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While January 1 just ensures a change of calendar and considered a new English year for administrative purposes, Chaitra Shukla Pratipada and Mesh Sankranti of Sun are the days which actually tell us the future.
I will tell in brief. Please read in full.
RT after reading.
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Samvat begins when Chaitra Krishna Amavasya ends (Sun and Moon are at the same degree in Pisces). This Tithi is Chaitra Shukla Pratipada (CSP). The Lord of the day at Sunrise when CSP is operating is considered Raja (king) of the Samvat. It is called वर्षपति in Sanskrit.
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Varshpati or The King of Samvat should be very strongly placed in the chart prepared at the start of CSP for the year to be good and promise growth. A poorly placed king, even if a Shubh Graha, reduces growth and doesn't deliver its benefic results.
Importance of Varshpati.
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Raja (Varahpati/King) of the Samvat also decides Vaahan (vehicle) of the Samvat. Different vehicles set the tone of results of the year.
First 4 Pratipadas of Samvat also decide 4 Pillars (Stambhas) of the Samvat: Jal (water), Truna (grass), Vaayu (wind) & Anna (food grains).
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Knowing Samvat vaas (Residence) also helps in predicting rainfall for the Samvat. In medieval and ancient periods, these were heavily used but now the habit to use them for rain predictions has reduced, however, they still form the basis for such predictions.