On the Ionisation Fraction in Protoplanetary Disks I: Comparing Different Reaction Networks

We calculate the ionisation fraction in protostellar disk models using a number of different chemical reaction networks, including gas–phase and gas–grain reaction schemes. The disk models we consider are conventional α–disks, which include viscous heating and radiative cooling. The primary source of ionisation is assumed to be X–ray irradiation from the central star. For most calculations we adopt a specific disk model (with accretion rate Ṁ = 10 M⊙yr −1 and α = 10), and examine the predictions made by the chemical networks concerning the ionisation fraction, magnetic Reynolds number, and spatial extent of magnetically active regions. This is to aid comparison between the different chemical models. We consider a number of gas–phase chemical networks. The simplest is the five species model proposed by Oppenheimer & Dalgarno (1974). We construct more complex models by extracting species and reactions from the UMIST data base. In general we find that the simple models predict higher fractional ionisation levels and more extensive active zones than the more complex models. When heavy metal atoms are included the simple models predict that the disk is magnetically active throughout. The complex models predict that extensive regions of the disk remain magnetically uncoupled (“dead”) even when the fractional abundance of magnesium xMg = 10 . This is because of the large number of molecular ions that are formed, which continue to dominate the recombination with free electrons in the presence of magnesium. The addition of submicron sized grains with a concentration of xgr = 10 −12 causes the size of the “dead zone” to increase dramatically for all kinetic models considered, as the grains are highly efficient at sweeping up the free electrons. We find that the simple and complex gas–grain reaction schemes agree on the size and structure of the resulting “dead zone”, as the grains play a dominant role in determining the ionisation fraction. We examine the effects of depleting the concentration of small grains as a crude means of modeling the growth of grains during planet formation. We find that a depletion factor of 10 causes the gas–grain chemistry to converge to the gas–phase chemistry when heavy metals are absent. When magnesium is included a depletion factor of 10 is required to reproduce the gas–phase ionisation fraction. This suggests that efficient grain growth and settling will be required in protoplanetary disks, before a substantial fraction of the disk mass in the planet forming zone between 1 – 10 AU becomes magnetically active and turbulent. Only after this has occurred can gas–phase chemical models be used to predict reliably the ionisation degree in protoplanetary disks.