Energetic electrons and ions can penetrate the middle atmosphere causing excitation, dissociation, and ionization of ambient neutral constituents. X-ray bremsstrahlung resulting from these processes has a larger mean free path than the particles and penetrates even deeper into the atmosphere causing ionization and dissipation at lower altitudes. Secondary electrons from ionization collisions carry enough energy for further ionization but not enough energy to travel a significant distance in the atmosphere below about 100 km. These secondary electrons can be considered local to the volume of air where they are generated. This situation lends itself to an approach using range calculations to obtain the ionization and energy deposition rates at atmospheric layers.
With this approach the energy of a particle is related to a maximum range that this particle can travel in air. The range is defined in terms of number of ambient molecules encountered before all of the particle's initial energy is consumed. Given a density model, this range can be related to altitude or pressure level. In addition, an energy dissipation function is given that describes the rate of energy loss along the path to the maximum range as a function of the initial energy of the precipitating electron. For precipitating electrons below 32 keV, the energy dissipation function is independent of the energy of the incident particle. The function is obtained from laboratory experiments in which electrons are shot into a volume of rarified gas and determined from the resulting optical emission (Grun, 1957; Barret and Hays, 1976). Theoretical calculations extend the energy dissipation function to relativistic particles. Spencer (1959) gives the energy dissipation functions for electrons in energy intervals up to 10 MeV. A 10 MeV electron has a range that allows it to penetrate to approximately the 5 mbar pressure level. Figure 11 shows the energy dissipation functions for electrons with energies between 25 keV and 10 MeV as a function of fractional range. The fractional range is the atmospheric depth from the electron source divided by the range of the electron (both measured in units of column density).
The range measurements and energy dissipation functions of
Spencer (1959) are
based on
mono-energetic and uni-directional electrons. We generalize the energy
dissipation function for
incident electron spectra other than uni-directional by using a comprehensive
electron transport
model for arbitrary spectra and pitch angle distributions (
Lummerzheim et al., 1989). The
pitch angle distribution is characterized by a single parameter
where mu is the cosine of the pitch angle and fsubo is a normalization factor:
For x = 0 the distribution is isotropic; negative values of x result in a downward peak, and positive values give a pancake distribution. Figure 12 shows the energy dissipation function of Spencer (1959) for 25 keV electrons modified fo r values of peaking parameter x from -6 to +3. The largest effect of a changing angular distribution occurs at high altitudes, in the first half of the fractional range.
The PEM particle instruments sample the incident spectra at a number of energies to give a good energy resolution of the spectra. Normalized energy deposition profiles (lambda functions) for mono-energetic electrons are pre-computed as functions of the instrument energy channels and selected peaking parameters. At each energy step the pitch angle distribution is characterized. The energy step and resulting peaking parameter are used to look up the appropriate energy deposition profile. This profile is then normalized by the measured particle flux. The energy deposition profile of an arbitrary spectrum of particles is the superposition of the energy deposition profiles from the individual energy channels.
Elastic scattering of protons is mostly in the forward direction, and angular redistribution is not as important in this case as it is for electron transport. Rather than fitting the proton flux to a function describing its angular distribution, each measured pitch angle direction is treated independently. The precomputed energy dissipation functions for incident uni -directional and mono-energetic proton fluxes are obtained from parameterizing energy deposition profiles calculated with a transport model, which solves a pair of simplified coupled transport equations for protons and hydrogen Basu et al., 1987) for unit number flux. Using the energy dissipation function and range rather than the original transport calculation speeds up the calculation considerably without loss in accuracy.
In the procedure, the flux values from each proton spectrum are placed in a pitch angle array of bin size 10 degrees. Multiple entries into single bins are averaged. Unfilled bins are filled by interpolation if they are bounded by data values and by adjacent values if they occur near the edges of the energy-angle matrix. The energy and pitch angle bins are searched to look up the normalized proton energy deposition profile, and the contents of the pitch angle array are then used to scale the profile. The proton energy deposition profile is the superposition of the energy deposition profiles obtained from the individual energy steps.
Precipitating charged particles are guided by the geomagnetic field and can be treated as pseudo-particles at the center of their magnetic gyromotion as long as the collisional mean free path is long compared to the gyroradius. For the range calculation we must thus take the slanted geomagnetic field lines into account when calculating the mass of air penetrated. This is accomplished by converting the range to the vertical depth penetrated into a US standard atmosphere. As a separate task, the magnetic field direction as determined by the VMAG is used to correct the vertical depth to depth along a dipole field line such that the atmospheric mass between UARS and the altitude of energy deposition along the vertical depth equals the mass along the field line. This procedure provides a table relating the altitude of energy deposition to the fractional range of the precipitating particles. Using this method, changes in the atmospheric model or in the magnetic field model do not require recomputing the entire energy deposition profile.
The correct implementation of our energy deposition calculation has been verified by internal checks of energy conservation and comparison with energy deposition calculations with other parameterizations and methods (Lummerzheim, 1992). For electrons these include parameterizations by Lazarev (1967) for isotropic Maxwellian energy spectra with mean energy below 30 keV, and by Goldberg et al. (1984) for high energy electrons. The proton energy deposition calculations compare favorably with the results of Reid (1961) and Jackman et al. (1980).
Once the energy deposition profile in the atmosphere is determined, the ionization rate profile is obtained by the relation that, on average, every ionization collision results in 35 eV energy loss. The ionization rates for individual ion species in the middle atmosphere, below about 80 km altitude, are constant fractions of the total ionization rate based on a standard atmosphere density model. Brasseur and Solomon (1984) give the following fractions for diatomic nitrogen (+1), atomic nitrogen (+1), diatomic oxygen (+1), and atomic oxygen (+1) ionization rates, mu, including dissociation ionization, where q is the total ionization rate:
The altitude dependent mixing ratios of the neutral constituents at thermospheric altitudes require knowledge of the density profiles to obtain individual ionization rates (Vallance-Jones, 1974).
The primary data product resulting from the analysis of AXIS data is a tabulation of the average atmospheric energy deposition profile as a function of altitude for each of the 16 pixels every 65.536 seconds (a UARS "minute"). The energy deposition table for each pixel is labeled with the latitude and longitude of the center of the average field-of-view footprint at 100 km altitude, and the tabulation is reported at altitudes and pressures corresponding to the established UARS standards.
The energy deposition profile is obtained from the AXIS data set by comparing the measured x-rays to the results of a series of models and selecting the model that most closely corresponds to the observations. The models were obtained from a self-consistent calculation of the altitude profile of atmospheric energy deposition and the spectrum of escaping x-ray flux produced by a distribution of electrons incident on the top of the atmosphere. Initially, the range of electron distributions has been limited to include exponentials in energy which are isotropic over the downward hemisphere. These calculations were made using the coupled electron- photon cross-section code, CEPXS/ONELD (Lorence, 1993 and references therein).
The escaping x-ray flux is used in an instrument response function model to synthesize the pulse-height distribution, Msubi(Esubo), which is a close approximation of the distribution measured by AXIS. Msubi (Esubo) is related to a specific atmospheric energy deposition profile, D(h,Esubo), by the e-folding energy of the electron energy spectrum, Esubo. All models are normalized to unit total electron energy input.
UARS minute averages of the measured 32-channel x-ray spectrum, Xsubi, suitably corrected for background, Bsubi, are used to determine the e-folding energy, Esubo, and the electron spectrum intensity scale factor, A, by minimizing a goodness of fit parameter, xi^2, for each pixel, where
The resulting value of Esubo is used to look up an energy deposition profile, D(h,Esubo), which is then scaled by the intensity factor, A, to obtain the final result.
Once the best-fit parameters are obtained, the statistical uncertainty of the energy deposition profile result is calculated directly as
The general procedure for assessing sources of systematic error in energy deposition from the x- rays is illustrated in Figure 13. An arbitrary electron spectrum is input to the definitive energy deposition code, which calculates the x-ray spectra and the complete energy deposition profile. The x-ray spectrum is input to a code that simulates instrument response. The count rates are then fed into the analysis software as if it were real data. The resulting energy deposition profile computed using the analysis software is compared with the profile calculated from the definitive energy deposition code. The errors are expressed as fractional differences at each altitude. By varying input spectra and the way in which the data are handled in the analysis, a number of different error sources have been investigated which were beyond the known statistical errors and instrumental uncertainties.
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