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Norouz - The ultimate formular

Two weeks ago Norouz was celebrated, the traditional feast of the New Year that goes back to the Iranian Achaemenidian empire. It is still the one of the most important traditional events of the seasonal cycle for people in Iran, Afghanistan, Tadshikistan and Kurdistan. But in fact, it is highly regarded by everybody who likes mathematics, celestrial mechanics or astronomy. 
In contrast to the christian new year which is very ill-defined (just think about the arbitrary adjustment by inserted lap-years or pope Gregors order to cancel 11 days of the year 1582. The reason for this highly imprecise calculations of the beginning of each New Year is the complete asynchronity between the earth' revolution around the sun (determining the seasons and in particular the points of spring and fall equinox) and the earth' rotation around its own axis. And since christian New Year always starts at 0:00 am on January 1st, all these artifical and highly unscientific adjustments have to be made.

It has a certain irony that although the United Nations a few years ago adopted the tradition of Iranian New Year at the spring equinox and included it into their official list of word-wide days to celebrate, they fall back into the very primitive and un-scientific approach by setting a fixed the date for the "International Day of Norouz", which is statically commemorated each your on March 21st, irrespectively if the the people in Iran, Afghanistan and so on celebrate "real" Norouz at 20th or 21st of March.

Quite different is the precise approach which the Achaemenidian astronomers used to define Norouz: The Achaemenidian emperors (the Shahs) were simply obsessed by understanding precisely the regular movement of stars and planets, including our earth. And they employed excellent scientists like Ibn-Sina and Omar Khayyam to make precise observations, find the general laws of the celestrial mechanics and allow precise predictions of astronomical events.
And for them, the most important was the moment of spring equinox, the beginning of their new year. This is roughly equivalent to the beginning of spring in the western world, which there always happen on March 21.
Iranian Norouz took place this year on March 20th, at precisely 17:57:07 GMT. And here starts a bit of confusion, of course, if you consider the common definition of Norouz as the spring equinox, or moment of the year when day and night have exactly the same duration. So how can day and night have the same duration at 17:57:07 GMT ???  In fact, it will happen very very rarely that any day-and-night pair at around the 20th or 21st March will be of equal length. The best one could do with measurement of day and night duration would be to say that a particular pair of night-and-day or day-and-night have the smallest difference in their length. But then, how can a precise time as this years 17:57:07 GMT can be related to a measurement of day or night length ??

So in fact, by measuring the length of day and night would not have satisfied nor the Achaemenidian rulers neither their skilled astronomers. It would not have been much different than the error prone definition of christian New Year, also linked to the midnight point of the earth rotation around its own axis. The Achaemenidians wanted to know precisely when the earth passes a definite position on its way around the sun, and they understoud that the clock time on earth is absolutely irrelevant for this and should be introduced only at a very late stage of calculation.

What is really relevant, though, is the orientation of the axis of earth rotation relativ to the direction of the link between earth and sun. And this direction is of course adequately described by a vector. 

So we have to vectors here, the earth own angular momentum A and the radius between sun and earth. direction R. The absolute length of both A and R is irrelevant for the later calculation, the only thing that matters is their direction.

A, the earth angular momentum is in first instance (for a period of several thousand years) fixed in space and as we all know orientated to the Polar star. For convenience, we can shift A from the earth to the center of the sun and still have it pointing to the Polar star.

Assuming that coordinate origin for all our calculations is at the center of the sun, than this point O has the  coordinates                               O:   [Xo=0; Yo=0; Zo=0].
The position of the polar star is   P: [Xp; Yp; Zp] 
and hence the unit vector of A will be defined as

                                         A  : [Xa; Ya; Za) = (Xp;Yp; Zp) / Sqrt(Xp*Xp+Yp*Yp+Zp*Zp)

                                               abs(A) = 1    angel(A)=23.4 degrees  (obliquity of the ecliptic)

The second vector that defines the trajectory point of Norouz is the earth radius around the sun. 
                                         R :  [Xr(t); Yr(t); Zr=0]
Here the bracketted t denotes that X and Y of the radius tip move during one year on a quasi-eliptical path around the sun (not taking into account disturbances by other planets).
Therefore, X and Y of the radius vector are interrelated by the formular


with a and b being the smaller and the larger radius of the ellipse. Conveniently, we also set this vectors at unit length.

And now the precise definition of Norouz is very simple. It is the point on the earth trajectory at which the vector A is orthogonal with vector R, i.e. when the rotational axis of the earth stands orthogonal on the radius between sun and earth. At precisely this position the rays from the sun illuminate the southern and the northern hemisphere at exactly the same degree, and the Northern and the Southern pole are at an equal distance from the sun, and the radius between sun and earth stands also orthogonalon the equator.

Although the scheme is still using the Ptolaemaus model (with the earth resting in the center and the sun revolving around it), the conditions and the mutual positions of earth and sun are equivalent to the Galilaean model. In the scheme above, the earth is conviniently not only place at a resting point in the eliptical path of the sun, but the earth own rotational axis is set exactly verticaly.

The algebraic formular for two orthogonal vectors can be expressed as the scalar (or point) product and should be zero:
                                              A o R = 0

And the scalar products A o R is the sum of the single components products
                                   A o R = (Xp*Xr(t) + Yp*Yr(t) + Zp*Zr)  /  Sqrt(Xp*Xp+Yp*Yp+Zp*Zp) 

since Zr=0, we can reduce further to
                                  A o R = (Xp*Xr(t) + Yp*Yr(t)  /  Sqrt(Xp*Xp+Yp*Yp+Zp*Zp)  

and after on the condition that the scalar product should be zero
                                  A o R = (Xp*Xr(t) + Yp*Yr(t)  /  Sqrt(Xp*Xp+Yp*Yp+Zp*Zp)  =  0
                   -->          (Xp*Xr(t) + Yp*Yr(t))  = 0
                   -->          Xp*Xr(t)  =  - Yp*Yr(t)                                                              (II)

With equations (I) and (II) we have now a system of one linear and one quadratic equation for Xr(t) and Yr(t)
              Xp*Xr(t) + Yp*Yr(t)  = 0
With x and y in the lower equation should in fact be Xr(t) and Yr(t) as in the formular above, and Xp, Yp, a and b are fixed tabulated values.  
Using a formular solver program like from Wofram Research' Mathematica this system of equations will straightforward give you a value for t (or in fact two values, one for spring and another one for autumn equinox). Be aware that t is note the a time, but the trajectory parameter. It is, however unequivocally related to the celestian time on earth and hence can be directly converted to a day and time.

 But this is the point where the whole fuzz of the precise beginning of Norouz, or Sal Tahvil gets in. But in fact, the confusion is not caused by any of the above equations to calculate Noruoz, but the fuzz is due to the
astronomical unprecise definition of the normal, global calender and its weak association with the celestrial calender. So the Sal Tahvil can vary between afternoon of the 20th of march till midday of 21st of march (according to the global calender and global time), but in relation to the fix stars and to the sun, Nowrouz is always precisely at the same position.

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