Numerical Models for a Protosolar Accretion Disk

We will limit our discussion to the viscous accretion disks because their study has been developed more than that of other possible mechanisms. Following rather general assumptions that we will not review here, in particular on the viscosity behavior of the disk, there remains two parameters, the collapse time and the viscosity coefficient, that can be combined into one.

Larson (1984) has established the order of magnitude of the collapse time, by the following considerations. The center of the dense interstellar nodule collapses fast and its outer part falls down more slowly; in order to accrete one solar mass, the outskirts of the nodule will reach the accretion disk some 105 years later. The actual rate will vary because of density fluctuations, but it is useful to know that the average rate is about 10-5 solar masses per year. Cameron (1985) proposes accretion rates a few times larger to explain the high luminosities observed in very young (T Tauri) stars.

The viscosity coefficient is the second parameter. It sets the dissipation rate of the inner angular momentum, hence the lifetime of the disk evolution. By combining the uncertain rate of collapse and the uncertain coefficient of viscosity in one single variable, this variable can be adjusted to empirical data. Such an adjustment changes the temperature everywhere in the disk, but not its radial gradient. It is fortunate that the temperature gradient in the midplane of the disk, as a consequence of the virial theorem, reflects the shape of the gravitational potential well made by the protoSuns's mass.

Morfill (1988) can be used as an example of a rather evolved model. Its midplane temperature varies with r-0'9 (r being the heliocentric distance) except at temperature plateaus due to the latent heat of condensation of the two major constituents, namely silicates within 0.4 AU, and water from 4 to 8 AU. Figure 2.1 shows Morfill's model. The two unknown parameters have been adjusted to the aggregation temperatures for the different terrestrial planets and two satellites of the giant planets, derived by Lewis (1974) from their empirical mean densities.

The success of the model adjustment (solid line) comes from the approximate temperature gradient in r-1 deduced by Lewis from the empirical condensation temperatures of the planets from a gas of solar composition. The two dotted lines represent the model for an accretion rate M larger (or smaller) by a factor of 10. A change in the viscosity coefficient would produce the same type of shift.

Lewis's (1972a, b) model assumed not only thermochemical equilibrium between gas and dust in the solar nebula, but also that planets accreted from dust captured only at the exact heliocentric distance of the planet. In spite of the gross oversimplification, the model predicted rather well the uncompressed densities of most of the planets. Lewis' model could not accurately predict the range of condensation temperatures for each planet, because it also depends on the location of the selected abiabat, hence on the model used for the solar nebula. However, its major virtue was to demonstrate that, at the time when dust sedimented from gas, the temperature gradient predicted by theory was empirically confirmed (Fig. 2.1).

Fig. 2.1. midplane temperature of the accretion disk, as a function of the distance to the Sun, at the time of dust sedimentation. The solid line is the adjustment of the disk theoretical model (Morfill, 1988). For a given viscosity coefficient, the two dotted lines correspond to an accretion rate M 10 times smaller or 10 times later. The crosses are Lewis's (1974) aggregation temperatures of planets and satellites derived from their densities. The crosses are not error bars: they show the width of the zones from which planetesimals were presumably collected, and their corresponding temperature ranges. The two horizontal plateaus in the profile are produced by the condensation latent heat of the silicates (near 0.4 AU) and of water ice (from 4 AU to 7 AU). The success of the adjustment comes from the fact that the empirical gradient derived by Lewis from the planet's densities is also predicted by theory. The actual temperatures of condensation for the planets remain however somewhat uncertain because they depend on the pressure in the nebula, hence on the adiabat chosen by Lewis.

Fig. 2.1. midplane temperature of the accretion disk, as a function of the distance to the Sun, at the time of dust sedimentation. The solid line is the adjustment of the disk theoretical model (Morfill, 1988). For a given viscosity coefficient, the two dotted lines correspond to an accretion rate M 10 times smaller or 10 times later. The crosses are Lewis's (1974) aggregation temperatures of planets and satellites derived from their densities. The crosses are not error bars: they show the width of the zones from which planetesimals were presumably collected, and their corresponding temperature ranges. The two horizontal plateaus in the profile are produced by the condensation latent heat of the silicates (near 0.4 AU) and of water ice (from 4 AU to 7 AU). The success of the adjustment comes from the fact that the empirical gradient derived by Lewis from the planet's densities is also predicted by theory. The actual temperatures of condensation for the planets remain however somewhat uncertain because they depend on the pressure in the nebula, hence on the adiabat chosen by Lewis.

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