Inflow models

In this case, which may be regarded as the opposite extreme to the Simple model, it is supposed that, through feedback processes, accretion occurs at just such a rate as to compensate the loss of gas in star formation, i.e. 

Thus the mass of stars and that of the whole system steadily increase while z soon approaches 1 and the stellar metallicity distribution is very narrow (see Fig. 8.24). The accretion rate is constant in time if the star formation rate is any fixed function of the mass of gas. Other models in which the accretion rate is constant, but less than in the extreme model, have been quite often considered in the older literature (e.g. Twarog 1980), but are less popular now because they are not well motivated from a dynamical point of view, there is an upper limit to the present inflow rate into the whole Galaxy of about 1 M0yr 1 from X-ray data (Cox Smith 1976) and they do not provide a very good fit to the observed metallicity distribution function. 

Some specific GCE models and related observational data 

2 Analytical models with declining inflow rates 

This generates a series of abundance distribution functions with M as a parameter (Fig. 8.25). As M increases, the distribution becomes more like a Gaussian (i.e. a parabola on this logarithmic plot) on the low-metallicity side of the peak (although there is still always a low-metallicity tail), and the peak itself shifts to lower metallicities in units of the yield. 

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Errors in the use of the instantaneous recycling approximation need to be considered, depending on the assumed history of star formation and mass ejection. After long times, e.g. 15 Gyr, low-mass stars eject relatively metal-poor material and modify the distribution in a similar way to what happens in some of the inflow models discussed below. However, it is still rather doubtful whether any Simple model can explain the abundance distribution function as well as satisfying other constraints. 

Fig. 8.24. Schematic behaviour of gas mass, total mass and metallicity in the Simple model (left), the extreme inflow model of Larson (1972) (middle) and a model with time-decaying inflow (right). The abscissa is u = /( co(t )dt where o> is the (constant or otherwise) transition probability per unit time for gas to change into stars. The initial mass has been taken as unity in each case.

Other analytical inflow models less sophisticatedly refer to a specific star formation law, usually linear... 

A still simpler inflow model, based on a dynamical model due to Sommer-Larsen (1991b), assumes inflow according to the law... 

Clayton (1988) has given analytical solutions for inflow models of this type. 

Fig. 8.40. Magnesium abundance distribution functions for the thick and thin disks. Histograms Fe/H distributions from Wyse and Gilmore (1995) converted to Mg/H from the relations found by Fuhrmann (1998). Curves dashed, simple inflow model (Eqs. 8.46 to 8.52) for the thick disk dotted, prompt initial enrichment model plus inflow for the thin disk short-dash-dotted, inflow model without PIE long-dash-dotted, PIE without inflow. After Pagel in E. Vangioni-Flam et al. (2001). Copyright by World Scientific.

Fig. 8.41. Thin curves age-metallicity relation predicted by the two-inflow model of Chiappini, Matteucci and Gratton (1997). Thick curves sketch of suggested separate AMRs for the thick and thin disks. Data points are from Edvardsson et al. (1993). Adapted from Chiappini et al. (1997) by Pagel (2004).

Neglecting the variation in the denominator of Eq. (8.39), show that the abundance distribution function in Clayton s inflow model peaks in the neighbourhood of z = 1. 

Show that, in the simple inflow model described by Eqs. (8.46) to (8.52), the ratio of a secondary to a primary element is 4/3 times what it is in the Simple model at the same (primary) metallicity. 

Show that in Larson s extreme inflow model with initial abundance Z0 = 0, the abundance distribution function is given by... 

Z (7Li). The long-dashed curve indicates deuterium depletion in the simple inflow model. 

Using the simplest inflow model of Section 8.5.2, our assumptions are ... 

Considering now the simple inflow model, we again have Eqs. (9.27) to (9.29) together with... 

The discussion of the GCE of light elements formed by cosmic-ray spallation is based on Pagel (1994) and has a somewhat different outlook from many other treatments in the literature, e.g. Fields, Olive and Schramm (1995), which should be consulted to get another viewpoint. A numerical treatment of the Simple model by Vangioni-Flam et al. (1990) gave similar results to the ones derived more simply here, as did the discussion of inflow models by Prantzos (1994). 

Calculate the Solar-System Be/H ratio predicted by the simple inflow model, Eq. (9.48), with the same assumptions as in Problem 4, except that now u = 3. 

Show that, in Larson s extreme inflow model (see Section 8.5.1), assuming infalling material to be pristine with primordial abundances, the deuterium abundance evolves according to... 

Fig. 10.3. Observed A -rat.ios (estimated error limits shown by pairs of horizontal lines) and their theoretical variation with the age of the Galaxy according to three models a pure initial spike (continuous straight lines) Fowler s (1987) model (broken-line curves) and the simple inflow model (continuous curves).

This formula fits the same U, Th data for 13 < T <20 Gyr, and it can be seen from Fig. 10.3 that the upper limit could be infinity for slightly different models, basically because the beginning of nucleosynthesis is something that is now assumed to happen gradually. The inflow model gives 244,23s = 0.04, much the same as in Fowler s model. Thus Solar-System actinide data do not provide any upper limit to the age of the Galaxy, nor do they strongly constrain GCE models. 

D. D. Clayton, in W. D. Arnett and J. W. Truran (eds.), Nucleosynthesis Challenges and New Developments, University of Chicago Press 1985, p. 65, gives a description and simple algorithms for GCE and nucleo-cosmochronological calculations on the basis of his standard inflow models. Our simple inflow model shares some... 

Fig. 11.8. N/O vs. O/H in stars and H n regions, from various sources, fitted with an analytical model, a is the coefficient in a Twarog-type inflow model with inflow rate = aaf. After Henry, Edmunds and Koppen (2000).

The OLGA model consists of 202 km of piping, a reservoir inflow model for eight wells 22 valves, the pipeline surroundings and sources for methanol and glycol injection. 

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