Research on Jyotish
Detailed notes on ganita and other topics
Table of Contents
1. Ganita - Astronomical sunrise
With modern formulas, sunrise can be calculated at best to an accuracy of 2 minutes, but in unusual atmospheric conditions could be off by 5 minutes. This may seem surprising 1A detailed explanation of astronomical refraction physics is required. considering we are calculating the position of astronomical bodies to within arc-seconds, but in those calculations, we do not need to account for atmospheric refraction at the horizon. For jyotish calculations 2Jyotish works with what we see. If we use true positions, why don’t we also account for the journey of light to the earth?, we are interested in the apparent sunrise from the tip of the sun which appears before the sun is directly visible.
3As you can see in the diagram, the light hits the atmosphere and then “refracts” which causes some of the sun’s rays to be visible. Note that for far away astronomical bodies, the refraction is negligible, but we usually are using the true position for them regardless.
The mathematical formulas to calculate sunrise are approximations and there are a quite a few ways for the observed sunrise to be different. For instance, the average value for the atmospheric refraction is usually taken to be 34 arc-seconds, but it does not account for any seasonal or location variations. 4This value was actually derived by Newton in the 1700’s and little progress has been made towards an optimal version. Meeus indicates the change from temperature or pressure differences could account for roughly 20-40 seconds difference in sunrise time from winter to summer. 5Meeus - Astronomical Algorithms (1998) pg. 101 Also, sunrise also becomes more sensitive to changes in refraction as latitude increases. 6Astronomical Refraction and the Equinox Sunrise - Sampson (2000) And we still haven’t considered the effect of altitude on sunrise which changes sunrise by 1 minute for every 1.5 km. On Mt Everest, sunrise would occur 6 minutes earlier!7As far as I know, no jyotish software has also implemented a topological map in the atlas. Essentially, the atmospheric refraction, elevation, temperature, and pressure can change the sunrise/sunset time by 3-5 minutes.
1.1. Various formulas and their accuracy
Given what we know, what is the most accurate formula tested against observed sunrise and sunset times? 8I did a few weeks of research, but since it isn’t published, in this section I will liberally refer to “Evaluating the Effectiveness of Current Atmospheric Refraction Models in Predicting Sunrise and Sunset Times Models in Predicting Sunrise and Sunset Times” Wilson (2018) And given the ease of observing sunrise and sunset, it is remarkable that there are few published datasets available. The largest of dataset is 10 years of data accumulated by Rabbi Avraham Druk in Israel, but most are less than a year.
There are several main algorithms that are used in the world:
- The US Naval Observatory Solar-Lunar Almanac Core 2.0 - uses standard refraction and mid-accuracy ephemeris for the sun.
- Meeus - Astronomical Algorithms (1998)
- Schylter - http://www.stjarnhimlen.se/comp/riset.html
- Swiss Ephemeris - https://www.astro.com/swisseph/swephprg.htm
The Wilson paper contains a novel sunrise algorithm which was compared against existing algorithms and the observed data. The calculation against existing algorithms was generally within 100 seconds for the tests. 9There seemed to be more variance against the Schylter algorithm at larger latitudes suggesting that one may not be as robust. This algorithm is apparently the standard in php.
| Dataset | Dates | Include dip | Refraction |
|---|---|---|---|
| Mt. Wilson | Sept 1 - March 30 | yes | underestimated |
| Edmonton | April 1 - Aug 30 | no | underestimated |
| Edmonton | Sept 1 - March 30 | no | overestimated |
| Hawaii + Chile | All dates | no | underestimated |
| Hawaii + Chile | June | yes | underestimated |
| Hawaii + Chile | Months not June | yes | overestimated |
| Smiley | Jan 1 - Jun 1 | no | overestimated |
Note there is another paper where they predicted sunrise and sunset to within 15 seconds by using different refraction models for summer and winter months in Israel. With the above table, we can see the refraction is generally underestimated in the winter months and overestimated in the summer months with Smiley’s data being the exception. Without more data at a variety of altitudes and latitudes, it is very difficult to come to a conclusion other than the variations are greater in the summer time by up to a few minutes as compared to winter time. Also, none of the publicly used algorithms account for the dip / altitude either, so for high-altitudes the sunrise/sunset could be off by 1 min for every 1.5 km. 10When accounting for the dip, in most cases we are over-estimating the refraction, but this is not relevant for this section for us right now.
Comparing Swiss Ephemeris
As the Swiss Ephemeris calculation was not taken by Wilson, I did a sample comparison myself below. 11The data from Edmonton was taken at 113W 29’ 14’’, 53N 31’ 33’’
| Date | Observed (UTC) | SwissEph | Observed - Computed |
|---|---|---|---|
| 1990-12-29 | 15:49:16 | 15:50:40 | -84 |
| 1991-01-10 | 15:45:01 | 15:46:42 | -41 |
| 1991-02-09 | 15:07:22 | 15:05:19 | 123 |
| 1991-02-11 | 15:02:51 | 15:01:25 | 86 |
| 1991-03-29 | 13:15:23 | 13:15:25 | -2 |
| 1991-04-18 | 12:27:32 | 12:28:05 | -33 |
| 1991-05-07 | 11:46:51 | 11:48:09 | -78 |
| 1991-06-05 | 11:04:00 | 11:08:43 | -283 |
| 1991-07-29 | 11:42:38 | 11:44:14 | -96 |
| 1991-08-14 | 12:10:20 | 12:11:16 | -56 |
| 1991-09-17 | 13:10:58 | 13:10:36 | 22 |
| 1991-10-10 | 13:52:14 | 13:51:13 | 1 |
| 1991-10-18 | 14:07:20 | 14:06:18 | 62 |
| 1991-11-14 | 14:58:43 | 14:57:39 | 64 |
| 1991-12-17 | 15:42:10 | 15:45:40 | -210 |
Note the seasonality of the observed data vs the calculation. In summer and winter times, the refraction has been underestimated and the calculated sunrise is too late. But in spring and fall, the calculated sunrise is early. Considering that Swiss Ephemeris does not account for elevation, we compare the data given in Figure 4.2d of the Wilson paper and find roughly the same pattern of underestimating refraction in the the summer months and overestimating in the winter. Swiss Ephemeris appears to be in line with the rough accuracy of other algorithms in this rudimentary comparison.
Further work
There is quite a bit of work that could improve the sunrise accuracy:
- compare the Swiss Ephemeris formula against existing calculations and observed data to get a baseline
- create a nominal refraction value for summer and winter months rather than fixing it at 34 arc-seconds.
- create a custom sunrise / sunset formula which includes the dip
- by looking at millions rectified charts, it could be possible to do fitting of sunrise data.
1.2. Practical relevance for jyotish
For jyotish, there are a variety of bodies that have calculations that depend on the sunrise. Depending on their speed, they can be increasingly sensitive to any changes in sunrise. As we observed in the Ganita section, sunrise can easily vary by 2-5 minutes in either direction depending on the season based on the observed data. From the collected data, it is “mostly” under 3 minutes with outliers at the 5 minute mark. 12Given by the standard deviation of two of the datasets in Wilson’s paper.
There isn’t enough data to come to a conclusion, but I would wager that adjusting the sunrise earlier in the summertime and later in the wintertime may be reasonable to do in non-equatorial regions.13Also, if anyone says that their software is right or accurate, please refer them to this page. All software is generally wrong, it’s just a question of how.
I’ve included the following table to help adjust the expectations of where a particular body could be accounting for the variation in sunrise.14For the upagrahas, we’re assuming each vela is 90 minutes. A one minute change in sunrise causes a 0.125 minute change in the period, which is 0.0625°. A five minute change in sunrise causes a 0.625 change in the period which corresponds to 0.3125°. The variation is then compounded for the 8th vela.
| Body | Movement Speed | D1 change for 1 min | D1 change for 5 min |
|---|---|---|---|
| Bhāva Lagna | 1X | 0.25° | 1.25° |
| Horā Lagna | 2X | 0.5° | 2.5° |
| Ghāṭikā Lagna | 5X | 1.25° | 6.25° |
| Prāṇapada Lagna | 20X | 5° | 25° |
| Upagrahas | 1X | 0.0625° to 0.5° | 0.3125° to 2.5° |
| Varnada Lagna | NA | Depends on Horā Lagna | Depends on Horā Lagna |
As you can see, the Prāṇapada moves extremely quickly, which means it can almost move forward or backwards an entire sign in 5 minutes. A rasi changes every 120 minutes and a navamsa changes every 13.33 minutes. In 1 minute, the Prāṇapada can move up to two navamsas in either direction. And in 5 minutes, for those that would like to rectify trines to the moon, it can be pretty much anywhere you would like it in the D9. Whenever any of these bodies are near the edge of a sign, it is important to evaluate what position would make the most sense.
Sunrise is a very low-accuracy event which everyone believes can be calculated to high-accuracy. Unfortunately, none of the main formulas have been shown to be more accurate or have less standard deviation than the others 15Though the Schylter formula does give some concern at larger latitudes., so caution is required when using special points based on sunrise.
2. Ganita - Sphuta Dṛṣṭi and Dṛg bala
The graha aspects are used to calculate various strengths. The following table of aspects is from BPHS and gives the special aspects to 3/10, 4/8, 5/9, and 7 for all the graha.16BPHS - Chapter 9 - Aspects of Signs - Shlokas 10-15 (Sharma pg 161). For instance, Ma has full aspect on 4/8, Jp has full aspect on 5/9, and Sa has full aspect on 3/10. In addition, these graha have less 7H aspect. The full table for the pada strength is listed below for each bhava. However, this is likely as an approximation as more precise calculation for sphuta dṛṣṭi is given later.17BPHS - Chapter 28 - Evaluation of Planetary Aspects (Sharma pg 359).
| House | Main | Jp | Ma | Sa |
|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 |
| 3 | 15 | 45 | 30 | 60 |
| 4 | 45 | 15 | 60 | 30 |
| 5 | 30 | 60 | 45 | 15 |
| 6 | 0 | 0 | 0 | 0 |
| 7 | 60 | 30 | 15 | 45 |
| 8 | 45 | 15 | 60 | 30 |
| 9 | 30 | 60 | 45 | 15 |
| 10 | 15 | 45 | 30 | 60 |
| 11 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 |
The Sharma translation gives special sphuta rules for
For completeness, I’ve listed the rules from Sharma which are subsequently ignored. These also contradict the maximum dṛṣṭi given above from the earlier chapter.
| Graha | House | Rule | Virupa |
|---|---|---|---|
| Sa | 2 | 2*x | 0-120 |
| Sa | 3 | 60-x | 60-30 |
| Sa | 8 | 30+x | 30-60 |
| Sa | 9 | 2*(30-x) | 60-0 |
| Ma | 2 | x/2+15 | 15-45 |
| Ma | 4 | 60-x | 60-30 |
| Ma | 5 | 60-x | 60-30 |
| Ma | 7 | 60 | 60 |
| Ma | 8 | 60-x | 60-30 |
| Ma | 9 | 60-x | 60-30 |
| Jp | 4 | 45+x/2 | 45-60 |
| Jp | 5 | 60-x | 60-30 |
| Jp | 8 | 45+x/2 | 45-60 |
| Jp | 9 | 60-x | 60-30 |
Pt. Sanjay Rath has also given a table indicating the strength from aspects for calculation in dṛg bala. There is agreement with Sharma except for the 6H. The table is below and there is no adjustment of special dṛṣṭi for Jp/Ma/Sa based on the earlier chapter.
| House | Angle | Sharma | Virupa (Sharma) | Virupa (Rath) | Ma | Jp | Sa |
|---|---|---|---|---|---|---|---|
| 1 | 0-30 | 0 | 0 | 0 | |||
| 2 | 30-60 | x/2 | 0-15 | 0-15 | |||
| 3 | 60-90 | 15+x | 15-45 | 15-45 | +45 | ||
| 4 | 90-120 | 30+(30-x)/2 | 45-30 | 45-30 | +15 | ||
| 5 | 120-150 | 30-x | 30-0 | 30-0 | +30 | ||
| 6 | 150-180 | x*2 | 0-60 | 0 | |||
| 7 | 180-210 | 60-x/2 | 60-45 | 60-45 | |||
| 8 | 210-240 | 45-x/2 | 45-30 | 45-30 | +15 | ||
| 9 | 240-270 | 30-x/2 | 30-15 | 30-15 | +30 | ||
| 10 | 270-300 | 15-x/2 | 15-0 | 15-0 | +45 | ||
| 11 | 300-330 | 0 | 0 | 0 | |||
| 12 | 330-360 | 0 | 0 | 0 |
We are using Pt. Rath’s methodology for calculating dṛg bala and bhava bala.22As of 2021-10 this is the default. From various discussions and by analyzing the calculation methodology, the above table appears to be an approximation for the sphuta dṛṣṭi. The above table only accounts for the special aspects of Ma on 4/8, Jp on 5/9, and Sa on 3/10 and uses the regular rules for the other aspects.
Considering that there are many factors combined in bhava bala, if this table is an approximation for Jp/Ma for 3/10, Jp/Sa for 4/8, and Ma/Sa for 5/9 it could affect the overall result by 30-60 virupas23Nominally, the largest difference is only 30 virupas, but the error may get compounded as the figure is used several times.. The 7H aspect is treated as four pada also in this table, and while it is not prescribed by Parasara in the earlier chapter, there are opinions that it is true.24Rama’s chart could likely not form a dharma-karma yoga between Jp and Ma with the 7H aspect being only two pada and one pada.
More study is required for sphuta dṛṣṭi as there are several shlokas that appear to not be accounted for. If special rules apply for Jp/Ma/Sa for their 1, 2, and 3 pada aspects, the calculation table above would need to be adjusted for those special aspects.
Footnotes:
A detailed explanation of astronomical refraction physics is required.
Jyotish works with what we see. If we use true positions, why don’t we also account for the journey of light to the earth?
As you can see in the diagram, the light hits the atmosphere and then “refracts” which causes some of the sun’s rays to be visible. Note that for far away astronomical bodies, the refraction is negligible, but we usually are using the true position for them regardless.
This value was actually derived by Newton in the 1700’s and little progress has been made towards an optimal version.
Meeus - Astronomical Algorithms (1998) pg. 101
Astronomical Refraction and the Equinox Sunrise - Sampson (2000)
As far as I know, no jyotish software has also implemented a topological map in the atlas.
I did a few weeks of research, but since it isn’t published, in this section I will liberally refer to “Evaluating the Effectiveness of Current Atmospheric Refraction Models in Predicting Sunrise and Sunset Times Models in Predicting Sunrise and Sunset Times” Wilson (2018) And given the ease of observing sunrise and sunset, it is remarkable that there are few published datasets available. The largest of dataset is 10 years of data accumulated by Rabbi Avraham Druk in Israel, but most are less than a year.
There seemed to be more variance against the Schylter algorithm at larger latitudes suggesting that one may not be as robust. This algorithm is apparently the standard in php.
When accounting for the dip, in most cases we are over-estimating the refraction, but this is not relevant for this section for us right now.
The data from Edmonton was taken at 113W 29’ 14’’, 53N 31’ 33’’
Given by the standard deviation of two of the datasets in Wilson’s paper.
Also, if anyone says that their software is right or accurate, please refer them to this page. All software is generally wrong, it’s just a question of how.
For the upagrahas, we’re assuming each vela is 90 minutes. A one minute change in sunrise causes a 0.125 minute change in the period, which is 0.0625°. A five minute change in sunrise causes a 0.625 change in the period which corresponds to 0.3125°. The variation is then compounded for the 8th vela.
Though the Schylter formula does give some concern at larger latitudes.
BPHS - Chapter 9 - Aspects of Signs - Shlokas 10-15 (Sharma pg 161).
BPHS - Chapter 28 - Evaluation of Planetary Aspects (Sharma pg 359).
BPHS - Chapter 28 - Evaluation of Planetary Aspects - Shlokas 9-10 (Sharma pg 368)
Sharma pg 368-372
Shloka 11
Shloka 12
As of 2021-10 this is the default.
Nominally, the largest difference is only 30 virupas, but the error may get compounded as the figure is used several times.
Rama’s chart could likely not form a dharma-karma yoga between Jp and Ma with the 7H aspect being only two pada and one pada.