HARMONIC DECOMPOSITION OF TIME EPHEMERIS TE405

Wataru Harada and Toshio Fukushima

National Astronomical Observatory, 2-21-1 Osawa,Mitaka, Tokyo 181-8588, Japan;Toshio.Fukushima@nao.ac.jp

Received 2003 June 23; accepted 2003 August 6

ABSTRACT

By using Harada’s method of nonlinear harmonic analysis, we have analyzed the latest version of the timeephemeris of Earth, TE405, provided by Irwin & Fukushima, which is the location-independent part of thetwo relativistic time scales, Barycentric Coordinate Time (TCB) and Geocentric Coordinate Time (TCG).We decomposed TE405 for its full period, 1600–2200, into a combination of a quadratic polynomial, 463Fourier terms, and 36 mixed secular terms. The rms and the maximum of the residuals are 0.446 ns and2.95 ns, respectively. By correcting for the effect of very long periodic terms, we determined the value of DLC,the Newtonian contribution of the Sun, Moon, and major planets to the mean rate between TCB and TCG,to be 1.4808268559 � 10�8 � 6 � 10�18.

Key words: ephemerides — reference systems — time

On-line material:machine-readable table

1. INTRODUCTION

The time ephemeris of Earth is the central part of the rela-tion connecting Terrestrial Time (TT) and the solar systemBarycentric Coordinate Time (TCB). The former differsonly in rate from the third major time scale, GeocentricCoordinate Time (TCG). See IAU (1992, 2001) for rigorousdefinitions of these time scales; Seidelmann & Fukushima(1992) for an explanation of the IAU recommendations;Fairhead, Bretagnon, & Lestrade (1988) and Fairhead &Bretagnon (1990) for the analytical procedures to obtain atime ephemeris; and Fukushima (1995) and Irwin &Fukushima (1999) for numerical techniques to establishtime ephemerides.

Irwin & Fukushima (1999) provided TE405, the mostaccurate time ephemeris created from the latest lunar andplanetary ephemeris, DE405 (Standish 1998), in the form ofnumerical tables with suitable software to interpolate valuesfor any epoch. Of course, this should be used for workrequiring the highest precision, such as the analysis of pulsartimings, range and range-rate data for interplanetary mis-sions, and lunar laser ranging. However, there remains aneed for more compact representations to realize timeephemerides at moderate precision, say, at the level of 1 nsor so. This is the reason that the International EarthRotation and Reference Systems Service (IERS) Conven-tions (McCarthy 1996; McCarthy & Petit 2003) haveadopted as their approximation a version of the analyticalseries, which we created from the midsize extension ofFairhead & Bretagnon (1990) with some modifications. Forlater reference, we call it FB2C.1

Unfortunately, the existing analytical series expansionshave significant and systematic differences from the correcttime ephemerides. See Figure 3 of Irwin & Fukushima(1999) for several variations of the Fairhead & Bretagnon(1990) series. The residuals are large, even after the correc-tion of a few periodic terms. (See x 4 of Irwin & Fukushima

1999 for a description of our struggle to find suitablecorrections.) From the appearance of the residuals finallyobtained in Figure 4 of that paper, we conclude that it isdifficult to reduce such systematic differences by extendingthe analytical approaches.

Also, as was shown clearly in Fukushima (1995) andIrwin & Fukushima (1999), the inaccuracy of the analyticalapproximations of time ephemerides introduces significanterror in the determination of DLC, the Newtonian contri-bution of the Sun, Moon, and major planets to the meandifference in rate between TCB and TCG, defined as

DLC � 1

c2

�UE þ v2E

2

�; ð1Þ

where c is the speed of light in vacuum,UE is the Newtonianforce function of the Sun, the Moon, and the nine majorplanets acting on Earth, vE is the solar system barycentricvelocity of Earth, and the brackets denote a time average.This quantity is the major part of LC � d(TCB � TCG)/dTCB, which is one of the astronomical constants that playsa key role in the conversion between TCB-based andTT-based measurements of physical quantities.

Recently, we developed a purely numerical treatment todecompose complicated time series such as time ephemer-ides into a combination of a low-order polynomial and anumber of Fourier and mixed secular terms (Harada 2003).As an application, we analyzed the time variation of twoangles specifying the instantaneous orbital plane of theEarth-Moon barycenter around the Sun in DE405 andextracted the latest formula for planetary precession as theirpolynomial parts (Harada & Fukushima 2003).

Then, using the same approach, we decomposed TE405.The target of our analysis is an overall accuracy of 1 nsthroughout the period 1600–2200. In x 2, we provide theresult of the decomposition. In x 3, we discuss thedetermination of DLC as a by-product.

2. ANALYSIS

In order to cover the whole period of TE405 (Irwin &Fukushima 1999), we picked up 73,000 data points spaced

1 The abbreviation FB2C stands for Fairhead & Bretagnon secondversion, corrected. The routine was incorrectly referred to as the Fairhead,Bretagnon, & Lestrade (FBL) model in the 1996 IERS Conventions(McCarthy 1996).

The Astronomical Journal, 126:2557–2561, 2003November

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E

2557

every 3 days starting from JD = 2,305,450.5. Here JD isephemeris time (Teph), the argument to DE405. Then weanalyzed the data using our procedure of nonlinear har-monic analysis (Harada 2003). The result is a sum of aquadratic polynomial, 463 Fourier terms, and 36 mixedsecular terms.

The polynomial part is expressed as

PðtÞ ¼ P0 þ P1� þ P2�2 ; ð2Þ

where the coefficients are estimated as

P0 ¼ �36:07422901312205� 4:0� 10�12 ; ð3ÞP1 ¼ þ70:04807354556313� 2:0� 10�12 ; ð4ÞP2 ¼ þ1:5435585� 10�7 � 1:0� 10�12 ; ð5Þ

in units of seconds. The variable � was chosen such that therange �2 � � � +2 corresponds to the period of datafitting, from JD = 2,305,450.5 to JD = 2,524,447.5; that is,

� � JD� 2;414;949

54;749:25: ð6Þ

The transformation from JD to � allows for good separationof the coefficients. The presence of the quadratic term ismeaningful if considering the magnitude of the term deter-mined, 0.1 ls at J2000.0. On the other hand, the cubic andhigher order terms are estimated to be insignificant.

The Fourier andmixed secular terms are expressed as

FðtÞ ¼XJj¼1

ðSj sin!j� þ Cj cos!j�Þ ;

MðtÞ ¼XKk¼1

�ðS0k sin!k� þ C0

k cos!k�Þ ; ð7Þ

where the coefficients Sj, Cj, S0k, and C0

k and the frequencies!j and !k are listed in Tables 1 and 2, respectively. The num-ber of terms is J = 463 for the Fourier terms andK = 34 forthe mixed secular terms. The set of frequencies is commonto both tables, although their order of appearance is quitedifferent. The longest period detected in Table 1 is 242.15years (ID = 15). This is caused by the 8 : 13 near-commensurability of the mean motions of Venus and theEarth-Moon barycenter.

The residuals from TE405 after subtracting our decom-position are shown in Figures 1 and 2. No significantsystematic trend seems to remain.

Tables 3 and 4 compare our results with the existinganalytical solutions, FBL (Fairhead et al. 1988), FB(Fairhead & Bretagnon 1990), and FB2C (McCarthy 1996).It is interesting that our numerical analysis detected twowaves of similar frequency with periods of nearly 1 year,ID = 1 and ID = 2. The frequencies detected are close tothose of the mean anomaly and the mean longitude of theEarth-Moon barycenter (EMB). The difference in fre-quency, the rate of change of the longitude of perihelion ofthe EMB, is so small—some thousandths of an arcsecondper century—that the existing analytical treatments expandits effect into mixed secular terms. This is the reason for thesignificant differences seen in Table 4.

Figure 3 shows the manner in which the rms residualsdecrease for FB2C and our solution. We omit the curves forFBL and FB because they are almost the same as that forFB2C. Our decomposition is most successful, in the sensethat the rms residuals decrease more smoothly than in theexisting solutions.

3. DETERMINATION OF LC

In the previous section, we obtained a quadratic poly-nomial as the secular part of TE405. However, this betrayedour expectation that the secular part represents a linearmotion. Thus, we conclude that the appearance of a quad-ratic term is due to the effect of very long periodic (VLP)terms that were detected with our method of analysis.

TABLE 1

Harmonic Decomposition of TE405: Fourier Terms

ID

!

(revolutions day�1)

�!(revolutions day�1)

Period

(days)

S

(ns)

C

(ns)

�

(ns)

1................. 0.0027377363889662 6.6 � 10�15 365.2652622182 505079.2018 �1551857.1407 0.2803

2................. 0.00273806800820 3.4 � 10�13 365.22102337 21856.7326 �23134.7679 0.2822

3................. 0.00250699439627 2.0 � 10�13 398.88401884 20733.1083 �8526.5271 0.0025

4................. 0.00547555688052 3.3 � 10�13 182.62982594 �11108.6620 �8369.7220 0.0017

5................. 0.0002307755779 1.4 � 10�12 4333.21415 �3405.2830 �3354.5797 0.0050

Notes.—Listed are the 463 Fourier terms that we detected in TE405. They are given in order of decreasing magnitude of the totalamplitude, (S2 + C2)1/2, so one may truncate the series at any level desired. Table 1 is presented in its entirety in the electronic editionof the Astronomical Journal. A portion is shown here for guidance regarding its form and content.

Fig. 1.—Residuals from TE405 after subtracting our decomposition.The rms is 0.446 ns, and the absolute maximum difference is 2.95 ns. Nosystematic features remain.

2558 HARADA & FUKUSHIMA Vol. 126

Table 5 lists the periodic terms in FB2C with periodslonger than 300 years, half the duration of TE405. Note thatthere is a purely quadratic term, whose index we set to bej = 0. This is an approximation of the effect of much longerperiodic terms. Unfortunately, FB2C, as well as the originalFairhead & Bretagnon (1990) solution, did not provide thecorresponding linear and constant terms. Therefore, theycannot be used in the correction of DLC. Thus, hereafterwhen referring to the listed VLP terms, we exclude the purequadratic term, j = 0.

On the other hand, our result contains no periodic termwhose period is longer than 250 years. In other words, ourdetermination of the polynomial coefficients may haveabsorbed the effect of the VLP terms.

In order to remove the effect of this contamination, wetook the difference of the VLP terms from FB2C and thequadratic term in our determination, P2�

2, for the wholeperiod of TE405. Then we fitted a linear polynomial to thedifference by linear least squares. The determined

TABLE 2

Harmonic Decomposition of TE405: Mixed Secular Terms

No. ID

!

(revolutions day�1)

�!(revolutions day�1)

Period

(days)

S0

(ns)

C0

(ns)

�0

(ns)

1.................. 1 0.0027377363889662 6.6 � 10�15 365.2652622182 18267.9631 11066.8338 0.0323

2.................. 4 0.00547555688052 3.3 � 10�13 182.62982594 84.1446 61.4644 0.0015

3.................. 68 0.00022830683 1.8 � 10�10 4380.070 0.4391 25.4636 0.0036

4.................. 30 0.000016861603 5.5 � 10�11 59306.3 10.6848 5.5756 0.0018

5.................. 21 0.000338070974 1.9 � 10�11 2957.9588 �1.2765 10.9397 0.0016

6.................. 25 0.004783374558 5.1 � 10�11 209.057432 7.1772 1.2209 0.0015

7.................. 24 0.008213394359 2.4 � 10�11 121.7523421 3.6166 1.7897 0.0015

8.................. 83 0.00274906896 5.0 � 10�10 363.75952 1.9576 0.6918 0.0025

9.................. 93 0.0031533690 1.0 � 10�9 317.1212 �0.3096 0.4641 0.0021

10................ 210 0.0027593670 5.0 � 10�9 362.4020 �0.3758 �0.3626 0.0029

11................ 211 0.0006988393 7.3 � 10�9 1430.94 0.3397 �0.3638 0.0042

12................ 207 0.0025417367 5.2 � 10�9 393.4318 0.2282 0.4285 0.0023

13................ 10 0.0022762265272 5.2 � 10�12 439.323586 �0.0596 0.4373 0.0022

14................ 102 0.00041683952 8.4 � 10�10 2399.005 �0.1157 0.4073 0.0041

15................ 64 0.00102522272 2.7 � 10�10 975.3978 �0.4135 �0.0593 0.0016

16................ 232 0.0007038514 8.1 � 10�9 1420.75 0.2388 �0.3378 0.0039

17................ 182 0.0001413107 5.2 � 10�9 7076.6 �0.3382 0.2041 0.0031

18................ 231 0.0022649466 6.4 � 10�9 441.512 0.0005 �0.3714 0.0024

19................ 110 0.0030758177 1.2 � 10�9 325.1168 �0.3104 0.0076 0.0035

20................ 47 0.00137461952 1.2 � 10�10 727.47403 0.2902 0.0230 0.0015

21................ 106 0.00232427511 9.0 � 10�10 430.2417 �0.1608 �0.2403 0.0020

22................ 264 0.0030707576 9.7 � 10�9 325.653 �0.2467 �0.0329 0.0035

23................ 259 0.000283713 1.1 � 10�8 3524.7 0.1027 �0.1993 0.0027

24................ 292 0.002466336 1.3 � 10�8 405.460 0.1035 �0.1666 0.0032

25................ 290 0.002361683 1.5 � 10�8 423.427 0.1695 �0.0862 0.0029

26................ 176 0.0058914893 4.6 � 10�9 169.7364 0.0821 0.1366 0.0014

27................ 18 0.000032564808 1.3 � 10�11 30707 �0.0695 �0.1353 0.0018

28................ 32 0.002428451549 4.5 � 10�11 411.785032 0.0674 0.1354 0.0017

29................ 49 0.00411247464 1.3 � 10�10 243.162594 �0.0880 0.0725 0.0014

30................ 55 0.03112541369 1.4 � 10�10 32.1280870 �0.0129 0.1093 0.0014

31................ 29 0.000461554562 5.1 � 10�11 2166.5911 �0.0432 0.0956 0.0019

32................ 57 0.00052036722 1.7 � 10�10 1921.7198 �0.0070 0.0866 0.0015

33................ 61 0.00616289732 1.9 � 10�10 162.261344 0.0685 �0.0389 0.0014

34................ 33 0.001108699636 4.3 � 10�11 901.95754 �0.0681 0.0223 0.0015

35................ 303 0.002853388 1.9 � 10�8 350.461 �0.0579 �0.0171 0.0016

36................ 9 0.0034251185916 4.2 � 10�12 291.9606937 �0.0324 �0.0312 0.0015

Note.—Same as Table 1, but for the mixed secular terms. The column ‘‘ ID ’’ indicates the termwith the same frequency in Table 1.

Fig. 2.—Same as Fig. 1, but for the period 1960–2020, covering themodern and precise astronomical observations. The deviation is less than1 ns for more than 99% of this period. The absolute maximum differenceduring this period is 1.58 ns.

No. 5, 2003 HARMONIC DECOMPOSITION OF TE405 2559

coefficients, in the sense VLPminus P2�2, are

DP0 ¼ ð1:296� 0:032Þ � 10�6 ;

DP1 ¼ ð0:470� 0:027Þ � 10�6 : ð8Þ

Therefore, DLC is determined as

DLC � P1 þ DP1

54;749:25� 86;400

¼ 1:4808268559� 10�8 � 6� 10�18 : ð9Þ

Table 6 shows a comparison of our result with previousstudies.

The uncertainty in the correction DP1 is larger than thatin P1 by more than four digits. In order to refine the aboveestimate, we would have to reduce the error due to the VLPterms, say, by extending the period of analysis muchfurther.

Of course, it was desirable to change the base planetaryand lunar ephemeris from DE405 to its longer version,DE406, spanning 6000 years. Unfortunately, the nonlinearharmonic analysis was quite time-consuming, to the extentthat we could not complete the task with our computingfacilities. However, this is a challenging approach for thenear future, to clarify the nature of very long periodicvariations of the time ephemeris.

4. CONCLUSION

Using a nonlinear method of harmonic analysis, we havedecomposed TE405 into a quadratic polynomial, 463Fourier terms, and 36 mixed secular terms. The decomposi-tion reproduces the original TE405 at the level of 1 ns for1960–2020, as can be seen in Figure 2. By correcting for theeffect of the very long periodic terms listed in Table 5, wedetermined DLC precisely as DLC = 1.4808268559 � 10�8

� 6 � 10�18. This value is consistent with existing determi-nations if the uncertainties are taken into account.Although the detailed results of the decomposition are givenby equation (2) and Tables 1 and 2, we recommend the useof HF2002, a FORTRAN 77 routine to compute ourdecomposition. It is provided in the IERS Conventions(2003) (McCarthy & Petit 2003) and electronically availablefrom their Web site.2

TABLE 3

Characteristics of Series Expressions of the Time Ephemeris

Number of Terms

Series sin t sin t2 sin t3 sin t4 sin

rms

(ns)

FBL .................. 90 26 6 1 . . . . . .

FB..................... 93 27 5 1 . . . . . .FB2C ................ 478 205 85 20 4 6.19

This work.......... 463 36 0 0 0 0.45

TABLE 4

Comparison of Major Periodic Terms

Frequency (revolutions day�1) Amplitude (ls)

ID FBL FB ThisWork

Period

(days) Origin FBL FB ThisWork

Fourier:

1 ............ 0.0027378030913 0.002737803091985 0.002737736388966(7) 365.265 E 1656.6894 1656.674564 1631.98210(40)

2 ............ . . . . . . 0.0027380680082(3) 365.221 ? . . . . . . 31.82663(40)

3 ............ 0.0025069942337 0.002506994233926 0.0025069943963(2) 398.884 E�J 22.4175 22.417471 22.417927(2)

4 ............ 0.0054756061835 0.005475606183970 0.0054755568805(3) 182.630 2E 13.8399 13.839792 13.908796(5)

5 ............ 0.0002308088581 0.000230808858059 0.000230775578(2) 4333.214 J 4.7701 4.770086 4.780079(2)

6 ............ 0.0026448596135 0.002644859613830 0.002644858375(1) 378.092 E�S 4.6767 4.676740 4.676131(2)

7 ............ 0.0000929434781 0.000092943478155 0.000092886649(3) 10765.810 S 2.2566 2.256707 2.307314(3)

N/A...... 0.00000153519011 0.000001535172349 . . . 651392.660 4E�8M+3J 1.7307 1.694205 . . .

8 ............ 0.0338631919930 0.033863191992973 0.033863192290(3) 29.531 D 1.5555 1.554905 1.554908(3)

9 ............ 0.0034251186893 0.003425118688794 0.003425118592(4) 291.961 2V�2E 1.2768 1.276839 1.276751(2)

10 .......... 0.0022761853756 0.002276185375867 0.002276226527(5) 439.324 E�2J 1.1934 1.193379 1.190283(2)

11 .......... 0.0017125593447 0.001712559344397 0.001712559404(4) 583.921 V�E 1.1153 1.115322 1.115561(3)

Mixed secular:

1 ............ 0.0027378030913 0.002737803132527 0.002737736388966(7) 365.265 E 10.2157 10.215672 14.249069(32)

Notes.—We compared all the periodic terms in FBL (Fairhead et al. 1988), FB (Fairhead & Bretagnon 1990), and this work whose amplitudes can belarger than 1 ls for 1900–2100.We omit the results fromFB2C, since they are the same as FB except for the frequencies of the three largest terms, whose dif-ferences are much less than the deviations observed here. Underscores show the digits that are the same as ours. Given in parentheses are the uncertaintiesin the last digit. The first column refers to the identification of the terms in Table 1.

Fig. 3.—Manner of rms decrease. Plotted are the rms residuals fromTE405 as a function of the number of terms in the analytical series approxi-mation. The solid line shows our result, and the dashed line shows FB2C(McCarthy 1996). There were no significant differences between the resultfrom FB2C and those from FBL (Fairhead et al. 1988) and FB (Fairhead &Bretagnon 1990).

2 See http://maia.usno.navy.mil/conv2000.html.

2560 HARADA & FUKUSHIMA Vol. 126

REFERENCES

Fairhead, L., & Bretagnon, P. 1990, A&A, 229, 240Fairhead, L., Bretagnon, P., & Lestrade, J.-F. 1988, in IAU Symp. 128, TheEarth’s Rotation and Reference Frames for Geodesy and Geodynamics,ed. A. K. Babcock &G. A.Wilkins (Dordrecht: Kluwer), 419

Fukushima, T. 1995, A&A, 294, 895Harada,W. 2003,Master’s thesis, Univ. TokyoHarada,W., & Fukushima, T. 2003, AJ, submittedIAU. 1992, Trans. IAU, 21B, 41———. 2001, Trans. IAU, 24B, 56

Irwin, A.W., & Fukushima, T. 1999, A&A, 348, 642McCarthy, D. D., ed. 1996, IERS Conventions (1996) (IERS Tech. Note21) (Paris: Obs. Paris), chap. 11

McCarthy, D. D., & Petit, G., eds. 2003, IERS Conventions (2003) (draft),chap. 10

Seidelmann, P. K., & Fukushima, T. 1992, A&A, 265, 833Standish, E. M. 1998, JPL Planetary and Lunar Ephemerides, DE405/LE405 (InterofficeMemo. 312.F-98-048) (Pasadena: JPL)

TABLE 5

Very Long Periodic Terms in FB2C

j

A

(ls)

�

(rad century�1)

�

(rad)

Period

(yr) Origin

sin:

1 ................ �1.694205 0.3523118349 �5.025132748 1783.41591 4E�8M+3J

2 ................ �0.046551 0.0980321068 �0.921573539 6409.31376 10E�19M+3S

3 ................ �0.040184 0.7113547001 �3.565975565 883.27037 2J�5S

4 ................ +0.010962 0.3590428652 +2.196567739 1749.98194 4E�8M+J+5S

5 ................ �0.005123 0.1484472708 �2.999641028 4232.60412

6 ................ �0.005488 0.3455808046 �0.090675389 1818.15229

7 ................ +0.002409 0.2542797281 +5.325009315 2470.97374

8 ................ +0.002366 0.3932153263 +6.215885448 1597.89939

9 ................ �0.001388 0.7046236698 �1.166145902 891.70795 8E�16M+6J

10 .............. �0.001155 1.4227094002 �3.042700750 441.63519

11 .............. �0.000472 1.8159247265 �1.999707589 346.00472

12 .............. +0.000399 1.4977853527 +2.094441910 419.49838

13 .............. +0.000389 1.7252277143 +1.395753179 364.19455

t sin:

1 ................ �0.0210568 0.3523118349 �6.262738348 1783.41591 4E�8M+3J

3 ................ �0.0011081 0.7113547001 �5.154724984 883.27037 2J�5S

9 ................ �0.0003210 0.7046236698 �1.863796539 441.63519

8 ................ +0.0001782 0.3932153263 +5.180433689 1597.89939

14 .............. �0.0000140 1.1045700264 �4.957936982 568.83540

t2 sin:

0 ................ +0.00406495 0.0000000000 +4.712388980 11 ................ �0.00013374 0.3523118349 �1.502210314 1783.41591 4E�8M+3J

3 ................ �0.00004088 0.7113547001 �0.060926389 883.27037 2J�5S

14 .............. �0.00000775 1.1045700264 �0.245548001 568.83540

Notes.—Listed are periodic terms in the FB2C series with periods longer than 300 yr. Note that FB2Ccontains a purely quadratic term, with index j = 0 in the t2 sin part, which was obtained by expanding muchlonger periodic terms. See the original descriptions in Fairhead et al. 1988 and Fairhead & Bretagnon 1990.

TABLE 6

Comparison of DLC Determinations

Reference

DLC

(�10�8)

�

(�10�18)

Fukushima 1995 .................................. 1.4808268562 5

Irwin & Fukushima 1999...................... 1.48082685594 10

This work............................................. 1.4808268559 6

No. 5, 2003 HARMONIC DECOMPOSITION OF TE405 2561