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\write 0 {11. GENERAL RELATIVISTIC MODELS FOR TIME, COORDINATES
AND EQUATIONS OF MOTION \noexpand\dotfill\the\pageno\noexpand\break}
\hangindent=7.8pc
\hangafter=1
\noindent{\large CHAPTER 11} {\bf GENERAL RELATIVISTIC MODELS FOR TIME,
COORDINATES AND EQUATIONS OF MOTION}
\bigskip
The relativistic treatment of the near-Earth satellite orbit
determination problem includes corrections to the equations of
motion, the time transformations, and the measurement model. The
two coordinate systems generally used when including relativity
in near-Earth orbit determination solutions are the solar system
barycentric frame of reference and the geocentric or Earth-centered
frame of reference.
Ashby and Bertotti (1986) constructed a locally inertial E-frame
in the neighborhood of the gravitating Earth and demonstrated that the
gravitational effects of the Sun, Moon, and
other planets are basically reduced to their tidal forces, with
very small relativistic corrections. Thus the main relativistic
effects on a near-Earth satellite are those described by the
Schwarzschild field of the Earth itself. This result makes the
geocentric frame more suitable for describing the motion of a
near-Earth satellite (Ries {\it et al.}, 1988).
The time coordinate in the inertial E-frame is Terrestrial
Time (designated TT) (Guinot, 1991) which can be considered to be
equivalent to the previously defined Terrestrial Dynamical Time
(TDT). This time coordinate (TT) is realized in practice by
International Atomic Time (TAI), whose rate is defined by the
atomic second in the International System of Units (SI). Terrestrial
Time adopted by the International Astronomical Union in
1991 differs from Geocentric Coordinate Time (TCG) by a scaling
factor:
$${\rm TCG-TT}={\rm L_{G}}\times({\rm MJD}-43144.0)\times
86400 \ {\rm seconds},$$
where MJD refers to the modified Julian date. Figure 11.1 shows
graphically the relationships between the time scales. See
{\it IERS Technical Note 13}, pages 137--142 for copies of the
IAU Recommendations relating to these time scales.
\write 0 {\indent Equations of Motion for an Artificial Earth Satellite
\noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\bf Equations of Motion for an Artificial Earth Satellite}
\nobreak
The correction to the acceleration of an artificial Earth
satellite $\Delta\vec{a}$ is
$$\Delta\vec{a}={GM_\oplus\over c^2r^3}\biggl\{[2(\beta+\gamma)
{GM_\oplus\over r}-\gamma v^2]\vec{r}+
2(1+\gamma)(\vec{r}\cdot\vec{v})\vec{v}\biggr\},\eqno(1)$$
\halign{#\hfil\kern1pc&\hfil#\kern.2pc&#\hfil\cr
where\cr
& $c$& = speed of light,\cr
&$\beta,\gamma$& = PPN parameters equal to 1 in General Relativity,\cr
&$\vec{r},\vec{v},\vec{a}$& = geocentric satellite position, velocity,
and acceleration, respectively,\cr
&$GM_\oplus$& = geocentric constant of gravitation.\cr}
The effects of Lense-Thirring precession (frame-dragging),
geodesic (de Sitter) precession, and the relativistic effects of
the Earth's oblateness have been neglected.
\write 0 {\indent Equations of Motion in the Barycentric Frame
\noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\bf Equations of Motion in the Barycentric Frame}
\nobreak
The n-body equations of motion for the solar system frame of
reference (the isotropic Parameterized Post-Newtonian system with
Barycentric Coordinate Time (TCB) as the time coordinate) are
required to describe the dynamics of the solar system and artificial
probes moving about the solar system (for example, see
Moyer, 1971). These are the equations applied to the Moon's
motion for Lunar Laser Ranging (Newhall {\it et al.}, 1987). In
addition, relativistic corrections to the laser range
measurement, the data timing, and the station coordinates are
required (see Chapter 12).
\write 0 {\indent Scale Effect and Choice of Time Coordinate
\noexpand\dotfill\the\pageno\noexpand\break}
\noindent{\bf Scale Effect and Choice of Time Coordinate}
\nobreak
A previous IAU definition of the time coordinate in the
barycentric frame required that only periodic differences exist
between Barycentric Dynamical Time (TDB) and Terrestrial Dynamical
Time (TDT) (Kaplan, 1981). As a consequence, the spatial
coordinates in the barycentric frame had to be rescaled to keep
the speed of light unchanged between the barycentric and the
geocentric frames (Misner, 1982; Hellings, 1986). Thus, when
barycentric (or TDB) units of length were compared to geocentric
(or TDT) units of length, a scale difference, L, appeared. This
is no longer required with the use of the TCG time scale.
The difference between TCB and TDB is given in seconds by
$${\rm TCB-TDB}={\rm L_{B}}\times({\rm MJD}-43144.0)\times86400.$$
The difference between Barycentric Coordinate Time (TCB) and
Geocentric Coordinate Time (TCG) involves a four-dimensional
transformation,
$${\rm TCB-TCG}=c^{-2}\biggl\{\int_{t_0}^t[{v_e^2\over
2}+U_{ext}(\vec{x}_e)]dt+\vec{v}_e\cdot(\vec{x}-\vec{x}_e)\biggr\},$$
where $\vec{x}_e$ and $\vec{v}_e$ denote the barycentric position and
velocity of the Earth's center of mass and $\vec{x}$ is the barycentric
position of the observer. $U_{ext}$ is the Newtonian potential of all
of the solar system bodies apart from the Earth evaluated at the
geocenter. $t_0$ is chosen to be consistent with 1977 January 1,
$0^{\rm h} 0^{\rm m} 0^{\rm s}$ TAI and $t$ is TCB. An approximation
is given in seconds by
$$({\rm TCB-TCG})={\rm L_{C}}\times({\rm MJD}-43144.0)\times86400
+c^{-2}\vec{v_e}\cdot(\vec{x}-\vec{x}_e)+P.$$
with MJD measured in TAI. Table 4.1 lists the values of the rates
L$_{\rm B}$, L$_{\rm C}$, and L$_{\rm G}$. Periodic differences
denoted by $P$ have a maximum amplitude of around 1.6 ms. These can
be evaluated by the ``FBL" model of Fairhead, Bretagnon and Lestrade
(1995). A comparison with the numerical time ephemeris TE245
(Fukushima, 1995) revealed that this series provides the smallest
deviation from TE245 for the years 1980--1999 after removing a linear
trend. Users may expect the FBL model to provide the periodic part
in TCB-TT within a few ns for a few decades around the present time.
This is sufficient for all precision measurement including timing
observations of millisecond pulsars.
Software to implement the FBL model is available by anonymous
ftp to maia.usno.navy.mil and is located in the directory
/conventions/chapter11. The files of interest are called fbl.f
and fbl.results. This model is based on earlier works (Fairhead
{\it et al.}, 1988; Fairhead and Bretagnon, 1990).
\vfil\eject
\halign{\kern2.5pc\hfil#\hfil\kern3pc&\hfil#\hfil\cr
TDT&TT\cr
Terrestrial Dynamical Time & Terrestrial Time \cr
\vrule width1pt height1pc & TDT $\equiv$ TT $\simeq$ TAI + 32$^{\rm s}$
\kern-.6pc .184\cr
\vrule width1pt height1cm & \vrule width1pt height1cm\cr
\vrule width1pt height1pc & TCG\cr
\vrule width1pt height1pc & Geocentric Coordinate Time\cr
\vrule width1pt height1pc & TCG - TT $={\rm L_{G}}\times \Delta T$\cr
\kern7.1pc\vrule width1pt height1cm \vrule width3cm height1pt
& \vrule width3cm height1pt \vrule width1pt height1cm \kern7.1pc\cr
\kern14.2pc\vrule width1pt height1cm &
\vrule width1pt height1cm \kern14.2pc\cr}
\centerline{4-dimensional}
\centerline{space-time}
\centerline{transformation}
\halign{\kern2.5pc\hfil#\hfil\kern3pc&\hfil#\hfil\cr
\kern7.1pc\vrule width3cm height1pt \vrule width1pt height1cm &
\vrule width1pt height1cm \vrule width3cm height1pt \kern7.1pc\cr
\vrule width1pt height1cm & \vrule width1pt height1cm\cr
\vrule width1pt height1pc & Linear transformation\cr
\vrule width1pt height1pc & L$_{\rm C}\times\Delta T$\cr
\vrule width1pt height1cm & \vrule width1pt height1cm\cr
TDB&TCB\cr
Barycentric Dynamical Time & Barycentric Coordinate Time\cr}
\centerline{TCB = TDB $+ {\rm L_{B}}\times\Delta T$}
\centerline{$\Delta$T = (date in days - 1977 January 1, 0
$^{\rm h}$)TAI$\times$ 86400 sec}
\hrule
\noindent Fig 11.1 Relations between time scales.
\noindent{\bf References}
\nobreak
\item{}\kern-2pc
Ashby, N. and Bertotti, B., 1986, ``Relativistic Effects in Local
Inertial Frames," {\it Phys. Rev. D}, {\bf 34} (8), p. 2246.
\item{}\kern-2pc
Fairhead, L. and Bretagnon, P., 1990, ``An Analytic Formula for the
Time Transformation TB-TT," {\it Astron. Astrophys.},
{\bf 229}, pp. 240--247.
\item{}\kern-2pc
Fairhead, L., Bretagnon, P., and Lestrade, J.-F., 1988, ``The
Time Transformation TDB-TDT: An Analytical Formula and Related
Problem of Convention," in {\it The Earth's Rotation and
Reference Frames for Geodesy and Geophysics}, A. K. Babcock
and G. A. Wilkins (eds.), Kluwer Academic Publishers,
Dordrecht, pp. 419--426.
\item{}\kern-2pc
Fairhead, L., Bretagnon, P., and Lestrade, J.-F., 1995, Private
Communication.
\item{}\kern-2pc
Fukushima, T., 1995, ``Time Ephemeris," {\it Astron. Astrophys.},
{\bf 294}, pp. 895--906.
\item{}\kern-2pc
Guinot, B., 1991, ``Report of the Sub-group on Time," in Reference
Systems, J. A. Hughes, C. A. Smith, and G. H. Kaplan (eds),
U. S. Naval Observatory, Washington, D. C., pp. 3--16.
\item{}\kern-2pc
Hellings, R. W., 1986, ``Relativistic Effects in Astronomical
Timing Measurement," {\it Astron. J.}, {\bf 91} (3), pp. 650--659.
Erratum, {\it ibid}., p. 1446.
\item{}\kern-2pc
Kaplan, G. H., 1981, {\it The IAU Resolutions on Astronomical Constants,
Time Scale and the Fundamental Reference Frame}, U. S. Naval
Observatory Circular No. 163.
\item{}\kern-2pc
Misner, C. W., 1982, {\it Scale Factors for Relativistic Ephemeris
Coordinates}, NASA Contract NAS5-25885, Report, EG\&G, Washington
Analytical Services Center, Inc.
\item{}\kern-2pc
Moyer, T. D., 1971, {\it Mathematical Formulation of the
Double-precision Orbit Determination Program}, JPL Technical
Report 32-1527.
\item{}\kern-2pc
Newhall, X X, Williams, J. G., and Dickey, J. O., 1987, ``Relativity
Modeling in Lunar Laser Ranging Data Analysis," in {\it Proceedings
of the International Association of Geodesy (IAG)
Symposia}, Vancouver, pp. 78--82.
\item{}\kern-2pc
Ries, J. C., Huang, C., and Watkins, M. M., 1988, ``Effect of
General Relativity on a Near-Earth Satellite in the Geocentric
and Barycentric Reference Frames," {\it Phys. Rev. Let.}, {\bf 61},
pp. 903--906.
\vfil\eject