Photoelectrons in the upper atmosphere: A formulation incorporating effects of transport

Abstract An efficient scheme is described for computation of photoelectron energy spectra. The method, similar to one used in radiative transfer theory, incorporates effects due to spatial and angular redistribution of electrons. These effects play a dominant role at heights above 400 km.

Absorption of solar radiation at wavelengths below 1000 A plays a major role in both the chemistry and thermodynamics of the atmosphere above an altitude of about 120 km. A significant fraction of the absorbed energy appears initially in photoelectrons. The ultimate fate of this energy depends on a complex sequence of collision processes and may involve significant spatial transport of the fast electrons.
Photoelectrons are released with an average initial energy of about 10 eV. Energy is lost to neutral species, through inelastic collisions which may contribute in part to the excitation of day airglow (Dalgamo et al., 1969). Collisions with ambient electrons represent a selective heat source which maintains an electron temperature exceeding the corresponding temperatures for ions and neutrals. Considerable attention has been focused in recent years on attempts to model the spatial and energy distribution of photoelectrons.
Initial efforts were directed primarily towards lower altitudes where the photoelectron energy distribution is set mainly by local production and loss (Hanson and Johnson, 1961;Hanson, 1963;Dalgarno, McElroy and Moffett, 1963). Hanson (1963) first drew attention to the potential importance of transport, and recent literature (Nisbet, 1968;Nagy and Banks, 1970;Banks and Nagy, 1970;Cicerone and Bowhill, 1971;Cicerone et QL, 1973;Mantas, 1975;Mantas and Bowhill, 1975;Swartz, 1976;Oran and Strickland, 1976;Lejeune and Wormser, 1976) has emphasized attempts to devise a unified theory for both local and non-local regimes. A variety of approaches have been devised to treat the complexities introduced by transport, including models based on Monte Carlo techniques, analogies with molecular diffusion, and theories for radiative transfer. The most complete models for transport are perhaps those described by Mantas (1975), Oran and Strickland (1976) and Lejeune and Wormser (1976). Victor et al. (1976) and Oran and Strickland (1976) give up to date descriptions of the atomic processes which affect the electron energy dist~bution, incorporating very recent laboratory and theoretical info~ation on relevant cross sections. Their models make use of extreme ultraviolet solar radiative fluxes as measured by Hinteregger (1977) on the Atmospheric Explorer Satellites (Dalgarno et al., 1973), and were used to analyze direct measurements of the photoelectron spectrum obtained by Doering et al. (1975) andDoering et al. (1976) with instrumentation on the same satellites. Peterson et al, (1977) have recently reported measurements of the photoelectron spectrum which arises due to electrons released from the magnetic conjugate ionosphere.
We shall now develop a model for photoelectrons which allows for a relatively complete description of the relevant transport and collisional physics, The model can be applied to study processes at either low or high altitude, and may be used to investigate interhemispheric transport. It will be applied elsewhere to an analysis of data from Atmospheric Explorer.

THE TRANSPORT EQUATION
The photoelectron spectrum may be characterized by a function #(I; E, 6, t) which defines the number of electrons moving with energy E, in direction 4 at time f, and position r. The function 4 is analogous to the specific intensity which appears in the theory of radiative transfer, and has units of electrons cm-* s-l sr-' eV_'. In general i$ is a function of 7 variables. We shall restrict attention here to steady state solutions. We shall further assume that are confined gyrate around well defined field line. radius of is taken be small to scales spatial inhomogeneity the background The direction motion along lines may defined by single quantity, pitch angle T'he photoelectron at an rary position a given line may specified then terms of parameters: x, displacement from reference position; the pitch and E, the energy. The change in photoele~on intensity, d+, along a differential path length, ds, is given by where the individual terms on the right hand side account for collisional loss of electrons from the initial beam (E, x, a), scattering of electrons from other pitch angles (E, a'), and production of electrons {E, a) either by photoioni~tion or by inelastic scattering from (E', a'). The quantity ~(8 x) denotes the inverse of the collisional mean free path; ,@, x) is the inverse of the mean free path limited to elastic scattering; P(E, a, a') defines the probability of an elastic scatter from pitch angle ar into (Y'. The change of 4 along the trajectory s may be written in the form dcb_dxad,: +a4 ds ds ax ds ap' (2) where @=cosLy, whete B is the strength of the magnetic field. The second term on the right hand side of (4) allows for the change in pitch angle along a trajectory as constrained to satisfy the adiabatic invariance of the magnetic moment. The transfer equation may be recast now in the form -SO3 7, IL), (5) where 7 is a dimensionless path length analogous to the optical depth parameter in theories of radiative transfer, dT(E, x) = -~(5, x) dx, ii(E, T) is a scattering efficiency similar to the single scattering albedo in transfer theory, and S is given by (8) The dimensionless path length 7 is uniquely related to the displacement x along field lines for any given value of E, and may be simply located in terms of a latitude-altitude coordinate system. The intensity I# at energy E depends implicitly on the intensity at higher energies E' through the source term S in (5).

METHOD OF SOLUXTON
It is convenient to introduce further quantities i, h, PC, R-, S' and S-defined as follows: s*(T, P) =%S(7, + r*)* S(7, -P)) with O<p51 in (9), (lo), (11) and (13), and with 0 < ~'5 1 in (12). Equation (12) reflects the assumption of reciprocity, i.e. scattering from CL + cl' is equivalent to scattering from ~'3 CL. With these definitions we have J 0 and aj (l-~2~dlog~ where (15) and (16) may be obtained by summing and diff erencing (5) for positive and negative values of p. The intensity $J is retrieved by where O< f* EG 1. Other quantities of physical significance include the mean intensity, J, given by For the limiting case of transport in a constant strength magnetic field, equations (15) and (16) reduce to a form identical to the equation of transfer for a plane parallel inhomogeneous atmosphere. With the further assumption that the functions P and S be even in p, (15) and (16) may be combined to give The integral in (20) may be replaced by a suitable quadrature formula. The second order integrodifferential equation reduces then to a set of coupled ordinary second order differential equations and may be solved by s~aigh~o~ard numerical techniques, as described for example by Auer (1967) and Prather (1974).
For the more general problem in which P may contain odd terms in CL, equations (15) and (16) can no longer be combined, and we must solve directly for both j and h. Replacing integrals by quadrature formulae as before, we have where subscripts k and 1 indicate quantities to be evaluated at either *L or JL~, the pivotal points of the quadrature formula, and g, denotes corresponding weights.
We selected a modified Gaussian quadra~re formula (Sykes, 1951) which allows an exact evaluation of polynomial integrands of order 2m -1, with m quadrature points distributed over the range 0c#.&51: Derivatives of j and h with respect to j.~,, evaluated at the quadrature points, may be expressed (Chandrasekhar, 1960, p.  The dependent variables jk and hk in (25) and (26) are functions of T and E. The spatial (7) dependence of jk and hk may be evaluated using a finite difference representation for (25) and (26) over an appropriate 7 grid. If we ignore, for the moment, complications introduced by the angular derivatives in (15) and (16), we may note that spatial derivatives of j or k may be evaluated directly in terms of local values for h or j. Thus, j and h need only be evaluated at alternate points along the grid rj: where i is an even integer. The matrices AkI and BU are defined by the farm of the right hand sides of (25) and (26). AnguIar derivatives of j and h are evaluated as averages of values at adjacent points.
Values of j at the boundaries, ro, may be related to h at adjacent interior points usi& series expansion: Thii condition, applied at both ends of a field line, provides the necessary boundary conditions. If the incident intensity, +*, is specified at some high altitude, then we can introduce an alternate boundary condition using (30), with h(r0, CL) = j(ro, cl)--cb*.
Solution of the transfer equation now requires inversion of a matrix equation. The matrix has dimensions nm x nm, where n gives the number of spatial grid points and m is the number of angular quadrature points. However, the matrix is block tri-diagonal with block sizes m x m, and the system of equations may be solved directly using an efficient scheme described by Auer (1967). The method has the advantage that computational time increases only linearly as a function of the number of spatial grid points.

ENERGY DEPENDENCE
The interdependence of photoelectron intensity at a specific energy, and electron intensities at other energies, arises through the source function 5 in equation (1). An electron may lose energy due to an inelastic collision with an atmospheric species j. Suppose that the process a has cross section 0, and involves a discrete energy loss AR,. In this case the photoelectron intensity at energy E + AIS, produces electrons of energy E. The rate at which electrons are inserted at energy E is then given by where n](r) denotes the number density of species j at T and R(p, EL') defines the probability of scattering from p' to p which may be related to the diff~rentiai scattering cross section. In addition to the discrete excitation of an atmospheric species j, we must also allow for ion~ng collisions which produce a continuum of electron energies, &fXj-+ e,+X]++e,.
In this case the higher energy scattered electron is arbitrarily designated by the subscript p Mornay). Energies of the incident (c), primary (p), and secondary (s) electrons are constrained to satisfy the relation E,+E,+AE, where hEi, denotes the ionization potential for the relevant reaction (j, a). The source of primary electrons at E due to (j, a) is given by where the differential cross section is normalized such that the total ionization cross section for electrons incident at EC is given by For the electron ionization reaction (34) the total number of electrons is not conserved and, in effect, we produce a new (secondary) electron for each reaction. The secondary electrons are included in the source term s' in a manner parahel to that for primary electrons. The integration in (36) for this case extends over all incident energies greater than 2E+AE,.
The total source of electrons due to the inelastic collisions of higher energy electrons with atmospheric species is calculated by summing the individual source terms Sk over all species and processes.
Electrons may arrive at energy E also as a consequence of collisions with ambient electrons. Energy degradation here is caused by a continuous succession of infinitesimal scatters with thermal electrons in the surrounding plasma. If we restrict ourselves to a set of discrete energy levels, El < E2<. . . , then the rate at which electrons cascade through successive levels varies inversely with their width. The source term due to this process for electrons in the energy interval Ei is given by where 2, denotes electron speed at Ei.+,, and dE/dt (E+t, 7) is the energy loss rate at Ei+l, a function of the plasma temerature and density. The source terms from inelastic collisions with neutral species and continuous loss to the ambient plasma are derived ultimately from photoproduction at higher energies. Direct photoproduction at E may be incorporated in S in a fairly straightforward fashion by adding a primary photoelec~on source calculated from measured values for the incident solar flux (Hinteregger, 1977) and absorption/ionization cross sections for major atmospheric gases as summarized for example by Victor et al. (1976).
The inverse mean free path K is defined in terms of the totality of collision processes which may affect an electron at energy E, and is given by where Q," denotes the cross section for elastic scattering of electrons with energy E by atmospheric species j and the summation over a includes all possible inelastic collisions, excitation plus ionization. The third term on the right hand of (39) arises from collisions with ambient electons and depends on the energy interval width AE as. described above. The inverse scattering mean free path ,y is given by x(E, '2) = c hi@.
(40) I' In calculating the source terms and mean free paths, the eollisional processes were treated in a manner similar to Victor et al. (1976) who computed photoelectron intensities in the limit of negligible transport. This limit, often called the local approximation, is valid at lower altitudes where the mean free path of the photoelectron is small. As may be seen from equation (5), when transport terms are dropped and the intensity is taken as isotropic, _ pal(~, ?) = S(E~ 7) %E, d 1cj(E, T) = K(@ T)-x(5,?)' Thus, the local approximation is determined only by the source term S and by inelastic opacity sources.
Up to this point we considered the energy space as a continuum of states and introduced the concept of energy intervals only in connection with the treatment of continuous loss. We propose to divide the energy spectrum into a number of energy intervals, each characterized by a mean energy Et. Re-dis~bution of inelastically scattered electrons assumes that electrons are uniformly distributed in the various energy bins. However, cross sections for ail electrons in a given interval are assigned values appropriate to the mean energy Ei.
Use of discrete energy bins introduces some modification to the source terms as given in equations (33) and (36). In order to obtain the contribution due to an inelastic collision process (j, a), we sum equation (33) over all higher energy bins which overlap with bin E downgraded in energy BE,. Each term in the sum is weighted by the fractional overlap in energy space, and is divided by the width of the lower bin of energy E. The integral over higher energies in the source term due to electron ionization is replaced by a sum, where each contributing bin is weighted according to both energy width and relevant differential cross section. The cross section is normalized (equation 37) also by a summation over the range of primary energy bins. Complications arise if the energy loss associated with an inelastic process is smaller than the energy bin size. We must be careful in this case to introduce an effectively elastic processs to allow for an energy loss process in which a fraction of the electrons remain within the original energy bin. The primary production of photoelectrons (eV-') from photoionization is readily expressed as a uniform source over a given energy bin by summing the total number of electrons produced within the energy interval and dividing by the bin width.
Although use of discrete energy intervals ignores the continuity of the 4(E) vs E relationship, there are also distinct advantages: small scale structure in the primary photoionization is automaticaily averaged; the mean free path for continuous loss processes is easily determined (