## 1 Introduction

[2] Interplanetary plasma is a major noise source in the error budget of deep space tracking experiments. The reduction of its effect plays a key role when dealing with the scientific requirements of challenging experiments such as gravitational waves detection or relativity tests. The concept of multifrequency tracking of interplanetary spacecraft (S/C) for the removal of interplanetary plasma and ionospheric effects from Doppler observables was first presented by *Bertotti et al.* [1993] and applied to Ulysses tracking data [*Bertotti et al.*, 1995]. Nowadays, multifrequency calibrations are also widely used in Global Navigation Satellite Systems applications for removing the ionospheric range delay affecting pseudorange and phase observables [*Petit and Luzum*, 2010]. In the next future, the BepiColombo Mercury Orbiter Radioscience Experiment will heavily rely on multifrequency radio tracking in order to achieve its ambitious error budget goals [*Iess and Boscagli*, 2001; *Iess et al.* 2009].

[3] The full multifrequency link configuration proposed in *Bertotti et al.* [1993] was successfully tested for the first time using Cassini's tracking data acquired during the cruise phase radio science experiments [*Tortora et al.*, 2002, 2003, 2004; *Bertotti et al.*, 2003; *Armstrong et al.*, 2003], showing a dramatic improvement in data stability, if compared to the raw X band and Ka-band observables for data acquired at low solar elongations angles (or SEP, Sun-Earth-Probe angle). The residuals obtained by the calibrated observables using a simple orbital model (the six parameter fit, described by *Bertotti et al.* [1995] based upon the Doppler formulation introduced by *Curkendall and McReynolds* [1969]) showed a fractional frequency stability (the fluctuation of the received sky frequency around the carrier reference frequency, in terms of Allan standard deviation, ASDEV) compatible with the one achieved during solar oppositions, a benign scenario for interplanetary plasma noise.

[4] Using the formalism presented by *Bertotti et al.* [1993], we see that a phase-coherent carrier sent in a round-trip loop from Earth and back is subjected to a phase scintillation caused by the refractivity index of the interplanetary medium, that consists of fully ionized plasma and is crossed during the uplink and downlink legs. This refractive index can be described as , assuming that the carrier frequency *f* is much higher than the electron plasma frequency: , with *e* electron charge, *n*_{e} electron number density, and *m*_{e} electron mass.

[5] The signal received by the ground station shows a total fractional frequency shift *y* and, under the assumption that the signal is affected only by dispersive noises, this is the combination of three components: the true observable *y*_{nd} (the “non-dispersive” Doppler shift), the uplink dispersive contribution, and the downlink one:

where *P*_{↑} and *P*_{↓} are unknowns, related to the total electron content of the medium and scaled with the uplink and downlink carrier frequencies, consistently with the electrostatic plasma dispersion relation.

[6] In a phase-coherent radio link, as the ones used for two-way tracking of deep space probes, the downlink reference frequency is not generated by the on-board oscillator, but keeps the ground station signal stability by simply scaling its frequency by a certain turn-around ratio *α* and transponding it back to Earth.

[7] Thus Equation (1) should be rewritten as

Since the goal of the calibration is to retrieve the *y*_{nd} component from the received observable *y*, we have a total of three unknowns.

[8] The full multifrequency link calibration, as implemented for Cassini's radio science experiments, exploits a radio system capable of generating three different observables, with two phase-coherent links working at X band and Ka-band, respectively, and a third “cross-link” generated receiving the up-link at X band and scaling the carrier to Ka-band for the downlink leg. So for Cassini, we have three independent observables acquired at the ground station (X/X, Ka/Ka, and X/Ka), and as Equation (2) holds for each one, a 3 × 3 set of equations can be compiled:

Furthermore, Equation (3) can be simplified using , and to get:

Equation (4) is easily inverted, and the three unknowns are solved for, to get in particular the non-dispersive (plasma-free) observable:

Despite its simplicity, this calibration scheme is extremely effective in terms of link stability and allows eliminating almost entirely its dependency upon the S/C SEP angle, as shown by *Tortora et al.* [ 2003, 2004]. The spectrum of solar plasma noise is strongly dependent on the SEP, because it represents the solar wind velocity component perpendicular to the carrier optical path, that is the main source of the phase scintillation [*Armstrong et al.*, 1979; *Asmar et al.*, 2005]. The price to be paid for these excellent results is the on-board hardware complexity: a radio system handling three different phase-coherent links, i.e., *five* different carriers spread between two bands, is quite demanding and expensive.

[9] A different, simpler calibration algorithm was still proposed by *Bertotti et al.* [1993] and its performance preliminarily assessed using qualitative considerations. The mathematical formulation of this simpler calibration scheme, and the comparison between the two algorithms are presented in the following section.