Modeling Ionization, Electron-Ion Recombination, and Vertical Diffusion in An Isothermal Martian Ionosphere

Introduction: We have recently shown [1] that the altitude profiles of CO2 (BuXg) and CO(aX) Martian ultraviolet dayglow emissions above 130 km follow the altitude profile of ambient CO2, reflecting the temperature of the neutral atmosphere. In contrast, the O(SP) ultraviolet dayglow emissions above 130 km follow the altitude dependence of ion-electron recombination in the Mars ionosphere, reflecting the square of the electron density. Thus Mars dayglow and radio science ionospheric occultation measurements return strongly correlated information. The high altitude O(S) dayglow emissions teach us about the topside ionosphere plasma scale height and temperature, while the CO2 (Bu) and CO(a) dayglow emissions teach us about the altitude profile of the primary ionizable atmospheric constituent, CO2. Relatively few Mars missions have included spectrometers capable of recording ultraviolet dayglow emissions, while every Mars orbiter or near flyby mission implicitly records information about the electron density altitude profile in the Mars ionosphere by refraction and time-delay just before radio transmission to Earth is blocked by the planet body and just after it becomes unblocked. Here we report development of a parameterized analytical model of the time and altitude variation of the electron density in Mars ionosphere based on a new eigen-solution expansion. We outline analytical and numerical mathematical formulas, tools, and procedures for systematic automated analysis and fitting of electron density altitude profiles, versus Mars astronomical variables: solar zenith angle, solar activity, and heliocentric distance. The primary targeted dataset is the several thousands of electron density altitude profiles recorded by the Mars Global Surveyor (MGS) orbiter achieved in the Planetary Data System (PDS). Eigen-Solutions of the Diffusion Equation: Consider an isothermal one-dimensional atmosphere with a vertical diffusion coefficient given by D(z) = D0 exp(z/HD). We write the partial differential equation for diffusion of a trace minor chemical species with isothermal scale height H, as (1) /t N(t,z) = /z { D0 exp(z/HD) [ /z N(t,z) + N(t,z)/H ] } , which has the well-know time-stationary or t   solution of the form (2) N0(t,z)  constant  exp( z/H ) . We make an independent variable substitutions for time as  = t {D0 / (4HD )}. We also make the much more important substitution for altitude,  = exp{ -z / (2HD) }, as suggested in the literature [2-5]. Using several definitions and equations in chapters 9-11 on Bessel functions in the handbook by Abramowitz and Stegun [6], we write the general solution of equation (1) as (3) N(t,z) =  0  g() exp(–  )  J( ) d , where  = HD/H -1 [5]. Ferraro and Ozdogan [2] were the first to find a Bessel function solution and L'vova et al. [5] were the first to implement Bessel function solutions with the order depending on the ratio of chemically unrelated scale heights. Bessel functions of integer and half-integer orders , are easily calculated from published formulas [6] and FORTRAN subroutines [7]. Subroutines for arbitrary orders are also available [8]. The non-zero values of g() in equation (3) can be determined from the fact that diffusion cannot create or destroy population. Thus 0 = 0 and J+1(j) = 0, for j > 0, defining the eigen-solutions (4) Nj(t,z)  exp(j 2 )  J(j ) , the orthonomality relationships (5)  0  e Nj(0,z) Nk(0,z) dz = jk , and the complete solution of equation (1) for arbitrary initial conditions: (6) N(t,z) = j cj Nj(t,z), where (7) cj =  0  e N(0,z) Nj(0,z) dz .