Abstract
The transport equation is first obtained for the various diffusion processes using a simple approach which gives a tool sufficient for a first reading and which allows to carry out exploratory calculations. In a second section, the transport equations currently used in stellar evolution calculations are described and fully developed. Complications arising from the simultaneous presence of more than one state of ionization are analyzed and lead the reader to the embrionic research area of ambipolar diffusion.
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Notes
- 1.
We first neglect magnetic fields and rotation. In this chapter we will also frequently use the atomic mass symbol A i of an atom as a shorthand notation for the atom species itself.
- 2.
v D thus defined will be positive in the direction of increasing r.
- 3.
See § 5.2 and 5.3 of Spitzer (1962).
- 4.
The electric field term, \((Z_{i} + 1)m_{p}g/(2kT)\), is thus excluded.
- 5.
Which is equivalent to Eq. (6.62, 3) of Chapman and Cowling (1970).
- 6.
- 7.
- 8.
This velocity r i is equal to the local heat flow q i transported by particles of species i, divided by the thermal energy density of these particles—their partial pressure p i —[Burgers 1969, Eq. (2.17b)].
- 9.
The coefficients K ij , \(\Omega _{ij}^{(l)}(r)\), \(\sigma _{ij}^{(a\,b)}\), and a ij , are symmetric with respect to i and j: \(\{i \leftrightarrow j\}\).
- 10.
If one assumes that there are two dimensional currents similar, for instance, to meridional circulation, one does not need to assume that currents, Eq. (2.19), are negligible. These would then be linked to magnetic fields.
- 11.
To simplify the discussion we first assume a perfect non-degenerate gas and negligible radiation pressure; then P gas = P.
- 12.
Using the integrals of Paquette et al. 1986a (see also § A.1), one has
$$\displaystyle{K_{ij} = \frac{m_{p}g} {Nv_{0}}N_{i}N_{j}a_{ij}(Z_{i}Z_{j})^{2}F_{ ij}^{(1)}(1)\,.}$$ - 13.
The concept of Archimede’s law has been applied to particles in a solution by Landau and Lifshitz (1958), § 88.
- 14.
- 15.
The important \(\text{H}^{+} + \text{H}\) scattering cross-section was redetermined by Glassgold et al. (2005).
- 16.
As noted by Geiss and Burgi (1986a); see their second paragraph after their Eq. (56a).
- 17.
Equation (8) of Babel and Michaud (1991a).
- 18.
From Eq. (12) of Babel and Michaud (1991a).
- 19.
In other contexts, drift velocity is also used for the advective part of the diffusion velocity.
- 20.
See Geiss and Burgi (1986a) for the required expressions.
- 21.
A more detailed discussion may be found in Babel and Michaud (1991a).
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Michaud, G., Alecian, G., Richer, J. (2015). Atomic Transport: Diffusion Equations. In: Atomic Diffusion in Stars. Astronomy and Astrophysics Library. Springer, Cham. https://doi.org/10.1007/978-3-319-19854-5_2
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