Astronomical Calculations in GCAL

Algorithms for calculation of tithi, naksatra etc are given in this document. All descriptions are given "as they are" in application, that means this document does not aim to define calculation algorithms for values like tithi, naksatra etc. rather it describes how these algorithms are implemented.

This documents is for certification of algorithms used in GCAL program.

Content

Mathematical operations

Mathematical operations

/ divide

* multiple

floor returns whole of part of real number

rmdr return remainder after subtraction of whole part of number

put_in_360 puts given angle into range 0..360 degrees or equivalent 0..2*PI

Calculation of Tithi

input values:

m – tropical longitude of the Moon /real number/

s – tropical longitude of the Sun /real number/

output value:

tithi – number of tithi /integer/

tithi_elapsed – amount of elapsed time from tithi (in percents) /real number/

tithi = floor ( put_in_360(m – s – 180.0) / 12.0 )

tithi_elapsed = rmdr( put_in_360(m – s – 180.0) / 12.0 ) * 100.0

Paksa

input value:

tithi – moon's tithi /integer/

output value:

paksa – moon's paksa /integer/

value 0 is for Krsna Paksa

value 1 is for Gaura Paksa

paksa = floor( tithi / 15)

Naksatra

input values:

m – tropical longitude of the moon /real number/

a – ayanamsa value calculated for the same moment as longitude of the moon /real number/

output values:

naksatra – number of naksatra /integer/

naksatra_elapsed – elapsation of naksatra /real number/

We need convert range <0,360) to range <0,27). Therefore constant 3/40.0 is used.

naksatra_pos = (sidereal_moon_pos) * (3 / 40.0)

in other symbols:

naksatra_pos = put_in_360(m – a) * (3 / 40.0)

naksatra = floor( naksatra_pos )

naksatra_elapsed = rmdr ( naksatra_pos ) * 100.0

naksatra value 0 is for Asvini, value 26 is for Revati

Yoga

input values:

m – tropical longitude of the moon /real number/

s – tropical longitude of the Sun /real number/

a – ayanamsa value calculated for the same moment as longitude of the moon /real number/

output value:

yoga – number of yoga /integer/

yoga_pos = (sidereal_moon_pos + sidereal_sun_pos) * (3/40.0)

in other symbols:

yoga_pos = ((m – a) + (s – a)) * (3/40.0) = put_in_360(m + s – 2*a) * (3/40.0)

yoga = floor( yoga_pos )

Yoga value 0 is for Viskumba, value 26 is for Vaidhrti.

Rasi

input values:

s – tropical longitude of the Sun /real number/

a – ayanamsa value calculated for the same moment as longitude of the moon /real number/

output value:

rasi – integer number of rasi for sun in given position

sidereal_sun_pos equals (s – a)

rasi = floor ( put_in_360( s – a) / 30.0 )

rasi value 0 is for Mesa (Aries), value 11 is for Mina (Pisces)

Sankranti

Sankranti calculation is based on rasi.

1) If rasi is changing from one day (day1) to next day (day2), then exact time of change of rasi is calculated.

2a) If calculated time of sankranti is before noon for day1 for given location, then sankranti is mentioned in day1.

2b) If calculated time of sankranti is after noon for day1, then sankranti is mentioned in day2.

Masa

input values:

date – day, month, year

time of sunrise

tithi for given date

output values:

masa – calculated masa for given day

gyear – gaurabda year

Now we will calculate conjunctions of sun and moon, that is moment when sun and moon have the same longitude, in other words Gaura Pratipat Tithi begins.

Let us say that t1 is moment of sunrise for given day. Then we will calculate four previous conjunctions and two next conjunctions. Ordered by time, we have values c0, c1, t1, c2, c3, c4, c5 where c0 < c1 < c2 < c3 < t1 < c4 < c5

step 1

We will calculate sun rasi for each conjuction and we get values rasi0, rasi1, rasi2 ... rasi5. We are not calculating rasi for t1 (sunrise of given date) since we dont need it.

Then we are looking for ksaya month within values c0...c5. Than means

if [rasi(n+1)] equals [(rasi(n) + 2) modulo 12] , then there is ksaya month.

Normal is [rasi(n+1)] equals [(rasi(n)+1) modulo 12].

If we have found such index k, that [(rasi(k-1)+2) modulo 12] equals rasi(k)

then we are looking for adhika masa

if we have found such index ke, that rasi(ke+1) equals rasi(ke)

then corrected range is <k, ke>

otherwise let ke = 5 and corrected range is <k, ke>

If we have found no such index k, then skip to test 2.

Now, we will make decrement of each rasi in the interval k..ke

see examples for step1 effect.

step 2 is performing evaluation for masa. Following is decision table which reads from top to bottom and is similar to decision tree.

if rasi[3] equals rasi[4] ... note that time t1 is between rasi1 and rasi2 moments
yes, equals	no, rasis are different
t1 belongs to adhika masa rasi_g (for gaurabda year testing) is set to rasi[3]	test of paksa of t1
	paksa is krsna	paksa is gaura
	t1 belongs to masa according rasi[4] (next conjunction rasi) rasi_g set to rasi[4]	t1 belongs to masa according rasi[3] (previous conjunction rasi) rasi_g set to rasi[3]

step 3

After calculation of masa, we will calculate gaurabda year.

Basic formula is

GYprep = Year – 1486

But this is not true for time interval from 1st January to Gaura Purnima. In that interval is valid formula GYprep = Year - 1487.

So test for Gaurabda Year is:

if masa_g is in the interval from Kesava to Govinda (that means before Gaura Purnima)

and at the same time month of given date is from interval January – June (that means after 1st January)

then GaurabdaYear = GYprep – 1

otherwise GaurabdaYear = GYprep

Examples

4 Examples for STEP 1 of masa calculation algorithm.

old sequence						corrected sequence
c0	c1	c2	c3	c4	c5	c0	c1	c2	c3	c4	c5
aries	aries	taurus	gemini	cancer	leo	aries	aries	taurus	gemini	cancer	leo
aries	aries	gemini	cancer	cancer	leo	aries	aries	taurus	gemini	cancer	leo
aries	aries	gemini	cancer	leo	leo	aries	aries	taurus	gemini	cancer	leo
aries	taurus	taurus	gemini	leo	leo	aries	taurus	taurus	gemini	cancer	leo
taurus	cancer	leo	virgo	virgo	libra	taurus	gemini	cancer	leo	virgo	libra

normal sequence is aries – taurus – gemini – cancer – leo – virgo – libra – ...

Other representation of these examples is given by ksaya/adhika notation:

old sequence						corrected sequence
aries	taurus	gemini	cancer	leo	virgo	aries	taurus	gemini	cancer	leo	virgo
adhika	normal	normal	normal	normal	-	adhika	normal	normal	normal	normal	-
adhika	ksaya	normal	adhika	normal	-	adhika	normal	normal	normal	normal	-
adhika	ksaya	normal	normal	adhika	-	adhika	normal	normal	normal	normal	-
normal	adhika	normal	ksaya	adhika	-	normal	adhika	normal	normal	normal	-
-	normal	ksaya	normal	normal	adhika	-	normal	normal	normal	normal	normal

Therefore STEP 1 is, in general, handling these cases of ksaya-adhika sequences:

Case	Sequence	No. of changes in rasi	No. of changes in masa
1	adhika-ksaya-adhika	1	2
2	adhika-normal-ksaya-adhika	1	2
3	adhika-ksaya-normal-adhika	2	3
4	adhika-normal-normal-ksaya-adhika	1	2
5	adhika-normal-ksaya-normal-adhika	2	3
6	adhika-ksaya-normal-normal-adhika	3	4

No adjustment of months is required BEFORE the ksaya month, only after.

Algorithm does not handle single ksaya month.

Revision History

Date	Description
May 2, 2008	Initial version
May 9, 2008	Corrections of few small errors
June 7, 2008	Changed algorithm for masa calculation

Author: Gopalapriya Das