Stochastic abundance

The solar system abundances are a useful local galactic abundance standard because many nearby dwarf stars are similar in composition however, in detail there are some stochastic abundance variations (e.g [1,2,3,5]). The term cosmic abundances should be avoided because abundances generally decrease with galactocentric distance. There are also abundance differences between our galaxy and galaxies at high red-shift hence there is no generic cosmic composition that applies to all cosmic systems. 

Fig. 2. Stochastic accretion models for an open system. The infalling gas is assumed to be extragalactic material with standard Big Bang nucleosynthetic abundances (Xo = 0.758, Yo = 0.242, 2D=6.5xlCP5, SBBN) and zero metals, (a) Star formation rate vs. time for the thin disk. From the top to the bottom the curves refer to 44%, 10%, 5%, 1% and no mass added, (b) Metallicity vs. time for the thin disk. From the top to the bottom the curves refer to standard case (no mass added), 1%, 5%, 10%, 44% of mass added. The metallicity evolution curve illustrates the relatively weak dilution effects that are offset by continuing star formation. Details for the Deuterium abundances are shown in Fig. 3...

Normal gas-source mass spectrometers do not allow meaningful abundance measurements of these very rare species. However, if some demands on high abundance sensitivity, high precision, and high mass resolving power are met, John EUer and his group (e.g., Eiler and Schauble 2004 Affek and Eiler 2006 EUer 2007) have reported precise (<0. l%c) measurements of CO2 with mass 47 (A47-values) with an especially modified, but normal gas-source mass spectrometer. A47-values are defined as %o difference between the measured abundance of all molecules with mass 47 relative to the abundance of 47, expected for the stochastic distribution. 

The solar system abundances of the elements are the result of the Big Bang, which produced hydrogen and helium, 7.5 billion years of stellar nucleosynthesis, which produced most of the rest of the elements, and the physical processes that mixed the materials together to form the Sun s parent molecular cloud. The unique features of the solar system composition may also reflect the stochastic events that occurred in the region where the Sun formed just prior to solar system formation. 

Various quantitative and statistical validation processes have been described, accounting for the fact that SpCs tend to be small numbers and vary due to the partial stochasticity of the process. In the example datasets included in this chapter, the relationship between the SpC and protein abundance obtained experimentally is shown in Figure 2, demonstrating that many proteins in bacterial cells are low in abundance while a small subset are highly abundant. Several studies have compared label-free with labeling methods or assessed its statistical validity (93-95). Overall, SpC alone should not be used as a means for absolute quantification (92,96), but it is quite adequate for... 

The basic data for stochastic simulations of galaxies and their constituent populations and metallicity evolution is the initial mass function (IMF), which represents the mass distribution with which stars are presumed to form. Its derivation from the observed distribution of luminosity among field stars (refs. 57 and 58 and references therein) and from star clusters involves many detailed corrections for both stellar evolution and abundance variations among the observed population. The methods for achieving the IMF from the observed distribution are most thoroughly outlined by Miller and Scalo but can be stated briefly, since they also relate to an accurate testing of various proposed stochastic methods. It should first be noted that the problems encountered for stellar distributions are quite similar to those with which studies of galaxies and thdr intrinsic properties have to deal. 

Corrections for He I absorption by the underlying OB association. These are fairly uncertain and affect the He I line ratios as well as the total He abundance. B main sequence and supergiants have the largest He I line strengths, and so need special attention the B supergiant contribution is likely to be affected by stochasticity and uncertain stellar evolution. 

STOCHASTIC APPROACH in the early halo phases, mixing was not efficient, thus pollution from single SNe (Tsujimoto et al. 1999 Argast et al. 2000 Oey 2000) can be seen in very metal poor stars. This approach predicts a large spread in the abundance ratios at very low [Fe/H], even larger than observed. 

In Figure 5.2, the same modeling data presented in Figure 5.1 are modeled using demographic stochasticity with scramble competition (see Akcakaya 2005 for definition) and a carrying capacity. In this example, we see the population abundance over... 

FIGURE 5.2 Stochastic population projection for a hypothetical species average population abundance over time (dotted line) and associated 95% confidence limits (solid and dashed lines). 

We have previously focused attention on purely gas-phase chemical models, gas-phase models with accretion onto grains, and gas-phase models with both accretion and desorption. Gas-grain models, such as that of Aikawa et briefly mentioned above, differ from these models in that they include surface chemistry. For models with large numbers of surface reactions, either the simple rate equation treatment is used, or the rate coefficients /cab (see Eq. (1.74)) are modified in a semi-empirical manner to handle fractional average adsorbate abundances to an extent. The so-called modified rate treatment has been tested against stochastic methods in small systems of equations. 

Apart from stochastic methods, some modifications to the rate equations have been put forward to mimic the stochastic behaviour of the surface chemistry. Caselli et al. [56] made semi-empirical adjustments to the rates of a selection of reactions, for the case where the surface migration of atomic hydrogen is significantly faster than its accretion rate onto grains. While this method showed only limited success, the more recent modified-rate approach suggested by Garrod et al. [57] has been much more successful. In this approach, the expressions for the reactions are adjusted if the modified reaction rate is smaller than the classical rate. The modified reaction rate is determined by the accretion rate of one of the reactants multiplied by the product of the surface abundance of the other reactant... 

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