Bruno Bertotti

A test of general relativity using radio links with the Cassini spacecraft

According to general relativity, photons are deflected and delayed by the curvature of space-time produced by any mass. The bending and delay are proportional to γ + 1, where the parameter γ is unity in general relativity but zero in the newtonian model of gravity. The quantity γ - 1 measures the degree to which gravity is not a purely geometric effect and is affected by other fields; such fields may have strongly influenced the early Universe, but would have now weakened so as to produce tiny—but still detectable—effects. Several experiments have confirmed to an accuracy of ∼0.1% the predictions for the deflection and delay of photons produced by the Sun. Here we report a measurement of the frequency shift of radio photons to and from the Cassini spacecraft as they passed near the Sun. Our result, γ = 1 + (2.1 ± 2.3) × 10-5, agrees with the predictions of standard general relativity with a sensitivity that approaches the level at which, theoretically, deviations are expected in some cosmological models.

Mach’s principle and the structure of dynamical theories

A structure of dynamical theories is proposed that implements Mach’s ideas by being relational in its treatment of both motion and time. The resulting general dynamics, which is called intrinsic dynamics and by construction treats the evolution of the entire Universe, is shown to admit as special cases Newtonian dynamics and Lorentz-invariant field theory provided the angular momentum of the Universe is zero in the frame in which its momentum is zero. The formal structure of Einstein’s general theory of relativity also fits the pattern of intrinsic dynamics and is Machian according to the criteria of this paper provided the so-called thin-sandwich conjecture is generically correct.

The rotation of LAGEOS

In view of the need of an accurate modelling of nongravitational forces on laser-tracked satellites, it is important to understand their rotational dynamics, which determines the temperature anisotropy and the ensuing radiation recoil effects. We propose a model of the torques acting on LAGEOS due to eddy currents and gravity gradient. The electromotive forces induced in the spacecraft by its rotation in the magnetic field of the Earth dissipate angular momentum and produce a precession of the spin axis; the oblate spacecraft will precess in the gravitational field of the Earth at a rate proportional to the rotation period. Therefore the gravitational torques become more and more important with time and eventually may produce a chaotic dynamics. The predicted evolution of the spin period agrees very well with the few experimental data available and corresponds to an approximately exponential growth rate of about 3 years.

STOCHASTIC GRAVITATIONAL WAVE BACKGROUND: UPPER LIMITS IN THE 10^-^6 TO 10^-^3 Hz BAND

We have used precision Doppler tracking of the Cassini spacecraft during its 2001-2002 solar opposition to derive improved observational limits to an isotropic background of low-frequency gravitational waves. Using the Cassini multilink radio system and an advanced tropospheric calibration system, the effects of heretofore leading noises—plasma and tropospheric scintillation—were, respectively, removed and calibrated to levels lower than other noises. The resulting data were used to construct upper limits to the strength of an isotropic background in the 10-6 to 10-3 Hz band. Our results are summarized as limits on the strain spectrum Sh( f), the characteristic strain (hc = the square root of the product of the frequency and the one-sided spectrum of strain at that frequency), and the energy density (Ω = energy density in bandwidth equal to center frequency assuming a locally white energy density spectrum, divided by the critical density). Our best limits are Sh( f) < 6 × 10-27 Hz-1 at several frequencies in the millihertz band, hc < 2 × 10-15 at about 0.3 mHz, and Ω < 0.025 × h, where h75 is the Hubble constant in units of 75 km s-1 Mpc-1, at 1.2 × 10-6 Hz. These are the best observational limits in the low-frequency band, the bound on Ω, for example, being about 3 orders of magnitude better than previous constraints from Doppler tracking.

Doppler measurement of the solar gravitational deflection

Testing alternative metric theories of gravity with an accuracy much better than the present level has recently drawn great attention, in particular in relation to the search for a very weak scalar field, a possible remnant of an early inflationary cosmology. The gravitational deflection of electromagnetic waves is controlled by the dimensionless post-Newtonian parameter , which takes a value of unity in general relativity. In this work we claim that the accuracy in the measurement of can be substantially improved by measuring the Doppler frequency shift of a microwave beam transponded back to the ground by an interplanetary spacecraft near solar conjunction. In this kind of experiment, the dispersion due to the plasma in the solar corona is the crucial difficulty, which, however, can be essentially overcome using skilful combinations of carriers with different frequencies. The spacecraft Cassini, launched in 1997, adopts a sophisticated radio system, including a Ka-band link at 32-34 GHz, which makes this possible. We discuss the noise budget for two experiments to be carried out with Cassini in 2002 and 2003. In particular, we consider the contribution of the solar corona, the non-gravitational accelerations, and thermal noise due to solar radio emission. We estimate that an accuracy in of about 10-5 is achievable.

The Cassini gravitational wave experiment

Doppler tracking experiments using the earth and a distant spacecraft as separated test masses have been used for gravitational wave (GW) searches in the low-frequency band(~0.0001-0.1 Hz). The precision microwave tracking link continuously measures the relative dimensionless velocity, Δv/c, between the earth and the spacecraft. A GW incident of the systems produces a characteristic signature in the data, different from the signatures of the principal noises. For 40 days centered about its solar opposition in December 2001, the Cassini spacecraft was tracked in a search for low-frequncy GWs. Here we describe the GW experiment, including transfer functions of the signals and noises to the Doppler observable, and present noise statistics and compare them with the pre-experiment noise budget.

Relativistic effects for Doppler measurements near solar conjunction

The authors consider the relativistic corrections to the Doppler effect in a metric theory of gravity, in the weak field and slow motion approximation, for a source and a receiver at great distances from, but near alignment with, the perturbing body (i.e. near conjunction). They also work in the approximation of a thin screen and paraxial rays, and apply this formalism to an ideal experiment, related to that of the deflection of light rays by the solar mass. They then introduce a differential Doppler technique, which allows some new and interesting developments in the field of experimental gravitation. With a spacecraft equipped with a hydrogen maser on board, one can measure the difference between the fractional frequency shifts in both directions. Near conjunction it is possible to have information, for example, on the angular momentum of the Sun or on a possible effect of a privileged cosmological frame of reference. This technique also allows a drastic reduction of the effect due to the plasma of the solar corona. The maximum fractional frequency change induced by the angular momentum of the Sun is about 7*10-16, which is barely consistent with the stability of hydrogen masers currently available.

Accurate light-time correction due to a gravitating mass

This technical paper of mathematical physics arose as an aftermath of the 2002 Cassini experiment (Bertotti et al 2003 Nature 425 374-6), in which the PPN parameter γ was measured with an accuracy σγ = 2.3 × 10 −5 and found consistent with the prediction γ = 1 of general relativity. The Orbit Determination Program (ODP) of NASAs Jet Propulsion Laboratory, which was used in the data analysis, is based on an expression (8) for the gravitational delayt that differs from the standard formula (2); this difference is of second order in powers of m—the gravitational radius of the Sun—but in Cassinis case it was much larger than the expected order of magnitude m 2 /b, where b is the distance of the closest approach of the ray. Since the ODP does not take into account any other second-order terms, it is necessary, also in view of future more accurate experiments, to revisit the whole problem, to systematically evaluate higher order corrections and to determine which terms, and why, are larger than the expected value. We note that light propagation in a static spacetime is equivalent to a problem in ordinary geometrical optics; Fermats action functional at its minimum is just the light-time between the two end points A and B. A new and powerful formulation is thus obtained. This method is closely connected with the much more general approach of Le Poncin-Lafitte et al (2004 Class. Quantum Grav. 21 4463-83), which is based on Synges world function. Asymptotic power series are necessary to provide a safe and automatic way of selecting which terms to keep at each order. Higher order approximations to the required quantities, in particular the delay and the deflection, are easily obtained. We also show that in a close superior conjunction, when b is much smaller than the distances of A and B from the Sun, say of order R, the second-order correction has an enhanced part of order m 2 R/b 2 , which corresponds just to the second-order terms introduced

Solar Coronal Plasma in Doppler Measurements

Doppler tracking of an interplanetary spacecraft near solar conjunction is strongly affected by the plasma in the solar corona, the main competitive contribution in measurements of the gravitational deflection of light rays. With the simultaneous availability of carriers in X band and Ka band for interplanetary communications, the plasma contribution to the corona can be accurately eliminated and measured. If, as in the Cassini mission, three different observables are available, this can be done in two ways: one deals with the total plasma content in the electric approximation, even in the ionosphere and interplanetary space; another is limited to the corona, but has access to subtler effects, like the magnetic correction to the refractive index. This technique will allow important progress in the radio investigation of the solar corona.

The effect of the motion of the Sun on the light-time in interplanetary relativity experiments

In 2002, a measurement of the effect of solar gravity upon the phase of coherent microwave beams passing near the Sun was carried out by the Cassini mission, allowing a very accurate measurement of the PPN parameter γ. The data have been analysed with NASAs Orbit Determination Program (ODP) in the Barycentric Celestial Reference System, in which the Sun moves around the centre of mass of the solar system with a velocity v⊙ of about 15 m s−1; the question arises: what correction does this imply for the predicted phase shift? After a review of the way the ODP works, we set the problem in the framework of Lorentz (and Galilean) transformations and evaluate the correction; it is several orders of the magnitude below our experimental accuracy. We also discuss a recent paper (Kopeikin et al 2007 Phys. Lett. A 367 276), which claims wrong and much larger corrections, and clarify the reasons for the discrepancy.