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Astron. Astrophys. 355, 929-948 (2000)

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3. Modelling radial flows

We 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 N concentric rings or shells; in each ring k the gaseous component and its chemical abundances evolve due to:

  1. depletion by SF, which locks up gas into stars;

  2. stellar ejecta which shed back enriched material to the interstellar medium (ISM);

  3. infall of primordial protogalactic gas;

  4. gas exchange with the neighbouring rings because of radial flows.

The set of equations driving the chemical evolution of the k-th shell is:

[EQUATION]

where the various symbols are defined here below.

Primordial gas is accreted at an exponentially decreasing rate with time-scale [FORMULA]:

[EQUATION]

[FORMULA] is obtained by imposing that the integrated contribution of infall up to the present Galactic age [FORMULA]15 Gyr, corresponds to an assumed exponential profile [FORMULA]:

[EQUATION]

Indicating with [FORMULA] the surface gas density, we define the gas fraction:

[EQUATION]

and the normalized surface gas density for each chemical species i:

[EQUATION]

where [FORMULA] is the fractionary abundance by mass of i.

The 1st term on the right-hand side of Eq. (1) represents the depletion of species i from the ISM due to star formation; see PC99 for the various options concerning the SF rate, [FORMULA]. The 2nd term is the amount of species i ejected back to the ISM by dying stars; the returned fractions [FORMULA] are calculated on the base of the detailed stellar yields from PCB98 and keep track of finite stellar lifetimes (no instantaneous recycling approximation IRA). The 3rd term is the contribution of infall, while the 4th term describes the effect of radial flows. Full details on the first three terms can be found in the original static model by PCB98 and PC99. The novelty in Eq. (1) is the radial flow term, which we develop here below. We will adopt the simplified notation [FORMULA] and the like.

Let the k-th shell be defined by the galactocentric radius [FORMULA], its inner and outer edge being labelled as [FORMULA] and [FORMULA]. Through these edges, gas flows with velocity [FORMULA] and [FORMULA], respectively (Fig. 1). Flow velocities are taken positive outward; the case of inflow is correspondingly described by negative velocities.

[FIGURE] Fig. 1. Scheme of the gas flow through the k-th model shell

Radial flows through the borders, with a flux [FORMULA], contribute to alter the gas surface density in the k-th shell according to:

[EQUATION]

The gas flux at [FORMULA] can be written:

[EQUATION]

where [FORMULA] is the step function: [FORMULA] or 0 for [FORMULA] or [FORMULA], 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 [FORMULA]. An analogous expression holds for [FORMULA].

Let us take the inner edge [FORMULA] at the midpoint between [FORMULA] and [FORMULA], and similarly for [FORMULA] (Fig. 1). Writing Eq. (6) separately for each chemical species i, in terms of the [FORMULA]'s we obtain the radial flow term of Eq. (1) as:

[EQUATION]

where:

[EQUATION]

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 k from k-1, the second term is the gas being lost from k to k-1 and k+1, and the third term is the gas being gained in k from k+1. The coefficients (9) are all [FORMULA] and depend only on the shell k, not on the chemical species i considered. If the velocity pattern is constant in time, [FORMULA], [FORMULA] and [FORMULA] are also constant in time.

Notice that in the case of static models the final surface mass density is completely determined by the assumed accretion profile, namely [FORMULA]. 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 [FORMULA]. Hence, [FORMULA] cannot be assumed in advance and is only known a posteriori (see Sect. 4); at the end of each simulation we need to check how much radial flows have altered the actual density profile [FORMULA] with respect to the pure accretion profile [FORMULA]. With the slow flow speeds considered ([FORMULA] km sec-1), the two profiles will not be too dissimilar anyways.

3.1. Boundary conditions

Eq. (8) needs to be slightly modified in the case of the innermost and the outermost shell, since the shell k-1 or k+1 are not defined in these two respective cases.

3.1.1. The innermost shell

Our disc models will extend down to where the Bulge becomes the dominating Galactic component ([FORMULA] kpc). As to the innermost edge, we assume that the first shell is symmetric with respect to [FORMULA]:

[EQUATION]

and that [FORMULA] always, since we cannot account for outflows from still inner shells, not included in the model. For [FORMULA], Eq. (8) then becomes:

[EQUATION]

with:

[EQUATION]

[EQUATION]

3.1.2. Boundary conditions at the disc edge

As to the outermost shell ([FORMULA]), 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 [FORMULA] 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:

[EQUATION]

(e.g. Lacey & Fall 1985, their model equation with the SF term dropped). Here, with no SF, [FORMULA] and abundances always remain the primordial ones ([FORMULA]). Let us assume the following simplifying conditions for the outer disc:

  1. the infall time-scale is uniform:

    [EQUATION]

  2. the inflow velocity is uniform and constant:

    [EQUATION]

  3. the infall profile is flat:

    [EQUATION]

    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:

[EQUATION]

where we indicate [FORMULA], [FORMULA] and [FORMULA] to alleviate the notation. Eq. (13) has a straightforward analytical solution (Appendix B):

[EQUATION]

where [FORMULA] 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 [FORMULA] (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 ([FORMULA] km sec-1), with typical values of [FORMULA] kpc and [FORMULA] Gyr, the gas actually drifting into the model disc shells will be just the gas originally accreted within [FORMULA] kpc. Therefore, the boundary condition (14) remains valid as long as the gas layer stretches out to [FORMULA] kpc, a very plausible assumption since observed gaseous discs extend over a few tens or even [FORMULA] kpc.

3.1.3. The outermost shell

We take a reference external radius [FORMULA] in the outer disc where the (total and gas) surface density [FORMULA] is given by the boundary condition (14); typically, [FORMULA] kpc and [FORMULA] kpc. We take the outer edge of the outermost shell at the midpoint:

[EQUATION]

and replace

[EQUATION]

in Eqs. (6) and (7), since the primordial abundances [FORMULA] remain unaltered in the outer disc, in the absence of SF. We thus write the radial flow term for the N-th shell as:

[EQUATION]

where:

[EQUATION]

3.2. The numerical solution

Using (8), (10) and (15), the basic set of Eqs. (1) can be written as:

[EQUATION]

where we have introduced:

[EQUATION]

We refer to PCB98 for further details on the quantities [FORMULA] and [FORMULA], appearing also in the original static model. Neglecting, for the time being, that the [FORMULA]'s and the [FORMULA]'s contain the [FORMULA]'s themselves, we are dealing with a linear, first order, non homogeneous system of differential equations with non constant coefficients, of the kind:

[EQUATION]

There is a system (17) for each chemical species i, but the matrix of the coefficients [FORMULA] is independent of i.

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 [FORMULA]'s over a short enough timestep [FORMULA], the various quantities [FORMULA], [FORMULA] and [FORMULA] will remain roughly constant within [FORMULA]; similarly to the method for the static model (see PCB98), within [FORMULA] we approximate them with the values [FORMULA], [FORMULA] and [FORMULA] they assume at the midstep [FORMULA]. Over [FORMULA], (17) can then be considered a system with constant coefficients [FORMULA], and [FORMULA] becomes a constant vector, which allows for the analytical solution:

[EQUATION]

where [FORMULA] indicates the matrix:

[EQUATION]

with [FORMULA] the eigenvalues of [FORMULA] and [FORMULA] the corresponding eigenvectors. The matrix [FORMULA] and the explicit expression of the solution (18) for the [FORMULA]'s are calculated in Appendix A, resulting in:

[EQUATION]

where we intend [FORMULA], [FORMULA], and for the outermost shell one should replace:

[EQUATION]

[FORMULA] being defined by (16).

Similarly to the case of an isolated shell (see PCB98) the system (20) does not provide the final solution for [FORMULA], since the [FORMULA]'s and the [FORMULA]'s on the right-hand side actually depend on the [FORMULA]'s; it just represents a set of implicit non-linear expressions in [FORMULA]. As in the original static model, we can neglect the dependence of [FORMULA] on the [FORMULA]'s and consider only that of [FORMULA] (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 i considered, so at each iteration we actually need to solve as many systems as the species included in the model. Full details on the mathematical development of the model and its numerical solution can be found in the appendices and in Portinari (1998).

We tested the code against suitable analytical counterparts and obtained the following conditions for model consistency (Appendix B).

  1. 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.

  2. 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 [FORMULA] kpc for the inner shells to [FORMULA] kpc for the outermost ones. With such a grid spacing, and velocities up to [FORMULA] km sec -1, suitable timesteps are of [FORMULA] Gyr (Appendix B; see also Thon & Meusinger 1998). This means that roughly [FORMULA] timesteps, times 35 shells, are needed to complete each model, which would translate in excessive computational times. This drawback was avoided by separating the time-scales in the code.

  1. The timestep [FORMULA] used to update the "chemical" variables ([FORMULA], [FORMULA], etc.) is the minimum among: [FORMULA] which guarantees that the relative variations of the [FORMULA]'s are lower than a fixed [FORMULA]; [FORMULA] which guarantees that the total surface mass density [FORMULA] increases by no more than 5%; [FORMULA] which is twice the previous timestep of the model, to speed up the computation when possible; [FORMULA] which guarantees the Courant condition [FORMULA], indispensable for the stability of a numerical algorithm describing flows. So, [FORMULA] 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.

  2. 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 [FORMULA] is subdivided in much shorter timesteps [FORMULA] Gyr, upon which the solution (20) and its Newton-Raphson iteration are successively applied to cover the whole [FORMULA]. Only then a new upgrade of all the [FORMULA]'s and [FORMULA]'s is performed.

This trick keeps the code roughly as fast as if it would evolve with a single time-scale [FORMULA], and yet it gives the same results as the "slow" version in which all quantities are upgraded at each [FORMULA] Gyr.

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Online publication: March 21, 2000
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