The production of heavy elements in the late evolutionary stages of massive stars leads to a population of solid-phase silicate grains that first finds its way into the interstellar medium (ISM), and later into the circumstellar material of later-generation protostars (Kessler-Silacci et al. 2006). The gas, infused with such grains, distributes some of the giant molecular cloud’s angular momentum and elongates into a PPD (Tscharnuter, W. M.et al.2009,Bate
& Lorén-Aguilar 2017).
As is true of the Universe in general, PPDs are most abundant in Hydrogen and Helium.
Owing to the CNO cycle in the interiors of an earlier generation of stars, the next most common elements are carbon, nitrogen and oxygen. Also to be found are refractory minerals, such as iron, magnesium and silicon, which are products of supernovae. Such are the primary materials available to compose planetary cores, and to do so, they should exist in their solid form. The fractional mass abundance of solid grains with respect to gas, I shall denoteεand
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will refer to this quantity as either the ‘condensible solid metallicity’ or the dust-gas ‘mass loading’. In each stage of stellar and planetary co-evolution, collisions between gas and solid grains (henceforth also ‘dust’ or ‘particles’) have a critical effect on the gas temperature and dynamics, since dust effectively absorbs and emits radiation as well as exchanges momentum with the gas. The research represented in this thesis focuses narrowly upon the momentum exchange between the gas and solid phases and how this process can mediate spontaneous particle-density enhancements on local scales in a PPD.
The mean free path of gas molecules in circumstellar contexts is large, yet so is the characteristic volume. It is therefore warranted to regard PPDs as a continuum fluid flow where the statistical mechanics of ideal gasses apply (Thompson 2006,Pringle & King 2007).
This can be true of both the gas and solid phases, and therefore a model of a PPD reduces to a set of definitions of its primative state variables.
1.1.1. Equations of State
Consider a reservoir mainly composed of gas, in orbit around a central stellar object, with thin vertical height with respect to its large radial extent. Its natural system of coordinates is cylindrical, containing a star at the origin,Zbeing the distance above the orbital plane,Φ the azimuthal coordinate, andRthe radius. Initially in equilibrium, the cylindrical gaseous feature may not have finite physical boundaries, but is likely more well described by a diminishing density profile, in the ˆR, ˆZ, directions, and perhaps isotropic in the ˆΦdirection.
There are just three properties which fix the circumstellar-disk model, namely the temper-ature and surface density profiles,T(R)andΣ(R)respectively, and the mass of the star,M?. In their general form, these are expressed as follows:
Σ(r) = M?H
2πR3, (1.1)
T(R)∝R−c, (1.2) where H is the disk scale height. Determing the power-law index, c, of the temperature profile requires assumptions upon the sources of cooling and heating in the disk, an estimate of the total mass of the system, and the central star’s luminosity. The simplest estimate of T(R) is to assume that the disc is thin and therefore that its constituent particles achieve thermal equilibrium with their surroundings.
Written in terms of the equilibrium temperature at the radial position of Earth,
T(R) =280 K
Note that the dependence on stellar mass comes from the fact that the luminosity of a star is proportional to its mass. Observed discs do not differ so much from this approximation, as it has been found thatc∼1 (Dullemondet al.2007). As in the hydrostatic equilibrium configuration of an atmosphere, the gas surface density in the±Z direction decreases from the midplane valueΣmp(R)with H as the e-folding factor and when reflected about theZ-axis takes a Gaussian form,
Σ(Z,R) =Σmp(R)exp(Z/H)2. (1.4) It is common to constrainΣby requiring that its profile could result in our own Solar System, assuming that the planets form at their current location. This is done by distributing the measured masses of the planets in concentric annuli, corresponding to the distances between their nearly circular orbits (‘minimum mass solar nebula’, MMSN1.Weidenschilling (1977b),Hayashiet al.(1985)). Subscripting the gas, rock, and rock+ice surface density profiles, respectively, withg,r, andr+i,
1This model cannot describe the diversity of exoplanetary systems, but as of yet a more general model has not been found. SeeRaymond & Cossou(2014) for discussion.
Σr(r) = 7 g cm−2
Σr+i, depending upon radius, which is at most 0.017. This is of course a global estimate, as there are several mechanisms capable of creating local particle enrichments, particularly close to pressure maxima arising from fluid instabilities, which will be discussed in section1.2.1.
The actual mass densities of observed PPDs are poorly constrained, primarily due to the fact that hydrogen atoms have no excitational transitions at the temperatures of PPDs, so the column densities cannot be determined from spectroscopy. The mass must be inferred from other less abundant species, such as CO, which has low-frequency ro-vibrational transitions, corresponding to sub-millimeter wavelength radiation. Translating these measurements into hydrogen mass densities depends strongly on non-equilibrium chemistry models, which themselves are highly non-linear. Similarly, modeling of the spectral energy distributions of point sources is subject to model degeneracies in the radiative transfer calculations, which require a priori estimates of the line-of-sight density.