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TDB2TDT

TDB2TDT

## Author

Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
craigm@lheamail.gsfc.nasa.gov
UPDATED VERSIONs can be found on my WEB PAGE:
http://cow.physics.wisc.edu/~craigm/idl/idl.html

## Purpose

Relativistic clock corrections due to Earth motion in solar system

## Major Topics

Planetary Orbits

## Calling Sequence

corr = TDB2TDT(JD, TBASE=, DERIV=deriv)

## Description

The function TDB2TDT computes relativistic corrections that must
be applied when performing high precision absolute timing in the
solar system.
According to general relativity, moving clocks, and clocks at
different gravitational potentials, will run at different rates
with respect to each other. A clock placed on the earth will run
at a time-variable rate because of the non-constant influence of
the sun and other planets. Thus, for the most demanding
astrophysical timing applications -- high precision pulsar timing
-- times in the accelerating earth observer's frame must be
corrected to an inertial frame, such as the solar system
barycenter (SSB). This correction is also convenient because the
coordinate time at the SSB is the ephemeris time of the JPL
Planetary Ephemeris.
In general, the difference in the rate of Ti, the time kept by an
arbitrary clock, and the rate of T, the ephemeris time, is given
by the expression (Standish 1998):
dTi/dT = 1 - (Ui + vi^2/2) / c^2
where Ui is the potential of clock i, and vi is the velocity of
clock i. However, when integrated, this expression depends on the
position of an individual clock. A more convenient approximate
expression is:
T = Ti + (robs(Ti) . vearth(T))/c^2 + dtgeo(Ti) + TDB2TDT(Ti)
where robs is the vector from the geocenter to the observer;
vearth is the vector velocity of the earth; and dtgeo is a
correction to convert from the observer's clock to geocentric TT
time. TDB2TDT is the value computed by this function, the
correction to convert from the geocenter to the solar system
barycenter.
As the above equation shows, while this function provides an
important component of the correction, the user must also be
responsible for (a) correcting their times to the geocenter (ie,
by maintaining atomic clock corrections); (b) estimating the
observatory position vector; and and (c) estimating earth's
velocity vector (using JPLEPHINTERP).
Users may note a circularity to the above equation, since
vearth(T) is expressed in terms of the SSB coordinate time. This
appears to be a chicken and egg problem since in order to get the
earth's velocity, the ephemeris time is needed to begin with.
However, to the precision of the above equation, < 25 ns, it is
acceptable to replace vearth(T) with vearth(TT).
The method of computation of TDB2TDT in this function is based on
(so-called FBL model) and Fairhead & Bretagnon 1990, in terms of
sinusoids of various amplitudes. TDB2TDT has a dominant periodic
component of period 1 year and amplitude 1.7 ms. The set of 791
coefficients used here were drawn from the Princeton pulsar timing
program TEMPO version 11.005 (Taylor & Weisberg 1989).
Because the TDB2TDT quantity is rather expensive to compute but
slowly varying, users may wish to also retrieve the time
derivative using the DERIV keyword, if they have many times to
convert over a short baseline.
Verification
This implementation has been compared against a set of FBL test
data found in the 1996 IERS Conventions, Chapter 11, provided by
T. Fukushima. It has been verified that this routine reproduces
the Fukushima numbers to the accuracy of the table, within
10^{-14} seconds.
Fukushima (1995) has found that the 791-term Fairhead & Bretagnon
analytical approximation use here has a maximum error of 23
nanoseconds in the time range 1980-2000, compared to a numerical
integration. In comparison the truncated 127-term approximation
has an error of ~130 nanoseconds.

## Parameters

JD - Geocentric time TT, scalar or vector, expressed in Julian
days. The actual time used is (JD + TBASE). For maximum
precision, TBASE should be used to express a fixed epoch in
whole day numbers, and JD should express fractional offset
days from that epoch.

## Keyword Parameters

TBASE - scalar Julian day of a fixed epoch, which provides the
origin for times passed in JD.
Default: 0
DERIV - upon return, contains the derivative of TDB2TDT in units
of seconds per day. As many derivatives are returned as
values passed in JD.

## Returns

The correction offset(s) in units of seconds, to be applied as
noted above.

## Example

Find the correction at ephemeris time 2451544.5 (JD):
IDL> print, tdb2tdt(2451544.5d)
-0.00011376314
or 0.11 ms.

## References

Princeton TEMPO Program
http://pulsar.princeton.edu/tempo/
FBL Test Data Set
ftp://maia.usno.navy.mil/conventions/chapter11/fbl.results
Fairhead, L. & Bretagnon, P. 1990, A&A, 229, 240
(basis of this routine)
Rotation and Reference Frames for Geodesy and Geodynamics*,
ed. A. K. Babcock and G. A. Wilkins, (Dordrecht: Kluwer), p. 419
(original "FBL" paper)
Fukushima, T. 1995, A&A, 294, 895 (error analysis)
Irwin, A. W. & Fukushima, T. 1999, A&A, 348, 642 (error analysis)
Standish, E. M. 1998, A&A, 336, 381 (description of time scales)
Taylor, J. H. & Weisberg, J. M. 1989, ApJ, 345, 434 (pulsar timing)

## Modification History

Original logic from Fairhead & Bretagnon, 1990
Drawn from TEMPO v. 11.005, copied 20 Jun 2001
Documented and vectorized, 30 Jun 2001

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