1.2 Modelling protoplanetary discs

Material in a protoplanetary disc will be in orbit around a central star (or protostar). A first approximation to the motion of the material is that it is in so-called Keplerian motion, that is it obeys Kepler’s laws of planetary motion. In particular Kepler’s third law, which is a consequence of Newton’s law of gravity, states that the square of the orbital period is proportional to the cube of the orbital radius (assuming circular orbits, which is generally the case), i.e. P² ∝ a³. As long as the orbiting particle has a mass that is small compared to that of the star, this may be expressed in the following equation:

where G is the gravitational constant (6.674 × 10^-11 N m² kg^-2) and M_* is the mass of the star. The Keplerian orbital speed is therefore the distance travelled in an orbit (the circumference of the orbit, 2πa) divided by the orbital period, P. This is therefore

Figure 4 shows a schematic view of a protoplanetary disc, comprised mostly of molecular hydrogen. The rest of this section will aim to derive and solve the differential equations that govern the vertical (out-of-disc plane) gas-density profile as well as the radial dependence of the orbital (in-disc plane) velocity, which turns out to differ from the pure Keplerian motion. The following two subsections therefore will look in turn at how these two properties may be quantified.

Figure 4 Schematic illustration of a protoplanetary disc.

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In the figure, a central yellow sphere labelled ‘M subscript asterisk’ is shown. A horizontal line is drawn across the sphere, such that the sphere lies in the middle of the line. This line is labelled ‘midplane (z equals 0)’. A right arrowhead is drawn at the right end of the line. This arrowhead is labelled ‘r-axis’. An upward arrow labelled ‘z-axis’ is drawn from the sphere. The protoplanetary disc is shown as a semicircular ring around the yellow sphere, lying on the horizontal plane facing away from the observer. The two cross-sections of the ring, lying on either side of the sphere are symmetrical about the z-axis. The height of the disc increases on either side of the horizontal line towards the periphery, forming two triangular cross-sections. A thin green layer covers the upper and lower surfaces of the disc, and the interior portion of the disc is shown as a blue region with several black dots. The black dots lying closest to the horizontal line are bigger, and the size decreases the further the dots are from the horizontal line. A point labelled ‘A’ is taken near the top-right corner of the right disc. A line of length d joins A and the centre of the sphere. This line forms an angle theta with the horizontal line. From A, a vertical line of length z is drawn to the horizontal line.

Figure 4 Schematic illustration of a protoplanetary disc.