◄Previous: 1.2 Modelling protoplanetary discs

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.

◄Contents►Next: 1.4 Radial dependence of the orbital velocity

OpenLearn - The formation of exoplanets
ou logoCreative Commons non-commercial share alike icon Except for third party materials and otherwise, this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence, full copyright detail can be found in the acknowledgements section. Please see full copyright statement for details.