Sunday, June 5, 2016

Quick Approximation Formulas: Position of the Sun, Phase of the Moon

Quick Approximation Formulas: Position of the Sun, Phase of the Moon

 Caution:  The formulas for the position for the Sun and Moon are quick but approximate!  This is good for a fast, ballpark answer.  If you want precise data, make sure to use precise calculations or visit Wolfram Alpha or an astronomy website.  The formulas presented in today’s blog only measure longitude (right ascension), latitude (declination) is not included.

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
266.55-299.67
266.564-299.693
266.578-299.707
Capricornus
299.68-327.84
299.694-327.863
299.708-327.877
Aquarius
327.85-351.53
327.864-351.553
327.878-351.567
Pisces
351.54-360.00;
0.00-29.04
351.554-360.000;
0.000-29.063
351.568-360.000;
0.000-29.077
Aries
29.05 – 53.42
29.064- 53.443
29.078-53.457
Taurus
53.43 – 90.39
53.444-90.413
53.458-90.427
Gemini
90.40 – 118.21
90.414-118.233
90.428-118.247
Cancer
118.22 – 138.14
118.234-138.163
118.248-138.177
Leo
138.15 – 174.11
138.164-174.133
138.178-174.147
Virgo
174.12 – 217.76
174.134-217.783
174.148-217.797
Libra
217.77 – 241.10
217.784-241.123
217.798-241.137
Scorpius
241.11 – 247.99
241.124-248.013
241.138-248.027
Ophiuchus*
248.00 – 266.563
248.014-266.577
248.028-266.592

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
266.55-299.67
26.55-30.00 Sagittarius
0.00-29.67 Capricorn (270°)
Capricornus
299.68-327.84
29.68 – 30.00 Capricorn
0.00 – 27.84 Aquarius (300°)
Aquarius
327.85-351.53
27.85 – 30.00 Aquarius
0.00 – 21.53 Pisces (330°)
Pisces
351.54-360.00;
0.00-29.04
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°.
1st 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°.
3rd 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 1st Quarter, Full Moon, and 3rd Quarter moons?

1st Quarter:
90 = 360/29.5 * d
7.375 = d 

About 7 days – hence the 1st 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.

3rd Quarter:
270 = 360/29.5 * d
22.125 = d

About 22 days – the 3rd 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/dates-of-suns-entry-into-each-constellation-of-the-zodiac
Retrieved June 5, 2016


This blog is property of Edward Shore, 2016.

No comments:

Post a Comment

HP Prime and Casio fx-CG50: Trapezoid Midsegment, Height, Area

HP Prime and Casio fx-CG50:  Trapezoid Midsegment, Height, Area The program TRAPEZ calculates the following: Midsegment leng...