The Kepler Equation, M = E- e Sin E

The Kepler equatio M = E- e Sin E can be used to compute planetary longitudes.

M is the Mean Anomaly of the planet and E, the Eccentric Anomaly. Eccentric Anomaly is an auxiliary angle used in Kepler's Equations.

e is Eccentricity of Orbit. The planet's orbit is elliptical and not circular and Eccentricity or e is always Zero at Circle, 0-1 at ellipse and one at Parabola !

The Kepler Equation can be put in a different way

E = M + e Sin E

Kepler's Laws of Planetary Motion

Johannes Kepler expounded the principles of Planetary Motion thus

1)The orbit of a planet is an ellipse, with the Sun as the focus of the ellipse.

3)The motion of a planet is fastest at Perihelion and slowest at Aphelion.

4) p1^2 / p2 ^2 = a1 ^ 3 /a2 ^3

The periods of revolutions ( sidereal period ) of two planets is equal to the cubes of their semi major axes.

Long before Kepler Aryabhata expounded the Helio centric Theory of Gravitation, that all planets circle around the Sun due to celestial gravity. If it is terrestrial gravity which is the characteristic of the earth, then it is Celestial Gravity, which is the characteristic of the Sun

R, the Magic Figure in Astronomy

R is the Radius of a Circle. It is 206265 secs and is an important figure is Astronomy

We talk about the Circles in Astronomy. The Ecliptic is a Circle, the Zodiac is a Circle and so on. And how do we measure the Radius of the Circle?

The Magic Figure of 206265 seconds is got by these derivations


Circumference = 2 Pi r

360 degrees = 2 Pi r

r = 360 degrees / 2 Pi

= 360 degrees / 2* 3.14

= 57.3 degrees

( One degree is 60 minutes and one minute is 60 seconds and hence one degree = 60*60 = 3600 seconds )

= 57.3 degrees * 3600 seconds

= 206265 seconds of arc

This is the how the Magic Figure in Astronomy, 206265 seconds, is derived !

Computation of Directional Strength

Directional Strength or Digbala is one of the Sixfold strengths. Here we give the formula for computing Directional strength

First find out the Digbala arc. For instance Jupiter has full directional strength when he is on the Ascendant. That means 180 degrees from that point, on the Descendant he is zero.


So the equation for Digabala arc is

Digbala arc = Longitude of the planet - its powerless point.

( if the difference is more than 180, subtract it from 360 degrees ).

The midpoint of the House or Bhava is taken into consideration. The midpoint of the 7th House becomes the powerless point of Jupiter

Divide the Digbala arc by 3 and you get the Digbala or Directional strength.

Definitions of some Vedic terms

Astrology is based on the Platonic Imitation Theory that the Earth
is an exact replica of Heaven. "As Above So Below" - was the form-
ula summed up by Hermes Trismegistes. We find that the pauper of
today is the prince of tomorrow and the prince of today is the
pauper of tomorrow. What is the base of all this ?

Astrology - the Science of Time - is the only science that has the
answer as it deals with natural causes and the corresponding cele-
stial causes. Since everything happens by Cause we can infer that
everything happense by two types of causes - natural & celestial.
We know the natural causes which has determined the present events.
This heavenly science deals with the celestial causes which has de-
termined the present events. Divine Providence coincides completely
with natural causes.

The Law of Correspondences of classical philosophy states that
everything on earth is an exact copy of that which is structured
in Heaven. Astrologers therefore analyse the heavenly archetype
( the horoscope ) to understand the earthly reflection.

GLOSSARY OF SANSKRIT TERMS USED
-------------------------------

LMT - Local Mean Time after the longitudinal correction is applied.

Nirayana - Sidereal viz based on the constellation of fixed stars.

Sayana - Tropical viz based on the Signs of the Zodiac.

Rasi - Zodiacal Sign . There are 12 signs of the Zodiac from Aries
to Pisces . 30 degrees of arc.

Sphutam - Celestial longitudes of planets.

Nakshatra - Star or asterism. There are 27 asterisms or stars which
tenant the Zodiac

Zodiac - A belt of the heavens 9 degrees to either sides of the
Ecliptic.

Eclipitic - The real path of the earth round the Sun.

Pada - Quarter of a star or asterism. A star is divided into 4
quarters or 4 padas.

Lord - Lord of the star. All stars have lords. For instance Alpha
Arietis or Aswathi is owned by Ketu. Revathi or Zeta Piscium is
owned by Mercury.

Ayanamsa - Precessional Distance. It was observed that the Tropi-
cal Zodiac precedes 1 degree in every 72 years. It takes 25920
years to precede 360 degrees and this is called the precession of
the Equinoxes.

Chitra Paksha Ayanamsa - The Indian Govt adopted this Ayanamsa for
all astronomical calculations. ( There are other Ayanamsas formulated
by scholars.)

Ahargana - No of days elapsed from the beginning of Kali Era which
started from 3102 BC as per the Hindu Calender.

Thithi - Lunation . When the Moon is within 12 degrees from the Sun
it is the First Lunation or Prathama. When Moon is within 24 degrees
from the Sun it is the Second Lunation or Dwitheeya.

Thriteeya - Third Lunation
Chathurthi - Fourth Lunation
Panchami - Fifth Lunation
Shasthi - Sixth Lunation
Sapthami - Seventh Lunation
Ashtami - Eighth Lunation
Navami - Ninth Lunation
Dasami - Tenth Lunation
Ekadasi - Eleventh Lunation
Dwadasi - Twelvth Lunation
Thrayodasi - Thirteenth Lunation
Chaturdasi - Fourteenth Lunation
Pournami - Full Moon. Moon 180 degrees from the Sun
Amavasya - New Moon. Conjunction of the Moon with the Sun.

Nithya Yoga - There are 27 Nithya Yogas. Period during which the
joint motion in longitude of Sun and Moon amounts to 13 degree 20
minutes of arc.

Shukla Paksha - Bright Fortnight - 15 days after the New Moon.
Krishna Paksha - Dark Fortnight - 15 days after the Full Moon.

Dasa - Period of Directional Influence. Saturn's period is 19
years Jupiter's 16, Mars 7, Moon 10 Sun 6 Venus 20 Rahu 18 Ketu 7.
When a person is born he is born in a certain star. It is ruled
by a planet. This planet is ruling at the time of birth. Say
a child is born in a star called Zeta Piscium or Revathi. Its
lord is Mercury and Mercury Dasa will be ruling at the time of
birth. After 17 years it will be replaced by Ketu Dasa for 7
years and after 7 years Venus dasa for 20 years.

KETU - Moon's Descending Node . Caput Draconis.

RAHU - Moon's Ascending Node . Cauda Draconis.

Ashtavarga - Benefic points of planets in a Rasi or Zodiacal
Constellation.

Sarvashtavarga - Total benefic points in the 12 signs.

Rasi Chart - Map of the heavens at the time of birth.

Navamsa Chart - 1/9th division of the Rasi Chart.

Bhava Chart - Chart based on the Ascending Degree. ex - Suppose
the ascending degree of a man is 20th degree of Cancer. This point
is taken as the mid-point of Bhava and 15 degrees to either side
is taken as the Ascendant House. So the Ascendant house or the
First House is from 5th degree of Cancer to the 5th degree of Leo.

LAGNA - Ascendant - The Eastern Horizon of the place. The Sign
rising at the eastern horizon. For instance if the date of birth
is say 05th JAN the sign rising at the Eastern Horizon at the time
of Sunrise is Sagittarius. This is known as Sagittarius Ascendant
or Dhanus Lagna.

GUL - Gulika a satellite of Saturn.

Bhava Sphutam - Longitudes of Houses ( Bhavas ).

Arambha _ Starting Point
Madhya - Mid-point
Anthya - End-point

Ayana - Solstice . There are 2 two types of Ayanas _ Uttarayana
and Dakshinayana ( Northern progress of the Sun and Southern
progress of the Sun). When the declination of the Sun is North
- Uttarayana. When the declination of the Sun is south - Dakshi-
nayana).

Rithu - 2 months. There are 6 Rithus in a Year.

Chandra Vela Chandra Kriya and Chandra Avasthas - Subtle States
of the Moon.

Saptha Varga - Points attained by planets.
Subha Varga - Benefic Points.
Papa Varga - Malefic Points.

Drekkana - Decanate . 10 degrees of a Sign.
Hora - 1 hour. 15 degrees of a Sign.
Sapthamsa - 1/7th Division of a Sign.
Dwadasamsa - 1/12th division of a Sign.
Trimsamasa - 1/30th division of a Sign.

Apahara or Bhukthi - Sub -period of a planet. In a ruling Dasa
there will be sub-periods of other planets.

Sevenfold Division Strength ( Sapta Vargaja Bala )

A planet on account of its tenancy of certain divisions or vargas gets a certain amount of strength.

A planet in its Moola Trikona gets 45 Shastiamsas.

In own Varga, it gets 30.
In bosom friend's house, it gets 22.5 Shastiamsas.
In friend's house, it gets 15 Shastiamsas.
In neutral varga, it gets 7.5 Shastiamsas.
In inimical Varga, it gets 3.75 Shastiamsas

and in the varga of bitterest enemy, 1.875 Shastiamsas.

Odd/even Strength of Planets - OjaYugmaRasyamsa Bala

Certain planets in odd signs gets a strength of 15 Shashtiamsas while some others get 15 Shashtiamsas strength in even signs.

The Sun, Mars, Jupiter, Mercury & Saturn, when they are in Odd Sings get a strength of 15 Shashtiamsas. Moon and Venus get the same strength in even signs.

This should be applied to both Rasi & Navamsa charts. Hence this strength is known as OjaYugmaRasyamsa Bala

Oja - Odd
Yugma - Even
Rasi -
Amsa - Navamsa

Computation of Quadrant Strength ( Kendra Bala )

Quadrants ( Kendras ) are 1,4,7, 10 houses.

Panaparas - These are the signs next to quadrants, viz, 2,5,8 & 11.

Apoklimas - These are houses next to the Panaparas, 3,6,9 & 12

Quadrant strength is calculated by the rule that a planet in a Quadrant gets 60 Viroopas or Shashtiamsas, in a Panapara 30 Shastiamsas and in an Apoklima 15 Shashtiamsas.

Computation of Decanate Strength

Planets are classified into 3 types

Masculine planets - Sun, Jupiter and Mars
Feminine Planets - Venus & Moon
Hermaphrodite " - Saturn and Mercury

A male planet in a first Decanate of 10 degrees
A hermaphrodite planet in the middle decanate
& female planet in last Decanate

gets a strength of 15 Viroopas or Shastiamsas

Computation of Exaltation Strength ( Uchaa bala )

An exalted planet gets 100% strength. In Vedic Astrology 100% strength is 60 Shashtiamsas or Virupas or 1 Rupa.

When a planet occupies its deep exaltation point or Uchabhaaga, it gets a strength of 1 Rupa. Similarly, when it occupies its deep debilitation point, it gets zero strength.

Exaltation strength is given by the formula

Exaltation Strength = Planet's longitude - Its debilitation Point/180 * 60 ( in Virupas or Shastiamsas )

where the difference is more than 180, it is to be subtracted from 360

Computation of Lunation ( Thidhi )

Lunation or Thidhi has been defined as

Lunation ( Thidhi) = Longitude of Moon - Longitude of Sun.


From 0 to 12 degrees is Prathama and from 12 to 24 degrees is Dwitheeya and so on. It becomes easy to find out which Tidhi the time falls on. We give the degrees and the nomenclature of the Thidhis or Lunation.


12PRATHAMA 1
24DWITHEEYA 2
36THRITHEEYA 3
48CHATHURTHI 4
60PANCHAMI 5
72SHASHTI 6
84SAPTHAMI 7
96ASHTAMI 8
108NAVAMI 9
120DASAMI 10
132EKADASI 11
144DWADASI 12
156THRAYODASI 13
168CHATHURDASI 14
180POURNAMI 15
192PRATHAMA 16
204DWITHEEYA 17
216THRITHEEYA 18
228CHATHURTHI 19
240PANCHAMI 20
252SHASHTI 21
264SAPTHAMI 22
276ASHTAMI 23
288NAVAMI 24
300DASAMI 25
312EKADASI 26
324DWADASI 27
336THRAYODASI 28
348CHATHURDASI 29
360AMAVASYA 30

Computation of Decanates ( Drekkanas )

According the Drekkana or Decanate Theory, there are 36 decanates. Each Zodiacal Sign of 30 degrees is divided by 3 and we get 3 decanates per sign.


The first decanate is owned by the lord of that Sign
The second decanate is owned by the 5th lord from that Sign
The Third decanate is owned by the 9th lord from that Sign

For example, if Aries is the Sign, the first decanate is owned by Mars, the second the fifth lord from Aries, Sun and third decanate the ninth lord from Aries, Jupiter.

So Drekkanas are one of the Seven Vargas ( Sapta Varga ) division and is important.

If Mars is in Aries, say in the 13the degree of Aries, then the decanate ruler of Mars is the Sun, as the Sun is the fifth lord from Aries, as Mars is in the second decanate

Computation of Horas

The word " hour " is derived from the Sanskrit, Hora. A Zodiacal Sign is two hours and a day is made up of 24 Horas or Hours!

Ardha Rasi Bhaveth Hora Chaturvimsathi Smritah

In other words, 30 degrees is one Zodiacal Sign and 15 degrees one Hora or Hour.

The first 15 degrees of Aries is Odd and the next 15 degrees is Even. The Odd Horas are ruled by the Sun and the Even Horas are ruled by the Moon.

By this calculation, it is easy to understand who the lord of the Hora is.

So there are two Hora lords, Sun and Moon - Soorya Hora and Chandra Hora.

So from the longitude of a planet, it is possible to know the Hora lord, as 24 horas are ruled by the Sun and Moon.

For example, Mars is in the 5th degree of Aries. We say the Hora lord of Mars is the Sun. Suppose Jupiter is in the 25th degree of Aries. Moon is the Hora lord of Jupiter

Bhava Sphuta Ganitha, computation of House Division

Once the Ascending degree is obtained, Bhava Sphuta Ganitha or the computation of House Division becomes easy.

We have said that 180 degrees away from the Ascendant is the Descendant & 180 degrees away from the MC is the IC, the Patala Lagna.

So we have now 4 mathematical points on the Circle. Divide the quadrants by 3 and we get the starting and the end of 12 houses !

This type of Bhava Ganitha is attributed to Sripathi in the Indian system and in the Western system, there are many methods of computation, namely the sytems of Regiomontanus, Companus & Placidus !

Trignometry & Astronomy, well developed in India !

Western scholars Thibeau and Whitney argued that the early Indians did not have any conceptions about Maths and Astronomy and the calculations of the Cosmological Cycles ( One Cosmological Cycle = 4.32 billion years ) were looted from the Greeks and the Romans.

On the other hand, another Western scholar, the Rev Ebenezor Burgess who translated the Surya Siddhanta into English, differed from them saying that the early Indians were pioneers of Astronomy and Maths.

In Aryabhateeyam, Maha Bhaskareeyam and Brahma Sphuta Siddhanta, to mention 3 major astronomical treatises, Bhuja was Sin, Koti was Cos and Sparsa was Tan. Bhuja chapa was Arc Sine of an angle (Asin), Koti Chapa was arc cosine of an angle ( Acos )& Sparsa Chapa was arc tangent of an angle(Atan). Sin was Bhujajya/ Trijya, Opposite Side/Hypotenuse and Cos was Koti Jya / Trijya, Adjacent Side/Hypotenuse. All trignometric funtions which we know as Sin, Tan and Cos had Sanskrit names and Spherical Astronomy and Spherical Trignometry were well developed in India !

Planetary Hours or Kala Hora

Kala Hora has been defined as 1/12th of a day

Dina Dwadasamso Matha Kalahora

If Thursday, then the ruler of the first hour is Jupiter ( Jove or Thor's day, Thursday)

Pathi tasya poorvasya varadhi nadha

Nisayam thu vareswarath panchamadya

The ruler at night is the fifth lord

meaning on a Sunday, after sunset, the first hour is ruled by Venus

as per the rule

Arka Sukra Budhas Chandra Manda Jeeva Dharasutha

Sun, Venus, Mercury, Moon, Saturn, Jupiter and Mars.

Why is Monday after Sunday ?

The answer is given by the sequential order of Planetary Hours or Kala Hora

On Sunday, at surise, the first hour or Kala Hora is ruled by the ruler of Sunday, the Sun. So after 21 hours, the 22nd hour is ruled by Sun, the 23rd hour by Venus and the 24th hour by Mercury and the 25th hour, that is next day is ruled by Moon and so the next day is Moon's day or Monday !

Planetary Hour = Duration of day /12

Duration of Day = Sunset - Sunrise
Duration of Night = Sunrise - Sunset

The duration of Night and Day varies and it is not exactly 12 hours. So in the program formulae should be given for finding out Planetary Hours !

Parinathi Kriya, Reduction to Ecliptic !

We have said before that the mean longitude of the planet should be subject to a trignometric correction to get the Vikshepa Vritteeya Sphuta or Reduction to Aphelion. Then it is reduced to Ecliptic by the formula

Sin h = 1 - Cos l Sin Y Cos Y / Cos l

The longitude thus obtained is subjected to another trig correction, Reduction to Perihelion or Sheegra Kriya. The term Trignomety, it must be mentioned, is derived from the Sanskrit Trikonamithi and Geometry, from Jyamithi. Brahma Gupta in his mathematical treatise Brahma Sphuta Siddhanta defined Jya as 5 degrees of 1/6 th of a Zodiacal Sign. Bhujamsa, he said, is radius multiplied by modern sine. In hoary times, Astronomy was sister to Astrology and both were united by Mathematics. Astronomy and Maths ( Ganitha ) is one the Three Skandas ( branches ) which constitue Jyothisha or Astro Physics. Samhita ( Philosophy ) and Hora ( Astrology ) were the other branches.

The longitude of Gulika

The Longitude of Gulika is the longitude of the Ascendant at the time of Gulika's rise.

Gulika rises at different times on different days. Saturday he rises at 4 Nadis, that is 4*24 minutes, after sunrise.

Once the calculation of the Asc is carried out, the computation of Gulika's longitude becomes easy

14 trignometric corrections for the Moon's longitude

In the Western system, the Moon's mean longitude is subjected to 6 trignometric corrections. ( There are actually 300 ).

In the Indian Sidereal System, the Moon is subjected to 14 trignometric corretions, the Chatur Vidha Jya samskara. All the major perturbations of the Moon- the Parallactic Equation, the Evection, the Variation, the Annual Equation - are taken into consideration.

Formuala for computing Sunrise and Sunset

Sunrise = 6 Hours + Equation of Time - C, the Ascensional Difference - R, the refraction correction

Sunset = 18 Hours + Equation of Time + C + R


Computation of Sunrise and Sunset are necessary for an Astrology program. Sunrise varies from latitude to latitude. Also the Ascendant.

The Northern & the Southern Celestial Hemispheres

From Aries to Libra is the Northern Celestial Hemisphere. When the Sun is in the Northern Hemisphere, days have more duration.

From Libra to Aries is the Southern Celestial Hemisphere. When the Sun is in this Hemisphere, nights have more duration.

The Great Circle called the Zodiac, " the oblique line that beareth all planets", as Dante called it, is divided into 4 quadrants of (360/4) 90 degrees.

The First 90 degrees is Odd
The Second 90 degrees is Even
The Thirdt 90 degrees is Odd
The Fourth 90 degrees is Even

"Oja yatra guanira kotir apara meshadi jookadhi kau"

The distance traversed by a planet is called Bhuja & the untraversed distance is Koti.

We give below the equation of Bhuja

In the First Quadrant, if a planet traverses 42 degrees, the bhuja is 42 degrees


In the first 90 degrees, the same longitude
In the Second 90 degrees, longitude = 180- longitude
In the Third 90 degrees, longitude = longitude - 180
In the Fourth 90 degrees, longitude = 360 - longitude

This equation of Bhuja should be applied for all astro mathematical calculations.

Vishu Vat Vritta, the Celestial Equator

The Zodiac Software was developed according to the principles of Vedic Spherical Astronomy & Trignometry.

The longitudes measured along the Ecliptic are called true longitudes and the longitudes measured along the Vishuvat Vritta, the Celestial Equator, are known as Right Ascension.

When we say the Right Ascension of East Point, we mean the East Point along the Celestial Equator. The Right Ascension of the Meridian Cusp( RAMC ), is the longitude of the MC or Mid Heaven measured along the Celestial Equator.

The formula for computing the MC , the Madhya Lagna is as follows

tan K = tan R /cos w

where is the K is the longitude of the MC, and R, Sidereal Time converted into degrees.


180 degrees from the Madhya Lagna or the MC is the IC, the Imum Caeli or Patala Lagna.

Kshithija, the Celestial Horizon!

The formula for computing the Ascendant is


Tan L = Sin E / cos E cosw - Sin w Tan A


The Ascendant is a point on the Celestial Horizon. The Eastern Celestial Horizon

First compute E, the Right Ascension of East Point, the East Point lying on the Celestial Equator and this E is subjected to the trignometric correction referred to as above.

The Celestial Horizon is known as Kshithija. w is the maximum declination of the Sun, The Soorya Parama Kranthi. A is the latitude of the place.

The Eastern Point on the Celestial Horizon is Udaya Lagna
180 degrees from it is the Astha Lagna or the Descendant

Ganitha Nirnayam - A Sanskrit Astronomical & Astrological Treatise

Like Laplace's Celestial Mechanics, Ganitha Nirnayam or Mathematical Formulae is a treatise on Astro Maths to compute the longitudes of planets.

This book uses the Sidereal system.

The Western method is to find the True Anomaly of the planet and then add to it The Argument of Perihelion ( a mathematical quantity ) to arrive at a planet's longitude

Theta = v + w

The Indian method is Longitude Corrected Thrice. First the mean longitude of the planet is calculated and then three trignometric corrections are given to find the true longitude of the planet.

First the Mean longitude is reduced by a mathematical process called Manda Kriya. Mandochha refers to Aphelion. By Manda Kriya the mean longitude is reduced to Manda Sphuta, the longitude obtained after the first trignometric correction

Formulae for computing Asc Difference and the Sun's declination

The Ascensional Difference is the difference between Oblique Ascension and Right Ascension.


The formula for computing the Ascensional Difference

Sin C = Tan Phi Tan Delta


where C is the Asc difference Phi is the Latitude of the place and Delta the Sun's declination

The Sun's declination can be found out by the formula

Sin J = Sin L Sin w

where J is the Sun's declination, L the mean longitude of the Sun and w Sun's maximum declination

Astrology Software Development

In order to develop the software, I had to research on a lot of books. They are


Celestial Mechanics by Laplace
Astronomia Nova - Kepler
Bryon's Astronomy

the Encyclopaedia Brittanica Astronomical tables by Newcombe

and Indian Books

Siddhanta Sekhara by Sripathi
Ganitha Nirnayam by Puliyur

To study Astronomy was difficult and later to implement the formulae was also difficult. But then I persevered and developed the Vedic Astrology software

Algorithms developed in 1998

The algorithms for my Vedic Astrology Software were developed in 1998. Since I was conversant with Maths and Astrology, the process was easy for me. There were some bugs initially. But they were debugged and now my Zodiac software is one of the most accurate in the world !