## 3. Modelling radial flowsWe formulate our chemical model with radial flows as a
multi-dimensional extension of the static model of PCB98 and PC99, an
open model where the disc forms gradually by accretion of
protogalactic gas. The disc is divided in -
depletion by SF, which locks up gas into stars; -
stellar ejecta which shed back enriched material to the interstellar medium (ISM); -
infall of primordial protogalactic gas; -
gas exchange with the neighbouring rings because of radial flows.
The set of equations driving the chemical evolution of the
where the various symbols are defined here below. Primordial gas is accreted at an exponentially decreasing rate with time-scale : is obtained by imposing that the integrated contribution of infall up to the present Galactic age 15 Gyr, corresponds to an assumed exponential profile : Indicating with the surface gas density, we define the gas fraction: and the normalized surface gas density for each chemical species
where is the fractionary
abundance by mass of The 1st term on the right-hand side of Eq. (1) represents the
depletion of species Let the
Radial flows through the borders, with a flux
, contribute to alter the gas surface
density in the The gas flux at can be written: where is the step function: or 0 for or , respectively. Eq. (7) is a sort of "upwind approximation" for the advection term to be included in the model equations (e.g. Press et al. 1986), describing either inflow or outflow depending on the sign of . An analogous expression holds for . Let us take the inner edge at the
midpoint between and
, and similarly for
(Fig. 1). Writing Eq. (6)
separately for each chemical species The terms on the right-hand side of Eq. (8) evidence the
contribution of the 3 contiguous shells involved: the first term
represents the gas being gained in shell Notice that in the case of static models the final surface mass
density is completely determined by the assumed accretion profile,
namely . Therefore, in static models
the radial profile for accretion can be directly chosen so as to match
the observed present-day surface density in the Disc (see PCB98 and
PC99). The inclusion of the term of radial gas flows alters the
expected final density profile and .
Hence, cannot be assumed in advance
and is only known ## 3.1. Boundary conditionsEq. (8) needs to be slightly modified in the case of the
innermost and the outermost shell, since the shell ## 3.1.1. The innermost shellOur disc models will extend down to where the Bulge becomes the dominating Galactic component ( kpc). As to the innermost edge, we assume that the first shell is symmetric with respect to : and that always, since we cannot account for outflows from still inner shells, not included in the model. For , Eq. (8) then becomes: with: ## 3.1.2. Boundary conditions at the disc edgeAs to the outermost shell (), we need a boundary condition for the gas inflowing from the outer disc. We assume a SF cut-off in the outer disc, while the gaseous layer extends much further. In fact, in external spirals HI discs are observed to extend much beyond the optical disc, out to 2 or even 3 optical radii. A threshold preventing SF beyond a certain radius is expected from gravitational stability in fluid discs (Toomre 1964, Quirk 1972) and has observational support as well (Kennicutt 1989). For the Galactic Disc we assume no SF beyond the last shell at kpc, which is the empirical limit for the optical disc and for HII regions and bright blue stars tracing active SF. Gas can though flow in from the outer disc; extended gas discs might actually provide a much larger gas reservoir for star-forming spirals than vertical infall, at least at the present time when the gravitational settling of the protogalactic cloud is basically over. If no SF occurs in the outer disc, the evolution of the gas (and total) surface density can be expressed as: (e.g. Lacey & Fall 1985, their model equation with the SF term dropped). Here, with no SF, and abundances always remain the primordial ones (). Let us assume the following simplifying conditions for the outer disc: -
the infall time-scale is uniform: -
the inflow velocity is uniform and constant: -
the infall profile is flat: in accordance with observed extended gas discs in spirals, showing a much longer scale-length than the stellar component.
With these assumptions, Eq. (12) becomes: where we indicate , and to alleviate the notation. Eq. (13) has a straightforward analytical solution (Appendix B): where is the time when radial inflows are assumed to activate. Eq. (14) is our boundary condition at the outermost edge. Notice that (14) is the solution of (13) in the idealized case of
an infinite, flat gas layer extending boundless to any
(see also Appendix B). Of
course, this does not correspond to gaseous discs surrounding real
spirals; but since we will consider only slow inflow velocities
( km sec ## 3.1.3. The outermost shellWe take a reference external radius in the outer disc where the (total and gas) surface density is given by the boundary condition (14); typically, kpc and kpc. We take the outer edge of the outermost shell at the midpoint: and replace in Eqs. (6) and (7), since the primordial abundances
remain unaltered in the outer disc,
in the absence of SF. We thus write the radial flow term for the
## 3.2. The numerical solutionUsing (8), (10) and (15), the basic set of Eqs. (1) can be written as: where we have introduced: We refer to PCB98 for further details on the quantities and , appearing also in the original static model. Neglecting, for the time being, that the 's and the 's contain the 's themselves, we are dealing with a linear, first order, non homogeneous system of differential equations with non constant coefficients, of the kind: There is a system (17) for each chemical species We solve the system by the same numerical method used for the original equation of the static model - see Talbot & Arnett (1971) and PCB98 for details. We just need to extend the method to the present multi-dimensional case (17). If we consider the evolution of the 's over a short enough timestep , the various quantities , and will remain roughly constant within ; similarly to the method for the static model (see PCB98), within we approximate them with the values , and they assume at the midstep . Over , (17) can then be considered a system with constant coefficients , and becomes a constant vector, which allows for the analytical solution: with the eigenvalues of and the corresponding eigenvectors. The matrix and the explicit expression of the solution (18) for the 's are calculated in Appendix A, resulting in: where we intend , , and for the outermost shell one should replace: being defined by (16). Similarly to the case of an isolated shell (see PCB98) the
system (20) does not provide the final solution for
, since the
's and the
's on the right-hand side actually
depend on the 's; it just represents
a set of implicit non-linear expressions in
. As in the original static model,
we can neglect the dependence of on
the 's and consider only that of
(see PCB98 for a detailed
discussion). We must finally find the roots of the system (20) by
applying the Newton-Raphson method, generalized to many dimensions
(cfr. Press et al., 1986). Such a system holds for each of the
chemical species We tested the code against suitable analytical counterparts and obtained the following conditions for model consistency (Appendix B). -
Rather small timesteps are needed for the numerical model to keep stable; the required timesteps get smaller and smaller the higher the flow velocities considered, and the thinner the shells. -
To describe gas flows in a disc with an exponential density profile, the shells should be equispaced in the logarithmic, rather than linear, scale (so that they roughly have the same mass, rather than the same width).
We modelled the Galactic Disc using 35 shells from 2.5 to 20 kpc,
equally spaced in the logarithmic scale, their width ranging from
kpc for the inner shells to
kpc for the outermost ones.
With such a grid spacing, and velocities up to
km sec -
The timestep used to update the "chemical" variables (, , etc.) is the minimum among: which guarantees that the relative variations of the 's are lower than a fixed ; which guarantees that the total surface mass density increases by no more than 5%; which is twice the previous timestep of the model, to speed up the computation when possible; which guarantees the Courant condition , indispensable for the stability of a numerical algorithm describing flows. So, is basically set by the requirement that the chemical quantities do not vary too much within it, and it can get relatively large (up to 0.2 Gyr), especially at late ages when the various chemical variables evolve slowly. -
It is only the numerical solution (20) which needs very short timesteps to keep stable. Therefore, once the chemical variables are upgraded, the main timestep is subdivided in much shorter timesteps Gyr, upon which the solution (20) and its Newton-Raphson iteration are successively applied to cover the whole . Only then a new upgrade of all the 's and 's is performed.
This trick keeps the code roughly as fast as if it would evolve with a single time-scale , and yet it gives the same results as the "slow" version in which all quantities are upgraded at each Gyr. © European Southern Observatory (ESO) 2000 Online publication: March 21, 2000 |