;+
; 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 ns and
;   rms error of 14 ns in the time range 1980-2000, compared to a
;   numerical integration (see Table 12).  In comparison the truncated
;   127-term approximation has a maximum error of ~130 ns and rms
;   error of 26 ns.
;
;
; 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 - Julian day of a fixed epoch, which provides the
;           origin for times passed in JD.  Either a scalar, which
;           applies to all items, or a vector of same size as 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.
;
;   NTERMS - number of terms to use in the computation, in case the
;            full series accuracy is not required. 
;            Default: all terms.
;
;
; 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
;   Added helpful usage message, CM, 15 Mar 2002
;   Add NTERMS keyword, CM, 07 Mar 2007
;   NTERMS was having no effect, now fixed, CM, 16 Jul 2008
;   Documentation about 'verification' was enhanced, CM, 27 Feb 2009
;   TBASE may be a vector, CM, 2012-04-09
;   Bug fix in case TBASE is not a vector, CM, 2013-09-29
;
;  $Id: tdb2tdt.pro,v 1.10 2013/09/30 02:27:05 cmarkwar Exp $
;
;-
; Copyright (C) 2001, 2002, 2007, 2009, 2012, 2013, Craig Markwardt
; This software is provided as is without any warranty whatsoever.
; Permission to use, copy and distribute unmodified copies for
; non-commercial purposes, and to modify and use for personal or
; internal use, is granted.  All other rights are reserved.
;-
; T**1                          
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   0.000103d,     76.266071276d, 5.594294322d, $
   0.000104d,    949.175608970d, 5.674287810d, $
   0.000101d,  13517.870106233d, 2.196632348d, $
   0.000100d,  11933.367960670d, 4.056084160d  ]

i2terms = n_elements(fbldata)/3
; T**2                          
fbldata = [ fbldata, $
   4.322990d,   6283.075849991d, 2.642893748d, $
   0.406495d,      0.000000000d, 4.712388980d, $
   0.122605d,  12566.151699983d, 2.438140634d, $
   0.019476d,    213.299095438d, 1.642186981d, $
   0.016916d,    529.690965095d, 4.510959344d, $
   0.013374d,     -3.523118349d, 1.502210314d, $
   0.008042d,     26.298319800d, 0.478549024d, $
   0.007824d,    155.420399434d, 5.254710405d, $
   0.004894d,   5746.271337896d, 4.683210850d, $
   0.004875d,   5760.498431898d, 0.759507698d, $
   0.004416d,   5223.693919802d, 6.028853166d, $
   0.004088d,     -7.113547001d, 0.060926389d, $
   0.004433d,  77713.771467920d, 3.627734103d, $
   0.003277d,  18849.227549974d, 2.327912542d, $
   0.002703d,   6062.663207553d, 1.271941729d, $
   0.003435d,   -775.522611324d, 0.747446224d, $
   0.002618d,   6076.890301554d, 3.633715689d, $
   0.003146d,    206.185548437d, 5.647874613d, $
   0.002544d,   1577.343542448d, 6.232904270d, $
   0.002218d,   -220.412642439d, 1.309509946d, $
   0.002197d,   5856.477659115d, 2.407212349d, $
   0.002897d,   5753.384884897d, 5.863842246d, $
   0.001766d,    426.598190876d, 0.754113147d, $
   0.001738d,   -796.298006816d, 2.714942671d, $
   0.001695d,    522.577418094d, 2.629369842d  ]
fbldata = [ fbldata, $
   0.001584d,   5507.553238667d, 1.341138229d, $
   0.001503d,   -242.728603974d, 0.377699736d, $
   0.001552d,   -536.804512095d, 2.904684667d, $
   0.001370d,   -398.149003408d, 1.265599125d, $
   0.001889d,  -5573.142801634d, 4.413514859d, $
   0.001722d,   6069.776754553d, 2.445966339d, $
   0.001124d,   1059.381930189d, 5.041799657d, $
   0.001258d,    553.569402842d, 3.849557278d, $
   0.000831d,    951.718406251d, 2.471094709d, $
   0.000767d,   4694.002954708d, 5.363125422d, $
   0.000756d,   1349.867409659d, 1.046195744d, $
   0.000775d,    -11.045700264d, 0.245548001d, $
   0.000597d,   2146.165416475d, 4.543268798d, $
   0.000568d,   5216.580372801d, 4.178853144d, $
   0.000711d,   1748.016413067d, 5.934271972d, $
   0.000499d,  12036.460734888d, 0.624434410d, $
   0.000671d,  -1194.447010225d, 4.136047594d, $
   0.000488d,   5849.364112115d, 2.209679987d, $
   0.000621d,   6438.496249426d, 4.518860804d, $
   0.000495d,  -6286.598968340d, 1.868201275d, $
   0.000456d,   5230.807466803d, 1.271231591d, $
   0.000451d,   5088.628839767d, 0.084060889d, $
   0.000435d,   5643.178563677d, 3.324456609d, $
   0.000387d,  10977.078804699d, 4.052488477d, $
   0.000547d, 161000.685737473d, 2.841633844d  ]
fbldata = [ fbldata, $
   0.000522d,   3154.687084896d, 2.171979966d, $
   0.000375d,   5486.777843175d, 4.983027306d, $
   0.000421d,   5863.591206116d, 4.546432249d, $
   0.000439d,   7084.896781115d, 0.522967921d, $
   0.000309d,   2544.314419883d, 3.172606705d, $
   0.000347d,   4690.479836359d, 1.479586566d, $
   0.000317d,    801.820931124d, 3.553088096d, $
   0.000262d,    419.484643875d, 0.606635550d, $
   0.000248d,   6836.645252834d, 3.014082064d, $
   0.000245d,  -1592.596013633d, 5.519526220d, $
   0.000225d,   4292.330832950d, 2.877956536d, $
   0.000214d,   7234.794256242d, 1.605227587d, $
   0.000205d,   5767.611978898d, 0.625804796d, $
   0.000180d,  10447.387839604d, 3.499954526d, $
   0.000229d,    199.072001436d, 5.632304604d, $
   0.000214d,    639.897286314d, 5.960227667d, $
   0.000175d,   -433.711737877d, 2.162417992d, $
   0.000209d,    515.463871093d, 2.322150893d, $
   0.000173d,   6040.347246017d, 2.556183691d, $
   0.000184d,   6309.374169791d, 4.732296790d, $
   0.000227d, 149854.400134205d, 5.385812217d, $
   0.000154d,   8031.092263058d, 5.120720920d, $
   0.000151d,   5739.157790895d, 4.815000443d, $
   0.000197d,   7632.943259650d, 0.222827271d, $
   0.000197d,     74.781598567d, 3.910456770d  ]
fbldata = [ fbldata, $
   0.000138d,   6055.549660552d, 1.397484253d, $
   0.000149d,  -6127.655450557d, 5.333727496d, $
   0.000137d,   3894.181829542d, 4.281749907d, $
   0.000135d,   9437.762934887d, 5.979971885d, $
   0.000139d,  -2352.866153772d, 4.715630782d, $
   0.000142d,   6812.766815086d, 0.513330157d, $
   0.000120d,  -4705.732307544d, 0.194160689d, $
   0.000131d, -71430.695617928d, 0.000379226d, $
   0.000124d,   6279.552731642d, 2.122264908d, $
   0.000108d,  -6256.777530192d, 0.883445696d  ]

i3terms = n_elements(fbldata)/3
; T**3                          
fbldata = [ fbldata, $
   0.143388d,   6283.075849991d, 1.131453581d, $
   0.006671d,  12566.151699983d, 0.775148887d, $
   0.001480d,    155.420399434d, 0.480016880d, $
   0.000934d,    213.299095438d, 6.144453084d, $
   0.000795d,    529.690965095d, 2.941595619d, $
   0.000673d,   5746.271337896d, 0.120415406d, $
   0.000672d,   5760.498431898d, 5.317009738d, $
   0.000389d,   -220.412642439d, 3.090323467d, $
   0.000373d,   6062.663207553d, 3.003551964d, $
   0.000360d,   6076.890301554d, 1.918913041d, $
   0.000316d,    -21.340641002d, 5.545798121d, $
   0.000315d,   -242.728603974d, 1.884932563d, $
   0.000278d,    206.185548437d, 1.266254859d, $
   0.000238d,   -536.804512095d, 4.532664830d, $
   0.000185d,    522.577418094d, 4.578313856d, $
   0.000245d,  18849.227549974d, 0.587467082d, $
   0.000180d,    426.598190876d, 5.151178553d, $
   0.000200d,    553.569402842d, 5.355983739d, $
   0.000141d,   5223.693919802d, 1.336556009d, $
   0.000104d,   5856.477659115d, 4.239842759d  ]

i4terms = n_elements(fbldata)/3
; T**4                          
fbldata = [ fbldata, $
   0.003826d,   6283.075849991d, 5.705257275d, $
   0.000303d,  12566.151699983d, 5.407132842d, $
   0.000209d,    155.420399434d, 1.989815753d  ]

    nterms = n_elements(fbldata)/3
    fbldata = reform(fbldata, 3, nterms, /overwrite)
    const0 = reform(fbldata(0,*), nterms)
    freq0  = reform(fbldata(1,*), nterms)
    phase0 = reform(fbldata(2,*), nterms)

    texp = dblarr(nterms) +   0
    texp(i1terms:i2terms-1) = 1
    texp(i2terms:i3terms-1) = 2
    texp(i3terms:i4terms-1) = 3
    texp(i4terms:*        ) = 4

  endif

  if n_elements(tbase) EQ 0 then tbase = 0D
  t = ((tbase(0)-2451545D) + jd(0))/365250.0D
  if t EQ 0 then t = 1d-100
  if n_elements(nt0) EQ 0 then begin
      nt = n_elements(const0)
  endif else begin
      nt = round(nt0(0))>1
  endelse

  ph = freq0(0:nt-1) * t + phase0(0:nt-1)
  sint = sin( ph )
  sinf = const0(0:nt-1) * t^texp(0:nt-1)

  dt = total(sinf*sint)*1d-6
  if arg_present(deriv) then $
    deriv = total(sinf*(texp(0:nt-1)*sint/t + freq0(0:nt-1)*cos(ph)))*(1d-6/365250.0D)

  return, dt
end

function tdb2tdt, jd, deriv=deriv, tbase=tbase0, nterms=nt

  if n_params() EQ 0 OR n_elements(jd) EQ 0 then begin
      message, 'USAGE: ', /info
      message, ' TDB = TDT + TDB2TDT(JD)', /info
      message, '     ;; All units in seconds', /info
      message, '     ;; JD is julian day number referred to TDT or TDB', /info
      message, '', /info
      message, 'Other timescales (all units in seconds; JD=Julian date):', $
        /info
      message, ' TT  = TAI + 32.184d     ;; TAI to Terrestrial Time', /info
      message, ' TDT = TAI + 32.184d     ;; TAI to Terrestrial Dynamical Time',$
        /info
      message, ' TDB = TDT + TDB2TDT(JD) ;; TDT to Barycentric Dynamical Time',$
        /info
      message, '     ;; (JD referred to TDT or TDB)', /info
      message, ' TAI = UTC + TAI_UTC(JD) ;; UTC to TAI (JD referred to UTC)', $
        /info
      message, ' UTC = TAI + TAI_UTC(JD, /INV) ;; UTC to TAI (JD referred to TAI)', /info
      return, 0
  endif

  sz = size(jd)
  n = n_elements(jd)
  ntb = n_elements(tbase0)

  ;; Expand dimensions of TBASE to match JD
  if ntb GT 0 then begin
      if ntb GT 1 AND ntb LT n then $
        message, 'ERROR: JD and TBASE dimensions must match'
      tbase = jd*0 + tbase0
  endif else begin
      tbase = jd*0
  endelse

  if n EQ 1 then $
    return, tdb2tdt_calc(jd, deriv=deriv, tbase=tbase)

  result = reform(double(jd), sz(1:sz(0)))
  if arg_present(deriv) then begin
      deriv = reform(double(jd), sz(1:sz(0)))
      for i = 0L, n-1 do begin
          result(i) = tdb2tdt_calc(jd(i), deriv=dd, tbase=tbase(i), nterms=nt)
          deriv(i) = dd
      endfor
  endif else begin
      for i = 0L, n-1 do begin
          result(i) = tdb2tdt_calc(jd(i), tbase=tbase(i), nterms=nt)
      endfor
  endelse

  return, result
end