## Name

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

the analytical formulation by Fairhead, Bretagnon & Lestrade, 1988

(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)

Fairhead, L. Bretagnon, P. & Lestrade, J.-F. 1988, in *The Earth's

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)

## See Also

JPLEPHREAD, JPLEPHINTERP, JPLEPHTEST

## Modification History

Original logic from Fairhead & Bretagnon, 1990

Drawn from TEMPO v. 11.005, copied 20 Jun 2001

Documented and vectorized, 30 Jun 2001