arguments exist to suggest that the mixing-length parameter is the
same in all stars and at all evolutionary phases. Because of this, all stellar
models in literature are hampered by this basic uncertainty. The aim of this
study is to present a new theory of stellar convection that does not require
the mixing length parameter. We present a self-consistent analytical
formulation of stellar convection that determines the properties of stellar
convection as a function of the physical behaviour of the convective elements
themselves and the surrounding medium. This new theory is formulated starting
from a conventional solution of the Navier-Stokes/Euler equations, i.e. the
Bernoulli equation for a perfect fluid, but expressed in a non-inertial
reference frame co-moving with the convective elements. In our formalism the
motion of stellar convective cells inside convective- unstable layers is fully
determined by a new system of equations for convection in a non-local and time
dependent formalism. We obtain an analytical, non-local, time-dependent
solution for the convective energy transport that does not depend on any free
parameter. The predictions of the new theory are compared with those from the
standard mixing-length paradigm for the most accurate calibrator, the Sun, with
very satisfactory results.
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