Quick Approximation
Formulas: Position of the Sun, Phase of the Moon
Quick Sun Formula
To approximate where
the sun is:
S ≈ 360/365.25 *
n ≈ 0.985626283 * n
Where n is the
number of days from the last vernal equinox.
This date varies between March 19 to March 21 year to year. It also depends on your location on Earth, so
for most accurate results, check when is the last (or next) vernal equinox in
your area.
For us living on
the Pacific Time Zone, the vernal equinox occurred on March 19, 2016, 9:30 PM.
[*] At the time, we were in daylight
savings time, hence the difference between the Pacific Time Zone and Universal
time is 7 hours. In terms of Universal
Time, the vernal equinox occurred on March 20, 2016, 4:30 AM.
To check which constellation
the sun is in front of, first we have to take the rate of precession into
account.
Rate of Precession
The rate of
precession of the equinoxes is estimated.
One estimated equation of longitude (right ascension) is:
(I)
P = 5028.79695*t
+ 1.1054348*t^2 – 0.00007964*t^3 + 0.000170663*t^4
– (5.6 * 10^8)*t^5
[Captiatne et
al, Eqn 39]
The equation is
a basic IAU (International Astronomical Union) expression for precession, after
adjustments for obliquity of the equator on the moving elliptic and other
linear corrections, The coefficients are in terms of arcseconds. 1 degree had 3600 arcseconds The term t is in terms of Julian Century,
which consist of 36,525 days.
If we want to express
P in terms of years (Y), we need to make the substitution Y = t/100 and (I)
becomes:
(II)
P = 50.2879695*Y
+ 0.000110543*Y^2 – (7.964*10^11)*Y^3 +
(1.70663*10^12)*Y^4 – (5.6 * 10^18)*Y^5
Note that the P
is still expressed in arcseconds. To
express dP/dY in terms of degrees, divide each coefficient by 3600. As a result, (II) is now:
(III)
Pd = 0.01396888*Y
+ (3.070638889*10^7)*Y^2 – (2.212222222*10^14)*Y^3 + (4.740638889 *
10^16)*Y^4 – (1.555555556*10^21)*Y^5
I’ll label (III)
this as Pd to distinguish this from P which are represented in degrees and
arcseconds, respectively. This the
amount of longitude (right ascension) that occurs in degrees in terms of Julian
years (365.25 years).
If we want an
approximate year or do a year accounting, we can approximate the rate as:
(IV)
Rate of
Precession ≈ 0.0139691871 degrees/year
Using this
approximation, it takes about 71.586 years (around 71 years, 7 months, 2 days)
for the vernal equinox to move a full degree.
Pretty much a single human lifetime.
It doesn’t seem
much, but taking centuries and millennia into account, this can have an effect on
where the zodiac constellations are.
Determine the Zodiac Constellation where the
Sun appears
The following
are the range of constellations where the sun appears for 2016, 2017, and 2018. 2016 is from EarthSky and the 2017 and 2018
entries are calculated. All results are
rounded to 3 decimal places (degrees).
Note: This is
for constellations in astronomy (not astrology). The amounts in degrees.
Constellation

2016

2017

2018

Sagittarius

266.55299.67

266.564299.693

266.578299.707

Capricornus

299.68327.84

299.694327.863

299.708327.877

Aquarius

327.85351.53

327.864351.553

327.878351.567

Pisces

351.54360.00;
0.0029.04

351.554360.000;
0.00029.063

351.568360.000;
0.00029.077

Aries

29.05 – 53.42

29.064 53.443

29.07853.457

Taurus

53.43 – 90.39

53.44490.413

53.45890.427

Gemini

90.40 – 118.21

90.414118.233

90.428118.247

Cancer

118.22 – 138.14

118.234138.163

118.248138.177

Leo

138.15 – 174.11

138.164174.133

138.178174.147

Virgo

174.12 – 217.76

174.134217.783

174.148217.797

Libra

217.77 – 241.10

217.784241.123

217.798241.137

Scorpius

241.11 – 247.99

241.124248.013

241.138248.027

Ophiuchus*

248.00 – 266.563

248.014266.577

248.028266.592

Currently, the
sun is the constellation Sagittarius on New Year’s Day (January 1). The sun is about 279° to 281° longitude
around New Year.
FYI: Western (tropical)
astrology uses the calendar that dates back to the Age of Aries. Where S = 0° is
designated as the sign of Aries. Each
sign is 30° apart. The examples in this
blog entry work with the astronomical constellations, not astrological
signs.
The following
table shows the constellations as they are positioned in 2016 versus the
western astrological zodiac. Notice how
far off astrology is!
Astronomy in
2016 vs. Western Astrology (amounts are in degrees)
Constellation

2016

Western
Astrology

Sagittarius

266.55299.67

26.5530.00
Sagittarius
0.0029.67
Capricorn (270°)

Capricornus

299.68327.84

29.68 – 30.00
Capricorn
0.00 – 27.84
Aquarius (300°)

Aquarius

327.85351.53

27.85 – 30.00
Aquarius
0.00 – 21.53
Pisces (330°)

Pisces

351.54360.00;
0.0029.04

21.54 – 30.00
Pisces
0.00 – 29.04 Aries
(0°)

Aries

29.05 – 53.42

29.05 – 30.00
Aries
0.00 – 23.42
Taurus (30°)

Taurus

53.43 – 90.39

23.43 – 30.00
Taurus
0.00 – 30.00
Gemini (60°)
0.00 – 0.39
Cancer (90°)

Gemini

90.40 – 118.21

0.40 – 28.21
Cancer

Cancer

118.22 – 138.14

28.22 – 30.00
Cancer
0.00 – 18.14
Leo (120°)

Leo

138.15 – 174.11

18.15 – 30.00
Leo
0.00 – 24.11
Virgo (150°)

Virgo

174.12 – 217.76

24.12 – 30.00
Virgo
0.00 – 30.00
Libra (180°)
0.00 – 7.76
Scorpio (210°)

Libra

217.77 – 241.10

7.77 – 30.00
Scorpio
0.00 – 1.10
Sagittarius (240°)

Scorpius

241.11 – 247.99

1.11 – 7.99 Sagittarius

Ophiuchus

248.00 – 266.54

8.00 – 26.54
Sagittarius

Examples
Example 1:
June 6, 2016 and
I live in the Pacific Time Zone. For us,
the Vernal Equinox occurred on March 19.
Hence n = 79.
S ≈ 360/365.25 *
79 ≈ 77.86447°. Hence, the sun is in the
constellation Taurus.
Example 2:
It is 2017 and
the sun about 276° longitude. The vernal
equinox occurs on March 20. About what
day is it? Hence S = 276.
276 = 360/365.25
* n leads to n ≈ 280.025 (280 days from
the vernal equinox)
We would be 280
days from the vernal equinox. This would
be December 25, 2017. The sun would be
in front of the constellation Sagittarius.
Remember that
this is an approximation.
Quick Phase of the Moon
The following
formula describes the approximate elongation of the moon:
M ≈ (360/29.5 *
d) ≈ (12.20338983 * d)
M is given a
range of 0° to 360°. Should M exceed 360°,
subtract successive multiples of 360° until M is in the 0° to 360° range. In other words, M = (360/29.5 * d) MOD 360
Where d = number
of days from the last New Moon.
Elongation is
the longitudinal “distance” between the sun and the moon.
Elongation

Comments

M is about 0°. (or 360°)

New Moon. The moon is about in the same constellation
the sun is.

M is about 90°.

1^{st}
Quarter. The moon is approximately 90°
away from the sun.

M is about 180°.

Full
Moon. The moon is approximately 180° away
from the sun.

M is about 270°.

3^{rd}
Quarter. The moon is approximately 270°
from the sun.

For 0° < M
< 180°, the Moon is waxing. For 180° <
M < 360°, the Moon is waning.
Example:
It is June 5, 2016
in the Pacific Time Zone, 79 days from the last Vernal Equinox. (n = 79)
The New Moon is on June 5, 2016.
Approximately, when are the next 1^{st} Quarter, Full Moon, and
3^{rd} Quarter moons?
1^{st}
Quarter:
90 = 360/29.5 *
d
7.375 = d
About 7 days –
hence the 1^{st} quarter occurs approximately on June 12, 2016.
Full Moon:
180 = 360/29.5 *
d
14.75 = d
About 15 days –
the Full Moon is on June 20.
3^{rd}
Quarter:
270 = 360/29.5 *
d
22.125 = d
About 22 days –
the 3^{rd} Quarter Moon would be on June 27.
Next New Moon
About 29.5 days
or 30 days – the next New Moon would be on July 4.
Summary
Approximate
Longitude of the Sun: S = 360/365.25*n,
n is the number of days from the Vernal Equinox
Approximate
Phase of the Moon: M = 360/29.5*d, d is
the number of days from the last known New Moon
Eddie
Sources:
Quick Sun
Formula:
“Solstices &
Equinoxes for Los Angeles (Surrounding 10 Years)” timeandate.com http://www.timeanddate.com/calendar/seasons.html (my laptop was located in Azusa, CA) Retrieved June 5, 2016
Rate of
Precession Section:
N. Capitaine,
P.T. Wallace, and J. Chapront. “Expressions
for IAU 2000 precession quantities” Astronomy & Astrophysics. September 19, 2003. PDF file: http://syrte.obspm.fr/iau2006/aa03_412_P03.pdf
“Axial
Precession” Wikipedia https://en.wikipedia.org/wiki/Axial_precession Retrieved June 5, 2016
Zodiac Section:
McClure,
Bruce “Dates of sun’s entry into each
constellation of the zodiac” (2016) EarthSky http://earthsky.org/space/datesofsunsentryintoeachconstellationofthezodiac
Retrieved June
5, 2016
This blog is property
of Edward Shore, 2016.