11 General Relativistic Models for Propagation

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1 Tehnial No General Relativisti Models for Propagation 11.1 VLBI Time Delay Historial Bakground To resolve differenes between numerous proedures used in the 1980s to model the VLBI delay, and to arrive at a standard model, a workshop was held at the U. S. Naval Observatory on 12 Otober The proeedings of this workshop have been published (Eubanks, 1991) and the model given there was alled the onsensus model. It was derived from a ombination of five different relativisti models for the geodeti delay. These are the Masterfit/Modest model, due to Fanselow and Thomas (see Treuhaft and Thomas, in Eubanks (1991), and Sovers and Fanselow (1987)), the I. I. Shapiro model (see Ryan, in Eubanks, (1991)), the Hellings-Shahid-Saless model (Shahid-Saless et al., 1991) and in Eubanks (1991), the Soffel, Muller, Wu and Xu model (Soffel et al., 1991) and in Eubanks (1991), and the Zhu-Groten model (Zhu and Groten, 1988) and in Eubanks (1991). At the same epoh, a relativisti model of VLBI observations was also presented in Kopeikin (1990) and in Klioner (1991). The onsensus model formed the basis of that proposed in the Standards (MCarthy, 1992). Over the years, there was onsiderable disussion and misunderstanding on the interpretation of the stations oordinates obtained from the VLBI analyses. Partiularly the Conventions (MCarthy, 1996) proposed a modifiation of the delay, erroneously intending to omply with the XXIIst General Assembly of the International Astronomial Union in 1991 and the XXIst General Assembly of the International Union of Geodesy and Geophysis in 1991 Resolutions defining the Geoentri referene system. It seems, however, that this modifiation was not implemented by analysis enters. In the presentation below, the model is developed in the frame of the IAU Resolutions i.e. general relativity (γ = 1) using the Baryentri Celestial Referene System (BCRS) and Geoentri Celestial Referene System (GCRS) (as defined in the Appendix). However two approahes are presented for its usage, depending on the hoie of oordinate time in the geoentri system. It is disussed how the Terrestrial Referene System (TRS) VLBI station oordinates submitted to the, and the resulting ITRF2000 oordinates (Chapter 4), should be interpreted in relation to the IAU and IUGG Resolutions. The step-by-step proedure presented here to ompute the VLBI time delay is taken from (Eubanks, 1991) and the reader is urged to onsult that publiation for further details Speifiations and Domain of Appliation The model is designed primarily for the analysis of VLBI observations of extra-galati objets aquired from the surfae of the Earth. 22 All terms of order seonds or larger are inluded to ensure that the final result is aurate at the pioseond level. It is assumed that a linear ombination of dual frequeny measurements is used to remove the dispersive effet of the ionosphere, so that atmospheri effets are only due to the troposphere. The model is not intended for use with observations of soures in the solar system, nor is it intended for use with observations made from spae-based VLBI, from either low or high Earth orbit, or from the surfae of the Moon (although it would be suitable with obvious hanges for observations made entirely from the Moon). 22 The ase of radio soures inside our galaxy has been onsidered in e.g. Sovers and Fanselow (1987); Klioner (1991) 109

2 No. 32 Tehnial 11 General Relativisti Models for Propagation The geoentri elestial referene system (GCRS) is kinematially nonrotating (not dynamially non-rotating) and, inluded in the preession onstant and nutation series, are the effets of the geodesi preession ( 19 milli ar seonds / y). If needed, Soffel et al. (1991) and Shahid- Saless et al. (1991) give details of a dynamially inertial VLBI delay equation. At the pioseond level, there is no pratial differene for VLBI geodesy and astrometry exept for the adjustment in the preession onstant The Analysis of VLBI Measurements: Definitions and Interpretation of Results In priniple, the observable quantities in the VLBI tehnique are reorded signals measured in the proper time of the station loks. On the other hand, the VLBI model is expressed in terms of oordinate quantities in a given referene system (see Chapter 10 for a presentation of the different oordinate times used). For pratial onsiderations, partiularly beause the station loks do not produe ideal proper time (they even are, in general, synhronized and syntonized to UTC to some level, i.e. they have the same rate as the oordinate time Terrestrial Time (TT)), the VLBI delay produed by a orrelator enter may be onsidered to be, within the unertainty aimed at in this hapter, equal to the TT oordinate time interval d T T between two events: the arrival of a radio signal from the soure at the referene point of the first station and the arrival of the same signal at the referene point of the seond station. that we model here only the propagation delay and do not aount for the desynhronization or desyntonization of the station loks. From a TT oordinate interval, d T T, one may derive a Geoentri Coordinate Time (TCG) oordinate interval, d T CG, by simple saling: d T CG = d T T /(1 L G ), where L G is given in Table 1.1. In the following, two different approahes are presented using two different geoentri oordinate system with either TCG or TT as oordinate time. The VLBI model presented below (formula (9)) relates the TCG oordinate interval d T CG = t v2 t v1 to a baseline b expressed in GCRS oordinates (see the definition of notations in the next setion). In the first approah, therefore, if the VLBI delay was saled to a TCG oordinate interval, as desribed above, the results of the VLBI analysis would be diretly obtained in terms of the spatial oordinates of the GCRS, as is reommended by the IUGG Resolution 2 (1991) and IAU Resolution B6 (1997), i.e. one would obtain TRS oordinates that are termed onsistent with TCG, here denoted x T CG. In the seond approah, if the VLBI model (formula (9)) is used with VLBI delays as diretly provided by orrelators (i.e. equivalent to a TT oordinate interval d T T without transformation to TCG), the baseline b is not expressed in GCRS but in some other oordinate system. The transformation of these oordinates to GCRS redues, at the level of unertainty onsidered here, to a simple saling. The TRS spae oordinates resulting from the VLBI analysis (here denoted x V LBI ) are then termed onsistent with TT and the TRS oordinates reommended by the IAU and IUGG resolutions, x T CG, may be obtained a posteriori by x T CG = x V LBI /(1 L G ) (see Petit, 2000). All VLBI analysis enters submitting to the have used this seond approah and, therefore, the VLBI spae oordinates are of the type x V LBI. For ontinuity, an ITRF workshop (November 2000) deided to ontinue to use this approah, making it the present onventional hoie for submission to the. that the use of spae oordinates onsistent with TT is also the present onventional hoie of SLR analysis results submitted to the. At the ITRF workshop, it was also deided that the oordinates should not be re-saled to x T CG for the omputation of ITRF2000 (see Chapter 4) so that the sale of ITRF2000 does not omply with IAU and IUGG resolutions. 110

3 11.1 VLBI Time Delay Tehnial No The VLBI Delay Model Table 11.1 Notation used in the model. t i the TCG time of arrival of a radio signal at the i th VLBI reeiver T i the TCB time of arrival of a radio signal at the i th VLBI reeiver t gi the geometri TCG time of arrival of a radio signal at the i th VLBI reeiver inluding the gravitational bending delay and the hange in the geometri delay aused by the existene of the atmospheri propagation delay but negleting the atmospheri propagation delay itself t vi the vauum TCG time of arrival of a radio signal at the i th VLBI reeiver inluding the gravitational delay but negleting the atmospheri propagation delay and the hange in the geometri delay aused by the existene of the atmospheri propagation delay δt atmi the atmospheri propagation TCG delay for the i th reeiver = t i t gi T ij the approximation to the TCB time that the ray path to station i passed losest to gravitating body J T grav the differential TCB gravitational time delay x i (t i ) the GCRS radius vetor of the i th reeiver at t i b x2 (t 1 ) x 1 (t 1 ) and is thus the GCRS baseline vetor at the time of arrival t 1 δ b a variation (e.g. true value minus a priori value) in the GCRS baseline vetor w i the geoentri veloity of the i th reeiver ˆK the unit vetor from the baryenter to the soure in the absene of gravitational or aberrational bending ˆk i the unit vetor from the i th station to the soure after aberration X i the baryentri radius vetor of the i th reeiver X the baryentri radius vetor of the geoenter X J the baryentri radius vetor of the J th gravitating body R ij the vetor from the J th gravitating body to the i th reeiver R J the vetor from the J th gravitating body to the geoenter R the vetor from the Sun to the geoenter ˆN ij the unit vetor from the J th gravitating body to the i th reeiver V the baryentri veloity of the geoenter U the gravitational potential at the geoenter, negleting the effets of the Earth s mass. At the pioseond level, only the solar potential need be inluded in U so that U = GM / R M i the rest mass of the i th gravitating body M the rest mass of the Earth the speed of light G the Gravitational Constant Vetor magnitudes are expressed by the absolute value sign [ x = (Σx 2 i ) 1 2 ]. Vetors and salars expressed in geoentri oordinates are denoted by lower ase (e.g. x and t), while quantities in baryentri oordinates are in upper ase (e.g. X and T ). A lower ase subsript (e.g. xi ) denotes a partiular VLBI reeiver, while an upper ase subsript (e.g. x J ) denotes a partiular gravitating body. The SI system of units is used throughout. Although the delay to be alulated is the time of arrival at station 2 minus the time of arrival at station 1, it is the time of arrival at station 1 that serves as the time referene for the measurement. Unless expliitly stated otherwise, all vetor and salar quantities are assumed to be alulated at t 1, the time of arrival at station 1 inluding the effets of the troposphere. The VLBI hardware provides the UTC time tag for this event. For quantities suh as X J, V, w i, or U it is assumed that a table (or numerial formula) is available as a funtion of a given time argument. The UTC time tag should be transformed to the appropriate timesale orresponding to the time argument to be used to ompute eah element of the geometri model. 111

4 No. 32 Tehnial 11 General Relativisti Models for Propagation The baseline vetor b is given in the kinematially non-rotating GCRS. It must be transformed to the rotating terrestrial referene frame defined in Chapter 4 of the present VLBI Conventions in aordane to the transformations introdued in Chapter 5. (a) Gravitational Delay 23 The general relativisti delay, T grav, is given for the J th gravitating body by T gravj = 2 GM J 3 ln R 1J + K R 1J R 2J + K R 2J. (1) At the pioseond level it is possible to simplify the delay due to the Earth, T grav, whih beomes T grav = 2 GM 3 ln x 1 + K x 1 x 2 + K x 2. (2) The Sun, the Earth and Jupiter must be inluded, as well as the other planets in the solar system along with the Earth s Moon, for whih the maximum delay hange is several pioseonds. The major satellites of Jupiter, Saturn and Neptune should also be inluded if the ray path passes lose to them. This is very unlikely in normal geodeti observing but may our during planetary oultations. that in ase of observations very lose to some massive bodies, extra terms (e.g. due to the multipole moments and spin of the bodies) should be taken into aount to obtain an unertainty of 1 ps (see Klioner, 1991). The effet on the bending delay of the motion of the gravitating body during the time of propagation along the ray path is small for the Sun but an be several hundred pioseonds for Jupiter (see Sovers and Fanselow (1987) page 9). Sine this simple orretion, suggested by Sovers and Fanselow (1987) and Hellings (1986) among others, is suffiient at the pioseond level, it was adapted for the onsensus model. It is also neessary to aount for the motion of station 2 during the propagation time between station 1 and station 2. In this model R ij, the vetor from the J th gravitating body to the i th reeiver, is iterated one, giving t 1J = min [ t 1, t 1 ˆK ( X J (t 1 ) X 1 (t 1 )) ], (3) so that and R 1J (t 1 ) = X 1 (t 1 ) X J (t 1J ), (4) R 2J = X 2 (t 1 ) V ( ˆK b) X J (t 1J ). (5) Only this one iteration is needed to obtain pioseond level auray for solar system objets. X 1 (t 1 ) is not tabulated, but an be inferred from X (t 1 ) using X i (t 1 ) = X (t 1 ) + x i (t 1 ), (6) whih is of suffiient auray for use in equations 3, 4, and 5, when substituted into equation 1 but not for use in omputing the geometri 23 The formulas in this setion are unhanged from the previous edition of the Conventions. The more advaned theory in Kopeikin and Shäfer (1999) provides a rigorous physial solution for the light propagation in the field of moving bodies. For Earth-based VLBI, the formulas in this setion and those proposed in Kopeikin and Shäfer (1999) are numerially equivalent with an unertainty of 0.1 ps (Klioner and Soffel, 2001). 112

5 11.1 VLBI Time Delay Tehnial No. 32 delay. The total gravitational delay is the sum over all gravitating bodies inluding the Earth, T grav = J T gravj. (7) t v2 t v1 = (b) Geometri Delay In the baryentri frame the vauum delay equation is, to a suffiient level of approximation: T 2 T 1 = 1 ˆK ( X 2 (T 2 ) X 1 (T 1 )) + T grav. (8) This equation is onverted into a geoentri delay equation using known quantities by performing the relativisti transformations relating the baryentri vetors X i to the orresponding geoentri vetors x i, thus onverting equation 8 into an equation in terms of x i. The related transformation between baryentri and geoentri time an be used to derive another equation relating T 2 T 1 and t 2 t 1, and these two equations an then be solved for the geoentri delay in terms of the geoentri baseline vetor b. In the rational polynomial form the total geoentri vauum delay is given by T grav ˆK b [ ] 1 (1+γ)U V 2 V 2 2 w ˆK ( V + w 2) V b 2 (1 + ˆK V /2). (9) Given this expression for the vauum delay, the total delay is found to be ˆK ( w2 w 1 ) t 2 t 1 = t v2 t v1 + (δt atm2 δt atm1 ) + δt atm1. (10) For onveniene the total delay an be divided into separate geometri and propagation delays. The geometri delay is given by ˆK ( w2 w 1 ) t g2 t g1 = t v2 t v1 + δt atm1, (11) and the total delay an be found at some later time by adding the propagation delay: t 2 t 1 = t g2 t g1 + (δt atm2 δt atm1 ). (12) The tropospheri propagation delay in equations 11 and 12 need not be from the same model. The estimate in equation 12 should be as aurate as possible, while the δt atm model in equation 11 need only be aurate to about an air mass ( 10 nanoseonds). If equation 10 is used instead, the model should be as aurate as is possible. that the tropospheri delay is omputed in the rest frame of eah station and an be diretly added to the geoentri delay (equation 11), at the unertainty level onsidered here (see Eubanks, 1991; Treuhaft and Thomas, 1991). If δ b is the differene between the a priori baseline vetor and the true baseline, the true delay may be omputed from the a priori delay as follows. If δ b is less than roughly three meters, then it suffies to add ( ˆK δ b)/ to the a priori delay. If this is not the ase, however, the a priori delay must be modified by adding (t g2 t g1 ) = ˆK δ b 1 + ˆK ( V + w 2) V δ b 2. (13) 113

6 No. 32 Tehnial 11 General Relativisti Models for Propagation () Observations Close to the Sun For observations made very lose to the Sun, higher order relativisti time delay effets beome inreasingly important. The largest orretion is due to the hange in delay aused by the bending of the ray path by the gravitating body desribed in Rihter and Matzner (1983) and Hellings (1986). The hange to T grav is δt gravi = 4G2 M 2 i 5 b ( ˆN1i + ˆK) ( R 1i + R, (14) 1i ˆK) 2 whih should be added to the T grav in equation 1. (d) Summary Assuming that the referene time is the UTC time arrival of the VLBI signal at reeiver 1, and that it is transformed to the appropriate timesale to be used to ompute eah element of the geometri model, the following steps are reommended to ompute the VLBI time delay. 1. Use equation 6 to estimate the baryentri station vetor for reeiver Use equations 3, 4, and 5 to estimate the vetors from the Sun, the Moon, and eah planet exept the Earth to reeiver Use equation 1 to estimate the differential gravitational delay for eah of those bodies. 4. Use equation 2 to find the differential gravitational delay due to the Earth. 5. Sum to find the total differential gravitational delay. 6. Compute the vauum delay from equation Calulate the aberrated soure vetor for use in the alulation of the tropospheri propagation delay: ki = ˆK + V + w i ˆK ˆK ( V + w i ). (15) 8. Add the geometri part of the tropospheri propagation delay to the vauum delay, equation The total delay an be found by adding the best estimate of the tropospheri propagation delay t 2 t 1 = t g2 t g1 + [δt atm2 (t 1 ˆK b, k 2 ) δt atm1 ( k 1 )]. (16) 10. If neessary, apply equation 13 to orret for post-model hanges in the baseline by adding equation 13 to the total time delay from equation step Laser Ranging In a referene system entered on an ensemble of masses, if a light signal is emitted from x 1 at oordinate time t 1 and is reeived at x 2 at oordinate time t 2, the oordinate time of propagation is given by t 2 t 1 = x 2(t 2 ) x 1 (t 1 ) + J 2GM J 3 ln ( ) rj1 + r J2 + ρ, (17) r J1 + r J2 ρ where the sum is arried out over all bodies J with mass M J entered at x J and where r J1 = x 1 x J, r J2 = x 2 x J and ρ = x 2 x

7 11.2 Laser Ranging Tehnial No. 32 For near-earth satellites (SLR), pratial analysis is done in the geoentri frame of referene, and the only body to be onsidered is the Earth (Ries et al., 1988). For lunar laser ranging (LLR), whih is formulated in the solar system baryentri referene frame, the Sun and the Earth must be taken into aount, with the ontribution of the Moon being of order 1 ps (i.e. about 1 mm for a return trip). Moreover, in the analysis of LLR data, the body-entered oordinates of an Earth station and a lunar refletor should be transformed into baryentri oordinates. The transformation of r, a geoentri position vetor expressed in the GCRS, to r b, the vetor expressed in the BCRS, is provided with an unertainty lower than 1 mm by the equation r b = r (1 U ) ( ) V r V, (18) 2 where U is the gravitational potential at the geoenter (exluding the Earth s mass) and V is the baryentri veloity of the Earth. A similar equation applies to the selenoentri refletor oordinates. In general, however, the geoentri and baryentri systems are hosen so that the geoentri spae oordinates are onsistent with TT (position vetor r T T ) and that the baryentri spae oordinates are onsistent with TDB (position vetor r T DB ). The transformation of r T T to r T DB, is then given by r T DB = r T T (1 U ) ( ) 2 L C 1 V rt T V 2 2, (19) where L C is given in Table 1.1. Referenes Eubanks, T. M. (ed.), 1991, Proeedings of the U. S. Naval Observatory Workshop on Relativisti Models for Use in Spae Geodesy, U. S. Naval Observatory, Washington, D. C. Hellings, R. W., 1986, Relativisti effets in Astronomial Timing Measurements, Astron. J., 91, pp Erratum, ibid., p Klioner, S. A., 1991, General relativisti model of VLBI observables, in Pro. AGU Chapman Conf. on Geodeti VLBI: Monitoring Global Change, Carter, W. E. (ed.), NOAA Tehnial Report NOS 137 NGS 49, Amerian Geophysial Union, Washington D.C, pp Klioner, S. A. and Soffel, M. H., 2001, personnal ommuniation. Kopeikin, S., 1990, Theory of relativity in observational radio astronomy, Sov. Astron., 34(1), pp Kopeikin, S. and Shäfer, G., 1999, Lorentz-ovariant Theory of Light Propagation in Time Dependent Gravitational Fields of Arbitrary Moving Bodies, Phys. Rev. D, 60, pp /1 44. MCarthy, D. D. (ed.), 1992, Standards, Tehnial, 13, Observatoire de Paris, Paris. MCarthy, D. D. (ed.), 1996, Conventions, Tehnial 21, Observatoire de Paris, Paris. Petit, G., 2000, Importane of a ommon framework for the realization of spae-time referene systems, Pro. IAG Symposium IGGOS, Springer-Verlag, pp

8 No. 32 Tehnial 11 General Relativisti Models for Propagation Rihter, G. W. and Matzner, R. A., 1983, Seond-order Contributions to Relativisti Time Delay in the Parameterized Post-Newtonian Formalism, Phys. Rev. D, 28, pp Ries, J. C., Huang, C., and Watkins, M. M., 1988, The Effet of General Relativity on Near-Earth Satellites in the Solar System Baryentri and Geoentri Referene Frames, Phys. Rev. Lett., 61, pp Shahid-Saless, B., Hellings, R. W., and Ashby, N., 1991, A Pioseond Auray Relativisti VLBI Model via Fermi Normal Coordinates, Geophys. Res. Lett., 18, pp Soffel, M. H., Muller, J., Wu, X., and Xu, C., 1991, Consistent Relativisti VLBI Theory with Pioseond Auray, Astron. J., 101, pp Sovers, O. J. and Fanselow, J. L., 1987, Observation Model and Parameter Partials for the JPL VLBI Parameter Estimation Software Masterfit -1987, JPL Pub , Rev. 3. Treuhaft R. N. and Thomas J. B., 1991, Inorporating atmospheri delay into the relativisti VLBI time delay, JPL Tehnial Memorandum IOM Zhu, S. Y. and Groten, E., 1988, Relativisti Effets in the VLBI Time Delay Measurement, Man. Geod., 13, pp