HOW OFTEN IS THAT STAR OCCULTED?
------------------------------
John Pazmino
NYSkies
nyskiesastronomy@earthlink.net
2006 January 16
Introduction
----------
As a result of a renewed correspondence with a Midwest NYSkies
astronomy friend a couple years ago, the problem arose: How often is
any particular star occulted by the Moon? With the new round of
occultations of the Pleiades and Antares starting in 2005-2006, the
question surfaced again.
That is, many of us know casually that occultations of, say,
Regulus, come in waves or seasons. There are a bunch of occultations,
nearly every month, for a while, then there are none at all. Is there
a pattern here?
I gathered 100 years of occultations, from 1950 thru 2049, of the
six brightest stars (other than the Sun) hidden by the Moon to see
what's what. The stars are Aldebaran, Antares, Elnath, Nunki, Regulus,
and Spica. A seventh star, Pollus, was examined for being included in
catalogs of stars vulnerable by the Moon, but it had no occultations
in this 100 year period.
The occultation events were generated by computer and output as
ASCII text. It was a simple, tho tedious, task to segregate the list
into the several stars. I used several programs and none of them
allowed me to single out a particular star to study.
Yes, for any one star there is a run of almost monthly
occultations, then a hiatus with no occultations at all. But! A closer
inspection revealed something very curious.
Saros
---
I should have expected this but I didn't at first. The mechanics
of occultations is a special case of solar eclipses. There is a
saros involved with them just as for eclipses!
A Saros interval contains 223 cycles of lunar phases, from new to
new (solar eclipse) or full to full (lunar eclipse). These 223 cycles
sum to 18y 10/11d 8h. The day count toggles for the number of leapdays
within the Saros, whether 4 or 5.
A solar eclipse occurs if the Moon crosses her nodal point at new
Moon. This may be either the ascending or the descending node, so we
have really two interleafed sets of eclipse operating simultaneously.
What most readers forget is that the Saros also contains 241
sidereal cycles of the Moon. The sidereal cycle is ignored because it
does not contribute to the production of eclipses.
It is the repetition of the Moon's place in the stars that makes
for recurring occultations. This is accomplished by the 241 whole laps
of the lunar orbit during one saros interval. The lunation cycle does
not factor into the occurrence of the occultation. (It may be of
importance for assessing the observing conditions of the event.)
Like for eclipses, the Saros does unravel after a while the slivers
of inexactitude add up to drag the Moon and Sun out of line. No more
eclipses are produced. A Saros begins when these slivers accumulate to
push the Sun and Moon into alignment and start making eclipses.
Rule-of-19
--------
Many of us know the 'rule-of-19', by which a one eclipse is
followed by an other on the same calendar date but 19 years later.
This works because 19 calendar years is quite 12 lunations longer than
one Saros. The Moon is new again and stands in front of the Sun.
This rule applies also to occultations. Knowing when a specific
one takes place, an other will occur 19 years later on the same date.
By adding 13 sidereal periods to the 241 of a Saros, we have 254
cycles. This is almost exactly 19 calendar years of 365.25 days!
19 calendar years brings the Moon round to her initial location
among the stars to repeat the occultation. The calendar date may
toggle, like for the saros interval, due to the leapdays added during
the interval, 4 or 5.
The table here shows the rule-of-19 for Antares. Dates are
'dd/mm/yyyy' in Universal Time. These occultations are global. A given
location on Earth will miss many of them for daylight or the Moon's
absence from the local sky. I backfilled 1949 to show the start of the
earlier series.
OCCULTATIONS OF ANTARES, 1950-2049, SHOWING THE RULE-OF-19
--------------------------------------------------------
13/03/2042
10/04/2042
07/05/2042
03/06/2042
30/06/2042
28/07/2042
25/08/2023 24/08/2042
21/09/2023 20/09/2042
18/10/2023 18/10/2042
14/11/2023 14/11/2042
12/12/2023 11/12/2042
07/01/2005 08/01/2024 08/01/2043
04/02/2005 05/02/2024 04/02/2043
03/03/2005 03/03/2024 03/03/2043
30/03/1986 30/03/2005 30/03/2024 31/03/2043
26/04/1986 26/04/2005 26/04/2024 27/04/2043
24/05/1986 24/05/2005 24/05/2024 24/05/2043
20/06/1986 20/06/2005 20/06/2024 20/06/2043
18/07/1986 18/07/2005 17/07/2024 18/07/2043
14/08/1986 14/08/2005 14/08/2024 14/08/2043
10/09/1967 10/09/1986 10/09/2005 10/09/2024 10/09/2043
07/10/1967 07/10/1986 08/10/2005 07/10/2024 08/10/2043
04/11/1967 04/11/1986 04/11/2005 04/11/2024 04/11/2043
01/12/1967 01/12/1986 01/12/2005 01/12/2024 01/12/2043
29/12/1967 29/12/1986 29/12/2005 28/12/2024 28/12/2043
25/01/1968 25/01/1987 25/01/2006 25/01/2025 25/01/2044
20/02/1949 21/02/1968 21/02/1987 21/02/2006 21/02/2025 21/02/2044
19/03/1949 19/03/1968 21/03/1987 21/03/2006 20/03/2025 19/03/2044
16/04/1949 16/04/1968 17/04/1987 17/04/2006 16/04/2025 16/04/2044
13/05/1949 13/05/1968 14/05/1987 14/05/2006 14/05/2025 13/05/2044
10/06/1949 10/06/1968 11/06/1987 10/06/2006 10/06/2025 09/06/2044
07/07/1949 07/07/1968 08/07/1987 08/07/2006 07/07/2025 07/07/2044
03/08/1949 03/08/1968 04/08/1987 04/08/2006 04/08/2025 03/08/2044
30/08/1949 31/08/1968 01/09/1987 01/09/2006 31/08/2025 30/08/2044
27/09/1949 27/09/1968 28/09/1987 28/09/2006 27/09/2025 26/09/2044
24/10/1949 24/10/1968 25/10/1987 25/10/2006 25/10/2025 24/10/2044
21/11/1949 21/11/1968 22/11/1987 21/11/2006 21/11/2025 20/11/2044
18/12/1949 18/12/1968 19/12/1987 19/12/2006 18/12/2025 17/12/2044
14/01/1950 14/01/1969 15/01/1988 15/01/2007 14/01/2026 14/01/2045
11/02/1950 11/02/1969 12/02/1988 11/02/2007 11/02/2026 10/02/2045
10/03/1950 10/03/1969 10/03/1988 11/03/2007 10/03/2026 09/03/2045
06/04/1950 06/04/1969 06/04/1988 07/04/2007 06/04/2026 06/04/2045
04/05/1950 04/05/1969 04/05/1988 04/05/2007 04/05/2026 03/05/2045
31/05/1950 31/05/1969 31/05/1988 01/06/2007 31/05/2026 30/05/2045
27/06/1950 27/06/1969 27/06/1988 28/06/2007 27/06/2026 27/06/2045
25/07/1950 25/07/1969 25/07/1988 25/07/2007 24/07/2026 24/07/2045
21/08/1950 21/08/1969 21/08/1988 22/08/2007 21/08/2026 20/08/2045
17/09/1950 17/09/1969 17/09/1988 18/09/2007 17/09/2026 16/09/2045
15/10/1950 15/10/1969 15/10/1988 15/10/2007 14/10/2026 14/10/2045
11/11/1950 11/11/1969 11/11/1988 11/11/2007 11/11/2026 10/11/2045
08/12/1950 08/12/1969 08/12/1988 09/12/2007 08/12/2026 07/12/2045
05/01/1951 05/01/1970 05/01/1989 05/01/2008 04/01/2027 04/01/2046
01/02/1951 01/02/1970 01/02/1989 01/02/2008 31/01/2027 31/01/2046
28/02/1951 01/03/1970 28/02/1989 29/02/2008 28/02/2027 27/02/2046
28/03/1951 28/03/1970 28/03/1989 27/03/2008 27/03/2027 27/03/2046
24/04/1951 24/04/1970 24/04/1989 23/04/2008 23/04/2027 23/04/2046
21/05/1951 21/05/1970 21/05/1989 20/05/2008 21/05/2027 20/05/2046
18/06/1951 18/06/1970 17/06/1989 17/06/2008 17/06/2027 17/06/2046
15/07/1951 15/07/1970 15/07/1989 14/07/2008 14/07/2027 14/07/2046
12/08/1951 12/08/1970 11/08/1989 10/08/2008 11/08/2027 10/08/2046
08/09/1951 08/09/1970 07/09/1989 07/09/2008 07/09/2027 07/09/2046
05/10/1951 05/10/1970 05/10/1989 04/10/2008 04/10/2027 04/10/2046
01/11/1951 01/11/1970 01/11/1989 31/10/2008 01/11/2027 31/10/2046
29/11/1951 29/11/1970 28/11/1989 28/11/2008 28/11/2027 28/11/2046
26/12/1951 26/12/1970 25/12/1989 25/12/2008 25/12/2027 25/12/2046
23/01/1952 22/01/1971 22/01/1990 21/01/2009 22/01/2028 21/01/2047
19/02/1952 19/02/1971 18/02/1990 17/02/2009 18/02/2028 18/02/2047
17/03/1952 18/03/1971 18/03/1990 17/03/2009 16/03/2028 17/03/2047
13/04/1952 14/04/1971 14/04/1990 13/04/2009 12/04/2028
11/05/1952 12/05/1971 11/05/1990 10/05/2009 10/05/2028
07/06/1952 08/06/1971 07/06/1990 07/06/2009 06/06/2028
05/07/1952 05/07/1971 05/07/1990 04/07/2009 04/07/2028
01/08/1952 02/08/1971 01/08/1990 31/07/2009 31/07/2028
28/08/1952 29/08/1971 28/08/1990 27/08/2009 27/08/2028
25/09/1952 25/09/1971 25/09/1990 24/09/2009
22/10/1952 22/10/1971 22/10/1990 21/10/2009
18/11/1952 19/11/1971 18/11/1990 17/11/2009
15/12/1952 16/12/1971 15/12/1990 15/12/2009
12/01/1953 12/01/1972 12/01/1991 11/01/2010
08/02/1953 09/02/1972 08/02/1991 07/02/2010
08/03/1953 07/03/1972 07/03/1991
04/04/1953 03/04/1972 04/04/1991
01/05/1953 01/05/1972
28/05/1953 28/05/1972
25/06/1953 24/06/1972
22/07/1953 21/07/1972
18/08/1953 18/08/1972
15/09/1953 14/09/1972
12/10/1953
08/11/1953
06/12/1953
02/01/1954
29/01/1954
26/02/1954
The columns are the events within one season, like 1949-1954,
2005-2010. and 2042-2049. Each event in a column belongs to its own
Saros. About 65 Saros at once generate occultations of Antares.
Each row has events within a rule-of-19 series. See how closely
the rule-of-19 holds true?
Seeing the action of Saros is tricky. Look at 14/08/2005. 19 years
is 13 lunar sidereal periods longer than the Saros. So, index left one
column to back up 19 years. Then step down 13 rows to take away tho
extra 13 cycles. There's the previous event in the instant Saros on
04/08/1987.
The next one in this Saros is found by indexing right one column
to advance 19 years. Then step up 13 rows to remove the extra 13
cycles. The next occultation is on 25/08/2023.
In this way you 'do stairs' to trace a Saros, ascending or
descending 13 rows per column away from a given event. Just like for
eclipses, the Saros peters out eventually as we recede far enough in
time from a given occultation.
Phase
---
The rule-of-19 conveys the phase of the Moon. From the ecliptic
longitude of the occulted star and of the Sun, we can get the
elongation of the Moon from the Sun. The latter, as a fraction of 360
degrees or of 29.53 days, is the phase or age of the Moon.
Because by the rule-of-19 the date of each occultation is the
same, the longitude of the Sun is the same. That with the the fixed
longitude of the star (and Moon) yields a constant elongation of the
Moon from the Sun. Hence, not only do we get a repeat occultation by
the rule-of-19, we get one with the Moon in just about the same phase
thruout the series!
Look at the Antares occultation of 14/08/2005. On August 14th the
solar longitude is 142 degrees. We obtain this by lookup tables or a
planetarium program. The longitude of Antares is 250 degrees, read off
of a starchart or planetarium program. So we have
(elong Moon/Antares) = (long Antares) - (long Sun)
= (250deg) - (142deg)
= (108deg)
(approx colong) = (elong Moon) - (90deg)
= (108deg) - (90deg)
=(18deg), ignoring libration
(phase) = (elong Moon) / (360deg)
= (108deg) / (360deg)
= (0.300)
(age) = (phase) * (29.53days)
= (0.300) * (29.53days)
= (8.6day)
This is the age/phase of the Moon for every occultation in a row of
the table above! If you like, you may add a column at the far right to
give the lunar phase for each row.
Retrograding nodes
----------------
The cause of this periodicity is cunning. The Moon's orbit is not
rigidly fixed in space. She waggles around the ecliptic so that her
nodal points march westward thru the zodiac. As the one and then the
other passes near the star, for one on the ecliptic, occultations
occur.
Hence, we should see TWO cycles, one for each node, just like for
eclipses. Indeed we do! But the second cycle is 9-1/2 years offset and
this interval is clumsy to look up on a calendar. It's the way our
calendar is laid out with nonequal months and all that.
So far so good. It turns out that Regulus is quite on the
ecliptic, only 0.46 degree north of it. Yes, indeed, we do have
occultation seasons for it spaced 9-1/2 years apart.
What happens for stars off of the ecliptic? Now watch closely. The
Moon's path is inclined to the ecliptic by 5.15 deg. Consider a star a
couple degrees south of the ecliptic, like Spica at ecliptic latitude
-2.05 deg.
This is sketched out in the figure below. The horizontal line is
the ecliptic, the sine curve is the lunar orbit, and the asterisks are
stars near the ecliptic. The orbit slides up the ecliptic from left to
right (north is up) with a 18.61 year period.
The star 1, on the ecliptic, will be hit by the Moon at intervals
of 9-1/2 years, just 1/2 of the 19-year period for the nodes to return
to the star after one revolution. (Because of the crude figures here,
the period is rounded. The '19' is NOT from the rule-of-19.) This is
indicated by the upper dimension lines.
|<-------19y------->|<-------19y------->|
|<--9.5-->|<--9.5-->|<--9.5-->|<--9.5-->|
| 3* | | - | |
| 2* \ | | / \ | |
|/ | | \| |/ | | \| |/
---------1*--|---|--\---------/--|---|--\---------/----------
/ | | \ / | | \ /
| | \ / | | \ /
| | - | | -
3y|<->|<-----16y----->|<->|3y
Star 2, far from the ecliptic, will be hit also twice during the
cycle, but the intervals between the occultation seasons is no longer
equal. In the lower dimension lines, we have seasons spaced 3 and 16
years apart, which repeats in the next cycle. Note that the sum of the
two, 3 + 16, still is 19 years.
Occultation flurry
----------------
Why do we get a flock of occultations and not just one or a
couple? First off, the Moon is a large disc on the sky and she plows
thru a corridor 0.5 deg wide. There is some leeway where the Moon is
as she passes near the star.
Second, taking in the whole Earth, the Moon has a large parallax
or perspective offset. From the north pole to the south pole, the Moon
can displace on the sky by a full two degrees. Hence, the 'collision
cross section' of the Moon on the sky is this two degrees plus the
diameter of the Moon, or 2.5 degrees. This straddles the lunar orbit
as seen from the center of the Earth.
Hence, just because the Moon missed a star from your location, it
may well have hit it from an other place on the globe. In other words,
the orbit of the Moon may clear the star by +/- 1.25 degree and still,
from some place on Earth, occult that star.
The result is that as the Moon circulates around the Earth she has
many months of chance to hit a star and we do get a flurry of
occultations. When the orbit drifts too far up the ecliptic, the
occultations peter out.
So far so good. We got this lawn-mover swath of the Moon in the
sky which in general cuts over a given star twice, once on the
ascending arc and once on the descending arc. A plot of this swath
shows that near the northern and southern extremes of the swath, the
orbit can pass over a star ONLY ONCE in the 19-year round. The two
arcs merge into one and we get a single wave of occultations every 19
years.
This is the case with star 3 in the diagram above; the dimension
lines are omitted for the clutter. Yet it is obvious that we get an
occultation season only once every 19 years. This situation applies to
a star more than 3.9 degree off of the ecliptic, the inclination of
5.15 deg, minus the 1.25 degree halfwidth of the Moon's swath.
In the limit, a star at latitude 6.4 degree from the ecliptic will
be nicked only by the utter outer edge of the Moon on only a couple
instances in each season. This distance is the 5.15 degree inclination
plus the halfwidth of the Moon's swath. We would get a few
occultations and then must wait a full 19 years to see any again.
Occultation season
----------------
The results of this analysis are summarized in the first chart
below. The latitude of the star is vertical, whether north or south.
Horizontally is the years within the 19-year cycle.
I plotted the occurrence of occultations in each halfyear. 'x'
means an event takes place; '.' means there are no events. For
latitude 5.0 degrees, as example, occultations occur in both halfyears
of the first four years and in only the first halfyear of the fifth
year. The duration of this season is 5-1/2 years.
The stars I examined are labeled at their approximate latitude.
Pollux, as I noted earlier, had no occultations during the period of
study. This chart was calculated from the geometry of the Moon's
corridor.
OCCULTATION SEASONS VERSUS LATITUDE FROM ECLIPTIC
-------------------------------------------------
lat
6.4 x. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
6.0 xx xx x. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
5.5 xx xx xx x. .. .. .. .. .. .. .. .. .. Aldebaran, Elnath
5.0 xx xx xx xx x. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
4.5 xx xx xx xx xx x. .. .. .. .. .. .. .. Antares. .. .. ..
4.0 xx xx xx xx xx xx .. .. .. .. .. .. .. .. .. .. .. .. ..
3.5 xx xx x. .. xx xx x. .. .. .. .. .. .. Nunki .. .. .. ..
3.0 xx xx .. .. .. .x xx x. .. .. .. .. .. .. .. .. .. .. ..
2.5 xx x. .. .. .. .. xx x. .. .. .. .. .. .. .. .. .. .. ..
2.0 xx x. .. .. .. .. .. xx x. .. .. .. .. Spica .. .. .. ..
1.5 xx x. .. .. .. .. .. .x xx .. .. .. .. .. .. .. .. .. ..
1.0 xx x. .. .. .. .. .. .. .x xx .. .. .. .. .. .. .. .. ..
0.5 xx x. .. .. .. .. .. .. .. xx x. .. .. Regulus. .. .. ..
0.0 xx x. .. .. .. .. .. .. .. .x xx .. .. .. .. .. .. .. ..
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18
year within the 19-year cycle
The duration of a season widely with latitude from the ecliptic.
The shortest is at the extreme latitudes where the Moon reaches only
rarely. Next are the latitudes at and near the ecliptic, because the
swath sweeps quickly across the star at a steep angle. These seasons
last only 1-1/2 years. Stars like Regulus and Spica suffer these short
seasons.
The longest seasons are in the upper latitudes, 4.0 to 4.5
degrees. The swath passes at a shallow angle and takes longer to
complete its crossing of the star. We can expect occultations over a
5-1/2 to 6 year span, like for Antares.
Some authors cite the interval between the end of one season and
the start of the next. The two intervals are less, sometimes
substantially, than 19 years. In the chart above, for a latitude of
3.0 degrees. The first interval is 3-1/2/ The second is 11-1/2. The
sum, 15 years, is four short of 19.
By adding back the length of each season, in this case two years
each, we get the full 19 years. The trouble is that authors usually
miss out the season length and give only the duration of the gaps.
Occultation timeline
------------------
The second chart demonstrates the sequence of occultations for the
six stars from 1950 to 2049, enough to bring out the pattern. Each
year mark spans two years to keep the chart within the width of a
letter page. For example, 1960 is actually 1960-1961. A 'x' means that
occultations occurred in only one of the two years at each mark. a 'X"
means they occur in both years at each mark. The '.' marks the gaps
between seasons. And I tossed in ':' to extend the year marks into the
body of the chart.
SEQUENCE OF OCCULTATIONS FOR THE SIX BRIGHTEST STARS
----------------------------------------------------
each mark covers two years; eg, 1960 = 1960 and 1961
x = in only one year, X = in both years, . = no occultations
star latit
Aldebaran -5.47 ..:...xXx...:...XX....:..XXx....:.xXx:....:xXX.:....:XX
Elnath +5.39 .xXx...:..xXX....:..XXx....;.xXx:....:xXX.;....:XX..:..
Antares -4.57 .xXXx..:..xXXx...:..XXx....:.xXXx....:xXXx:....:XXX.:..
Nunki -3.45 XxxX...:.xXxX....:xXxXx....:XxXX:....xXxX.;...xXXx..:..
Spica -2.05 .xX..Xx:...Xx.xx.:..xx:xX..:.xX.:X...:.X..xx...:xx.xX..
Regulus +0.46 ..xx...xx..xx...xx...X:...X:..xx:..Xx:.xx.:.xx.:.X..:..
year 1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050
Ecliptic coordinates
------------------
To help track occultations, I give here a table of ecliptic
coordinates for several zodiac stars, more than the seven I examined,
and of the Sun at 10-day marks thruout the year. The star positions
are for epoch 2000 and are good for a lifetime. The Sun positions are
for 2002, midway between leapyears 2000 and 2004.
SOME STARS OCCULTED BY THE MOON
------------------------------
star lon lat comments
--------- ----- ---- ------------
Alcyone 60.0 +4.1 eta Tauri, in Pleiades
Aldebaran 69.8 -5.5 alpha Tauri
Elnath 82.7 +5.4 beta Tauri
Pollux 113.7 +3.1 beta Geminorum
eps Cnc 127.4 +1.2 in Presaepe
Regulus 149.8 +0.5 alpha Leonis
Porrima 190.1 +2.8 gamma Virginis
Spica 203.8 -2.1 alpha Virginis
Z'elgenubi 225.1 +0.3 alpha Librae
Graffias 243.2 +1.0 beta Scorpii
Antares 249.8 -4.6 alpha Scorpii
Kaus Bor's 276.3 -2.1 lambda Sagittarii
Nunki 282.4 -3.5 sigma Sagittarii
It happens that Capricornus, Aquarius, Pisces, and Aries have no
bright stars occulted by the Moon. Neither does Cancer, but I included
the Presaepe cluster for being almost on the ecliptic. Taurus, Gemini,
Scorpius, Sagittarius have many bright stars, of which I included a
couple. Aldebaran can stand in for the Hyades cluster; Alcyone, the
Pleiades.
SUN COORDINATES DURING 2002
---------------------------
Jan Feb Mar Apr May Jun jul Aug Sep Oct Nov Dec
--- --- --- --- --- --- --- --- --- --- --- ---
0 280 311 340 11 40 70 98 128 158 188 219 249
10 290 322 350 20 50 79 108 138 168 198 229 259
20 300 332 360 30 59 89 118 147 178 208 239 269
30 310 --- 10 40 69 98 127 157 188 218 249 279
Linear interpolation is sufficient for intermediate dates.'0' of a
month is the last day of the previous month, Due to leapdays, the
longitude of the SUn can shift by at most one degree from year to
year. This is negligible for computing lunar phase.