The results of reduction

We followed the approach outlined in the previous section to build the gas-phase and gas-grain reduced networks for e- and CO. The relative accuracy of calculated abundances of these species is requested to be better than $30\%$.

The sizes of the reduced networks and the corresponding computational time speed gains are tabulated in Tables $3$,$4$. As can be clearly seen, the reduction is more efficient for the gas-phase chemistry than for the gas-grain chemistry. One reason is that in the latter case it is necessary to keep almost all surface species in the network since accretion and desorption processes become very important after about $10^3-10^4$ years of the evolution. Another reason consists in the fact that dust surface reaction rates are much larger compared to the rates of gas-phase reactions, therefore a lot of these reactions are deemed to remain in the network.

The most interesting results are illustrated in Figures $3$,$4$. In Figure $3$ we plot relative abundances of CO molecule in respect to hydrogen nuclei obtained with the full and reduced networks for the conditions of dense clouds (``DENS'' model). It can be clearly seen from the dramatic decrease of the CO gas-phase abundances in the case of gas-grain chemistry (red and black lines) that accretion onto dust grains proceeds very efficiently under such low ($10$ K) temperatures and high ($10^7$ cm$^{-3}$) density. One may understand the effects of the gas-grain interactions, discussed above by looking on the shape of the curves. For the pure gas-phase chemistry (blue lines), behaviour of the abundance evolution curve for CO is simple: it grows exponentially till chemical equilibrium is reached around $t=10^3$ years and then it becomes a constant. As indicated on the figure, only $8$ species and $9$ reactions are necessary to reproduce the evolution of CO abundances accurately in this case. The corresponding computational speed-up factor is more than $10^4$.

On the contrary, for the gas-grain chemistry the behaviour of the CO evolution curves are not trivial: the equilibrium can be reached only after $t\sim 10^6$ years since the gas-dust interaction becomes of great importance. Obviously, in this case one needs to hold more chemical species and reactions in the reduced networks in order to follow the evolution of CO abundances with a reasonable accuracy. The corresponding computational time gain is the order of ten.

Surprisingly, there is almost no difference between abundances computed with the high and low metalicities for the same chemical models. It reflects the fact that CO-chemistry is sensitive mainly to the amount of C, O, and H available in the gas-phase, which is the same for the both metalicity cases.

We present the evolution of the ionization degree computed with the full and reduced networks for the conditions of dark diffuse clouds (``DIFF'' model) in Figure $4$. One may point out that compared to Figure $3$ there is no great difference for the fractional ionization degree calculated with the pure gas-phase and gas-grain chemical networks. It implies that the gas-grain interaction is not efficient anymore under such low ($n=10^3$ cm $^{-3}$) density since the typical timescale of collisions between dust grains and gas species are far too large.

In contrast to Figure $3$ there is a difference between ``high metals'' and ``low metals'' cases. In the high metalicity case metal ions, like Na$^+$, Mg$^+$, dominate the regulation of the ionization degree. For the ``low metals'' initial abundances, typical dominant ions are more complicated chemical species, like H$_3^+$ and HCO$+$ since the metals are heavily depleted from the gas phase. Typical computational speed gain can be as large as $500$ for the gas-phase chemistry and as small as $\sim 5$ for the gas-grain.

A possible application for reduced chemical networks would be the modelling of the evolution of magnetized protostellar clouds or protoplanetary accretion discs, when it is necessary to compute the ionization fraction self-consistently, accurately and rapidly.

The results of this work were reported in Budapest, 15-18 May 2002, during the conference ``Interaction of Stars with their Environments II'' [15].

Table: Reduction made for the pure gas-phase chemistry (``GAS'')

Model	Important	Reduced	Speed
	species	network	gain
HM-DIFF	e-	$58\times111$	$550$
HM-DENS	e-	$131\times313$	$130$
LM-DIFF	e-	$73\times169$	$490$
LM-DENS	e-	$157\times386$	$55$
HM-DIFF	CO	$158\times588$	$50$
HM-DENS	CO	$8\times9$	$\sim 10^4$
LM-DIFF	CO	$95\times313$	$240$
LM-DENS	CO	$8\times9$	$\sim 10^4$

Table: Reduction made for the gas-grain chemistry (``DUST'')

Model	Important	Reduced	Speed
	species	network	gain
HM-DIFF	e-	$210\times683$	$15$
HM-DENS	e-	$298\times1060$	$8$
LM-DIFF	e-	$357\times1648$	$2$
LM-DENS	e-	$291\times924$	$7$
HM-DIFF	CO	$298\times1293$	$8$
HM-DENS	CO	$388\times1563$	$3$
LM-DIFF	CO	$306\times1310$	$5$
LM-DENS	CO	$270\times798$	$16$

Figure 3: Evolution of abundance of CO molecule computed with the full (solid lines) and reduced (dashed lines) networks.

Figure 4: Ionization degree computed with the full (solid lines) and reduced (dashed lines) networks.