TIME DILATION AND THE LENGTH OF THE SECOND: WHY TIMESCALES DIVERGE

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1 The Astronomical Journal, 134:64Y70, 2007 July # The American Astronomical Society. All rights reserved. Printed in U.S.A. A TIME DILATION AND THE LENGTH OF THE SECOND: WHY TIMESCALES DIVERGE Steven D. Deines 330 Alburnett Road, Marion, IA 52302, USA; sddeines@inov.net and Carol A. Williams Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA; cw@math.usf.edu Received 2006 January 14; accepted 2007 February 11 ABSTRACT We show that the timescale divergence between Universal Time ( UT1) and international atomic time ( TAI), which is compensated for by the occasional addition of a leap second, is due to the fact that the Système Internationale (SI) second is shorter than the UT second. Celestial mechanicians saw the necessity of introducing a timescale that eliminated the discrepancy between the observed and calculated longitudes of the Moon, Sun, and planets. This timescale, called ephemeris time (ET), was measured and used to calibrate the length of the SI second. It has been shown that ET and TAI are equivalent for all practical purposes. We show that the length of the ET second (and consequently the length of the SI second) was shorter than the length of the UTsecond at the beginning of the tropical year , even though it was intended that the ET second would equal this length. We further show that this difference in the lengths of the UT and SI seconds is due to time dilation. The ET (or equivalently the SI) second is a measure of the scale of coordinate time, while the UT second is a measure of proper time for an observer moving with the Earth. Our calculation of the time dilation effect matches both the difference between the SI and UT seconds and also the leapsecond insertion rate to within 0.2% since atomic time began in 1958 up to 2000, when UT was redefined. The deceleration of Earth s rotation contributes less than 1% of this timescale divergence according to the measurements from paleontological records of tidal friction. One possible method to convert from the TAI timescale is to use a multiplicative scalar to obtain a UT timescale. This method would necessitate the insertion of a leap second into the UT timescale only once in approximately 14 decades to account for tidal friction. Key words: celestial mechanics ephemerides relativity time Online material: color figure 1. HISTORICAL INTRODUCTION Throughout the 20th century, timekeeping advanced to unprecedented levels of complexity. Several new timescales were defined to upgrade the independent time parameter used in ephemerides for improved accuracy (Standish 1998). Many relate to the oldest timescale, mean solar time, based on both the rotation of the Earth and the apparent motion of the Sun and realized today as Universal Time (UT) (Seidelmann 1992, p. 79). When the stellar meridian transits for UT are adjusted for polar motion, the resulting timescale UT1 is obtained. ( In this paper, UT designates the technical timescale UT1.) For more precise time, the international atomic time (TAI) was approved in 1967 as an official timescale using the Système Internationale (SI) second in atomic clocks maintained by international timing laboratories. Soon after TAI was introduced, it became apparent that UT and TAI were diverging. As an approximation to UT1, Coordinated Universal Time (UTC) is determined from reading those atomic clocks, but it is offset from TAI by the insertion of an extra second, a leap second, whenever necessary to keep UTC within 0.9 s of UT1. The divergence between TAI and UT1 is the subject of this paper. Although UT depends on the rotation and orbit of the Earth, the Moon s orbit is often used for validating timescales. There are continuing problems in establishing an accurate lunar theory, one being an ad hoc secular acceleration in the lunar mean longitude. This problem goes back to the 17th century, when Halley found that quadratic terms had to be added to the Moon s mean longitude to match the times recorded for ancient eclipses ( Halley 1695). In 1787, Laplace announced that the acceleration term was due to perturbations from the Earth s orbital eccentricity ( Laplace 1786). Adams then determined that Laplace had not included many higher order terms, which reduced Laplace s final result to about half of Halley s empirical value (Adams 1853). To explain the source for the remaining observed effect, Ferrell (1864) and Delaunay (1865) independently assigned this discrepancy to tidal interactions between the Earth and the Moon. Newcomb considered variations of the Earth s rotation to explain some of the lunar residuals (Newcomb 1878), but he could not obtain verification of these variations from inner planet data (Newcomb 1912). In the latter 19th century, Chandler discovered minute variations in Earth s axis of rotation, or polar wobble (Chandler 1891, 1892). Newcomb then announced that these or possibly other irregular variations of the Earth s rotation might be an explanation of the residuals in the lunar mean longitude (Newcomb 1912). To reconcile discrepancies between theory and observation, E. W. Brown set a goal to establish a relatively high order, purely gravitational theory of the Moon, but was forced to add Newcomb s Great Empirical Term, GET ¼þ10:71 00 sin (140:0 T þ 240:7 ), to reduce residuals in the longitude of his lunar tables ( Brown 1919). Between 1928 and 1937, Brown and Brouwer wrote many papers on the reduction of observed lunar occultations and referred to an irreducible gap between observations and Brown s theory (Brown 1926). In all these cases, calculated positions (without ad hoc adjustments) were ahead of observed positions by amounts proportional to the frequencies. Astronomers came to believe that discrepancies in ephemerides were not due to errors in the expressions for the mean longitude but were due to unmodeled irregularities and 64

2 TIME DILATION AND THE LENGTH OF THE SECOND 65 a deceleration of the Earth s rotation, on which UT depends. Spencer Jones (1939) examined residuals in the mean longitudes of the Sun, Moon, and two planets and concluded that the error was due to a slow deceleration in Earth s rotation (Spencer Jones 1926). Since Newton s theory of gravitation requires a uniform (nonaccelerating) timescale for the computation of orbital motions and since the Earth s rotation was assumed to be decelerating, astronomers thought a timescale determined by the orbits themselves would be the uniform scale they needed. Thus, astronomers proposed and established in the mid 20th century a timescale called ephemeris time (ET), based on orbital motions, to be used for all dynamical calculations. 2. LENGTH OF A SECOND OF EPHEMERIS TIME The definition of the basic unit of ET starts with Newcomb s expression for the geometric mean longitude of the Sun relative to the mean equinox of date (Seidelmann 1992; Newcomb 1898), L S ¼ :04 00 þ :13 00 T þ 1: T 2 ; where T is the time measured in Julian centuries of 36,525 UT days from the epoch of 1900 January 0, Greenwich Mean Noon, JD 2,415, The mean motion in this expression was used to define the ephemeris second as a fraction of the tropical year at , namely, the inverse of (360 ; 60 ; 60) ;602;768:13 00 century 1 36;525 days century 1 ; 86;400 s day 1 ¼ 31;556;925:9747 s: ð2þ It was thought that this definition would assure that the length of the UT and ET seconds would be equal at the epoch of We will show that the length of the ET second is not equal to the length of the UT second at We will further show that if the time span between two events is measured by two different clocks, one counting UT seconds (UT), and the other ET seconds (ET), then the relation between these two time counts is ET UT ET ¼ 2:292 ; 10 8 : To show that the length of the ET second is shorter, it is necessary to reassess the scientific history that took place in the first half of the 20th century leading to the development and establishment of ephemeris time. In the early 20th century, the best theories were from S. Newcomb for the Sun and inner planets (Newcomb 1898) and from E. W. Brown for the Moon (Brown 1919). In 1926 and again in 1939, Spencer Jones analyzed the residuals between these theories and observations and presented formulae for the correction of the calculated mean longitudes. In particular, his correction of the Sun s mean longitude (L S )isgivenby L S ¼ 1:00 00 þ 2:97 00 T þ 1:23 00 T 2 þ 0:0748B; where T is the same as in equation (1) and B is an unknown function assumed to model irregular fluctuations in the Earth s rotation that was hoped to be empirically determined from lunar residuals. The coefficient of B, , is the ratio of the mean motion of the Sun to that of the Moon; L S is the expression that one could subtract from calculated solar mean longitudes to bring calculations into agreement with the observed values at the Universal Time, T. ð1þ ð3þ ð4þ Because the behavior of the analogous residuals in the mean longitudes of Mercury, Venus, and the Moon were all correlated with those of the Sun, Spencer Jones and other astronomers and geophysicists became convinced that the error was not in the theories but was connected to the value of the independent variable, time, as Newcomb and others before him had suspected (Spencer Jones 1932). ( Tidal friction was often blamed, since it produces a deceleration in Earth s rotation, but we will show later that this effect is 100 times too small to produce these residuals. The residuals are in fact caused by time dilation.) In 1948, Clemence presented the formula that defined ephemeris time (although he called it Newtonian Time). Multiplying equation (4) by , the number of seconds of time needed for the Sun to travel through an angle of 1 00, he obtained t ¼ 24:349 s þ ð72:3165 s ÞT þ ð29:949 sþt 2 þ 1:821B; ð5þ where T is measured in Julian centuries of UTas in equation (1). The scale factor is obtained by dividing the length of the tropical year at by the number of arcseconds in 360, 24:349 s arcsec 1 ¼ 31;556;925:9747 s (360 ; 60 ; 60) 00 : ð6þ The term proportional to T in equation (5) is in fact the source of the leap second. The International Astronomical Union ( IAU) approved expression (5) in 1952 and agreed on the name ephemeris time (Oosterhoff 1954). All ephemerides were to be rewritten using the independent variable ET. All terms involving B were dropped from consideration. Although ET should be calibrated by combining observations from all solar system bodies, the lunar theory dominated the calibration. Brown s theory (then considered the most precise) gives for the mean longitude of the Moon L M ¼ :71 00 þ 1336 r :06 00 T þ 7:14 00 T 2 þ 0: T 3 ; where the superscript r signifies a revolution of 360 and T represents Julian centuries of UT as in equation (1). It was understood that Newcomb s GET would be added to the mean longitude of equation (7) to match observations. The GET has a negative value from about 1850 to 1975, reducing the value of the calculated mean longitude to better match observations, at least in this period of time. In the 1948 paper, Clemence suggested the following steps be taken to adjust the calculated mean longitude, L M,inequation(7), which were later endorsed (Oosterhoff 1954): 1. Substitute T t for T in L M. 2. Add Spencer Jones s and de Sitter s corrections for the supposed tidal friction effect. 3. Remove the GET. (It does not appear in eq. [7], but was included in computations.) The substitution of T t for T in L M produces the following adjustment: L ¼13: :71 00 T 16:44 00 T 2 ; which has the effect of reducing L M when T is positive. One may also obtain equation (8) by multiplying equation (4) by the negative of 13.37, the ratio of the lunar to the solar mean motion. ð7þ ð8þ

3 66 DEINES & WILLIAMS Vol. 134 The Spencer Jones and de Sitter corrections will be discussed in a future paper. Their corrections (Spencer Jones 1939; de Sitter 1927) are 4:65 00 þ 12:96 00 T þ 5:22 00 T 2 : ð9þ With the GET removed from consideration and the B-term ignored, the final change to L M, given by Clemence (1948), is the sum of equation (8) and the terms in equation (9): L ¼8: :75 00 T 11:22 00 T 2 : ð10þ The Improved Lunar Ephemeris (ILE; USNO & Nautical Almanac Office 1954) was the document used to provide data for the determination of ephemeris time. In that document, one will find two expressions for the Moon s mean longitude. The first is L M given in equation (7), and the second is L ¼ L M þ L ¼ :99 00 þ 1336 r :31 00 T 4:08 00 T 2 þ 0: T 3 ; ð11þ where T in this expression is understood to be measured in Julian centuries of ET. It is clearly stated in the ILE that the determination of ephemeris time would be based on L,notL M,inalmost all calculations. It fell to Markowitz and Hall of the US Naval Observatory to obtain, from lunar observations, the values of ET needed to calibrate the atomic clocks and to define the SI second (Markowitz et al. 1958). Markowitz designed the dual-rate Moon camera for this purpose ( Markowitz 1959a). Clemence and Sadler described the method of determining ET from observations (Sadler & Clemence 1954). The Improved Lunar Ephemeris (USNO & Nautical Almanac Office 1954) included values of the right ascension () and declination ()ofthe Moon that were calculated using Brown s theory with L for the mean longitude. These tabulated coordinates were tagged with the value of the independent time variable used for the computations, and this value of time was assumed to be the ephemeris time of the calculation (Eckert et al. 1954). Observed values of and were obtained from the reduction of the photographic plates taken by the Moon camera. These observed values were matched to calculated values by interpolation in the tables, and the time tag for the matching calculation was also interpolated, giving a precise value for the ET of the calculation best matching the observation. The camera was designed so that the UT of an observation could also be determined with the same level of precision as ET. Thus, the value of T ¼ ET UT could be measured. Markowitz and Hall submitted their data to the National Physical Laboratory (NPL) for the final calibration of the SI second. In 1967, the IAU approved the result as the definition of the SI second, which is the basic time unit in TAI (Markowitz et al. 1958). Markowitz wrote, A determination of the frequency of cesium in terms of the second of Ephemeris Time (E.T.) was made jointly by the National Physical Laboratory, Teddington, and the U.S. Naval Observatory, Washington. The frequency is 9;192;631; cycles s 1 of E.T. The second of E.T. is identical with the prototype unit of time defined by the International Bureau of Weights and Measures in 1956 ( Markowitz 1959b). So, the current SI second matches the ET second and provides continuity between the ET and TAI timescales (Seidelmann 1992). In 1986 Markowitz (1988) reported that the SI second and the ET second were still consistent to less than a part in ;both seconds are equivalent for all practical purposes. There seems to be little or no discussion in the literature regarding the length of the SI second compared to that of the UT second. Some suggest that the divergence between the two timescales is due to the deceleration of Earth s rotation rate by tidal friction, implying that the UT second is growing larger. In a later section, we consider measurements of the length of day that suggest that tidal friction is not large enough to explain the observed divergence between the timescales. Analysis of the method used to calibrate ET gives a reasonable explanation of the divergence in terms of the lengths of the seconds. The table of calculated lunar positions in the ILE as used by Markowitz and others was based on L (eq. [11]). Consider the frequencies of L and L M (eq. [7]) ( n and n M, respectively) and note that n < n M. From an observed pair (, ), say ( 1, 1 ), one obtains ET 1 and can calculate L(ET 1 ). Since UT 1 is also observed, L M (UT 1 ) can be calculated. Consider a second observation and compute L M (UT 2 ) and L(ET 2 ). If the time interval between UT 1 and UT 2 is equal to the sidereal period of the lunar orbit, then it should be true that L M (UT 2 ) L M (UT 1 ) ¼ 2. Since L also represents the lunar mean longitude, then it must be true that L(ET 2 ) L(ET 1 ) ¼ 2. If this were not the case, the calculations used to calibrate ET would not have fit the lunar orbit. Considering only the linear terms in the mean longitudes, the preceding paragraph implies that, approximately, n M (UT 2 UT 1 ) ¼ n(et 2 ET 1 ): ð12þ Since n M > n, the time interval ET 2 ET 1 must be numerically larger than the interval UT 2 UT 1. Since both intervals represent an identical span of time between these two events, the unit of ET must be smaller than the unit of UTin order to make the time count ET 2 ET 1 larger. For arbitrary time intervals, from equations (12), (7), and (11) we obtain equation (3): ET UT ET ¼ n M n n M ¼ 2:292 ; 10 8 : 3. DIVERGENCE BETWEEN UNIVERSAL AND INTERNATIONAL ATOMIC TIMES Between 1958 and January 1999, there was a steady but gradual divergence between UT and TAI timescales. Since the epoch of TAI was set at 0 hr on 1958 January 1 UT1, UT has trailed behind TAI, accumulating a total of 32 s by 1999 (USNO 2000, p. K9). In Figure 1 we show both the fractional seconds inserted into the UTC timescale between 1958 and 1972 and the leap seconds inserted between 1972 and Between 1972 and 1998 leap seconds were inserted at midnight of either December 31 or June 30 to keep UTC within 0.9 s of UT1. The original definition of UT has exactly 86,400 s dividing the mean solar day. The average length of day (LOD) in SI seconds is 86, s over 41 years between 1958 and 1999, with the total insertion of 32 leap seconds. The average leap-second rate is s yr 1, or 1.28 yr between leap-second insertions. After 1999 January, no leap seconds were inserted until the International Earth Rotation Service (IERS) did so on 2005 December 31, 7 years later. Prior to this, the longest delay between consecutive leap-second insertions was 2.5 yr, but most leap seconds occurred between 12 and 18 months after the previous one. As of 2005 March, the difference between TAI and UT1 was 0.6 s ( IERS 2005), and immediately afterward, the IERS reported the difference as +0.3 s (IERS 2006). Since 1999 the divergence 1 See

4 No. 1, 2007 TIME DILATION AND THE LENGTH OF THE SECOND 67 Fig. 1. Observed divergence between UTC and TAI, in seconds. [See the electronic edition of the Journal for a color version of this figure.] between TAI and UT has suddenly narrowed. If this is due to changes in Earth s rotation, it would mean the inferred Earth rotation rate has increased since We suggest that the narrower divergence between UT and TAI is due to the new definition of UT1. In 2000, IAU resolution B1.8 redefined UT1 in terms of the angle between the celestial ephemeris origin (CEO) and the terrestrial ephemeris origin ( TEO); is measured with a high level of precision by very long baseline interferometry ( VLBI). This new definition of UT1 was approved by the IAU (Petit 2002). The formula that gives UT1 from is (T u ) ¼ 2(0: þ 1: T u ; 36;525); ð13þ where T u ¼ (Julian UT1 date 2;451;545:0)=36525 (Capitaine et al. 1986). In that paper it is stated in x 6.2 that barycentric dynamical time (TDB), which is based on the SI second, is equal to UT1. Since we show that the UT second is slightly larger than the SI second, the assumption that they are equal will cause small errors in the coefficients of equation (13). Ignoring the difference artificially causes UT to run at the same rate as TAI (or TDB, which has the SI second as its basic time unit). This may explain why no leap seconds were necessary for 7 years. The solitary leap second in 2005 may well be an artifact of decadal time variations reported by the Jet Propulsion Laboratory (Dickey1995). No viable physical explanation accounts for a sudden increase in the Earth s rotation rate after 1999 as the absence of leap seconds would suggest. If Earth has increased its rate of rotation! so that the LOD determined by the Earth s rotation rate is shorter and closer to 86,400 SI seconds, the conservation of Earth s rotational angular momentum requires that Earth s moment of inertia must decrease, by the law I! ¼ I 0! 0. In short, Earth s total moment of inertia I must have decreased to make the rotational velocity! increase. The only notable physical change in Earth s moment of inertia since 1995 has been the Sumatra earthquake of 2004 December 26, which reportedly increased the length of the day by 2.68 s, which is too small to detect but is inferred from calculations. 2 The prior UT LOD is larger than the TAI LOD by 2.14 ms, nearly 3 orders of magnitude larger. No notable seismic activity could be responsible for this significant change in LOD. 2 JPL news release, NASA Details Earthquake Effects on the Earth (2005 Jan. 10). Also, global warming could not account for an increase in!, since rising atmospheric temperatures and expansion would cause an increase in both the atmosphere s moment of inertia and the Earth s overall moment of inertia. This would cause! to be less than previous values, so global warming would increase the divergence between TAI and UT. If one were to use the divergence of the timescales to measure the effects of global warming, it would predict that the overall contracting atmosphere is getting colder, contrary to all other conclusions on this subject. Other possibilities have been offered, but none produce a viable explanation for the Earth spinning faster since ET was calibrated from the theories of the orbits of celestial bodies orbiting the solar system barycenter. Therefore, ET is the coordinate time tied to the reference frame whose origin is the solar barycenter. The calibration of atomic time at mean sea level to ET incorporates the gravitational shift between the geoid and Earth s center of mass. It can be shown that atomic clocks on Earth s ellipsoid (albeit Earth s geoid) will beat at the same rate (Ashby 1990), because the sum of special and general relativity effects for dynamical and gravitational potentials is a constant at this surface. So, Earth s own gravity and rotational dynamics are incorporated in the current definition of the SI second (i.e., the ET second). TAI with the SI second is still the closest realization of ET over the long term. In the original atomic clock calibration, the cesium atomic clocks were calibrated by NPL at nearly sea level over a 4yearperiod.Anyrelativisticannualfluctuationinthesolargravitational potential affecting the Earth, or monthly gravity changes from the lunar potential, were averaged out in the final calibration. So, the atomic clocks do follow a coordinate timescale very precisely. Other small relativistic effects (planetary perturbations to the Earth s velocity, different gravitational potentials from third bodies, etc.) are insignificant compared to tidal friction, which takes 140 years to accumulate a 1 s lapse in UT1. This is discussed in x 5ofthispaper. 4. RELATIVISTIC TIME DILATION WITH A POST-NEWTONIAN METRIC Two of the timescales found in Einstein s theory of general relativity are proper time, that is, the time kept by an observer, and coordinate time t, the uniform independent variable of the equations of celestial mechanics. According to Einstein, an inertial frame is a coordinate system where Newton s mechanical laws hold and is equivalent to Einstein s definition of a resting frame (Einstein 1905). In celestial mechanics, the barycenter of the solar system usually represents the origin of the nonaccelerating reference frame to which orbital motions are referred. We assume that the observer is on the surface of the Earth and adapt the work of Nelson (1990) to establish the proper time of this observer. Nelson develops a post-newtonian metric with signature (+, +, +, ), which sets g 00 < 0. Following Ohanian & Ruffini (1994, p. 322), we take ds 2 ¼c 2 d 2,wheres is the spacetime arc length and c is the speed of light in a vacuum. Nelson establishes the components of the metric tensor for an observer undergoing translational and rotational accelerations and subject to both scalar and vector gravitational potentials. To second order in 1/c, Nelson obtains the following components: g ij ¼ 1 2=c 2 ij þ O 1=c 4 ; g 0j ¼ ðw < rþ j =c þ O 1=c 3 ; g 00 ¼ 1 þ 2W = r=c 2 þ 2=c 2 ðw < rþ 2 =c 2 2ðw < rþ = U=c 2 þ O 1=c 4 ; ð14þ

5 68 DEINES & WILLIAMS Vol. 134 where ij is the Kronecker delta; W is the time-dependent translational acceleration of the observer s frame relative to an inertial frame; w is the time-dependent angular velocity vector of the observer s rotating frame compared to the inertial frame; and U are the scalar and vector Newtonian gravitational potentials, respectively; r is the position vector from the observer to the origin of an inertial frame; (w < r) j is the jth component of that vector. From equation (14) we establish the metric cd 2 ¼ 1 2 c 2 ij dx i dx j þ 2 c (w < r) j dx j cdt " # 2W = r þ 2 w < r 1 þ ð Þ2 2ðw < rþ= U c 2 dt 2 : c 2 ð15þ Consider the inertial reference frame to have its origin at the solar system barycenter and the observer to be on the surface of the Earth. Further, consider the masses of the Moon and the planets to be zero. We adopt the following list for simplification: (1) the origin of the inertial frame is at the center of mass of the Sun if the Earth s mass is ignored, (2) the gravitational potential of the Earth is ignored, (3) the center of the Earth is located at the Earth-Moon barycenter, (4) the scalar potential is evaluated using the Sun s gravity field only, (5) a Keplerian ellipse is used to model W (= r), and (6) the acceleration due to the Earth s rotation on its axis is small compared to the Earth s orbital motion. Therefore, terms in w are ignored. The differential relationship between and t is obtained from equation (15) by dividing both sides of the equation by c 2 dt 2, adopting the list of simplifications, and taking the square root of both sides: d dt ¼ 1 þ c 2 þ W = r c 2 v 2 2c 2 ; where v ¼ jj: ṙ ð16þ From our assumptions, W ¼ r ¼r/r 3 and W = r ¼ ¼/r, where is the reduced mass. Substituting this into equation (16) and rewriting as a definite integral gives 0 ¼ 1 v 2 2 t 0 2c 2 rc 2 Z t dt: ð17þ Integration is best done using the eccentric anomaly E of an elliptical orbit. We adopt the following: v 2 ¼ 1 þ e cos E a 1 e cos E ; rffiffiffiffiffi a 3 t t perihelion ¼ E e sin E; r ¼ að1 e cos EÞ; ð18þ where a is the semimajor axis and e is eccentricity. Taking the definite integral over one anomalistic period with 0 ¼ 0and t 0 ¼ t perihelion ¼ 0gives t ¼ 5 pffiffiffiffiffi a 2 E 2 : ð19þ c 2 0 Using the accepted values for the parameters in equation (19), one obtains t ¼0:77875 s of ephemeris time per anomalistic year. Since leap seconds are determined for the tropical year, we multiply this result by the ratio of the length of the tropical year to the anomalistic year, / , yielding t ¼0:77871 s of ephemeris time per tropical year: ð20þ The observed divergence rate for TAI and UT1 between 1958 and 1999 (see Fig. 1) is s yr 1, only 0.2% different from this calculated rate. This calculation shows that the length of a second of coordinate time is 2:468 ; 10 8 s shorter than a second of proper time. This compares well with the results of x 2, where the ET second is 2:292 ; 10 8 s shorter than the UT second. This also explains the LOD being 2.14 ms longer in UT than the TAI day of 86,400 SI seconds. These calculations support our assumptions that ET should be understood as coordinate time and UT as proper time for an Earth-based observer. Metrics related to timescales considered by the IAU in resolution A4, proposed in 1991 and approved in 2000, contain corresponding terms, but also contain the more complex additions of planetary perturbations (Petit 2002) that we ignore as too small for consideration in our derivation. For example, the IAU resolution defining the relationship between TCB ( barycentric coordinate time) and TCG (geocentric coordinate time) is Z t TCB TCG ¼ c 2 v 2 E t 0 2 þ U ext dt þ v E = ðx x E Þ þ Oc 4 ; ð21þ where x is the solar system barycentric position of the observer and x E and v E are the barycentric position and velocity of Earth s geocenter, respectively. The term U ext is the Newtonian potential of all solar system bodies except the Earth; without the planets, U ext of equation (16). From the solution of equation (21), Petit gets the final result of htcg=tcbi ¼ 1 L c ; ð22þ where L c ¼ 1: ; 10 8 and the brackets designate a sufficiently long-term average taken at the Earth s geocenter. It is interesting to note that result (22) is approximately 3/5 of our result. The dot-product term in equation (21) is small when averaged in Earth s geocentric frame. However, in a frame centered at the solar system barycenter x ¼ 0, and the dot-product term is significantly larger. Placing the dot product (with x ¼ 0) inside the integral would contribute the remaining 2/5 of our result. 5. EARTH ROTATION RATES FROM PALEONTOLOGICAL DATA Tidal friction coupled with gravity can transfer Earth s angular momentum into any satellite s orbital angular momentum, especially the Moon s. Paleontological records give an apparent deceleration in Earth rotation rate that is independent of the timescale divergence between UT and TAI. From several coral ( Wells 1970; Scrutton 1970; Mazullo 1971), bivalve ( Pannella 1972), and sedimentary geological records (Williams 1994), the LOD variation ranges from 370 to 620 days yr 1. The following table shows the results of calculating the change in LOD per year from several reports of the paleontological record. We compute the length of the year in UT seconds at by multiplying mean solar days in Earth s tropical year by 86,400 UTs day 1.TheLOD

6 No. 1, 2007 TIME DILATION AND THE LENGTH OF THE SECOND 69 Reference TABLE 1 Paleontological Tidal Record Millions of Years Ago Days in Year Ancient LOD (s) Loss in LOD (s yr 1 ) Williams (1994) , Wells (1970) , Mazullo (1971) , Wells (1970) , Mazullo (1971) , Scrutton (1970) , Wells (1970) , Pannella (1972) , Pannella (1972) , Wells (1970) , Wells (1970) , Pannella (1972) , Pannella (1972) , Pannella (1972) , Mean in UTseconds for each entry is obtained by dividing this length of a year by the number of days reported in each of the records. Each record yields a value for the rate of change of LOD per year that is equal to the difference between the ancient LOD and current LOD divided by the elapsed millions of years. The changing LOD is interpreted as a deceleration in the Earth s rate of rotation. Paleontological data, taken from Lambeck (1980, p. 360), are shown in Table 1 above. Note that a negative value for the loss in LOD represents a gain or increase in the LOD as time goes forward. The Earth s orbital year is a stable time span over the millions of years shown in the table. If all of the Earth s rotational angular momentum were to be transferred into its orbital angular momentum, the maximum increase in the Earth s orbital period would be less than 16 s. Theories from celestial mechanics have not found any large-scale secular perturbations in the semimajor axis of the Earth s orbit. Thus, it can be said that over the time span of this table, Earth s orbital period has not had any large-scale secular changes that would influence these LOD results. Also, Stephenson and Morrison compiled a precise lunar ephemeris based on ancient eclipses and reported that over the past millennium the LOD grew larger by 1.7 ms century 1 (or 17 syr 1, which is close to the mean value in the table; Stephenson & Morrison 1984, 1995). Figure 1 can be used to determine an equivalent LOD increase, which turns out to be 2137 s yr 1 (divide s yr 1 by ). This is 2 orders of magnitude greater than effects of tidal friction from all observed ancient sources. Using the mean value of 19.7 s yr 1 for the tidal rate of increase in LOD, it would take 139 years to accumulate a total loss of a second of UT. Stephenson wrote, For example, the calculated path of the total eclipse of the Sun witnessed in Babylon in 136 B.C. would be in error by 48.8, corresponding to a time difference of 11,700 s, assuming a uniform rate of rotation (Stephenson 1997). Similar statements have been made that the Earth has lost over 3 hr in rotational time in the past 2000 years ( Nelson et al. 2001). The paleontological evidence does not support their tidal friction rates. If one accounts for time dilation and uses the paleontological tidal rate, then the Babylon eclipse probably occurred 15.4 s earlier, not11,700. All of these data strongly suggest that tidal friction cannot be the cause of the currently observed divergence between UT and TAI. The large time differences in ancient eclipse analyses have ad hoc linear and quadratic adjustments in mean longitude that are actually caused by time dilation and will be discussed in a later paper. 6. CONCLUSION Astronomers have been very precise in documenting formulas that describe UT and ET. Residuals between observation and theory in celestial mechanics have been well documented over many centuries and have led to an understanding that they are related to timescale issues. We showed that relativistic time dilation can explain the divergence between TAI and UT1. Residuals in longitude also show the need for quadratic terms in time. These apparent secular accelerations are modeled by empirical terms applied to artificial geodesic satellites and GPS satellites, as well as to the Moon and other solar system bodies. A separate paper will examine whether or not these apparent accelerations are attributable to time dilation. The atomic clocks maintain the ultraprecise SI second for defining TAI. Establishing UT from TAI can be accomplished easily by scalar multiplication. The scalar to convert from a number of ET seconds (or SI seconds) to the equivalent number of UT seconds is 1:0 2:468 ; This proposed scalar is derived from equation (17) using the average values given for the parameters. It can also be approximated from equation (3). Other metrics can be examined with higher order terms or planetary perturbations, which should not appreciably change the value given in this paper. Tidal friction is at least 3 orders of magnitude larger than most of these higher order contributions. Timing institutions could easily broadcast the time for either timescale: one for scientific (coordinate) time and one for civil (proper) time. The common epoch between TAI and UT1 is 1958 January 0, and the UT1 timescale could be calculated precisely from TAI using the equivalent UT seconds elapsed since this epoch. Defining two different seconds will usually not entail two different values of physical quantities that depend on time units, because the level of uncertainty for those defined physical quantities is greater than the precision of the difference between the UTand SI seconds. Only the speed of light in a vacuum would be impacted, as it is exact using the SI second in its definition. Using the current value for the speed of light, the derived speed of light in UT could be specified when necessary. As shown in x 5, 1 year of UT will lose a full second after 1.39 centuries due to tidal friction. Very accurate clocks calibrated to the UT second would require a single leap-second adjustment every 14 decades for precise civil timekeeping. Financial support from the US Office of Naval Research, contract N , is gratefully acknowledged by both authors. The impetus for this paper arose from work on timescales and GPS satellites by the first author. Thanks are given to Rockwell Collins, Inc., for granting an academic leave of absence to the first author, and thanks are given to the University of South Florida for a sabbatical granted to the second author. Work on lunar theory as a consultant to the National Institute of Standards and Technology andtotheunitedstatesnavalobservatorygaveusefulinsightsto the second author, for which she is very grateful. Special thanks are given to Dennis Ross for his reviews and comments addressing relativity and to Jim Ray for information regarding the determination of UT1. Several personal commitments intervened and regretfully forced both authors to postpone this report many times until its current publication.

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