1.3 Vertical gas-density profile

In the direction perpendicular to the disc plane (vertically, corresponding to the z-axis in Figure 4), the profile of the gas density is such that the vertical pressure gradient and the vertical component of the stellar gravity are in hydrostatic equilibrium. Since, as mentioned earlier, , any disc contribution to the gravitational force can be ignored. Therefore we may write:

Referring to Figure 4, the vertical component of the stellar gravity at a point A located a distance d from the star is , where G is the gravitational constant.

Figure 4 (repeated) 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 (repeated) Schematic illustration of a protoplanetary disc.

The angle θ is such that , hence . Now, the distance d is given by but for geometrically thin discs, , so we have simply , and therefore

Note that the Keplerian angular speed ω_K at this same point in the disc is

so and we may write

This equation can be simplified by recognising that, for an ideal gas, the pressure and density are related by the sound speed c_s such that

where k_B is the Boltzmann constant, T is the temperature of the disc and is the mean molecular mass. The sound speed may be assumed to be constant for a given disc. Hence, and the expression of hydrostatic equilibrium becomes

This differential equation has the following solution, which gives the density in terms of the disc scale height H and the density at the midplane ρ₀:

where

and

Here, is the surface density of the disc (i.e. its mass per unit surface area).

The shape of a disc can be described by its aspect ratio , where is the Keplerian speed at a radius r. Normally, for protoplanetary discs,

This is because the speed of sound (Equation 3) and the temperature profile of the disc is driven by the stellar irradiation, so that usually . (The reason for this latter dependence is that, from the Stefan-Boltzmann law, the temperature of the disc where the flux received from the star .) Hence, and this result, combined with the fact that (Equation 2), leads to Equation 8.