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Rasinamas and Charakhandas - ascensional differences

 
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kepler



Joined: 30 Jul 2013
Posts: 11

Posted: Thu Apr 11, 2019 11:26 pm    Post subject: Rasinamas and Charakhandas - ascensional differences Reply with quote

Good evening

Actually, my question is not exactly regarding the title, but it has to do with that: the ascensional differences.

I'm trying to get to the different values each sign takes to rise according to the latitude.

I believe that there must be a trignometric function for this - to calculate the Charakhandas for instance.

So if I know that Aries takes 1h51m36s to rise at 0º latitude (in the equator), or Gemini takes 2h8m44s, etc., how do I calculate - accuratly - the number of minutes or seconds I must subtract each one to know how long will they take to rise at 10 degrees North, or 50 degrees south for instance?

Does anyone has the trignometric formulas? I made some advances, but I'm getting too much shorter times for latitudes of 30 or 40.

Clear skies

Kepler
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astrojin



Joined: 15 Nov 2005
Posts: 469

Posted: Mon Apr 15, 2019 3:24 pm    Post subject: Reply with quote

Hello,

I could just give you my Excel file that calculates the rising times for different latitudes rigorously (that you could also see the formulas in action) - PM me and provide your email if you are interested.

Calculating the rising times of the signs using spherical trigonometry:

The calculation using spherical trigonometry is relatively simple as we are calculating for the degrees on the ecliptic (ecliptic latitude is always zero).

I am assuming you are just interested in changes of the rising times of each sign for different geographical latitudes.

First:
We need to calculate the obliquity of the ecliptic but the value is almost constant with a couple of centuries.

Obliquity = 23.43929° = 23° 26' 21.448'' - calculated for Jan 1, 2000 (@12UT).
Mean Obliquity decreases 46.8 seconds of arc per century - hence, can be assumed constant when working within a couple of centuries

Taking this value of obliquity, we could then calculate the rising times of each zodiac sign for any latitude.

For latitude 0 deg:
Ari - 1h 51m 39s
Gem - 2h 8m 43s

These numbers are slightly different than yours probably due to a very slightly different value of obliquity (which can be safely ignored).

One important thing to note however is that the above numbers are duration measured according to the sidereal clock and not the normal/regular clock. Sidereal clock is faster than the regular clock (Sidereal clock is 1.00273790934 times faster than a regular clock). This is why the stars rises 3 minutes and 55.90947 seconds earlier every night.
If we were to measure the rising times according to the regular clock, then:
Ari - 1h 51m 21 s
Gem - 2h 8m 22s

List of all rising times at 0N (using sidereal clock):
Ari = 1h 51m 39 s
Tau = 1h 59m 38s
Gem = 2h 8m 43s
Can = 2h 8m 43s
Leo = 1h 59m 38s
Vir = 1h 51m 39 s

List of all rising times at 0N (using regular clock):
Ari = 1h 51m 21 s
Tau = 1h 59m 18s
Gem = 2h 8m 22s
Can = 2h 8m 22s
Leo = 1h 59m 18s
Vir = 1h 51m 21 s

Do remember that Ari-Pis, Tau-Aqu, Gem-Cap, Can-Sag, Leo-Sco and Vir-Lib are pairs of SIGNS of EQUAL RISING TIMES.

Once we have the rising times for each sign of the zodiac, we can then calculate for any latitude by using the following correction:

Rising times (sidereal clock):
Ari: 27.91288 – abs[arcsin(0.20283xtan(phi))]
Tau: 29.90840 - abs[arcsin(0.36673xtan(phi))] + abs[arcsin(0.20283xtan(phi))]
Gem: 32.1787 - abs[arcsin(0.43328xtan(phi))] + abs[arcsin(0.36673xtan(phi))]
Can: 32.17872 - abs[arcsin(0.36673xtan(phi))] + abs[arcsin(0.43328xtan(phi))]
Leo: 29.90840 - abs[arcsin(0.20283xtan(phi))] + abs[arcsin(0.36673xtan(phi))]
Vir: 27.91288 + abs[arcsin(0.20283xtan(phi))]
Where phi is the geographical latitude and “abs” is taking the absolute value (i.e. always take positive value).

Example for rising times of Tau at 20N:
29.90840 – abs[arcsin(0.36673xtan(20))] + abs[arcsin(0.20283xtan(20))]
= 29.90840 – abs[arcsin(0.36673x0.36397)] + abs[arcsin(0.20283x0.36397)]
= 29.90840 – abs[arcsin(0.133479)] + abs[arcsin(0.073824)]
= 29.90840 – abs[7.671] + abs[4.234] = 26.4714

Multiply this by 4 to get rising times in minutes according to sidereal clock = 105.8856 minutes
= 1h 45m 53s – voila!

To get the rising times according to the regular clock you must divide the minutes obtained from the sidereal clock by 1.00273790934
105.8856/1.00273790934 = 105.5965 minutes = 1h 45m 36s.
The rising times for Tau is the same as Aqu as they are signs of equal rising times.

You can then do the other 5 and by equivalent pairing, get all the rising times!

Once you get for Northern latitude, you just have to invert Ari to Virgo and Libra to Pisces for southern latitudes.

The following are the values for 20N:
Ari-Pis: 1h 34m 43s, 1h 34m 27s
Tau-Aqu: 1h 45m 53s, 1h 45m 36s
Gem-Cap: 2h 3m 6s, 2h 2m 46s
Can-Sag: 2h 14m 20s, 2h 13m 58s
Leo-Sco: 2h 13m 23s, 2h 13m 1s
Vir-Lib: 2h 8m 35s, 2h 8m 14s

First column are values for sidereal clock and second column are values for regular clock.

The following are the values for 20S:
Vir-Lib: 1h 34m 43s, 1h 34m 27s
Leo-Sco: 1h 45m 53s, 1h 45m 36s
Can-Sag: 2h 3m 6s, 2h 2m 46s
Gem-Cap: 2h 14m 20s, 2h 13m 58s
Tau-Aqu: 2h 13m 23s, 2h 13m 1s
Ari-Pis: 2h 8m 35s, 2h 8m 14s

The same can be done for any latitudes as long as it is smaller than (90 - obliquity)!!!

Hope this is clear!
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