Variation in the Equation of Time
Kevin Karney
Analemma Variation
over Three Millennia
Equation of Time Components
We have settled on the straightforward concept that our
time should be based on the perceived daily spin of the
earth and on its annual orbit around the sun. Furthermore
noon should occur on average when the sun is overhead
and the equinoxes on average occurring on the same date.
Thus, our civil time is based on the concept of a Mean
Sun, which by definition rotates at a Uniform rate around
the Equatorial plane. Sundials however read the Real
Sun, which in fact appears to rotate at a Non-uniform
rate around the Ecliptic plane, which has an obliquity of
23.4° to the equatorial plane. The difference between the
uniform and non-uniform rotational rates gives rise to the
so-called Eccentricity effect. The angle between the ecliptic
and equatorial planes gives rise to the so-called Obliquity
effect.
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Difference/Error Classes
Four classes of difference or error may be distinguished:
•Astronomical Differences:
Orbital Eccentricity, Rotational Obliquity, Luni-Solar
Precession, Planetary Precession, Solar Aberration,
Nutation Effects.
•Sun Finding Differences:
Locational Parallax, Atmospheric Refraction, Penumbral
Effects.
•Calendar Alignment Errors:
Leap Year, Leap Second, Summer Time
•Human Errors:
Longitude set-up, Gnomon set-up, Plate Direction &
Angle, Engraving
This article covers only those items marked in italics which
are almost entirely responsible for the Equation of Time.
Degrees from Meridian
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3000
A.D.
The Midday Sun over the Parthenon
Jan 12, 2002 - Dec 21, 2002
© 2001-2005, Anthony Ayiomamitis
used with his kind permission
see http://www.perseus.gr/Astro-SolarAnalemma.htm
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Degrees of Declination
I have set myself the somewhat daunting task of designing
a heliochronometer that is accurate to within 30 seconds for
the next 250 years. It must operate anywhere in the world
where the sun normally shines. The aim is to do better than
both my father’s Longine watch and Pilkington & Gibbs’
engineering marvels. The Longine was within 30 seconds
for 60 years. Pilkington & Gibbs is still doing well after
nearly a century. One of the first steps in the design task
was a rigorous study of the differences or errors between
Civil Time and Sun Time. I use the word ‘difference’ for
those items that are natural and therefore cannot be errors.
The word ‘error’ is used when our carelessness, inattention
or inability is to blame.
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Fig i
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Each Curve covers 366 days
and is separated from the next by
a Julian Century of 36525 Days
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The Eccentricity effect is an offshoot of the operation of
Johannes Kepler laws proposed in 1609. In his 1st law, he
states that planets move in ellipses with the Sun at one
focus; in his 2nd law, that a line joining the earth to the
sun sweeps out equal areas in equal times. The laws imply
that the earth is moving fastest when it is closest to the sun
at perihelion (around 2nd January) and slowest at aphelion.
Compared with a uniformly moving mean ecliptic sun, the
real sun races ahead after perihelion. But by aphelion, the
mean sun has caught up as the real sun slows down. In the
second half of the year, the mean sun is ahead of the real
sun, which only catches up again at perihelion. This is
shown in Fig ii-left on the next page.
The eccentricity of our orbit around the sun is only 0.0167
– the orbit is very nearly circular. However, it does give rise
to a difference that is nearly - but not quite – a sine curve
with an annual period, of magnitude 7 1/2 minutes and
phased with perihelion/aphelion.
Page The Eccentricity Effect
Pole
Mar
Autumnal
Equinox
b
Fe
M
ay
Apr
The Obliquity Effect
22nd Sept
Jun
Jan
Winter
Soltice
21st Dec
Apelion
Perihelion
Earth
2nd Jan
Jul
Dec
at Focus
Summer
Soltice
21st June
Vernal
Equinox
ug
A
N
ov
20th Mar
Oct
n.b. Equal Angles in Equal Times
since both are Mean Suns
Sep
Tthe Eccentricity Effect
The Obliquity Effect
Illustrated ellipse has eccentricity of 0.2
Earth’s orbit in reality has eccentricity of 0.0167
Real Sun
Mean Ecliptic Sun
Mean Ecliptic Sun
Mean Sun
Note that : at Equinoxes, Mean Ecliptic Sun is Coincident with Mean Sun,
at Soltices, the Projection of the Mean Ecliptic Sun is Coincident with Mean Sun.
Thus, at these times, Obliquity Effect is zero.
Jan
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10
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Equation of Time Components
Perihelion
c
Aphelion
Eccentricity Effect
Obliquity Effect
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Equation of Time
a
0
b
c=a+b
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Page Feb
Fig ii
Equation of Time Changes
Referring to Fig iii, there are four values that dictate the
shape of the Equation during a given year.
• The value of the eccentricity of the earth’s orbit;
• The date/time of Perihelion;
• The value of the earth’s obliquity;
• The date/time of the Vernal Equinox.
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Magnitude in Minutes of Time
A table showing values of the
Equation of Time was first
published by Thomas Tompion in
1683, using data probably supplied
by the Astronomer Royal, John
Flamsteed. Since then, numerous
methods have been proposed to
calculate the values. See references
for a few examples.
Ecliptic Plane
Equatorial Plane
Eccentricity Effect
Obliquity Effect
moving according to Kepler’s 2nd Law
each cheese segment of equal area
moving at uniform rate around the Ecliptic plane
projected onto Equatorial plane
moving at uniform rate around the Equatorial plane
The Obliquity effect can be viewed as the result of
projecting the path of the real sun (in the ecliptic) onto the
plane of the mean sun (in the equator). Fig ii-right, which
for simplicity shows a ‘mean ecliptic sun’ rather than the
real sun, illustrates how this manifests itself. At both the
equinoxes and the solstices, there is no difference, but at all
other times, there is a difference that turns out to be nearly
– a sine curve of biannual period, of magnitude 10 minutes
and phased with the equinoxes.
The sum of the eccentricity and
obliquity effects, together with a
number of relatively minor other
effects, add up to give the familiar
shape of the Equation of Time (Fig
iii) and its alternative guise as the
Analemma (Fig i).
23.4° =
Obliquity
Solar Longitude = Mean Hour Angle
Vernal
Equinox
1st Point of Aries
Summer
Soltice
Autumnal
Equinox
Winter
Soltice
Fig iii
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82.5
21
82.0
19
81.5
17
81.0
15
80.5
13
n.b. No Leap Years in
2100, 2200 & 2300
80.0
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79.5
9
79.0
7
78.5
5
78.0
2000
2050
2100
2150
2200
2250
Year
A) aagainst the backdrop of the stars, the
whole axis of the earth gyrates like an out-of-balance
spinning top in a 25,770 year cycle. This is a major
astronomical effect: e.g. North no longer points at the Pole
star; the pyramids are no longer aligned with the correct
stars; the astrologer’s 1st Point of Aries - the Vernal
Equinox - is now in the constellation of Pisces).
2300
2350
2400
2450
Days from 1st Jan to Perihelion
Equinox & Perihelion Changes
Days from 1st Jan to Vernal Equinox
All of these are changing slowly with time
as a result the gravitational pull of the Sun,
the Moon and other planets, each of which
moves in a plane that is not the same as the
equator. In the case of the Sun and Moon
(and marginally Jupiter), the pull acts
unevenly due to the equatorial bulge of the
Earth’s shape. The varying pulls causes a
general short-term periodic wobble in the
earth’s axis called Nutation, which has only
a minor effect on the Equation of Time. It
also causes a number of longer-term largerscale effects called Luni-Solar Precession
(see A below) & Planetary Precession (see
B, C & D below).
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2500
Fig iv
by 1/4 day, until it is pulled back “into line” by the 4-year
and 100-year leap year cycles. If a given leap-year is taken
as a base-line and considering mid December when the rate
of change of the Equation is greatest at 30 seconds/day, the
Equation will vary with 7.5 secs in the 1st year following,
by 15 secs in the 2nd, by 22.5 secs in the 3rd. It will then
partially ‘self-correct’ in the following leap year.
However, on the one hand, it has only a very minor effect
long term on the Equation of Time, because we have
chosen the length of our civil year - which averages at
365.242500 days - to match the mean equinox-to-equinox
‘tropical’ year of 365.242191 days. This ensures that the
Equinoxes do on average stay at the same time of the year
: see horizontal dotted line in Fig iv.
B) the ellipse of the earth’s orbit around the sun is itself
turning against the background of the stars in a 21,000 year
cycle. This is a significant effect in relation to the Equation
of Time, since it is shifts the time of perihelion in relation to
that of the vernal equinox - the lower saw-tooth in Fig iv.
But, on the other hand, however, the saw-tooth shape of the
curve which echoes our leap years does cause significant
short-term variation - see the upper saw-tooth in Fig iv.
On each successive year, the time of the equinox is shifted
C) the obliquity of the Earth ranges between 22.1°
& 24.5° in a 41,000 year cycle, mainly under the
influence of Jupiter - see the lower line in Fig v.
0.01695
23.45
0.01690
23.44
0.01685
23.43
0.01680
23.42
0.01675
23.41
0.01670
23.40
0.01665
23.39
0.01660
23.38
0.01655
23.37
0.01650
23.36
0.01645
2000
Obliquity - Degrees
Obital Eccentricity
Obliquity & Eccentricity Changes
23.35
2050
2100
2150
2200
2250
Year
2300
2350
2400
2450
2500
Fig v
D) the eccentricity of the Earth’s orbit changes
in a complicated manner in cycles of 96,000 and
413,000 years. - see the upper line in Fig v.
This effect (the Milankovitch cycle) is believed
by some to be one of the astronomical effects
that causes climate change. The Northern
hemisphere will warm as perihelion - when the
earth receives the greatest amount of the sun’s
radiation - is moving towards its summer.
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Eccentricity & Obliquity Effects on the Equation-of-Time
for Years 2000 & 2500
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Equation of Time - Minutes
Adding the Changes
Fig vi shows these four items together over
a 500 year period. Darker colours are for the
year 2000; lighter for the year 2500. One
sees:
• almost no change in the obliquity effect
(the green lines),
• a slight reduction in the magnitude of the
eccentricity effect (the height of the red
lines),
• a significant shift in the position of the
eccentricity effect, as perihelion gets
nearer to the vernal equinox (the position
of the red lines).
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2000 Eccentricity Effect
2500 Eccentricity Effect
2000 Obliquity Effect
2500 Obliquity Effect
2000 Equation of Time
2500 Equation of Time
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The blue lines shows the result of adding the
two effects to give the Equation of Time providing significant change.
Jan
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Fig vii illustrates the effect over three millennia. The same
data, in analemma form, is shown in Fig i.
Important observations are:
• variability is mainly controlled by the period between
perihelion and the equinox;
• in the long term, since our calendar fixes the average
date of the equinox, the change in date of perihelion, as
a result of planetary precession, is the prime variable.
• in the short term, this period is mostly changed by the
vagaries of our leap year cycles;
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Is The Variation important?
From Fig vii, it is concluded that:
• old dials have old tables which become progressively
more out-of date;
• new dials, which incorporate a table or curve, should
ensure that the Equation used is appropriate for the
expected lifetime of the dial and duly dated;
• designs for heliochronometers must seek the means to
overcome the variability or accept that accuracy will
degrade. Figs viii (on the following page) show the
error introduced if Equation of Time values for the Year
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Equation of Time Variation over Three Millennia
Equation of Time in Minutes
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3000
A.D.
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Fig vii
Each Curve covers 366 days and is separated from the next by a Julian Century of 36525 Days
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Page Fig vi
Subsequent articles will trace
strategies to eliminate the
effect of leap year cycles and
how to combat the longer-term
perihelion drift. They will also
cover the other sources of error
or difference
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40
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Error if Year-2000 values of EOT are used for Years 2001-2024
Magenta Lines are Leap Years
Red Lines are Leap Years + 1
Green Lines are Leap Years + 2
Blue Lines are Leap Years + 3
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Error in Seconds
2000 are used in successive
years. These show bands
reflecting the various leap
and non-leap years cycles and
the progressive change within
each band. Note that errors of
almost 40 seconds are reached
within 200 years.
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Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
Year
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
Fig viii
Error in Seconds
Figs 8. Error bands if Year 2000 E-o-T values are used during period 2000 - 2024 (above)
References
and 225 years later (below). Note the banding resulting from the leap-year effect (refer to
1) Anthony
Ayiomamitis’ Fig iv). In each band, the darker lines represent the earlier years. The horizontal axis reprewebsite: http://www.perseus.
sents one year.
gr/Astro-Solar-Analemma.
htm. The photo is used with
his kind permission.
50
Year 2225
Error if Year-2000 values of EOT are used for Years 2225-2248
2)Admiralty
Manual
of
Year 2226
40
Year 2227
Navigation, Volume III Chap
Magenta Lines are Leap Years
Year 2228
Red Lines are Leap Years + 1
IX, HM Stationary Office,
Year 2229
30
Green Lines are Leap Years + 2
Year 2230
Blue Lines are Leap Years + 3
London (1955).
Year 2231
20
Year 2232
3) Peter Duffet-Smith, Practical
Year 2233
Year 2234
10
Astronomy
with
your
Year 2235
Year 2236
Calculator, Cambridge Univ.
0
Year 2237
Press, Cambridge (1988).
Year 2238
Year 2239
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4) Jean Meeus, Astronomical
Year 2240
Year 2241
Algorithms,
Willman-Bell,
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Year 2242
Year 2243
Richmond (1998).
Year 2244
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5)For the Equation of Time with
Year 2245
Year 2246
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ultimate precision, see NASA/
Year 2247
Year 2248
JPL’s: http://ssd.jpl.nasa.gov/
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Fig viii
cgi-bin/eph
Notes
1) For an animation of E-o-T build-up and changes,
and for a coloured version of this document, see
http://precisedirections.co.uk
2)Readers who would like a print of Figures i & vii in
colour, please e-mail the author.
December 2005
Author’s address
Freedom Cottage, Llandogo,
Monmouth, NP25 4TP
01594 - 530 595
[email protected]
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