Relativistic Corrections to Lunar Occultations

Journal of the Korean Physical Society, Vol. 56, No. 5, May 2010, pp. 1694∼1699

Costantino Sigismondi∗ International Center for Relativistic Astrophysics & University of Rome La Sapienza, Piazzale Aldo Moro 5 00185 Roma, Italia; Universit´ede Nice-Sophia Antipolis, Laboratoire H. Fizeau UMR 6525, Parc Valrose, 06108 Nice, France; IRSOL, Istituto Ricerche Solari Locarno, Via Patocchi - Prato Pernice CH - 6605 Locarno Monti, Switzerland

(Received 17 January 2008)

During a lunar occultation, the light of a star is bent by the solar gravitational field, while the lunar profile is much less shifted by relativistic light bending. Relativistic corrections of the order of milliseconds in the absolute timing of the occultations have to be considered for this effect. The accuracy of the absolute timing predictions is studied with the aid of the WinOccult program in the cases of occultations of Saturn (22 May 2007), Venus (18 June 2007), and Pleiades (7 August 2007), including the effect of the irregularities of lunar limb’s profile down to a sub-arcsecond length scale.

PACS numbers: 95.10.Gi, 95.10.JK, 95.10.Km, 71.15.Rf Keywords: Lunar occultations, Astrometry, Ephemerides, Relativistic eﬀects DOI: 10.3938/jkps.56.1694

I. INTRODUCTION is the case for quasar 3C273, whose optical counterpart was identificated in 1962 when the Moon occulted its ra- dio flux and two Saros cycles were later re-examined in Lunar occultations have been used for several purposes 2001 [5]. in astronomy. The geometry of an occultation is deter- Nowadays, accuracies of millarcsecs are common when mined outside the atmosphere; therefore, atmospheric dealing with lunar occultations. refraction and seeing do not affect the timing of the phe- Relativistic corrections in lunar orbit are also known nomenon. experimentally from Lunar Laser Ranging [6], and rela- Ancient occultations recorded by Ptolemy in the Al- tivistic effects due to gravitational light bending by the magest were used recently for recovering the secular ac- solar gravitational field are of the order of milliarcsec- celeration of the Moon [1]. This acceleration is aMoon ∼ onds. They will be here evaluated. −2300/cy2, and after 23 centuries it implies a deviation from the mean motion of 3.4◦. First observations of lunar occultations were made by Timocharis in the third century before Christ (β Scorpii on 21 Dec. 294 B.C.) II. SOLAR ECLIPSES AND LUNAR [2]. The accuracy of timing was ∼10 minutes, and at the EPHEMERIDES average angular velocity of the Moon relative to fixed 00 0 ◦ stars of vm = 0.5 /s, it corresponds to ∼5 = 0.08 . The more diffused software for occultations’ predic- The discovery of close multiple systems is possible from tions in 2007 was WinOccult , while now it has been up- the accurate timing of occultations’ light curves. Identi- graded to Occult4 [7]. WinOccult was based upon DE200 fications of multiple stellar systems during lunar occulta- lunar ephemerides and stars’ positions of the Hyppar- tions became common in the last century. Spica’s occul- cos catalogue. Occultations can be used to check the tation of 10 June 1753, was observed by G. B. Audiffredi accuracy of lunar ephemerides. The lunar ephemerides in Rome. He reported a non-instantaneous disappearing used in the Baily beads module of Occult4 can be ad- of the star [3], which may have been first noted evidence justed in order to reproduce the observed timed sequence of its multiplicity [4]. of beads near the centrality of the solar eclipse (techni- More recently, with precise timing and knowledge of cally speaking solar eclipses are lunar occultations of the lunar orbit, occultations were used in astrometry to de- Sun). The amount of corrections should not exceed 0.500 tect accurately the position of a star or a quasar. This in right ascension and declination of the Moon, but the author expects no corrections at all because DE200 lu- ∗E-mail: [email protected] nar ephemerides are issued up to relativistic effects. A -1694- Relativistic Corrections to Lunar Occultations – Costantino Sigismondi -1695-

the Occultation module of WinOccult uses the average lunar limb, introducing an uncertainty in the absolute occultation timing of ±3 s. Dealing with planetary occultations of Saturn (22 May 2007) and Venus (18 June 2007), I have considered the local Watts’ proﬁle of the Moon to correct the predictions made with VSOP87 or DE200 ephemerides with mean limbs.

III. LUNAR LIMB PROFILE ACCURACY

Fig. 1. Dependance of lunar longitude corrections on libration phases for the WinOccult program. On 7 August 2007 Using Watts profiles should reduce the uncertainty to 00 the correction in longitude was 0.5200. the subarcsecond level of ±0.2 , i.e., ±0.4 s, which is the intrinsic uncertainty on Watts limb’s features. Morrison and Appleby [10] have extensively studied lunar occultations records and they found systematic corrections to be fortiori this affirmation remains valid with the DE421 applied to the Watts profiles. These corrections depend ephemerides used for the period of our interest (years on the Watts angle (read counterclockwise from the lunar 1900-2050) in Occult 4. North pole) on the lunar limb. Their maximum ampli- tude is 0.200. After that, systematic correction random errors up to 0.200 are still possible. 1. Correction to Lunar Longitudes An example of this kind of uncertainty is evident in the case of grazing occultations [11]: the spatial resolution of the Watts profile is 0.01◦ for a Watts angle corresponding There is a difference in lunar longitude among French to ∼ 0.0800 at mean lunar distance from Earth. Since 100 VSOP87 and American (issued by Jet Propulsion Labo- at the same distance corresponds to 2 km, details on ratories) DE200/LE200 lunar ephemerides of +0.5000 in lunar limb are sampled each 160 m, with an accuracy lunar longitude and –0.2500 in latitude. It deals with of ±200 m. It is, therefore, possible to discover new the difference between the center of the figure and the lunar limb’s features during grazes [12] or to miss the center of mass of the Moon, and it gives a systematic occultation by a few meters, as in the case of Electra’s difference between eclipse predictions made by NASA graze of 7 August 2007. and those made by other National institutions like the Bureau of Longitudes. In the occultations’ prediction module of WinOccult, viewing lunar polynomials, that empirical longitude correction appears to be 0.500 + IV. OCCULTATION DATA AND 0.0300 ·sin(2π/T ) with T = 27.2 days (draconitic month), PREDICTIONS and it follows the libration phases of the Moon (Fig. 1). 1. Electra’s Graze on 7 August 2007

2. Corrections to the Mean Lunar Limb At the location of 41◦54055.500 N and 12◦14039.300 E, we were at the predicted southern limit for grazing oc- The irregularities of the lunar limb at a subarcsecond cultation of Electra, the 3.7 magnitude star of Pleiades. scale are treated by deﬁning three values of mean lunar The mean lunar limb would occult the star for the same limbs [8]: average, upper and lower. These mean limbs longitude at latitude 41◦5303400 N, being at 1021.500 N of are used for general predictions: the average one for pre- that location we were 81.500 N, i.e. 2.51 km inside the dicting the location of the umbral path, the upper one mean limb shadow. for the beginning of totality, and the lower one for its The Moon at that time was moving toward northern end (in this way the observers will know the time of each latitudes with an angle of 13.92◦ with respect to celes- event within 3 s of accuracy). These predictions are reg- tial parallels. The corresponding position angle is PA ularly published by IMCCE and NASA on the occasion = 76.08◦. The lower part of the lunar limb shown in of solar eclipses. Fig. 2 has Watts angle (WA) of 171.69◦; the lunar north True lunar limbs have to be used for accurate local pole (WA = 0◦) was at PA = 347.40◦ = −12.6◦ that ◦ predictions, or data reductions in the case of observa- night, so the PA of graze is PAg = 159.09 . A hori- tions of Baily beads. The software that has an access zontal line corresponds to motion perpendicular to PAg, to Watts’ charts of lunar limbs [9] for various libration i.e., at 69.09◦. The star appears to move with an angle phases is the Baily beads module of WinOccult, while of 7◦ with respect to the horizontal. The motion of the -1696- Journal of the Korean Physical Society, Vol. 56, No. 5, May 2010

Table 1. Pleiades occultations of 7 August 2007.

Star name obs.UTC(calc.) WA h v residuals h : m : s (s) degrees arcsec arcsec/s arcsec

V0624 Tau 0 : 19 : 10.5(09) 276 –0.90 0.58 1.58 V1187 Tau 0 : 22 : 02.3(02) 315 0.90 0.41 –0.92 SAO 76119 0 : 36 : 05.6(04) 267 0.55 0.59 0.19 Celaeno 0 : 38 : 28.8(28) 239 –0.46 0.52 0.69 SAO 76136 0 : 41 : 40.9(31) 228 0.21 0.44 4.05* Taygeta 0 : 56 : 08.7(07) 268 0.02 0.59 0.78 SAO 76152 0 : 57 : 14.1(13) 233 –0.41 0.48 0.77 Maia 1 : 03 : 36.1(35) 236 –0.90 –0.50 1.28 Asterope 1 : 15 : 08.7(07) 272 –0.63 –0.59 1.43 Average 0.73* ± 0.80

indicated); these times are taken from Occultation predictions module of WinOccult. They are calculated relative to the mean lunar limb. The height of Watts profile with respect to the mean lunar limb with Morrison and Appleby [8] corrections are in the table, with the velocity of the lunar profile corresponding to each Watts angle (PA = WA − 12.6◦ for that night). The final average was obtained by eliminating the data on SAO 76136. The effect due to the rounding processes op- erated on the calculated timings can be estimated by adding −0.5 ± 0.5·random(x) seconds to the predicted Fig. 2. Geometry of the graze of Electra at the observer’s values. The observed data have been obtained from an location. The plot of the profile includes the corrections to audio record, synchronized to UTC, of the occultation. the Watts charts determined by Morrison & Appleby [8] as The personal equation (PE) [10] has been considered a correction to the profile height above the mean limb. The as 0.34 s. The final result is rather independent of PE, profile deduced from past grazes is independent of the datum which can range between 0.3 and 0.8 s, yielding similar used for the Watts charts (and thus is not subject to these results. The observed time (OT) and the real time (RT) corrections). For the longitude 12◦14039.300 E on the x-axis, of the event is given by the equation OT = RT + PE. there is the time in minutes before and after graze. On the y-axis the km are in (–) or outside (+) the shadow of the The calculated time (CT) is given by CT(mean limb) Moon. The inclined line is the grazing star’s path relative to + hMoon/vMoon. The final differences between OT and the Moon. CT includes PE and uncertainties on the Watts profiles hMoon at each WA. The reapparition events on the dark lunar limb are systematically in delay with respect to the calculated timings. It means that the real lunar longi- Moon is eastward, so a star near the limb moves towards tude is 0.700 ± 0.800 less than the calculated one.The em- ◦ ◦ rising PA, from 160 to 180 . pirical correction to the VSOP87 longitude of the Moon of +0.5200 for that night shown in Fig. 2, therefore, seems not to be justified. Moreover the correction to the Uni- versal Time Coordinated due to the de-rotation of the 2. Pleiades, 7 August 2007 Earth ∆UT 1 = −0.16 s (IERS bulletin B for 7 August 2007) has the effect to shift positively the real lunar lon- 00 In the Table 1, the reapparitions of Pleiades from the gitude of ∼0.03 does not change the situation. dark limb of the Moon have been recorded during a series of occultations on 7 August 2007. The Universal Time Coordinated (UTC) of the reapparition of the stars out 3. Venus 18 June 2007 of dark lunar limb is indicated with 1/10 s of accuracy. The calculated timings are in parentheses and are The reapparition of the Venus limb during lunar occul- rounded to the closest integer second (only seconds are tation of 18 June 2007 has been accurately videorecorded Relativistic Corrections to Lunar Occultations – Costantino Sigismondi -1697- with UTC synchronization and it occurred at 15 : 52 : V. RELATIVISTIC CORRECTIONS TO 29.0 UTC at east longitude 12◦27007.100, lat. 41◦52044.800, OCCULTATION PREDICTIONS and altitude 64 m above sea level. From WinOccult Occultation predictions moduel, the tabultated time of Thanks to the optimal astrometry available for bright reapparition (of the center of the planet) is 15 : 53 : 04, stars we can check light gravitational bending during lu- with an 81-s partial stage of the disk on the lunar limb. nar occultations with maximum exactness. Using IM- Therefore, the reapparition of the first limb is calculated CCE, NASA JPL or USNO ephemerides, the coordinates for 15 : 52 : 23.5 UTC, 40.5 s before the apparition of of stars from the Hypparcos catalogue and the correc- the center of Venus above the mean lunar limb. tions to the mean lunar limb from theWatts charts, and At the Watts angle WA = 256◦, the true lunar limb is now from the profiles made by the Kaguya team [13] 1.30 arcsec above the average lunar limb. The libration (their accuracy in elevation is at meter level, whit sam- phases are 6.22◦ in longitude and −2.59◦ in latitude. The pling steps of 1.5 km in the polar lunar regions and 10 Moon’s velocity (orbital motion + parallax) relative to km near the equator), the evidence of relativistic effects Venus is 0.31 arcsec/s, so Venus is expected to appear measured with an apparatus capable of millisecond ac- 4.2 s later at 15 : 52 : 27.7 UTC because of the limb’s curacy can be obtained. mountain, still 1.3 s before than observed. The gravitational light bending during a lunar occul- Again, the longitude correction, posed to +0.5000, gives tation is properly a differential effect, but we can neglect, an extra 1.6 s of advance of lunar ephemerides with re- in the first approximation, the contribution due to the spect to the observed Moon, whithout this correction the lunar gravitational field, so we consider only the solar prediction for the emersion of Venus would be 29.3 s and field. This effect for a star at χ degrees of elongation O-C = –0.3 s. This final difference would correspond to from the Sun is given by the equation (see Fig. 3): 0.09300 which is within the uncertainty of Watts’ profile: there the mountain is 93 milliarcsec smaller than Watts’ 4GM δχ = atlas, but it includes the rounding error to the near in- c2r tan(χ/2) teger second in the prediction of reapparition’s time. Considering also in this case the correction to the Uni- At χ = 75.2◦, the elongation of the Pleiades during versal Time Coordinated due to the de-rotation of the the occultation of 7 August 2007, and at the observer’s Earth ∆UT 1 = −0.15 s (IERS bulletin B for that day) position r = 1 AU was given by δχ = 10.64 milliarc- has the effect to shift positively the real lunar longitude sec. This bending effect shifts radially away from the of ∼ 0.0300 does not change the situation. Here the agree- Sun the stars’ apparent position; in practice, it shifts ment between ephemerides and observations is greather the star backwards by 10 milliarcsec in the lunar lon- than in the previous case of the Pleiades occultation. gitude direction in this case. That is a delay with respect to ephemerides of 17.2 milliseconds, well measur- able with time accuracy of photoelectric photometry or with new commercial videocameras as SANYO HD1010 which achieve 3.3 ms, or faster CMOS detectors as Phan- 4. Saturn 22 May 2007 tom v7.3 with its 222222 frame per second of acquisition rate [14]. Relativistic corrections vary with solar elongations and Applying the same considerations as those for the case affect systematically, to a level <10 milliarcsecs, all Mor- of Venus, there is a difference of O-C = +0.3 s in the dis- rison and Appleby corrections to Watts dats and all ob- appearance of Saturn’s disk at East Long. 12◦27007.100, servational corrections to Watts profiles. This correction Lat. 41◦52044.800, and altitude 64 m above sea level is still smaller than the uncertainty of <200 milliarcsecs while at Specola Vaticana (East Long. 12◦3900.900, Lat. for each feature of Watts map, but it has to be consid- 41◦44050.200, altitude 350 m), O-C = +1.0 s. The dif- ered when more accurate lunar profiles, as the Kaguya’s ference between the two locations may be due only to one, become available. rounding errors in the predictions. In our cases the temporal accuracy has been down to At both locations, UTC timings were accurate at lev- 1/25 s (video records with standard camcorder). At the els better than 0.1 s. The calculated times included angular velocity of the Moon, that corresponds to 77 the effect of a lunar limb mountain of h00 = 1.3200 at milliarcsecs of accuracy in the case of Venus’ occultation. Watts angle of 125◦. The observations show the lunar In case of planetary occultations, in particular Venus, ephemerides in advance of 1 s with respect to the real the observer’s position indicated in Fig. 3 can be posed in Moon. At the angular speed of Moon relative to Sat- the planet, and the calculation follows with correspond- urn (0.4100/s), this correspond to a shift of 0.4100 again ing elongation angles. In the case of Saturn, the effect is compatible with the above longitude correction of 0.500. similar to a star with the same elongation (1.4 milliarcsec Also the case of Saturn seems to reject such correction: or 2 millisec of time), while for Venus is much smaller, there is a similar advance of the lunar ephemerides with being Venus at 0.72 AU from the Earth. The Moon, for respect to the observed Moon always of ∼= 0.500. the same reason, undergoes a negligible bending effect, -1698- Journal of the Korean Physical Society, Vol. 56, No. 5, May 2010

and 7.3 in declination, 18.2 milliarcsec in total. For com- parison the relativistic precession for Venus’ perihelion is 86.3 milliarcsec per year, which multiplied by the orbital eccentricity gives the observable eﬀect of 0.59 milliarcsec per year [16]. For reference, solar ephemerides VSOP87 and the one used by US Naval Observatory in the Nau- tical Almanac are close, within 0.1 s [17], or 4 milliarc- seconds.

VII. CONCLUSIONS

As devised in the last European Symposium on Occul- tation Project held in Poland (28 August - 4 September 2009 [18]) it is important to observe accurately lunar occultations in order to establish unambiguously the longitude corrections to the ephemerides. Relativistic light bending is detectable during lunar occultations modu- lated by the elongation from the Sun. The uncertainties in lunar limb features are presently larger than this effect. This problem can be partially overcome by observ- ing the same star occulted after a sidereal month (there Fig. 3. Geometry of a lunar occultation, with χ and r is a small change in libration phase). Reduction proce- shown. dures have to consider explicitly the corrections to lunar longitude.

ACKNOWLEDGMENTS

Thanks to Paolo Colona and Denise Leone who ob- servted with me the occultation of the Pleiades. Thanks to Prof. Giovanni Bernardini for his observations of Sat- urn occultation and to Denise Leone for Venus ingress data, to Micol Benetti, Paolo Fermani, Marco Innocenti, Alessandra Mastrobuono Battisti, Irene di Palma, Sonia Paoloni and Luca Vignoli for the cooperation in Venus egress absolute timing. Fig. 4. Relativistic corrections versus lunar phases. Nega- tive corrections correspond to phases of the crescent Moon.

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