Q:

What is the exact time of sun stay in each zodiac every year on sidereal year duration?

I tried to reverse engineer from astro-chart times at which sun stays in a zodiac.
Here is the list.
Sun_Stay_zodiac  timeofstay_sec   timeofstay_days
  1  Aries         2672700     30 day 22 hour 00 minute
  2  Taurus        2714700     31 day 10 hour 05 minute
  3  Gemini        2734080     31 day 15 hour 28 minute
  4  Cancer        2719440     31 day 11 hour 24 minute
  5  Leo           2680080     31 day 00 hour 28 minute
  6  Virgo         2586960     29 day 22 hour 00 minute
  7  Libra         2626020     30 day 09 hour 27 minute
  8  Scorpio       2548020     29 day 11 hour 47 minute
  9  Sagittarius   2533080     29 day 07 hour 38 minute
 10  Capricorn     2544240     29 day 10 hour 44 minute
 11  Aquarius      2576520     29 day 19 hour 42 minute
 12  Pisces        2622540     30 day 22 hour 25 minute

Since the orbit of earth is elliptical and therefore also variation is orbital speed the stay of sun in each zodiac is different i.e Maximum at Gemini and after 6 months minimum at Sagittarius. However when I add them up sums up to 31558380 seconds which is 230.495 seconds than time in seconds in sidereal year ie 31558149.504 Now what I am looking for is more exact time duration of sun stay in each zodiac that includes seconds. I hope this data that I have in my question will help others too.

A:

A sidereal zodiac is not the same as the IAU constellations but corresponds more closely to them than the Western tropical zodiac does, e.g. the eastern edge of Virgo remains near Spica despite precession.
I assumed equal 30° blocks of ecliptic longitude with the J285.25 equinox as origin (Lahiri ayanamsa).
Your local tradition may differ slightly, but any difference should affect all signs equally and have little effect on Sun residence times.
I computed these statistics for the years 1970-2070 using Skyfield and the DE430 ephemeris:
     mean  stdev   mean
       seconds     days
 1  2667075 ±186  30.8689
 2  2702097 ±174  31.2743
 3  2717490 ±139  31.4524
 4  2708635 ±122  31.3499
 5  2678191 ±139  30.9976
 6  2635085 ±167  30.4987
 7  2591328 ±182  29.9922
 8  2558345 ±168  29.6105
 9  2544241 ±125  29.4472
10  2552351 ±101  29.5411
11  2580755 ±128  29.8698
12  2622569 ±167  30.3538

Gravitational interactions with other planets tend to make any given year slightly longer or shorter than the mean sidereal year of 31558150 seconds.
For the same 100 years as above, I computed a standard deviation of ±320 seconds in sidereal year length.

