Scale height

Planetary examples

Approximate atmospheric scale heights for selected Solar System bodies:

Scale height for a thin disk

For a disk of gas around a condensed central object, such as, for example, a protostar, one can derive a disk scale height which is somewhat analogous to the planetary scale height. We start with a disc of gas that has a mass small relative to the central object. We assume that the disc is in hydrostatic equilibrium with the z component of gravity from the star, where the gravity component is pointing to the midplane of the disk: $${\displaystyle \frac{dP}{dz}=-\frac{G{M}_{\ast }\rho z}{(r^2+z^2{)}^{3/2}},}$$ where

G = Newtonian constant of gravitation ≈ 6.674×10⁻¹¹ m³⋅kg⁻¹⋅s⁻²‍

r = the radial cylindrical coordinate for the distance from the center of the star or centrally condensed object

z = the height/altitude cylindrical coordinate for the distance from the disk midplane (or center of the star)

M_* = the mass of the star/centrally condensed object

P = the pressure of the gas in the disk

${\displaystyle \rho }$ = the gas mass density in the disk

In the thin disk approximation, ${\displaystyle z\ll r}$, and the hydrostatic equilibrium, the equation is $${\displaystyle \frac{dP}{dz}\approx -\frac{G{M}_{\ast }\rho z}{r^3}.}$$

To determine the gas pressure, one can use the ideal gas law: $${\displaystyle P=\frac{\rho k_{\text{B}}T}{\overline{m}}}$$ with

T = the gas temperature in the disk, where the temperature is a function of r, but independent of z

${\displaystyle \overline{m}}$ = the mean molecular mass of the gas

Using the ideal gas law and the hydrostatic equilibrium equation, gives $${\displaystyle \frac{d\rho }{dz}\approx -\frac{G{M}_{\ast }\overline{m}\rho z}{k_{\text{B}}T{r}^3},}$$ which has the solution $${\displaystyle \rho ={\rho }_0\exp \left(-{\left(\frac{z}{h_{\text{D}}}\right)}^2\right),}$$ where ${\displaystyle {\rho }_0}$ is the gas mass density at the midplane of the disk at a distance r from the center of the star, and ${\displaystyle h_{\text{D}}}$ is the disk scale height with $${\displaystyle h_{\text{D}}=\sqrt{\frac{2k_{\text{B}}T{r}^3}{G{M}_{\ast }\overline{m}}}\approx 0.0306\sqrt{\frac{(T/100\text{K})(r/1\text{au}{)}^3}{(M_{\ast }/M_{\odot })(\overline{m}/2\text{Da})}}\text{au},}$$ with ${\displaystyle M_{\odot }}$ the solar mass, ${\displaystyle \text{au}}$ the astronomical unit, and ${\displaystyle \text{Da}}$ the dalton.

As an illustrative approximation, if we ignore the radial variation in the temperature ${\displaystyle T}$, we see that ${\displaystyle h_{\text{D}}\propto r^{3/2}}$ and that the disk increases in altitude as one moves radially away from the central object.

Due to the assumption that the gas temperature T in the disk is independent of z, ${\displaystyle h_{\text{D}}}$ is sometimes known as the isothermal disk scale height.

Disk scale height in a magnetic field

A magnetic field in a thin gas disk around a central object can change the scale height of the disk. For example, if a non-perfectly conducting disk is rotating through a poloidal magnetic field (i.e., the initial magnetic field is perpendicular to the plane of the disk), then a toroidal (i.e., parallel to the disk plane) magnetic field will be produced within the disk, which will pinch and compress the disk. In this case, the gas density of the disk is $${\displaystyle \rho (r,z)={\rho }_0(r)\exp \left(-{\left(\frac{z}{h_{\text{D}}}\right)}^2\right)-{\rho }_{\text{cut}}(r)\left[1-\exp \left(-{\left(\frac{z}{h_{\text{D}}}\right)}^2\right)\right],}$$ where the cut-off density ${\displaystyle {\rho }_{\text{cut}}}$ has the form $${\displaystyle {\rho }_{\text{cut}}(r)=({\mu }_0{\sigma }_{\text{D}}r{)}^2\frac{B_z^2}{{\mu }_0}{\left(\frac{{\mathrm{\Omega }}_{\ast }}{{\mathrm{\Omega }}_{\text{K}}}-1\right)}^2,}$$ where

${\displaystyle {\mu }_0}$ is the permeability of free space

${\displaystyle {\sigma }_{\text{D}}}$ is the electrical conductivity of the disk

${\displaystyle B_z}$ is the magnetic flux density of the poloidal field in the ${\displaystyle z}$ direction

${\displaystyle {\mathrm{\Omega }}_{\ast }}$ is the rotational angular velocity of the central object (if the poloidal magnetic field is independent of the central object, then ${\displaystyle {\mathrm{\Omega }}_{\ast }}$ can be set to zero)

${\displaystyle {\mathrm{\Omega }}_{\text{K}}}$ is the keplerian angular velocity of the disk at a distance ${\displaystyle r}$ from the central object.

These formulae give the maximum height of the magnetized disk as $${\displaystyle H_{\text{B}}=h_{\text{D}}\sqrt{\ln \left(1+{\rho }_0/{\rho }_{\text{cut}}\right)},}$$ while the e-folding magnetic scale height is $${\displaystyle h_{\text{B}}=h_{\text{D}}\sqrt{\ln \left(1+\frac{1-1/e}{1/e+{\rho }_{\text{cut}}/{\rho }_0}\right)}.}$$