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.