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## The Tuned Inter-Range Vector (TIRV) Force Model and Reference System

This note was written in 1995 by Andrew Sinclair and has been finalized in consultation with Richard Eanes, Brion Conklin and Rolf Koenig. Use of TIRVs in the ILRS was phased out in 2006 and replaced with the predictions in the Consolidated Prediction Format (CPF).

Editor's note: Some of the information below might be out-dated.

The TIRV format, coordinate system and force model were adopted by CSR, University of Texas in about 1982 as a means of providing predictions for Lageos, following a system previously in use by NASA. A description is given in a paper “Lageos Ephemeris Predictions” by B.E. Schutz, B.D. Tapley, R.J. Eanes and B. Cuthbertson in the proceedings of the Laser Ranging Workshop, Austin, 1981. Even though the force model in particular is now somewhat dated the system has been retained unchanged, as it serves its purpose perfectly well, and the CSR software (IRVINT) and software written by the RGO that follows the same force model and reference system are in wide use through the SLR network, either used directly or as the basis of other software packages. The TIRV system has now been adopted for all satellites, but it is not clear where, if at all, the parameters for these other satellites have been documented. This note is intended to clarify the present situation, and provide a starting point for discussion of updating the system, should this be considered necessary.

**Force Model**

GEM10N *(normalized)* gravity field. Degree and order 7
for LAGEOS, ETALON, GPS, GLONASS. Degree and order 18 for other satellites.

GM = 0.39860044 * 10^{15}m^{3}s^{-2} for all satellites

a_{e}= 6378145 m.

Lunar and Solar perturbations are computed from simple precessing-ellipse orbit models. No drag or solar radiation pressure forces are included.

**Note**. An earlier suggestion that a slightly different value of GM should be used
for the CSR TIRVs for the ETALON satellites is incorrect.

**Reference System**

The TIRVs (inter range vectors) are referred to a pseudo body fixed reference frame, the true equator and Greenwich meridian of 0h of the day for which they apply. They are not referred to the CIO equator, which differs from the true equator by the polar motion transformation.

The action of the integrator used to generate satellite positions from the IRVs can be regarded as a black box by most users, but, in outline, what it does is the following :

- Convert the IRV velocity to an inertial frame:

V_{x}= V_{x}- ωy

V_{y}= V_{y}+ ωx

where ω = ω_{0}+ IRATE x 10^{-14}

ω_{0}= 7.2921151463 x 10^{-5}radians/sec

and IRATE is given with the IRVs. - Calculate θ
_{0}= GMST at 0^{h}UTC of the day required, ignoring the correction from UTC to UT1. Rotate IRV position and velocity through - θ_{0}about the z-axis to refer them to the mean equinox of 0^{h}. - Integrate numerically the force equations in the near-inertial frame of the true equator
of date and mean equinox of 0
^{h}of the day. At a general point in this integration at time T seconds from 0^{h}the Earth’s position relative to the integration frame is obtained by a positive rotation through the angle θ_{0}+ ωT. This is needed in order to calculate the force due to the gravity field, which is then rotated back through the same angle, to be used in the numerical integration. - The integration gives the positions at steps through the day of the satellite relative
to the true equator of date and mean equinox of 0
^{h}. The position at a general time T seconds from 0^{h}is converted to the true equator of date and Greenwich meridian by a positive rotation through the angle θ_{0}+ ωT, and these are the positions delivered to the user.

At this stage the user could apply polar motion if desired to refer the positions to the CIO equator. It may also be desirable to make a correction to the nominal “Greenwich meridian” to which these positions are referred. When forming the IRVs the prediction centre has to adopt predicted values of UT1-UTC. For the Lageos and Etalon satellites the predictions are generated for a year or more ahead, and over these periods the predicted values of UT1-UTC can be significantly in error, and this will affect the nominal Greenwich meridian to which the satellite positions are referred. The University of Texas supplies with their IRVs for the Lageos and Etalon satellites a file of the values of UT1-UTC used, and so the user can correct the reference frame of the final satellite positions if the discrepancy of the predicted UT1-UTC is significant. This effect is unlikely to be significant for lower satellites, for which the predictions are generated for shorter time periods.

**Comments**

There are some approximations introduced by the integrator program. In stage 2 the correction UT1-UTC is neglected in the rotation to the equinox. This has just a very small effect on the calculation of the lunar and solar perturbations. It does not affect the calculation of the gravity field force, as the same rotation is reversed before this is calculated. In stage 3 the gravity field force is calculated relative to the true equator, whereas the actual orientation of the Earth is described by the CIO equator. However the tuning program makes the same approximation, and thus the IRVs have been tuned to accommodate this approximation.

For several years the IRVs issued by ATSC (formerly Bendix) have been referred to the CIO equator instead of the true equator. The effect of using these with the Texas or RGO software was to cause an along-track run-off of the orbit over each day, which for example amounted to about 180 metres for Topex. From about November 1994 the ATSC IRVs have been changed to refer to the true equator, and so this problem has been removed.

The original version of the software package PC TIVAS announced by ATSC in CDDIS Bulletin August 1994 requires that the IRVs should be referred to the CIO equator. ATSC have produced a revised version of PC TIVAS that uses IRVs referred to the true equator, and it is expected that this will be placed in CDDIS at about June 1995.

The original IRV programs from the University of Texas used the FK4 expressions for
GMST. It would not matter much if a reconstruction program were to use the FK5 expression
instead, or if it were to make the correction from UTC to UT1 in calculating GMST
(provided the same _{0} is used at stages 2, 3 and 4), as these would only have a
small effect on the calculation of the lunar and solar perturbations. In fact it is known
that the original Texas program contains a slip in the subroutine RAOGU, where the
parameter AL1 has the value 1.720279266007D-2 instead of 1.7202791266D-2, this being the
coefficient of T in the GMST expression, usually written as 8640184.542^{s}/cy,
expressed as radians/day. It is unlikely that other reconstruction packages would
duplicate this slip, and so this small difference exists in the community between some IRV
reconstruction packages. The only significant consequence of this is if attempts are made
during testing of software to compare the outputs of IRV packages by comparing positions
referred to the equinox, and then these small differences of transformation to the equinox
must be taken into account.

As mentioned above, the lunar and solar positions should be computed from simple models of precessing ellipses. This is not in fact a particularly good model for the lunar motion, and the PC-TIVAS software uses a much more elaborate model in a subroutine called DIANA, which evaluates long series of terms from the “Improved Lunar Ephemeris”, with a claimed accuracy of 2 arcsec. This is not necessarily an advantage, if the IRVs in use have been tuned to the simpler lunar model, but the effect of using a different lunar model from that for which the IRVs have been tuned is fairly small, only about 10 metres for Lageos, and smaller for lower satellites.