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Comming soon: The Persian New Year

In precisely 8 hours, 27 min, 43 sec and 253 msec the 2015 Persian new year will start. Norooz, as it is called in the old indo-german Persian language, starts all around the earth at exactly the same moment. So how people know so precisely when Norooz takes place ?  Is it simply the beginning of the spring season, which by some coincidence indeed happens nearly at the time given above. Converted into Central European Time, Norooz takes place tonight at 11:45:11 pm, i.e. 15 min before the European begin of spring at March 21st. But this is pure coincidence, since the real astronomic begin of spring is only weakly bound to a fixed date, but on the spring solace. Only the United Nations made the funny mistake and declared an International Norooz Day, fixed on March 21st.
For every Persian who is aware of the great tradition of their astronomers of Archaemenid and Sasanid era, however this simplification is rediculous. To show them that people in Europe are well aware of the proper calculation of Norooz, I show here how to do it right.

In common wisdom, spring starts at spring solace. And already school kids learn that thats the point in the calender when day and night are equally long. But this still is a very poor definition. In fact, simply 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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