Modifications to Erich Grossman's AT Code

Introduction

Erich Grossman's AT (Atmospheric Transmission) Code (Grossman 1989) is a package in common use by the infrared, submillimeter, and millimeter astronomical community for calculating atmospheric transmission for varying site conditions. Users input in latitude, elevation, and zenith angle, and the code uses the AFCRL line catalog of atmospheric lines (from 0 to 17000 cm^-1 or about 6000 Å) to create a line-by-line transmission spectrum. The lines used are from the seven main atmospheric active gases: H2O, CO2, O3, CH4, N2O, CO, and O2. The atmospheric profiles are based on simplified US Standard Atmosphere and Supplements (1962 & 1966) profiles (which the code assumes to have a linear lapse rate in the troposphere and an isothermal stratosphere), and adjustments are made according to local weather conditions in the form of user supplied temperature and pressure. A choice of five different collisionally-broadened line shapes (simple Lorentzian, full Lorentzian, two versions of the Zhevakin-Naumov/Harmonic Oscillator, and Van Vleck-Weisskopf) is provided to give the user both conservative and optimistic estimates of optical depths.

Doppler Broadening

The AT code unfortunately does not consider Doppler broadening of its absorption lines, which can become significant at high altitudes, where pressures are lower, and the collisional width is correspondingly smaller, and also significant at higher frequencies where the Doppler width is greater. One important modification was thus to include in the choice of line shapes, an approximation to the Voigt profile (the convolution of collisionally and thermally broadened profiles). Whiting's empirical approximation to the Voigt profile (Whiting 1968) has the advantage of being exact in the pure Gaussian and the pure Lorentzian limits. Whiting derived a single term and a more accurate two term approximation, with the two term approximation tested out to ten Voigt widths (at half-heights) with maximum deviations of -5% and +2.5%. Similar fits were made to derive empirical relations for the Voigt width at half-height (error on the order of 1%) and the line intensity (also error of 1%). A new version of AT with the Whiting approximation, ATV, was written.

Nonresonant Pressure-Induced N2 Absorption and O2 Relaxation

Chamberlin & Bally (1994) have reported a discrepancy between measured 225 GHz dry air opacity at the South Pole and that predicted by the AT code. Measuring optical depth tau and comparing with balloon sounding values of the column of precipitable water vapor (PWV), they found an empirical relation of

,

with tau in units of nepers/airmass and PWV in units of mm of precipitable water. The dry air opacity of .028 nepers/airmass is much larger than that calculated from AT, which gives values between .003 and .007 nepers/airmass. Hans Liebe's MPM code however does give a closer (although still lower) value of .024 +/- .002. nepers/airmass. One of the important differences between MPM and AT is the inclusion of continua terms in MPM that take into account of nonresonant effects due to pressure-induced N2 absorption and an O2 relaxation spectrum. As seen in Fig. 1, where the imaginary component of the complex refractivity is plotted versus frequency, at 225 GHz, the nonresonant components (dashed for O2 and dot-dash for N2) dominate over the line component (dotted) in absorption, and is within an order of magnitude or two of the refractivity of some of the O2 line cores.

For millimeter astronomy, it would appear important that we consider nonresonant contributions to the opacity. Another modified version of the AT code is currently being worked on where these components are included. The complex index of refraction

is often rewritten in terms of a new variable N that is measured in parts-per-million [ppm], and in Liebe's (1992) formulation, is expressed as:

,

where N(0) is a frequency-independent term, N_D is the dry-air term, N_V is the water vapor term, and N_W is the suspended water droplet term. The dry-air term is the one we are currently interested in, and it can can be broken up into three components:

,

with the terms in order from left to right: oxygen lines, non-resonant oxygen, and non-resonant nitrogen. Let us introduce a definition for inverse temperature:

,

a water vapor pressure:

,

where U is the relative humidity, and a total pressure:

,

where p is the dry-air pressure. We can now write down Liebe's fit for the complex refractivity from non-resonant O2 relaxation, with an expression for the strength of the term,

,

and an expression for the frequency f (in GHz) dependent shape function:

,

where the relaxation frequency is defined:

.

A similar relation for the complex refractivity from non-resonant pressure-induced N2 is:


.

In the simplest scenario, the imaginary component N" of the complex refractivity can be used to derive the amount of attenuation that electromagnetic radiation will experience when travelling a pathlength z through a parcel of air. The real component N' of the refractivity will give the dispersion, and the two together give a correction factor to the field strength of a propagating EM wave:

,

where k is the wavenumber, and nu is the frequency. We can now define the attenuation factor to be:

,

with N" again being the imaginary component of the complex refractivity.

If we assume a plane-parallel atmosphere, with n layers, each with a thickness delta-z, then the total attenuation A that a ray of light will experience after passing through the entire atmosphere at a zenith angle of 0, is:

,

which can be re-written, in the limit that delta-z goes to 0,

.

Thus, to calculate the total optical depth and absorption over an entire airmass, we are integrating the contributions at each infinitesimal atmospheric layer.

The AT model uses a simple two-component atmosphere consisting of a troposphere and a stratosphere. The troposphere is assumed to have a linear lapse rate L, i.e., the decrease in temperature is linear with height above the ground. Using the equation of state (the ideal gas law) and hydrostatic equilibrium, one can derive a power-law expression for the dry-air pressure:

,

where p_0 is the pressure at sea-level, T_0 is the temperature at sea-level, g_0 is a standard gravity (equal to 9.828e2 cm/sec) used to remove local dependence of the gravitational acceleration, M_0 is the mean molecular weight (set to 28.9644 amu), and k is Boltzmann's constant. For the stratosphere, AT assumes an isothermal atmosphere, i.e.,

.

The pressure in the stratosphere will then fall off exponentially:

,

where p_t is the pressure at the tropopause, Tt is the temperature at the tropopause, and Ht is the height of the tropopause.

Because of this two-layer atmosphere, the above integral of the imaginary refractivity will have to be divided in two. In the actual code, we would like to do the integral over km and have the frequency units in GHz. If the speed of light c and frequency nu in the expression for the total attenuation are expressed in cgs units, then we need a factor of 10^9 and a factor of 10^5 to convert them to compatible units. Letting the height of the ground equal H_B and the top of the atmosphere equal to H_end (nominally set to 100 km), the total absorption becomes:

.

The resulting absorption coefficients for an airmass path, equivalent to from sea-level to 100 km and at a zenith angle of 0, are plotted in Fig. 2. The dotted line represents the contribution from non-resonant N2, and the dashed line represents the contribution from non-resonant O2. The solid line is the sum of the two. Unfortunately in our current tests, the total absorption from both components at 225 GHz is only ~.008 nepers/airmass, which is not enough to account for the discrepancy between Liebe's MPM results with that of the AT results in Chamberlin & Bally. More work needs to be done to ascertain whether there is a bug in our current code, or if the discrepancy is real, and is due to another factor not covered here.

References