Astronomy of Makar Sankranti

I am a member of Khagol Mandal. Last year I was told to deliver a slide show on "Astronomy of Makar Sankranti" in one of our sky observation programs. I expected that there would be a lot of material on internet on this subject, and I will have to just compile it. But to my great surprise, there is absolutely nothing available on internet! (At least I didn’t find anything). Also, I couldn’t find any books on this subject. So, I had to do all the work on my own. I had to look up Daate Panchang for getting Sankranti dates in the past and had to do some R & D through calculations to understand the astronomy behind Sankranti. So, this little work is my own original work and is not any kind of “copy-paste”. (Of course it is nothing new as the people who make panchang know this for centuries, but same thing is found out by me independently.) This blog is an effort to make available some material for those who are interested in the subject.

My article on the same subject got published in Marathi news-paper Loksatta dated 13th Jan 2009 (Thane vruttant supplement). This blog is translation of the same article.

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Astronomy of Makar Sankranti

In India, Moon is given more importance than Sun as far as religious festivals are concerned. Festivals like Diwali, Ganeshotsav, Holi, Naag panchami, etc are all based upon lunar calendar. Only exception is “Makar Sankranti”. This is the only Indian festival which is dependent on solar calendar. The reason is that, it is expected to be celebrated only in winters. On its occasion, we enjoy sweets like Tilgul, gul-poli in order to combat winter. Seasonal cycle is dependent upon Sun and that’s why Sankranti is celebrated based upon solar calendar. We are going to discuss the scientific reasoning behind Sankranti in this article.

Makar Sankranti word comes from the combination of two words “Makar (meaning Capricorns)” + “Sankraman (meaning Transition)”. It becomes clear that on this day, Sun enters the zodiac sign of Capricorns (Makar). It happens nearby 14th January or 15th January. Of course, this is only current state and in future Sankranti will slowly move forward. Right now, Sankranti comes on 15th January in a “leap year” and comes on 14th January in rest of the 3 years. For example, Sankranti was on 14th January in years 2001, 2002, 2003, 2005, 2006, 2007. And it was on 15th January in years 2004 and 2008. This means that the pattern of Sankranti is 14-14-14-15 in a span of 4 years. We are experiencing this pattern for last 40 years. As stated earlier, Sankranti is going to move forward and right now we are on the verge of such a transition. This is because in next 4 years – viz. 2009-2012 the pattern is going to change to 14-14-15-15. This means that in 2009 and 2010, Sankranti will be on 14th January, but in 2011 and 2012 Sankranti will be on 15th January. This new pattern will persist for another 40 years but in 2049-2052 Sankranti will take a jump of 1 day and the pattern will become 14-15-15-15. Again after 40 years, i.e. in 2089-2092, the pattern will be 15-15-15-15. In year 2100, something special will happen. This year will NOT be a leap year in spite of the fact that it is divisible by 4. In Gregorian calendar that we use, there is a rule that if a year is divisible by 100, then it has to be also divisible by 400 in order to be a leap year. This means that years 2100, 2200, 2300 and 1900 are not leap years, but 2000 and 2004 are. Because of the fact that 2100 won’t be a leap year, extra day won’t be taken in that year. As a result, Sankranti will jump by one day and in 2101-2104 Sankranti pattern will become 16-16-16-16. Similar jump will occur every 100 years (if it is not divisible by 400). This 100 years cycle and previously mentioned 40 years cycle – both these operate independently and keep pushing Sankranti forward. Average rate at which it moves forward is 1 day in 72 years. Table 1 states the dates of Sankranti in future.

Till now we have seen “what” happens, now we will try to understand “why” it happens. We have seen that when Sun enters Capricorns, the day is celebrated as Sankranti; but there is one more rule in this. If Sun’s entry into Capricorns occurs after Sunset, then Sankranti is celebrated on the next day. For example, if Sun enters into Capricorns on 14th January at 10:00 pm (i.e. post Sunset), then Sankranti will be considered on 15th January. Table 2 states the timings of Sun’s entry into Capricorns in years 2001-2008 (reference: Daate Panchang). If you observe it carefully, then you will understand that every year Sun’s entry into Capricorns moves forward by over 6 hours. In years 2001, 2002 and 2003, Sun’s entry into Capricorns was before Sunset, so Sankranti was on 14th January. But, in 2004 that was at 11:43 pm (after Sunset), so Sankranti was on 15th January. Meanwhile 2004 was a leap year, so one extra day was taken in it. So, Sankranti jumped 24 hours “backward”, so it came back on 14th January in 2005 and 14-14-14-15 pattern was resumed. If the extra day was not taken, then Sankranti wouldn’t have returned to 14th January (Same thing is going to happen in 2100.).



But, why does Sun’s entry into Capricorns moves forward by over 6 hours every year? The answer is that, Sun’s year is not exactly 365 days, but it is in fact 365.2564 days (This is called as sidereal year.). So, Sankranti moves forward by 0.2564 days or 6 hours 9 minutes every year. These 9 minutes are very crucial. In a span of 4 years, Sun’s entry into Capricorns moves 24 hours and 36 minutes forward, but leap year brings it backward by only 24 hours! As a result, Sankranti moves forward by net 36 minutes in 4 years. This difference goes on accumulating and in 40 years it becomes big enough to change the pattern of Sankranti.

Observe the year 2007 in Table 2. For that year, Sun’s entry into Capricorns is at 6:07 pm IST. On 14th January, sunset is at 6:19 pm IST at Mumbai. In 2011, Sun’s entry into Capricorns will move forward by 36 minutes, which means that it will be after sunset. So, Sankranti will be on 15th January and pattern will become 14-14-15-15. Now, interesting thing is that the city Kolkata is 15 degrees towards east of Mumbai. So, on 14th January sun sets at 5:19 pm IST at Kolkata. This means that, in Kolkata Sankranti was on 15th January in 2007 itself! Sankranti pattern for Kolkata was changed 4 years before Mumbai. The graph below will give a better idea about Sankranti transition every year and every 4 years too.

Fig 1: Graph showing forward transition of Sankranti

The main reason behind the forward movement of Sankranti is that Indians assume a fixed zodiac belt. That is the first point of Aries is assumed to be fixed in Indian system. It doesn’t take Earth’s precessional motion into account, so this system has flaw. The western zodiac belt takes precession into account. So, even if Sun’s entry into “Indian” Capricorns is on 14th January, its entry into “western” Capricorns is on 22nd December. We know that Sun is at its southernmost point on 22nd December. So, this day looks more appropriate for Sankranti celebration. Fact is that roughly 2000 years ago, Sankranti was on 22nd December only. Since then it has moved forward continuously and today it has reached 14th January. Sankranti is a seasonal festival, so its forward movement is not desirable. Some time in distant future, Sankranti will be in May and we will be eating Tilguls and fall ill because of the excessive heat. Of course, this will take thousands of years, but we are already 2000 years into the cycle. Today also we celebrate it near the end of winter. Peak of winter is in December. So thinking scientifically, it is appropriate to keep Sankranti on 22nd December forever (make it dependent upon zodiac belt with precession). This is my personal opinion.

Figures 2, 3, 4, 5 will give you a better idea about the Sankranti patterns.

Fig 2: 14-14-14-15

Fig 3: 14-14-15-15

Fig 4: 14-15-15-15

Fig 5: 15-15-15-15

Verbal proof of "Law of conservation of momentum".

Hmmm. The intension here is to demonsrate the law of conservation of momentum more intutively. Newton's 3 laws of motion can be understood quite intutively. When a force is applied, a body changes its motion. If there was no force, its motion wouldn't have changed. It can be imagined quite easily. But, when we go one step ahead, i.e. to Law of Conservation of Momentum, things tend to become more mathematical. And One makes a statement "the velocity of a body should be blah blah m/s because the momentum of the system should get conserved." I am sure, in many cases the student who makes this statement doesn't understand much from it. The student is perfect mathematically. But does he really understand the physics behind it?

In this little blog, I will try to expalin the proof of Law of conservation of momentum verbally and so more intutively. The raw material for the proof is going to be only Newton's 3 laws of motion! So, before starting with the actual proof, I will state the 3 laws.

1st law: Every body continues to be in its state of rest or of uniform motion along a straight line unless it is acted upon by an external unbalanced force.
This law defines force qualitatively.

2nd law: The rate of change of momentum is directly proportional to force applied and the change takes place in the direction of force.
This law defines force quantitatively.

3rd law: For every action there is an equal and opposite reaction.

Momentum = mass * velocity

Law of conservation of momentum: In a system of bodies, the total momentum remains constant if no external unbalanced force acts on the system.



Proof starts ------X--------X--------

To start with, consider one single body. What happens if no force acts on it? Well, 1st law states that its velocity should remain constant. And its mass is also constant (We won't consider relativity). So, mass times velocity that is the momentum remains constant.

But, what if there are a number of bodies in the system? They can affect each others' motions through collisions or by gravitational / electromagnetic forces. In this case, 3rd law comes to our rescue! According to it, the forces in a pair of interacting bodies are equal and opposite. We have extracted this important information that, 2 bodies - whatever may be their masses - apply equal and opposite forces on each other. The 2nd law now does rest of the work. According to it, rate of change of momentum of a body is directly proportional to force applied on it. So for a given amount of time "Change in momentum of a body is directly proportional to the force applied". In case of 2 bodies interacting, they apply force on each other for equal amounts of time. So, both the bodies experience equal and opposite forces for equal amounts of times. As a result, they are subjected to equal change in momentum in opposite directions. That is, if momentum of body A increases by some amount then for body B it must reduce by same amount. In other words, velocities of bodies change in inverse proportion of their masses. So, product of mass and velocity remains constant. As a result, the total momentum gets conserved.

What if there are more than 2 bodies? Principle of superposition applies here. That is net effect on a body is combined effect because of all bodies considered individually. Above 3 paragraphs apply to "all pairs" in the system. Superimposing them, the total momentum gets conserved.

Proof ends ------X--------X---------

Law of conservation of momentum is really a very handy law. It gives a simple solution to complex system. Working out individual interacting particles by analysing forces between them would become very tidious. Instead we can consider the whole system of particles at one go and work out the solution. But, in doing so there is a chance that the student does it totally mathematically and moves away from the underlying science behind it. I hope this blog clears the concept.

Ameya L. Gokhale