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%  Authors: L. Petrov and S. Kopejkin ( pet@leo.gsfc.nasa.gov and 
%                                       kopeikins@missouri.edu )
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%  Revision of 23-FEB-2001 13:33:59
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\noindent{\large CHAPTER 12}{\bf\  GENERAL RELATIVISTIC MODELS FOR
PROPAGATION}
\bigskip

\write 0 {\indent VLBI Time Delay \noexpand\dotfill\the\pageno\noexpand
\break}
\noindent{\bf VLBI Time Delay}

\nobreak

  Modeling VLBI time delay follows the first post-Minkowskian approximation of
  the equations of gravitation field developed by Kopeikin and Schaefer [1].

  The formalism of description of light propagation includes three main 
  constituents: (1) equations of the gravitational field, (2) equations of 
  light propagation, and (3) initial and/or boundary conditions. The equations 
  of gravitational field are formulated in harmonic coordinate system with 
  arbitrary origin in space. The equations below are obtained in the linear 
  approximation with respect to the universal gravitational constant $G$ 
  without expansion in the velocities of bodies, so-called first 
  post-Minkowskian approximation. The metric tensor of the gravitational field 
  is found immediately from the energy-momentum tensor of the source of 
  gravitational field described by equations of Einstein [2]. The metric tensor
  is a function of the retarded time argument $t'$ related to the running time 
  $t$ and to the coordinates $\vec{x}$ of photon, as well as the retarded 
  coordinates $\vec{x}_a(t')$ of the gravitating body by the light-cone 
  equation $ t' = t- {1\over c} | \vec{x} - \vec{x}_a(t') | $. 

  The relativistic model of light propagation assumes that the solar system
  consists of spherically symmetrical bodies: Sun, Mercury, Venus, Earth, Moon,
  Mars, Jupiter, Saturn, Uranus and Neptune whose motion is described by the
  numerical ephemerides DE403. The source is assumed to be a point-like and 
  a radio signal propagates in the medium which may have a refractivity 
  different from 1.

  The VLBI delay is modeled as the difference of two intervals: 
1) the interval of proper time measured by the clock of the second station 
between the events: arrival of the radio signal from the source to the 
reference point of the second station and clock synchronization; 
2) the interval of proper time measured by the clock of the first station 
between the events: arrival of the radio signal from the source to the 
reference point of the first station and clock synchronization.

Table 12.1.  Notation used in the formulas
\smallskip
\hrule

\tabitem{ $T_i$ } the BRS time of arrival of a radio signal at the reference 
                  point of the $i^{th}$ antenna. 

\tabitem{ $t_i$ } the GRS time of arrival of a radio signal at the reference 
                  point of the $i^{th}$ antenna. 

\tabitem{ $t_{ip}$ } the interval of proper time measured by the clock of the
                     $i^{th}$ VLBI station between the event: clock 
                     synchronization and arrival of a radio signal to the
                     reference point of the $i^{th}$ antenna. 

\tabitem{ $T'_{ia}$ } retarded BRS time of arrival of a radio signal at the 
                      reference point of the $i^{th}$ antenna with respect 
                      to the $a^{th}$ body.

\tabitem{ $T^{atm}_i$ } atmosphere path delay between the source and the 
                   reference point of the $i^{th}$ antenna. It includes 
                   contribution of ionospheric, neutral hydrostatic and 
                   non-hydrostatic constituents.

\tabitem{ $\Delta T_{grav}$ } the differential gravitational time delay.

\tabitem{ $\Delta T_{atm}$ } the differential atmosphere time delay.

\tabitem{ $\vec{X}_i(T_i)$ } the BRS vector from the geocenter to the reference
                             point of the $i^{th}$ antenna at the BRS time 
                             $T_i$.

\tabitem{ $\vec{x}_i(t_i)$ } the GRS vector from the geocenter to the reference
                             point of the $i^{th}$ antenna at the GRS time 
                             $t_i$.

\tabitem{ $\vec{b}$ } $\vec{x}_2(t_1) - \vec{x}_1(t_1)$ the GRS baseline vector
                      at the GRS time $t_1$ of arrival a radio signal to the 
                      reference point of the first antenna.

\tabitem{ $\vec{w}_i$ } the GRS velocity of the reference point of the 
                            $i^{th}$ antenna.

\tabitem{ $\vec{S}$ } the unit BRS vector from the barycenter to the source.

\tabitem{ $\rho$ } the annual parallax of the source.

\tabitem{ $\vec{X}_\oplus$ } the BRS vector from the barycenter to the 
                             geocenter.

\tabitem{ $\vec{V}_\oplus$ } the BRS velocity vector of the geocenter.

\tabitem{ $\vec{X}_a$ } the BRS vector from the barycenter to the center of 
                        mass of the $a^{th}$ gravitating body.

\tabitem{ $\vec{V}_a$ } the BRS velocity vector of the $a^{th}$ gravitating 
                        body.

\tabitem{ $\vec{R}_{ia}$ } the BRS vector from the $a^{th}$ gravitating body 
                           taken at the retarded moment of the BRS time 
                           $T'_{ia}$ to the reference point of the $i^{th}$ 
                           antenna taken at the moment $T_1$.

\tabitem{ $M_a$ } the mass of the $a^{th}$ gravitating body.

\tabitem{ $M_\oplus$ } the mass of the Earth.

\tabitem{ $M_\odot$ } the mass of the Sun.

\tabitem{ $c$ } the speed of light.

\tabitem{ $G$ } the universal gravitational constant.

\tabitem{ $a_\oplus$ } the equatorial radius of the Earth.

\tabitem{ $ \bar{D}_\oplus $ } the mean distance from the geocenter to the 
                               barycenter (astronomical unit).

\line{\dotfill}


  It is assumed that the BRS time and spatial coordinates are consistent with 
the BRS metric tensor defined in the Appendix B1.3.2 and the GRS time and 
spatial coordinates are consistent with the GRS metric tensor defined in the 
Appendix B1.3.3. 

  The expression for the VLBI time delay is obtained in three steps: first
the difference of barycentric time coordinates is computed. 

  Omitting the terms with contribution less than 1 psec, the difference of 
barycentric time coordinates between the events of arrival of radio signal 
from the point-like source located at the distance \hbox{$> 1$ parsec} from the 
Earth at the first antenna and at the second antenna, assuming the light is 
propagating through the medium which causes small additional delay, is 
expressed according to [1] as:

\par\vskip -1ex\par
$$
  T_{2} + T^{atm}_2 - T_{1} - T^{atm}_1 = -{1\over c} \vec{S}\cdot 
      \biggl( \, \vec{X}_2(T_2+T^{atm}_2)-\vec{X}_1(T_1+T^{atm}_1) \, \biggr) +
      \Delta t_{grav} \eqno(1)
$$

\par\vskip -3ex\par
$$
\Delta t_{grav}  = 
   2\sum_{a=1}^N {G M_a\over c^3} \bigl( 
                                  1 + \vec{S}\cdot \vec{V}_a(T'_{1a}) \bigr) \;
           \ln { {|\vec{R}_{1a}| + \vec{S} \cdot \vec{R}_{1a} }\over
                 {|\vec{R}_{2a}| + \vec{S} \cdot \vec{R}_{2a} }     } \eqno(2)
$$

where 
$ 
\vec{R}_{1a} = \vec{X}_{1}(T_{1}) - \vec{X}_{a}(T'_{1a}) \;, \quad\quad 
\vec{R}_{2a} = \vec{X}_{2}(T_{1}) - \vec{X}_{a}(T'_{1a})\;.
$

 $\vec{X}_a$ is a position of the $a^{th}$ gravitating body taken at a retarded 
time $T'_a$ which is a solution of the gravitational null-cone equation:
   
\par\vskip -1ex\par
$$   T'_{1a} = T_1 - {1\over c} \bigl| X(T_1) - X_a(T'_{1a}) \bigr|  \eqno(3)$$
\par\vskip -1ex\par

  Neglecting terms of $O \bigl( {\dot{X}_a^2 \over c^2} \bigr)$ and 
$ O \bigl( { \vec{X}_a \cdot \vec{\ddot{X}}_a \over c^2} \bigr) $, the retarded 
moment of time $ T'_{1a} $ can be computed precisely enough to provide the
accuracy of computation of the gravitation delay better than 1 psec with 
using the following expression:

\par\vskip -1ex\par
$$   
     T'_{1a} = T_1 - {1\over c} \, { \; | X(T_1) - X_a(T_1)| \; \over 
                                     1 - { \vec{V}_a(T_1) \over c } 
                                     { \vec{X}(T_1) \over | \vec{X}(T_1) | } }
     \eqno(4)
$$
\par\vskip -1ex\par

  At the second step the time difference and the vectors of station 
coordinates are transformed from the barycentric reference frame to the 
geocentric reference frame by using the expression listed in the 
Appendix~B1.3.4. At the third step the time difference of geocentric time 
coordinates is transformed to the differences of the intervals of proper time 
by integrating the expression for time transformation in the Appendix~B1.3. 
The final expression for the VLBI time delay defined above for a source located 
at the distance greater than 1 parsec is as follows:

\par\vskip -1ex\par
$$
  \eqalign{ t_{1p} - t_{2p} = &
%
            { 1 \over
                   {\ds 1 + { \vec{S} \cdot (\vec{V}_\oplus + \vec{w}_2 ) 
                              \over \ds c } 
                   } 
            } \;
%
            \Biggl( - 
                     { \ds \vec{S} \cdot \vec{b} \over {\ds c} } \;
                     \biggl[1 - 
                        2{ G M_{\odot}  \over | \vec{X}_{\oplus} | c^2 } -
                         { G M_{\oplus} \over | a_{\oplus} | c^2 } -
                         { { |\vec{V}_\oplus|^2} \over {2c^2} } - 
                         { {\vec{V}_\oplus \cdot \vec{w}_2} \over {c^2} }
                     \biggr] - \cr
%
            & { \ds \vec{V}_\oplus \cdot \vec{b}  \over \ds c^2 }
                \bigl( 1 + { \vec{S} \cdot \vec{V}_\oplus \over 2c } \bigr) +
                { \ds {1 \over c} \, \rho \, \vec{b} \, } 
                { \vec{R}_\oplus \over \bar{D}_\oplus } 
                + \Delta T_{grav} + \Delta T_{atm} 
            \Biggr) 
         }
         \eqno(5)
$$
\par\vskip -1ex\par

The differences between the current expression for time delay and the 
expression in the IERS Conventions 1996 are:

\item{---} the IERS Conventions 1996 has the coefficient $-2$ in front of the
           term $ G M_{\oplus}/| a_{\oplus} | c^2 $, while the current
           formula (5) has the coefficient $-1$. This error, which can be as 
           large as 28 psec, results in scaling the estimates of positions 
           of the VLBI stations obtained from the analysis with using that 
           formula by the factor of $ 1 + 7 \cdot 10^{-10}$.

\item{---} IERS Convention 1996 proposed to use positions of the bodies for
           computation of the gravitation delay at the moment of time when the 
           distance between the center of mass of the body and the photon is 
           minimum while the current expression requires to use the retarded 
           moment of time. This difference can result in the diffrence in VLBI 
           delay greater than 1 psec at the observations closer than 30 arcmin  
           from the Jupiter.

\item{---} IERS Conventions 1996 did not multiply atmosphere delay by the
           factor of $ { 1 \over {1 + { \vec{S} \cdot (\vec{V}_\oplus + 
           \vec{w}_2 ) \over c } } } $. This effect can exceed 10 psec.

\item{---} The last term in formula 10 in the IERS Conventions 1996 is 
           unnecessary since it cannot exceed 0.7 psec.


\noindent{\bf References}

\nobreak

\item{}\kern-2pc
  Kopeikin, S., Schaefer, G. 1999, Physical Review, D, vol. 60, 124002, 1999.

\item{}\kern-2pc 
  Landau, L.D. and Lifshitz, E.M., {\it The classical Theory of Fields}
     (Pergamon, Oxford, 1971).

\item{}\kern-2pc
  Eubanks, T. M., ed, 1991, {\it Proceedings of the U. S. Naval
     Observatory Workshop on Relativistic Models for Use in Space
     Geodesy}, U. S. Naval Observatory, Washington, D. C.

\end
