aa r X i v : . [ phy s i c s . pop - ph ] M a y G ERMAN V ERSION PUBLISHED IN : Sternzeit 45 , No. 2 / 2020, p82–91 (ISSN: 0721-8168) arXiv:2005.07131 [astro-ph.EP] v2: 30th May 2020
Hierarchical Eclipses
Emil Khalisi, Joachim Gripp
Sternzeit e.V., Heidelberg and Kiel, Germanye-mail: khalisi or gripp [email protected] Abstract.
The obscuration of a celestial body that covers another one in the background will be called a“hierarchical eclipse”. The most obvious case is that a star or a planet will be hidden from sight by the moonduring a lunar eclipse. Four objects of the solar system will line up then. We investigate this phenomenonwith respect to the region of visibility and periodicity. There exists a parallax field constraining the chancesfor observation. A historic account from the Middle Ages is preserved that we analyse from different viewingangles. Furthermore, we provide a list of events from 0 to 4000 AD. From this, it is apparent that Jupiter ismost often involved in such spectacles because its orbit inclination is small. High-inclination orbits reducethe probability to have a coincidence of an occultation of that object with a lunar eclipse.
Keywords:
Occultations, Lunar Eclipses, Celestial Mechanics, Planets, Solar System
Received: 6 February 2020. Accepted: 4 March 2020
Occultations of planets in the solar system by the moon re-ceive particular attention in astronomical almanacs. Theyoccur at semi-regular intervals of a few years’ time and, assuch, they are quite common. In the 21st century there area total of 59 cases for the classical naked-eye planets, 34of which happen during daytime. Lunar eclipses are morefrequent, and for a fixed location one can expect roughly 1per year on average.The impressive lunar eclipse of 27 July 2018 took placein the vicinity of Mars. This gave rise to the idea of a com-bination of both spectacles: a “hierarchical eclipse”. It willbe characterised by four bodies placed in a straight line —Sun, Earth, Moon, and a planet. In this paper we put the“double overlaps”, as they would be seen from the sun, toa test. We try to gain insight into the following questions:Did this kind of consecutive covering of celestial objectshappen in the historical past? In particular, are there reportsabout an eclipsed moon eclipsing another planet simultan-eously? How often does such a line-up occur? When willthe next opportunity be scheduled? Do exist cycles or, atleast, accumulations for these hierarchical eclipses?The analysis is based on an empirical list of eventscompiled for the years from 0 to +4,000 AD. We used thesimulation software packages
Guide 9.1 (2017),
Cartes duCiel 4.0 (2017), and
Occult 4.6.0 (2018). The next sectionpresents some few accounts from history that match ourconfiguration. In Section 3 we check the tolerance for visib-ility, and in the subsequent section we discuss the frequencyof the occurrences. Within the scope of our investigation,we achieved some instructive results.For the sake of clarity, we use the following tech-nical terms: The obscuration of a planet will be called “occultation” in order to distinguish it from the “eclipse”of the moon. The instant of disappearance of the planetbehind the moon’s disk is labelled “immersion”, while itsre-appearance is the “emersion”. The eclipse contacts withthe earth’s umbra are assigned U1 to U4, as defined in theastronomical textbooks: begin of the umbral eclipse (U1),begin of totality (U2), end of totality (U3), and end of thedeclining phase (U4), respectively.
One historic account about a hierarchical eclipse was passeddown by the medieval priest Roger of Hoveden. His lifetimecan only be limited to the years between 1174 to 1201 AD.In the relevant paragraph of his chronicle he reports abouta “bright star” being involved in a peculiar scene with theeclipsed moon. The record for the year 756 AD reads asfollows [3]:On the eighth day before the calends ofDecember (23rd November), the moon, onher fifteenth day, being about her full, ap-peared to be covered with the colour of blood,and then, the darkness decreasing, she turnedto her usual brightness; but, in a wondrousmanner, a bright star followed the moon, andpassing across her, preceded her when shin-ing, at the same distance at which it hadfollowed her before she was darkened.The procedure is not correctly described from the astro-nomical point of view. The direction of the course is inver-ted, and the lunar eclipse took place one year earlier, in 755AD. The “star” was Jupiter who was overtaken by the moon . Khalisi & J. Gripp (2020): Hierarchical Eclipses Figure 1: Jupiter was occulted by the lunar disk, whichsuffered an eclipse itself, on 23 November 755 AD. Simu-lated view from London. while the latter was darkened in the earth’s shadow (Figure1). The event goes back more than four centuries prior toRoger’s time. It appears only natural that scribal errors slipin, especially, when one of the copists in the chain of trans-mission failed to understand the meaning of the line himself.The original record does not seem to be preserved.Another record, that would be contemporary with Ro-ger’s life, closely missed a similar sight in Syria. The pat-riarch of the Orthodox Church in Aleppo, Michael Syrus(1126–1199), vividly depicts the nightmare of a solar ec-lipse on 11 April 1176. Thereafter he continues [4]:Fifteen days after [the solar eclipse], in thismonth of Nissan (April) at the decline ofMonday, at dusk, there was an eclipse of theMoon in the part of the sky where the eclipseof the Sun had taken place . . .Of course, it took place in the opposite part of the sky.The eclipse was partial (mag = 0.673) with the Northernrim unobscured. Above the non-eclipsed edge, Jupiter wasshining. The cleric must have regarded the bright spot as ausual star and did not mention it. Somewhat further to theSouth that spot was occulted, e.g. in Kenya and, even more,in South Africa.Another great occasion to watch a hierarchical eclipseoccurred shortly before the telescopic era. On 26 July 1580the eclipsed moon met even two planets, Saturn and Uranus.An observer in Japan could have seen the disappearance ofSaturn, while someone in Northern Australia the same ofUranus. Only on the small Indonesian island of Koror bothplanets were occulted, though not simultaneously. The par-tially eclipsed moon ran over Uranus first, and 10 minutesafter its emersion, the occultation of Saturn began. The sub-sequent lunation provided an occultation of both planets atthe same time, but without the eclipse then. However, inthat year no-one knew about the existence of Uranus, andtelescopes were not invented yet.
Figure 2: Geocentric views of Jupiter and Moon during thelunar eclipse of 755 at four different times. The circle de-notes the parallactic field of view for the occultation to bevisible from an unspecified geographic location somewhereon Earth. Compare Figure 3.
As regards modern times, the sole chance to have hada glimpse of Saturn being covered during an eclipse wason 14 December 1796. It was visible in East Asia, but anyremark about an observation is not known to us.
The occultation of any background object during a lunareclipse implies its opposition with the sun. As a matter offact, Mercury and Venus are excluded.At first, let us imagine a fictitious observer in the cen-ter of the earth looking through a vitreous sphere. Theshadow of the earth (umbra) has an apparent radius as largeas 1.3 ◦ at the distance of the moon. The geocentric observerwould see the maximum duration when the planet is placedexactly in the ecliptic and the moon traverses the shadowcentrally. Then we have the longest totality, and the mooncrosses over the planet alongside its diameter, also provid-ing the longest occultation possible. Such a configurationalmost never happens in practice. Usually, the moon passesthrough the shadow at some displacement from the center,and the planet will hide along a cord behind the moon’sface.For the other extreme, the moon just scratches both theplanet and the umbra with its diametrically opposite fringes.On one edge it would cover the shining dot for a momentand, on the other, it touches the umbra on its antipode. Thisadds an angle of 0.45 ◦ to 0.52 ◦ to the extent of the ter-restrial shadow, depending on the current distance of themoon, since the size of its disk varies between perigee andapogee. For the planet, it gives ≈ . ◦ of tolerance to stayabove or below the ecliptic (Figure 2).In contrast to that geocentric observer, the real viewerhas the advantage to move on the surface of the earth. Histopocentric position gains a parallax, as it would be seenfrom the moon (cosine of his geographical latitude). At the . Khalisi & J. Gripp (2020): Hierarchical Eclipses Figure 3: Region of visibility for the Jupiter occultation of 755 AD. poles this accounts for another ≈ ′ . On the whole, wefind the parallactic field of view to be 4 times larger thanthe diameter of the moon. To ensure the hierarchical eclipseto be seen from some spot on earth (topocentric), the planetmust stay inside this parallactic circle (geocentric). If theplanet is outside, there will be no point on the globe for thehierarchical eclipse.Figure 2 shows the field of tolerance for Roger’s oc-cultation event. From the geocentric view, four instants arepresented:(a) At 18:40 UT, the circle reached Jupiter. On the sur-face of the earth, both the planet and the moon wererising in Dakar, West Africa. The planet was stand-ing already in the umbra, and the occultation of theplanet (its immersion) could be seen during totalitythere. However, from the location of London, Jupiterresided in the penumbra, while the moon was stillrunning towards the planet.(b) At 19:37 UT, the geocentric observer would haveseen the planet closer to the darkened face, while inLondon its immersion occurred. But the totality hasalready ended at 19:25 UT, such that the occultationstarted during the decreasing partial phase. Jupiterentered the umbra, while covered, and returned forvisibility (emersion) after the lunar eclipse had fullyended, i.e. beyond the U4-contact. (c) Going even further eastward, the occultation ofJupiter would be observed from another angle, e.g.in Perm in the Ural: immersion as well as emersiontook place with Jupiter standing in the penumbra.The terrestrial shadow crept up slowly towards theplanet, while the moon was overtaking it. When themoon reached the planet, the eclipse was close tofinish (penumbra neglected).(d) For Beijing it was not until 22:40 UT as the mooncaught up with the shining dot, and the whole pro-cedure passed off sequentially instead of simultan-eously.Note that the fictitious observer at the center of the earthwould not have seen any occultation at all. It is a pure ef-fect of the extent of the spherical earth that the hierarchicaleclipse happened. The region of visibility is shown as thegrey-shaded area in Figure 3. East of the yellow line thetotal eclipse ended (U3-contact), and the red line marks theborder of the finish of the partial phase (U4-contact).A useful insight is that there are no fixed times for im-mersion and emersion, but they depend on the location ofthe observer. The occultation occurs at different times inspite of having deployed the same time frame for reference(like the UT). With regard to Earth’s shadow, the planetseems to stay at different positions in the sky. While theeclipse is in progress, the umbra of the earth moves on, too,reducing the deviation for an eastward observer. . Khalisi & J. Gripp (2020): Hierarchical Eclipses Table 1: Eclipsed moon occults a planet as visible for a “good” observing site. Times in [UT].
Date [AD] Ecl. U1 – U4 Magn. Planet Bri. I – E Best visibility2 Nov 8 21:26 – 23:59 0.46 Mars -1.8 21:58 – 23:02 W-Brazil195 Jul 10 23:41 – 03:36 1.71 Saturn 0.6 03:35 – 04:42 Central Pacific354 Dec 16 12:28 – 15:50 1.33 Saturn -0.4 15:27 – 16:22 E-Africa400 Dec 17 17:05 – 20:27 1.06 Jupiter -2.7 19:46 – 21:11 Seychelles412 Nov 4 18:22 – 22:06 1.60 Mars -1.9 20:06 – 21:05 S-Africa458 Nov 6 21:16 – 00:11 0.80 Jupiter -2.8 21:10 – 22:16 Caucasus480 Sep 5 03:25 – 06:19 0.60 Jupiter -2.9 03:21 – 03:58 Hudson-Bay* 502 Dec 29 12:52 – 16:34 1.64 Saturn -0.3 13:24 – 14:29 Hawaii513 Jun 4 08:02 – 11:50 1.34 Jupiter -2.7 07:23 – 08:39 Bolivia524 May 3 16:25 – 20:02 1.65 Jupiter -2.6 19:59 – 20:37 S-Pole755 Nov 23 16:49 – 20:37 1.40 Jupiter -2.8 19:37 – 20:53 Europe771 Feb 4 08:29 – 11:32 0.93 Saturn 0.3 11:27 – 12:25 Tasmania799 Jul 21 13:47 – 17:30 1.56 Jupiter -2.9 17:02 – 18:17 Kazakhstan* 810 Jun 20 18:04 – 21:36 1.84 Jupiter -2.8 20:11 – 21:34 Madagascar821 May 20 18:30 – 22:14 1.41 Jupiter -2.6 21:42 – 22:52 E-Brazil879 Apr 10 09:29 – 12:52 1.36 Jupiter -2.5 12:08 – 12:43 S-Pole959 Jun 23 06:37 – 09:44 0.94 Saturn 0.1 09:42 – 10:30 Antarctica1052 Dec 8 20:38 – 00:07 1.65 Jupiter -2.7 23:56 – 00:53 Caribbean1176 Apr 25 17:43 – 20:46 0.67 Jupiter -2.5 19:20 – 20:29 Indian Ocean1234 Mrc 17 02:06 – 04:51 0.65 Jupiter -2.5 02:55 – 03:47 Patagonia1312 Jun 19 18:05 – 21:06 1.55 Saturn 0.1 20:05 – 21:11 Namibia1407 Nov 15 11:05 – 14:25 1.19 Jupiter -2.8 11:16 – 11:44 N-Siberia1418 Oct 14 20:15 – 23:53 1.12 Jupiter -2.9 22:40 – 22:48 N-Canada1462 Jun 12 00:32 – 03:17 0.59 Jupiter -2.7 00:16 – 00:58 Antarctica* 1473 May 12 06:21 – 08:27 0.37 Jupiter -2.6 07:09 – 08:26 Polynesia1531 Apr 1 18:05 – 19:24 0.11 Jupiter -2.5 19:01 – 20:03 S-Africa1580 Jul 26 09:27 – 12:47 1.26 Saturn 0.3 10:52 – 11:58 Japan (+ Uranus!)1591 Dec 30 02:12 – 05:45 1.57 Saturn -0.4 05:14 – 06:11 Alaska1796 Dec 14 13:05 – 15:27 0.49 Saturn -0.3 14:52 – 15:55 Siberia2344 Jul 26 10:40 – 14:21 1.34 Saturn 0.1 12:18 – 13:44 N-Pacific2429 Jun 17 10:42 – 11:10 0.02 Saturn 0.1 10:58 – 11:58 New Zealand2488 Apr 26 07:42 – 11:02 1.38 Mars -1.6 08:07 – 08:51 Antarctica2829 Jan 11 01:41 – 05:33 1.81 Saturn -0.4 05:26 – 06:40 N-Pacific2932 Jun 9 22:13 – 23:50 0.21 Jupiter -2.6 22:32 – 23:38 E-Brazil* 2977 Jan 26 07:03 – 10:49 1.65 Saturn -0.4 07:37 – 08:48 S-Mexico2990 May 1 23:57 – 01:09 0.09 Jupiter -2.5 00:26 – 01:21 S-Chile3108 Jun 15 06:26 – 08:57 0.44 Saturn 0.0 06:17 – 07:47 French Polynesia3218 Jul 30 00:00 – 02:23 0.46 Jupiter -2.7 00:06 – 01:05 Madagascar3229 Jun 28 16:30 – 19:20 0.55 Jupiter -2.6 18:03 – 18:48 Madagascar3287 May 19 14:30 – 17:32 0.88 Jupiter -2.5 14:36 – 15:27 Fr. S-Antarctica3376 Aug 21 18:58 – 20:39 0.20 Saturn 0.2 19:06 – 20:33 Maldives3444 Dec 17 16:17 – 19:07 0.59 Saturn -0.4 17:13 – 17:30 Arctic3461 Jul 14 22:35 – 02:12 1.62 Saturn 0.0 22:10 – 22:41 Antarctica3584 Jun 6 12:52 – 16:32 1.16 Jupiter -2.5 14:26 – 14:58 Madagascar3805 Jan 28 08:12 – 11:19 0.93 Jupiter -2.7 08:29 – 09:17 Central Chile3815 Dec 29 12:28 – 15:37 0.89 Jupiter -2.9 13:19 – 13:58 Svalbard3826 Nov 27 15:22 – 18:25 0.65 Jupiter -2.9 17:40 – 18:36 Svalbard3870 Jul 26 00:20 – 03:57 1.48 Jupiter -2.7 00:33 – 01:34 Seychelles3881 Jun 25 08:28 – 12:04 1.82 Jupiter -2.6 09:26 – 10:10 S-Australia
Table 1 lists all hierarchical eclipses we were able to findbetween 0 and 4,000 AD. We make no claim for complete-ness. Dates before 1582 are Julian, thereafter Gregorian. Considered are only cases with at least one planetary con-tact (immersion or emersion) inside the time interval for theeclipse, which is given in column 2: between U1 and U4.The magnitude in column 3 denotes the maximum eclipse.If below 1.0, the eclipse is partial. Columns 4, 5, and 6 give . Khalisi & J. Gripp (2020): Hierarchical Eclipses Figure 4: Number of hierarchical eclipses per century. the name of the planet, its brightness, and the time of im-mersion (I) and emersion (E). These latter times correspondto a “good” site of visibility in the last column 7. It may notnecessarily be an ideal spot, but it would be very close toit. The star (*) in the first column indicates the hierarchicalcoverage for the fictitious position at the center of the earth.It is informative to discover that only four (*) of 49 in-cidents could be seen from the geocentric point. The othersare attributed to the extended surface of the earth. That is tosay, we observe them, because one is placed at a certain par-allax somewhere on the globe. As the case would be, onecould witness either the eclipsing hierarchy, or a simple oc-cultation of the planet, or even nothing. The lunar eclipse,though, is visible in the same way for anyone having themoon above the horizon.Figure 4 shows a histogram of the data for each cen-tury. Hierarchical eclipses seem to be irregularly distrib-uted. There are intervals of accumulation, but we live in anextraordinary long interval of shortage. Jupiter was promin-ent in the first millennium, while Mars is very rarely occul-ted, in general. A strict periodicity cannot be extracted forany planet, but some repetitions and gaps do catch attention.The reasons for it rest upon the characteristics of the plan-etary orbits, as Meeus etal. point out [1]: inclination andeccentricity. The period of eclipses, which is governed bythe draconitic period of the moon, is also essential.For obtaining a cycle, three periods need to be con-sidered. The synodic month of the moon has to be an in-teger, otherwise there is no full moon. The draconitic monthhas to be half its number, otherwise there will be no eclipse.And, thirdly, the synodic revolution of the planet must bean integer, too, in order to meet the opposition. All peri-ods are incommensurable and carry a minute displacementagainst the other on the long run. Here we sketch the qualit-ies briefly for each planet, but a more precise mathematicaltreatment is still pending.
Mars has an orbital inclination of 1.85 ◦ to the ecliptic, butit exhibits the largest departure at the extremes, as seen fromEarth. There are only two small windows, each of about 50 ◦ centered on the nodes with the ecliptic, in which the planetcrosses this reference plane within a favourable latitude ac-cessible for a lunar eclipse. So, the hierarchical eclipse canonly happen in the days of April/May and from October tothe beginning of December. For the other months the ec-liptic latitude of Mars will be too high or too low, and, thus,out of reach for the parallactic field.A secondary obstacle regards the ellipticity of its orbit.The velocity will not be uniform owing to the perihelion andaphelion. Therefore, this gives rise to an advance or retardas compared to a circular orbit. This means that the lunarnode could fail to catch the planet at a convenient instantalthough the conditions are fulfilled. The “period” for arecurrence, if it exists, will possibly be valid for a smallpiece of its orbit only, unless extremely long time scales areenvisioned. Jupiter displays the smallest deviation from the eclipticand can be occulted by an eclipsed moon in any season.This planet is most often involved in hierarchical eclipses,as Figure 4 confirms. However, there are episodes of abund-ances as well as paucities. Two slight periods of 10.9 and57.9 years flash up, and they would be more prominent, ifthe orbit was circular. On the other side, these two quasi-periods have to be merged with the advancing differencewith respect to the lunar node. The revolution of the lunarnode is controlled by the precession of the moon’s plane,and should comply with half its number of the draconiticperiod which is 18.61 years.Again, both series above only satisfy the conditions fora small arc of Jupiter’s orbit. If the incident comes aboutclose to perihelion, then the 11-year period can hold forone or two more chances as in 799–810–821. When the . Khalisi & J. Gripp (2020): Hierarchical Eclipses minute differences on subsequent “hits” have accumulated,the series tears off and there are no hierarchical eclipses forseveral centuries. See [1] for details. Saturn is an intermediate case. The inclination of its orbitis 2.49 ◦ but the planet’s outward distance makes the verticalelevation from the ecliptic appear ± . ◦ at maximum. Fora few weeks in spring and autumn the planet is beyond thethreshold for eclipses, and the hierarchy is suspended. Anasset is that Saturn changes its position along the orbit quiteslowly and stays inside the admissible belt for lunar eclipsesfor several years. The conditions seem to provide a largerstability, but we confess not having checked this in detail.We were not able to identify a period for Saturn, for all itshierarchical eclipses seem to proceed at random. The Moon itself is liable to the extent of the circular paral-lactic field. Its elliptical orbit around Earth brings the anom-alistic month as another period into consideration. This typerules the exact size of the circle and determines whether ornot the planet will be positioned inside or just slightly out-side the ring. The circle “pulsates” in the rhythm of theanomalistic month.For extremely long timescales, the eccentricity of eachorbit varies as well. This holds even for the earth itself.Taken all these factors into account, there will hardly bea cycle of a stable nature. Anyway, all periods turn out in-commensurable in the Solar System, while the purpose ofany search for periods is usually to find an approximationas good as possible.
In closing, let us change perspectives. If the moon is able tocover a background object, then the sun will do so as well.Taking planets as targets, they will hide behind the solardisk. One may call this an “anti-transit”. As a matter ofprinciple, it is unobservable, and our examination becomesjust an academic question.In order to have this state of affairs hierarchically, thesun needs to be covered, too, i.e. the moon will be the ob-vious object to trigger a solar eclipse while the planet isanti-transiting. To achieve that, the planet has to be in thesuperior conjunction and stay close to one of its nodes. Themaximum ecliptic latitude allowed is ± ′ , since this is theapparent radius of the sun. Mercury and Venus can joinour consideration again. The duration of the anti-transit de-pends on the relative speed between the sun and the planet.The sun traverses its own diameter in about 12 hours, how-ever, Mercury and Venus move faster than the sun at theirsuperior conjunction and may stay longer than a day behindit, if the passage is central.If you think, it is too weird, you’re wrong. It hap-pens from time to time, most recently at the total eclipseof 30 June 1954 [2]. Jupiter stepped behind the sun at 10UT that day and remained obscured for the next 17 hours. The moon entered the playground around midday (SouthernScandinavia) and caused an eclipse of the sun between 11and 14 UT. Thus, two celestial objects were deprived fromsight for the observer on Earth. The next opportunity is en-visaged for 14 May 2105, when Mercury will perform itsanti-transit and a partial solar eclipse will have its stage. We presented the rare phenomenon of an “eclipse-occulta-tion” when a planet at opposition is eclipsed by the moonwhich, in turn, is eclipsed by the earth. The example of Ro-ger de Hoveden showed that Jupiter’s disappearance occursat different times for different places, though the same timeframe is used. This effect is due to the parallax for the ob-servers on the surface of the earth. In contrast to that, thelunar eclipse for all observers occurs at the same instant.As known since Antiquity, lunar eclipses can be utilisedto measure the time difference between two places on Earth,if they are widely spaced in geographical longitude. In fact,this method was employed in old times for localising timezones or synchronising clocks.Occultations by the moon can be used to determine itsposition and speed in the sky. Hence, they reveal the sec-ular acceleration that is based on the exchange of angularmomentum between Earth and Moon. For an evaluation ofthose observations and results, the geographical position ofthe observer is relevant. The method fails to work when doc-uments from various (unknown) cultural regions are com-pared, because the occultation takes place at different times,even if a common time scale like the “UT” is used. In caseof known places of observation, still a correction procedurehas to be applied.
Acknowledgments
We thank Dr. Eckhard Fliege for critically reading the ma-nuscript. The results of this paper were published in thepopular German magazine for astronomy,
Sternzeit 45 , No.2 / 2020. They are also included in the Habilitation (ch. 7.7)by EK submitted to the University of Heidelberg, Germany,in February 2020.
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