THE ALMOST TOTAL LUNAR ECLIPSE OF 2015 APRIL 4

By Helio C. Vital*

A Sensitive, Complex and Hard-to-Predict Eclipse Movie on the Lunar Screen

Lunar eclipses must be regarded as complex phenomena with characteristics that still remain to be fully understood and predicted with greater accuracy. The reason for that lies in the fact that our dynamic atmosphere also contributes to the shadow cast by our planet, enlarging it by a fraction that varies significantly from one eclipse to another or even during a particular eclipse itself. Therefore, even if Earth`s shadow is accurately simulated by using a rigorous model for the umbra, the best that can be done, as far as predictions are concerned, is to rely on a mean value for the atmospheric enlargement derived from observation. In theory, the edge of the umbra is defined as where the most abrupt gradient of light occurs. In practice however, it is not simple to determine. For centuries, experienced observers have noticed that the border of Earth`s inner shadow (umbra), when projected on the sensitive lunar screen, is diffuse and ragged as analyses of thousand of mid-crater and limb contact timings have consistently shown. Its lack of definition is the major cause of errors (roughly half a minute) in contact timings made by a typical observer. Thus while predictions of mean contact times for solar eclipses are mostly found to agree with observations within a couple of seconds or even less, rigorous comparisons for lunar eclipses usually agree within ±0.3 minute in average, typically ranging between ±0.15 and ±0.5 minute.

Predicting, Observing, Analyzing and Improving Predictions

For a quarter of a century now, we have used programs we created and extensively tested for predicting circumstances of lunar eclipses. They have also been used in analyses of 2,140 contact timings made by our REA/Brazil group of observers in addition to 5,000 from other sources. The results have been consistent. The programs are versatile and allow to select from 7 different models for the umbra as well as for the penumbra and to enter any value, either for the atmospheric enlargement for Earth`s radius (i.e. the lunar parallax, an improvement of Danjon`s approach) or for the overall enlargement of Earth`s shadow (a less rigorous approach proposed by Chauvenet). Earth`s oblateness has always been taken into account properly in our calculations that are based on models for the umbra published by Byron Soulsby (BAA Journal) more than 20 years ago. We have also included an algorithm recently proposed by Dave Herald and Roger Sinnott (BAA Journal, Oct. Issue, 2014). Its predictions practically match those calculated by using the models cited by Soulsby (better than ±0.03%). Those recent algorithms closely reproduce mean observed contact times (better than ±0.3% enlargement or roughly ±0.3 min.).

By comparing calculations for contact times with observations, we have learned that the use of a rigorous model for the umbra is strictly necessary. In addition, predictions must be based on a mean value of enlargement of the Moon`s parallax (as in Danjon`s approach) derived from observations. Furthermore, multiplying the total size of the umbra by 1.02, as proposed by Chauvenet is not theoretically justifiable and leads to significantly larger dispersion in data.

From analyses of tens or hundreds of contact timings, we have derived the mean atmospheric contribution for each eclipse our group has observed. Our statistics of more than 7,000 crater and limb timings from 30 eclipses shows that Earth`s upper atmosphere enlarges between 1.1% and 1.6% the visual radius of our planet around a mean of 1.343±0.077 (1σ) %. Finally, since our calculations properly account for Earth`s oblateness, we have not observed any systematic deviation that could be associated to contacts occurring at large umbral angles (closer to the lunar poles).

A Critical Test of Lunar Eclipse Models

As the lunar disc skimmed the outskirts of Earth`s umbra, the eclipse on April 04 2015 became indeed a critical test both for calculations and observations for several reasons. Fred Espenak had initially predicted the value of 1.0008 for its umbral magnitude and then updated it to 1.0001. The difference between his predictions hint at probable uncertainties associated to his predictions (roughly ±0.0005, 1 sigma). His calculations are based on an improved version of Danjon`s simplified model (that mostly compensates for Earth`s oblateness) and that has provided predictions in good agreement with observations for many years now. Without such improvement, errors roughly ten times larger would probably arise, as in predictions based on Chauvenet or Danjon`s simplified models for Earth`s umbra. Thus the use of both models would be inadequate for this extreme case where variations in the order of 10-4 in umbral magnitude might make a difference between totality and partiality.

In addition, Herald and Sinnott had predicted magnitude 1.002 for the eclipse, using 1.36 % as a mean value for the atmospheric enlargement calculated from a precious compilation of more than 25,000 contact timings. That is a figure 0.1% larger for the radius of the umbra than that calculated by Espenak. Also, it would correspond to a mere 4-arc sec distance of minimum penetration of the lunar disc into the umbra, predicted by both Espenak and Herald to be quite a challenge to identify visually.

 

A Challenge for Experienced Observers: Partial or Not?

Most observers have reported that a thin sliver of light remained visible throughout the period of the so-called pseudo totality. In fact we had observed that same feature during a few bright eclipses of umbral magnitudes very close to one. When most of the lunar disc is in the umbra, areas adjacent to its edge become relatively bright due to enhanced contrast. However to complicate things further, during the eclipse on April 4, the umbra practically covered the whole disc of the Moon, not sparing a significant fraction that would provide the visual contrast needed for a more precise identification of its fuzzy border. It is interesting that some observers claimed that a few craters (tens of kilometers in diameter) remained uncovered by the umbra. However, that would require predictions to be off by tens of arc seconds (instead of only a few arc seconds at most), and the chance of that happening would be very small, considering the accuracy of the models used (Sinnot-Herald`s and Espenak`s). So that it seems reasonable to conclude that most of the area inside the bright sliver that lingered at mid-eclipse was in fact part of the not-so-dark outermost part of the umbra.

All will agree that at mid-eclipse the lunar limb and the border of the umbra practically intersected.  However, if the umbra happened to be only slightly smaller than predicted in radius, there would remain a very discrete “hair-thin bright sliver” next to the moon`s limb, hardly noticeable. Then the question that remains is: has anyone really seen the arc sec-thick sliver corresponding to the not-so-bright innermost part of the penumbra or simply confused it with the much thicker and not-so-dark outermost part of the umbra?”

 

Bow and Arrow Targeting at the Lunar Screen at Mid-Eclipse

A few hundreds of crater and limb timings would probably help us get a better picture of what really happened at mid-eclipse. Some observers probably made them, but unfortunately such observations have not been made available.

It is common knowledge that imaging techniques can produce large variations in exposure, contrast, dynamic range etc of digital photographs and videos. However, my feeling is that the current lack of information from visual observations of this particularly special eclipse justifies an attempt to obtain information from such sources.

Keeping that in mind, I decided to analyze the video from Griffith Observatory in Los Angeles, webcast during the April 4 lunar eclipse and posted. So let us make an attempt to find what the numbers and the smooth fitted curves might have in store for us.

Our analyses simply consisted in carefully measuring, at several times, both the length of the chord and the maximum thickness (arrow) of the arched residual sliver (bow) of light, corresponding to the residual area of the Moon`s disc inside the innermost penumbra. That was done a few minutes before as well as after mid-eclipse, when it was still possible to identify the border of the umbra in the images. The data points were then plotted and fitted to second-degree polynomials. Parabolic fittings were chosen for their simplicity and symmetry features that would facilitate determining the minima. Our initial goal was to get information on mid-eclipse circumstances from data gathered during the partial phase, when the border of the umbra had a better definition. Some data points were obtained from the video with some difficulty due changes in settings, that produced abrupt brightness variations in the image of the lunar disc. In addition, three minutes before mid-eclipse, there was a large increase in brightness that moved the border of the umbra significantly. Such period lasted until some 15 minutes after mid-eclipse, preventing consistent data to be taken.

Shown in Fig. 1 is how the maximum thickness or arrow of the bright sliver (corresponding to the area of the lunar disc still uncovered by the umbra) varied with time during the period centered at mid-eclipse. Zero thickness would correspond to zero partiality, or totality.

In Fig. 2 the variation of the chord with time is depicted. Its length is measured straightly from one extreme of the arched sliver (or bow) to the other. A non-zero minimum value at mid-eclipse would mean totality did not occur. Errors associated to the analysis of the chord are expected to be smaller than those for the arrow.

Table 1 summarizes the parameters obtained from the analyses of the arrow and the chord of the bright sliver, both expressed as parabolic functions of time. Least-squares fitting provided the corresponding equations, from which the corresponding minima were calculated. Our very simple approach has been intended only to provide us with further qualitative information on what happened during mid-eclipse. However the results turned out so consistent that they can also be used to provide approximate quantitative estimates of the umbral magnitude at mid-eclipse.

As determined from both fittings, mid-eclipse apparently occurred at 2:46 (elapsed time of the video). That probably corresponded to 12:00 UT, if both predictions and reports by experienced observers regarding the time of minimum illumination of the Moon are taken into account.

In the analysis of the arrow, the umbral magnitude at mid-eclipse was calculated as equal to one minus the interpolated minimum thickness of the sliver divided by the diameter of the Moon (measured directly on the screen). In addition, in the analysis of the chord length, the angles subtended at the center of the Moon and at the center of the umbra by the sliver with minimum chord length were both calculated. The corresponding arrows were determined from simple trigonometric relations and subtracted to yield the thickness of the residual bright sliver at mid-eclipse. Finally, the corresponding magnitude was calculated in the same way as for the arrow. Errors, expressed in standard deviations, were estimated from propagation of uncertainties in the parameters of the fitting curves.  

 

Evaluating Circumstances

Our very simple approach has been initially meant to provide qualitative information on mid-eclipse circumstances. However, considering the high statistic significance of the results from both methods, as indicated by the correlation coefficients approaching 1 (r2~1), rough quantitative estimates of the umbral magnitude at mid-eclipse can also be considered. Consistently, both seem to indicate that probably totality did not occur. Since the chord method is expected to yield much more accurate results than the arrow method, our analyses will only focus on it henceforth as other three effects will be discussed.

 

Table 1: Curve Fitting Parameters and Mid-Eclipse Magnitudes from

Analyses of Arrow and Chord of the Residual Arched Sliver of Light

FEATURE

THICKNESS

CHORD

Number of Data Points

16

25

Fitted Parabolas in the form:

f(x)=ax2+bx+c

a= 0.00180556

b= -0.00020245

c= 0.01087138

a= 0.01615765

b= 0.00211096

c= 0.3380456

Interpolated Elapsed Time of Minimum (Video)

2:45:49

2:46:44

Correlation Coefficient (r2)

0.9917

0.9947

Standard Deviation

0.030

0.125

Minimum of Function

0.011 cm

0.338 cm

Mid-Eclipse Umbral Magnitude

0.9992±.0006

0.99990

(+0.00005,-0.00010)

 

Earth`s Oblateness

Soon after the eclipse, there were several claims that the eclipse had not been total because Earth`s oblateness had not been taken into account in predictions. Such claim failed to acknowledge that predictions based on rigorous algorithms had also been made. Espenak mostly compensates for it in his predictions. Herald and Sinnott had also predicted a magnitude of 1.002.  In addition, we had posted ours on our webpages last year and the possibility of a partial eclipse had also been calculated. Based on our all-time average atmospheric enlargement, our calculations had predicted that a brief period of totality would likely occur, though not neglecting the chance of a deep partial eclipse.

Old simplified models for the umbra, like those proposed by Chauvenet or Danjon can lead to relatively large errors in predictions and should not be in use any longer, particularly if extreme circumstances are expected such as in the April 4, 2015 eclipse. For this particular eclipse, they would have led to errors of approximately 1/4 (=21/80 km) of the atmospheric contribution, due to the high (72.7°) umbral angle of closest approach of the limb to the border of the umbra, a figure much larger than the discrepancy found in this work. Thus, they should be replaced, either by an improved simplified model (such as the one used by Espenak) or a more rigorous algorithm (like that recently proposed by Herald and Meeus, JBAA, Oct. 2015), which properly accounts for the Earth`s oblateness among other additional features.

 

Moon`s Parallax

The results from the chord method indicate that the eclipse probably failed to reach totality by less than 0.0001 magnitude. That is a very small figure that would correspond to a distance of only 0.35 km at the moon`s distance from Earth. Would there be a way to test it? Maybe. A careful inspection of photos taken at mid-eclipse gives us the impression that, on average, the residual bright sliver was significantly less prominent in images obtained in Oceania than those from the Western North America. Earth`s radius was subtending at the center of the Moon an angle of 0.907°. That means that those on Earth watching the Moon on the horizon would be seeing an addition 0.21 km or 0.11 arc sec of the Moon`s disc towards the edge of the umbra than those seeing it at the zenith.  Then accounting for the difference in altitude of the Moon in the two regions i.e. 57° for Los Angeles compared to 21° for New Zealand that yields a difference of 0.04 arc sec. That means that American observers could have seen (0.11x0.39)=0.04 arc second beyond those in Oceania towards the edge of the umbra, corresponding to a thicker sliver. Then the angular distance to the edge of the umbra for observers in Oceania would be 0.18” (0.0001x14.83`x2x60) – 0.04” = 0.14”, a significantly smaller distance that maybe could explain the thinner sliver.  

Moon`s Oblateness

Much has been discussed about the effects of Earth`s oblateness (1/298) on lunar eclipse predictions but nothing was mentioned about those associated to the oblateness of the Moon (=1/825). In fact they are totally negligible for most eclipses, since the Moon`s radius is not included in the calculation of the size of umbra. However, in this case, it could have made a difference as our satellite acted as a very sensitive probe that monitored the extreme north of the umbra. This is not relatively frequent because, as the Moon`s path across the umbra is usually much deeper, contacts tend to occur further away from the poles. It is now known that the Moon is not perfectly spherical but ellipsoidal, so that a point on its poles is 2.17 km closer to its center than a point on its equator. Consequently if a mean radius was considered in predictions, then our analyses were made for to a point located 1.08 km closer to the lunar center. However, in spite of that, it apparently remained in the penumbra at mid-eclipse as concluded from our analyses. Then a correction must be added to account for that distance, equivalent to 0.56 arc sec (at the Moon`s distance) or (0.56/14.83x2x60) = -0.00031mag. Thus the umbral magnitude at mid-eclipse, relative to the mean radius of the Moon, would be: 0.99990-0.00031= 0.99959.

Analyzing Results to Improve Predictions

Magnitude 0.99959 corresponds to an atmospheric enlargement of 1.219% at mid-eclipse. Relatively speaking, such figure would rank very low in our statistics of 30 past eclipses. In fact it would be the lowest mid-eclipse enlargement so far, being (1.219%-1.343%)/0.077% = 1.6σ below our all-time mean. Thus its probability, assuming that the Normal Distribution holds, would be roughly 5% only. That explains why most predictions, that have to rely on a mean value for the varying atmospheric enlargement of Earth`s radius, failed to predict a total eclipse: simply because a total eclipse was far more likely than a deep partial eclipse.  But then why was the atmospheric enlargement so low at this eclipse, causing the radius of the umbra to be 0.2% smaller than its most probable value? That is indeed a very complex question and its answer probably lies at the top of Earth`s mesosphere, that needs to be much better understood. A summary of conclusions and results from our analyses is presented in Table 2 to facilitate comprehension.

 

Table 2: Summary of Discussions and Results

Step

Description

Result

1

Analysis of Video by Method of Chord

Mag= 0.99990

2

Earth`s Oblateness (accounted for properly)

Mag = 0.9990

3

Moon`s Parallax Caused Thicker Sliver from America

Possible

4

Moon`s North Pole 1km below mean radius

Mag=0.99959

5

Contact Predictions for Eclipse Based on 1.219%

Enlargement of Moon`s Equatorial Horizontal Parallax

1.6σ below

statistic mean

6

Lunar eclipses are mostly empirical due to unpredictable variations in Earth`s atmosphere

Good Observing!

 

The corrected value of enlargement has been entered in our programs in order to provide improved predictions for limb and crater contact times. The results, listed in Table 3, can then be directly compared with timings made during the eclipse.

 

Table 3: April 4, 2015 Improved Predictions for

Immersion and Emersion Contact Times

2015 APRIL 04 LUNAR ECLIPSE - LIMB AND CRATER CONTACT TIMES PREDICTIONS

BASED ON AN ATMOSPHERIC ENLARGEMENT OF EARTH`S RADIUS OF 1.219%

Immersions

Emersions

Feature

UTC

(hh:mm.d)

Umbral Angle (°)

Feature

UTC (hh:mm.d)

Umbral Angle (°)

Umbral Eclipse

Begins (U1)

10:15.7

43.8

Plato

12:30.3

48.4

Grimaldi

10:20.2

49.8

Pico

12:35.0

45.1

Billy

10:24.4

46.8

Aristarchus

12:35.7

30.7

Campanus

10:33.0

42.2

Aristoteles

12:37.9

49.6

Kepler

10:36.3

57.2

Eudoxus

12:43.4

46.1

Tycho

10:39.9

38.0

Grimaldi

12:43.5

15.8

Aristarchus

10:40.9

64.8

Kepler

12:46.9

23.1

Copernicus

10:46.8

59.0

Timocharis

12:47.0

34.2

Pytheas

10:52.1

64.5

Pytheas

12:47.7

30.4

Timocharis

11:00.0

68.3

Billy

12:52.8

12.7

Dionysius

11:06.5

57.9

Copernicus

12:54.0

24.9

Manilius

11:06.6

63.3

Manilius

13:04.9

29.2

Menelaus

11:11.5

64.7

Menelaus

13:06.7

30.5

Censorinus

11:13.7

57.2

Campanus

13:07.7

8.2

Pico

11:15.4

79.3

Plinius

13:10.5

30.6

Goclenius

11:15.5

53.2

Dionysius

13:15.8

23.8

Plinius

11:15.5

64.8

Proclus

13:18.1

32.3

Plato

11:19.8

82.7

Tycho

13:19.5

3.9

Langrenus

11:21.6

54.4

Censorinus

13:24.2

23.1

Taruntius

11:22.5

60.9

Taruntius

13:25.6

26.8

Eudoxus

11:26.6

80.4

Goclenius

13:33.3

19.1

Proclus

11:27.1

66.5

Langrenus

13:37.1

20.3

Aristoteles

11:30.7

83.9

Umbral Eclipse

Ends (U4)

13:45.1

9.7

 

Urgent Need for Improvements

These analyses of the 2015 April 4 lunar eclipse video from Griffith Observatory provided information on mid-eclipse circumstances. Based on consistent results from measurements of the residual bright sliver by using the chord method, it can be concluded that totality was probably missed by approximately only one part in 10,000 as expressed in eclipse magnitude. The findings provide support to many claims made by experienced observers that the eclipse was indeed deep partial rather than total.

Furthermore, comparative inspections of mid-eclipse photos show that apparently the residual bright sliver was more pronounced in images taken in Western North America than those from Oceania. The Moon`s parallax has been suggested as a possible explanation for that effect. Usually of negligible impact on eclipse circumstances, it could have made a difference this time, considering the magnitude of the eclipse was extremely close to 1.

In addition, a correction was made to the measured magnitude of the eclipse due to the fact that the thin residual sliver corresponding to the innermost penumbral shading was observed near the Lunar North Pole, a region known to be 1.1 km closer to the center of the Moon`s disc than the mean lunar radius considered in predictions. So the umbral magnitude initially measured as 0.9999 was corrected to 0.9996. This new result corresponds to an increase of 1.22% enlargement in Earth`s parallax, a value 1.6 standard-deviation lower than its all-time mean. Consequently, predictions based on a mean atmospheric enlargement failed to predict partiality. This corrected enlargement was then used to calculate improved predictions of contact times for the 2015 April 4 lunar eclipse.

The science of lunar eclipses is indeed in severe need of observations. In addition, predictions must be made using recently improved models for the umbra. Also, the relatively high degree of unpredictability of limb and crater contact times should be acknowledged by official sources and largely informed. That would require predicted contact times to be expressed with accuracy up to tenths of a minute of time, so as not to induce readers into thinking that uncertainties in prediction are always ±1 sec.

Finally, we would like to add (as experience has taught us) that, when it comes to further understanding lunar eclipses, predictions are great, observations are even better and making and comparing both would be just about the ideal thing to do.

________________________________________________________________________________________________

 

*Helio de Carvalho Vital is a physicist and a nuclear engineer with a PhD from Purdue University (1985). As an amateur astronomer, he has performed an extensive research on lunar eclipses as head of the Eclipse Section of the Brazilian Observational Astronomy Network since 1990. Since 2003 he has maintained the Lunissolar Eclipse Pages.