# 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.

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 ωK at this same point in the disc is

Equation label:(Equation 2)

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 cs such that

Equation label:(Equation 3)

where kB 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

Equation label:(Equation 4)

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:

Equation label:(Equation 5)

where

Equation label:(Equation 6)

and

Equation label:(Equation 7)

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

- What is the gas density at a height of z = H?

- The gas density is .

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,

Equation label:(Equation 8)

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.

- How does Equation 8 explain the shape of the disc shown in Figure 4?

- The aspect ratio of the disc increases with r, so the disc is expected to be thicker on the edge than in the centre, like the one in Figure 4.

OpenLearn - The formation of exoplanets
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