Expert: Philip Stahl - 7/7/2010

Question
Hi,

I have an interest in archeo-Astronomy and I would like to calculate the date of an event in past time according to a measurements of angles I made on a petroglyph.  
I have my latitude: 45,09 which is in Quebec and I have the degree of a rising summer solstice sun on the horizon which is 57,17 degrees of azimuth. In the same year I should have had a rising winter solstice sun azimuth of 124,03 degrees. How can I calculate the date of this astronomical event? I have searched for calculators on the Net but didn't find any giving this answer easily.

Thanks,
Denis

Answer
Hello,

Your question poses problems at several levels, but let's examine the major ones.

First, there's the matter of errors. How accurate - to what uncertainty (or estimated standard error) - are your petroglyph markers? In their detailed examination of Megalithic Man and his use of such to ascertain the Solar rise position, Roy and Clarke (Vol. II, 'Astronomy Principles and  Practice'. pp. 60-61) note that even a differential of 2 yards (barely more than 1 m)to establish the angle, resulted in a shift of 12 arcsec in the Sun's position.

Second, a check of the putative declination of the Sun for the winter solstice Sun, based on your values and using:

cos (A) = sin(d)/ cos (L)

where A is the azimuth of the Sun

d is the Sun's declination, and L is the observer's latitude, yields

d = -23.3 Deg

As we know, the value ought to be the same as the negative for the obliquity of the ecliptic, or -23.5 deg.

This is the first hint - ASSUMING no large errors- we are dealing with a different, earlier era. The problem is to track it down.

A difference of 0.2 deg is about 12' of arc, or 720".

The problem is that the maximum variation limit for nutation in obliquity (delta eta) is ~ 18", as computed from:

delta eta =

(9.”2100 + 0.”00091t) cos Z  -  (0.”0904  - 0.”0004t) cos 2Z – (0.”0024 cos (2w_m + Z)  +  0.”0002 cos (2w_s – Z) + 0.”0002cos 2 ( w_ m + Z) + (0.”5522 – 0.”00029) cos 2L_s


where t denotes the time interval measured from 1900 January 0.5 d ET in Julian centuries (1 JC = 36525 mean solar days),  Z is the longitude of the mean ascending node of the lunar orbit on the ecliptic measured from the mean equinox of date, w_m is the ‘argument’ of the point where the Moon is nearest the Earth (i.e. from the lunar perigee), L_s is the mean longitude of the solar perigee measured from the mean equinox of date, and L_s is the geometric mean longitude of the Sun measured from the mean equinox of date.


(cf. Eichhorn and Mueller, 'Spherical and Practical Astronomy Applied to Geodesy', Frederick Ungar Publ., 1969)

A similar check using your summer solstice parameters yields an equally large deviation from the normal obliquity (+23.5 deg) and this is: 22.4 deg.  The odd thing here (and this speaks to the possible errors in measurements) is that technically the absolute magnitude of the value ought to be the same as for the summer parameters, but it's off by:

23.3 - 22.4 = 0.9 deg

which is stupendous.

Now, a still controversial theory known as the Milankovitch theory postulates a cyclic change in the value of Earth's obliquity, from 22 to 25 deg, over a period of 41,000 years. As we see, your values (assuming no errors) fall into this range- though based on the obliquity range (25 deg - 22 deg = 3 deg, your 0.9 deg represents a 30% error).

The trick then (and it's made much more troublesome by the 0.9 deg deviation!)  is to pin down the year for the petroglyph pointers, and that means obtaining a graph of the varying Milankovitch obliquity over time. I did some searching on the net, but up to now haven't located any such graph that's useful.

The probable reason why you're unable to find any net calculators to do the job you want is that - as I showed- the variable changes in obliquity veer outside the normal secular change range. Alas, no computational software that I know of incorporates Milankovitch obliquity cycles to obtain solar sunrise and sunset positions (tied to azimiuth) in years past.

There is a program called 'Cybersky' which allows you to go to any point in the past (e.g. 4622 BCE) , and locate the Sun's azimuth (by accessing the horizon coordinates option) as well as other markers, but it's all 'seat of the pants'. You can't just plug in the key values (such as you gave) and 'Voila!" get the precise year for those values, or markers. You have to do it all trial and error. Also, I suspect it only has the longitudinal and nutational obliquity variations programmed in, nothing more. So, it probably wouldn't be much use to you anyway.

A last resort tactic would be to write the astrometry gurus at the U.S. Naval Observatory, since they appear to have done some work or research in the area of the Milankovtich cycles, e.g.

http://www.usno.navy.mil/USNO/astronomical-applications/astronomical-information...

At the very least, they ought to be able to provide you with a high resolution graphic showing the changing nutation in obliquity!

Sorry, I can't be more helpful to you!