Abstract. Correction due to finite speed of light is among the most inconsistent ones in absolute gravimetry. Formulas reported by different authors yield corrections scattered up to 8 µGal with no obvious reasons. The problem, though noted before, has never been studied, and nowadays the correction is rather postulated than rigorously proven. In this paper we make an attempt to revise the subject. Like other authors, we use physical models based on signal delays and the Doppler effect, however, in implementing the models we additionally introduce two scales of time associated with moving and resting reflectors, derive a set of rules to switch between the scales, and establish the equivalence of trajectory distortions as obtained from either time delay or distance progression. The obtained results enabled us to produce accurate correction formulas for different types of instruments, and to explain the differences in the results obtained by other authors. We found that the correction derived from the Doppler effect is accountable only for 2 3 of the total correction due to finite speed of light, if no signal delays are considered. Another major source of inconsistency was found in the tacit use of simplified trajectory models.
In the article (Rothleitner and Francis 2011 Metrologia 48 187-195) the correction due to the finite speed of light in absolute gravimeters is analyzed from the viewpoint of special relativity. The relativistic concepts eventually lead to the two classical approaches to the problem: analysis of the beat frequency, and introduction of the retarded times. In the first approach, an additional time delay has to be assumed, because the frequency of the beam bounced from the accelerated reflector differs at the point of reflection from that at the point of interference. The retarded times formalism is equivalent to a single Doppler shift, but results in the same correction as the beat frequency approach, even though the latter is explicitly combines two Doppler shifts. In our comments we discuss these and other problems we found with the suggested treatment of the correction.
Abstract. Correction due to finite speed of light is among the most inconsistent ones in absolute gravimetry. Formulas reported by different authors yield corrections scattered up to 8 µGal with no obvious reasons. The problem, though noted before, has never been studied, and nowadays the correction is rather postulated than rigorously proven. In this paper we make an attempt to revise the subject. Like other authors, we use physical models based on signal delays and the Doppler effect, however, in implementing the models we additionally introduce two scales of time associated with moving and resting reflectors, derive a set of rules to switch between the scales, and establish the equivalence of trajectory distortions as obtained from either time delay or distance progression. The obtained results enabled us to produce accurate correction formulas for different types of instruments, and to explain the differences in the results obtained by other authors. We found that the correction derived from the Doppler effect is accountable only for 2 3 of the total correction due to finite speed of light, if no signal delays are considered. Another major source of inconsistency was found in the tacit use of simplified trajectory models.
Relativistic treatment of the finite speed of light correction in absolute gravimeters, as evolved by Rothleitner and Francis in Metrologia 2011, 48 442-445, following the initial publication in Metrologia 2011, 48 187-195, leads to spurious conclusions. The double Doppler shift implemented in the gravimeters obliterates the difference between its relativistic and non-relativistic formulation. Optical heterodyning used in Michelson-type interferometers makes the quadratic Lorenz-like term of the double Doppler shift discernable against the linear term, while in relativistic experiments the quadratic term has to be detected against the unit. The disturbance of the registered trajectory caused by the finite speed of light includes tracking signal delay as intrinsic part not reducible to the Doppler shifts.
Purpose. A qualitative and quantitative study of the correlation of space-time changes of the total electron content of the ionosphere with variations in the energetic electron flux during a significant increase of the solar wind density and velocity. Determination of the conditions when the increase of the intensity of the flux of energetic electrons can be accompanied by the appearance of large-scale inhomogeneities of the ionosphere observed in magnetically conjugated regions of the Northern and Southern hemispheres. Methods. The research methodology is based on the construction of the time sequence of electron fluxes spatial distributions and their subsequent comparison with the maps of the total electron content (TEC) over North America and the TEC diurnal variations in magnetically conjugated regions. The degree of similarity has been estimated in this paper, and the corresponding correlation coefficients have been obtained. The TEC was calculated from the ground-based Global Navigation Satellite System (GNSS) observations, and the electron fluxes in the ionosphere were obtained from the in situ measurements by the POES satellites. The map-making region was selected by the presence of a dense network of GNSS receivers and the presence of stations in the magnetically conjugated region of the Antarctica, as well as by the favorable configuration of spatial distribution of energetic particles at the orbital height of POES satellites. The study is based on the two geomagnetic disturbances of the St. Patrick's Days in March 2013 and 2015. Results. The satellite and ground-based data during geomagnetic disturbances were processed by using the developed technique. It is found that the consistency of changes in the total electron content of the ionosphere and electron fluxes in time and space coincide with the variation range increase of the horizontal component of the geomagnetic field that has been observed according to the data of ground-based magnetometers and indicates the existence of ionospheric currents in the geospace. According to the analysis of the two events, the assumption is made that the presence of ionospheric currents formed by protons and electrons precipitated from the magnetosphere is one of the conditions for the consistency of changes in the total electron content and electron flux. Conclusions. It is shown that during the geomagnetic disturbances the space-time changes of the ionospheric inhomogeneities are partially consistent with the variations of the fluxes of energetic electrons that allows the possibility of using these observations of TEC as indicators of precipitation.
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