“…The frequency shift is modeled from the ephemerides of the spacecraft and planets, and these are given by NASA's SPICE toolkit Navigation and Ancillary Information Facility (NAIF) [33]. Then, the physical properties (e.g., the neutral number density, the pressure, and the temperature) can be inferred by making use of the ideal gas law together with hydrostatic equilibrium assumption (see e.g., Schinder et al [9] for Titan, Fjeldbo and Eshleman [5] for Mars, and Fjeldbo et al [14], Gramigna et al [34] for Venus). Therefore, the profiles φ(K), N (K), P (K), and T (K) that are needed for evaluating our analytical solutions in Eqs.…”