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 rho sub gas is such that the vertical pressure gradient d cap p sub gas postfix solidus d z and the vertical component of the stellar gravity g sub z are in hydrostatic equilibrium. Since, as mentioned earlier, cap m sub disc much less than cap m sub asterisk operator, any disc contribution to the gravitational force can be ignored. Therefore we may write:

d cap p sub gas of z divided by d z equals negative rho sub gas of z times g sub z full stop

Referring to Figure 4, the vertical component of the stellar gravity at a point A located a distance d from the star is g sub z equals left parenthesis cap g times cap m sub asterisk operator solidus d squared right parenthesis times sine of theta, where G is the gravitational constant.

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

The angle θ is such that sine of theta equals z solidus d, hence g sub z equals cap g times cap m sub asterisk operator times z solidus d cubed. Now, the distance d is given by d squared equals r squared plus z squared but for geometrically thin discs, z much less than r, so we have simply d almost equals r, and therefore

g sub z almost equals cap g times cap m sub asterisk operator times z divided by r cubed full stop

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

omega sub cap k equals left parenthesis cap g times cap m sub asterisk operator divided by r cubed right parenthesis super one solidus two
Equation label:(Equation 2)

so g sub z equals z times omega sub cap k squared and we may write

d cap p sub gas of z divided by d z equals negative z times rho sub gas of z times omega sub cap k squared full stop

This equation can be simplified by recognising that, for an ideal gas, the pressure cap p sub gas and density rho sub gas are related by the sound speed cs such that

equation sequence part 1 c sub s squared equals part 2 k sub cap b times cap t divided by m macron equals part 3 cap p sub gas divided by rho sub gas comma
Equation label:(Equation 3)

where kB is the Boltzmann constant, T is the temperature of the disc and m macron is the mean molecular mass. The sound speed may be assumed to be constant for a given disc. Hence, d cap p sub gas equals c sub s squared d rho sub gas and the expression of hydrostatic equilibrium becomes

d rho sub gas of z divided by d z equals negative z times rho sub gas of z times left parenthesis omega sub cap k divided by c sub s right parenthesis squared full stop
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:

rho sub gas of z equals rho sub zero times exp of negative z squared divided by two times cap h squared comma
Equation label:(Equation 5)

where

cap h equals c sub s divided by omega sub cap k
Equation label:(Equation 6)

and

rho sub zero equals one divided by Square root of two times pi times cap sigma divided by cap h full stop
Equation label:(Equation 7)

Here, cap sigma equals integral rho sub gas of z d z 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 multirelation rho sub gas of cap h equals rho sub zero times exp of negative one solidus two equals rho sub zero solidus normal e super one solidus two almost equals 0.607 times rho sub zero.

The shape of a disc can be described by its aspect ratio cap h solidus r equals c sub s solidus v sub cap k, where v sub cap k equals r times omega sub cap k is the Keplerian speed at a radius r. Normally, for protoplanetary discs,

cap h divided by r proportional to r super one solidus four full stop
Equation label:(Equation 8)

This is because the speed of sound c sub s proportional to cap t super one solidus two (Equation 3) and the temperature profile of the disc is driven by the stellar irradiation, so that usually cap t proportional to r super negative one solidus two. (The reason for this latter dependence is that, from the Stefan-Boltzmann law, the temperature of the disc cap t proportional to cap f sub asterisk operator super one solidus four where the flux received from the star cap f sub asterisk operator proportional to one solidus r squared.) Hence, c sub s proportional to r super negative one solidus four and this result, combined with the fact that omega sub cap k proportional to r super negative three solidus two (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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