Convection

The most ubiquitous method of energy transport within a star is radiation. Pho- tons diffuse from hotter regions (emission) to cooler regions (absorption), driven by the star tending towards local thermal equilibrium. The efficiency of radiative transfer is directly hindered by opacity, leading the way convection to take over in certain situations. Radiative and conductive heat transfer are mostly irrelevant for the stellar evolution stages being examined here. As such, only convection will be discussed in detail.

Convection is the transport of heat energy via the motion of fluid hotter or cooler than its environment and becomes a significant consideration in red giants

and other cool stars. The mixing length theory (MLT; B¨ohm-Vitense, 1958)

approximates local convective heat transport in stars by looking at a “parcel” of fluid moving either outwards in inwards in a star that is a slightly different temperature to its surroundings, a process commonly referred to as “hot bubbles rising”. These parcels are created by instabilities in the fluid, with their motion

1.1. RED GIANT EVOLUTION 9 being determined by competing gravitational and buoyancy forces. When these forces balance the parcel will simply oscillate, but if a perturbation causes the parcel to rise and an unbalanced buoyancy force is acting in the same direction,

the parcel will travel outwards in the star some characteristic distance l, after

which it dissolve, depositing its heat energy. This distance is known as the mixing length.

The MLT provides a first order approximation of convection which is used in most cases. More accurate convective models requires incorporating complex three–dimensional, time–dependent fluid dynamics, which is why the MLT is so commonly used (see Pasetto et al., 2014, and references therein).

Further simplifications of the MLT are often made by using the Boussinesq approximations (Boussinesq, 1903). This set of assumptions includes ignoring effects such as magnetic fields, rotation, acoustic phenomena, and shocks; assum- ing that the temperatures and densities vary little between the parcel and its

surroundings; and assuming that the parcel is of a size of order l3_{, which is much}

smaller than any scale lengths associated with the star.

While it is assumed that the movement of this parcel through the star is roughly adiabatic (that is, no heat is exchanged with its surroundings and none is produced internally through nuclear burning), there is always the possibility that the parcel will in fact radiatively release some of its heat into its environment along

the way. However, since tKH tdyn (Section 1.1.1) convection can be assumed

to be adiabatic over short timescales. This adiabatic nature of the temperature gradient is what forces RGB stars to share a common structure, with very similar minimum effective temperatures regardless of mass (Hayashi et al., 1962).

When determining the onset of convection, an important quantity to consider is the logarithmic slope of the temperature over pressure (both of which will be functions of radius):

∇ ≡ dlnT

dlnP. (1.11)

The Schwarzschild criteria (Schwarzschild, 1906) for local convection is when the

decrease in the parcel’s temperature as it traverses distance l is less than the

decrease in the temperature of the surroundings across the same radii:

∇>∇ad. (1.12)

moving outwards in the star (Cox & Giuli, 1968). For a detailed derivation and discussion of these quantities, see Hansen et al. (2004), and references therein.

Convective processes take over from radiative ones as the dominant method of energy transport in a few situations. Firstly, when the opacity increases to such a level that radiative cooling of that region is greatly inhibited. Since the opacity increases with decreasing temperature this case generally only occurs in cooler stars, or the outer regions of stars. Opacity dictates the radiative energy

transfer (∇rad) by definition. Increasing opacity inhibits radiative heat transfer

and so the energy builds up in a region. If there is not sufficient convection ∇

needs to increase to keep the radiative flux at a point where it balances the inward pressure due to gravity.

Ionisation zones are also expected to be highly convective because ionisa-

tion causes ∇ (through its dependency on pressure) to exceed the temperature–

pressure gradient present in the adiabatic case. Again, this is often limited to the outer regions of the stars since the core is generally fully ionised. However,

ionisation zones occur at small radii in cooler stars where ∇ad is relatively small

and opacities are high (Cox & Giuli, 1968).

Another cause of convection dominating energy transport is when the energy generation rate is very sensitive to temperature (for example the CNO cycle

and 3α process), causing the flux to increase rapidly as the stellar radius tends

to zero. In phases where the CNO cycle dominates, convection enforces a very different structure on the star based on the thermodynamics of the stellar material and can influence the course of future evolutionary phases by homogenising the composition of the star within convective regions.

Convection plays a significant role in setting the structure of an RGB star (recall that the hydrogen envelope is largely convective; Section 1.1.1). Convec- tion is not well modelled in one–dimension, and coupled with the relatively rapid progression through red giant evolution, there are far more uncertainties in the modelling of RGB stars than MS stars (see Section 1.3 for further discussion).