Scale height

, as can be confirmed from this diagram of air pressure p by altitude h: At an altitude of 0, 8.5, and 17 km, the pressure is about 1000, 370, and 140 hPa, respectively. In physics, a scale height, usually denoted by the capital letter H, is a distance (vertical or radial) over which a physical quantity decreases by a factor of e (the base of natural logarithms, approximately 2.718).

, as can be confirmed from this diagram of air pressure p by altitude h: At an altitude of 0, 8.5, and 17 km, the pressure is about 1000, 370, and 140 hPa, respectively.]] In physics, a scale height, usually denoted by the capital letter H, is a distance (vertical or radial) over which a physical quantity decreases by a factor of e (the base of natural logarithms, approximately 2.718). H = \frac{k_\text{B}T}{mg}, or equivalently, H = \frac{RT}{Mg}, where k B = Boltzmann constant = R = molar gas constant = 8.31446 J⋅K −1 ⋅mol −1 T = mean atmospheric temperature in kelvins = 250 K for Earth m = mean mass of a molecule M = mean molar mass of atmospheric particles = 0.029 kg/mol for Earth g = acceleration due to gravity at the current location The pressure (force per unit area) at a given altitude is a result of the weight of the overlying atmosphere. If at a height of z the atmosphere has density ρ and pressure P, then moving upwards an infinitesimally small height dz will decrease the pressure by amount dP, equal to the weight of a layer of atmosphere of thickness dz. Thus: \frac{dP}{dz} = -g\rho, where g is the acceleration due to gravity. For small dz it is possible to assume g to be constant; the minus sign indicates that as the height increases the pressure decreases. Therefore, using the equation of state for an ideal gas of mean molecular mass M at temperature T, the density can be expressed as \rho = \frac{MP}{RT}. Combining these equations gives \frac{dP}{P} = \frac{-dz}{{k_\text{B}T}/{mg}}, which can then be incorporated with the equation for H given above to give \frac{dP}{P} = - \frac{dz}{H}, which will not change unless the temperature does. Integrating the above and assuming P 0 is the pressure at height z = 0 (pressure at sea level), the pressure at height z can be written as P = P_0\exp\left(-\frac{z}{H}\right). This translates as the pressure decreasing exponentially with height. In Earth's atmosphere, the pressure at sea level P 0 averages about , the mean molecular mass of dry air is , and hence m = × = . As a function of temperature, the scale height of Earth's atmosphere is therefore H/T = k B /mg = / ( × ) = . This yields the following scale heights for representative air temperatures: T = 290 K, H = 8500 m, T = 273 K, H = 8000 m, T = 260 K, H = 7610 m, T = 210 K, H = 6000 m. These figures should be compared with the temperature and density of Earth's atmosphere plotted at NRLMSISE-00, which shows the air density dropping from 1200 g/m 3 at sea level to 0.125 g/m 3 at 70 km, a factor of 9600, indicating an average scale height of 70 / ln(9600) = 7.64 km, consistent with the indicated average air temperature over that range of close to 260 K. Note: Density is related to pressure by the ideal gas laws. Therefore, density will also decrease exponentially with height from a sea-level value of ρ 0 roughly equal to . At an altitude over 100 km, the atmosphere is no longer well-mixed, and each chemical species has its own scale height. Here temperature and gravitational acceleration were assumed to be constant, but both may vary over large distances. \frac{dP}{dz} = -\frac{GM_*\rho z}{(r^2 + z^2)^{3/2}}, where G = Newtonian constant of gravitation ≈ 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 \rho = the gas mass density in the disk In the thin disk approximation, z \ll r , and the hydrostatic equilibrium, the equation is \frac{dP}{dz} \approx -\frac{GM_*\rho z}{r^3}. To determine the gas pressure, one can use the ideal gas law: P = \frac{\rho k_\text{B} T}{\bar{m}} with T = the gas temperature in the disk, where the temperature is a function of r, but independent of z \bar{m} = the mean molecular mass of the gas Using the ideal gas law and the hydrostatic equilibrium equation, gives \frac{d\rho}{dz} \approx -\frac{GM_* \bar{m}\rho z}{k_\text{B} Tr^3}, which has the solution \rho = \rho_0 \exp\left(-\left(\frac{z}{h_\text{D}}\right)^2 \right), where \rho_0 is the gas mass density at the midplane of the disk at a distance r from the center of the star, and h_\text{D} is the disk scale height with h_\text{D} = \sqrt{\frac{2k_\text{B}Tr^3}{GM_* \bar{m}}} \approx 0.0306 \sqrt{\frac{(T/100~\text{K})(r/1~\text{au})^3}{(M_* / M_\odot)(\bar{m}/2~\text{Da})}}~\text{au}, with M_\odot the solar mass, \text{au} the astronomical unit, and \text{Da} the dalton. As an illustrative approximation, if we ignore the radial variation in the temperature T , we see that 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, h_\text{D} is sometimes known as the isothermal disk scale height. \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 \rho_\text{cut} has the form \rho_\text{cut}(r) = (\mu_0 \sigma_\text{D} r)^2 \frac{B_z^2}{\mu_0} \left(\frac{\Omega_*}{\Omega_\text{K}} - 1\right)^2, where \mu_0 is the permeability of free space \sigma_\text{D} is the electrical conductivity of the disk B_z is the magnetic flux density of the poloidal field in the z direction \Omega_* is the rotational angular velocity of the central object (if the poloidal magnetic field is independent of the central object, then \Omega_* can be set to zero) \Omega_\text{K} is the keplerian angular velocity of the disk at a distance r from the central object. These formulae give the maximum height of the magnetized disk as H_\text{B} = h_\text{D} \sqrt{\ln\left(1 + \rho_0/\rho_\text{cut}\right)}, while the e-folding magnetic scale height is h_\text{B} = h_\text{D} \sqrt{\ln\left(1 + \frac{1 - 1/e}{1/e + \rho_\text{cut}/\rho_0} \right)}.

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