Expert: Philip Stahl - 3/15/2009

Question
Dear Dr. Stahl
My question was on which date, in latitude  13° 32' South, the tip of the noon shadow will be precisely halfway between the tips of the noon shadows of the two solstices. I would also be happy to know how it is calculated. I am not a student but a medical doctor and the question relates to my noncurricular interests. I would be very grateful for your reply.
Thank you.

Answer
Hello,

Your question-problem is actually not so simple as first appears because you do not ask for a *specific* length (for which a specific shadow variation) can be computed using the dates of the solstices, for example. "Halfway between the tips" of "noon shadows" is not very information intensive when one has no idea what the lengths of the shadows are originally.

As a matter of comparison, someone else asked (about four years ago) concerning minimum shadow lengths for given locations.

A very basic relation (along with some simple observations)also  allows one to determine one's latitude.

That is:

 tan (ALT)  =  H/ L_m

where 'ALT' is the max. altitude of the Sun at the location of the
observer, H is the height of a vertical object (assumed to be planted on flat ground), and L_m is the (minimum) length of the object shadow measured.

(Note here, that ALT can easily be computed at the solstices from Cuzco using the geometry of the situation, and knowing it is at latitude of (-13.5) approximately.)

For example, a latitude of (-13.5) means that at the equinoxes the Sun will be directly on the celestial equator and so have a declination D= 0 degrees. For a location on the equator (0 deg latitude) this means directly overhead at noon and zero (essentially shadow length). For Cuzco the Sun would be 13.5 degrees *north* of the local zenith because you already have (-13.5 declination) overhead.

Similar working can be done for the solstices, knowing the Sun's declinations on those dates (e.g. (-23.5) for Dec. 21, and (+23.5) for June 21). However, the shadow length angles are still minimal at local noon.

Here's the core problem in a nutshell: what you are really asking for is the minimum shadow angle for a date that occurs "precisely halfway" between the minimal shadow angles of the two solstices.

The relation by geometry is given by:

tan (S) = sin(lat) *  tan (H)

where S = shadow angle

lat = latitude (e.g. -13.5 deg for Cuzco)

H = hour angle.

To fix ideas here, when H = 2h, that is an hour angle of 2hrs. but this is of no use because NOON shadows have the minimum hour angles. Thus, H = 2h is inapplicable because this applies to a time of 2 p.m. local noon (or 10 a.m. if negative) but not noon.

so the ONLY value H can assume for NOON shadows is H = 0

But tan(H) = tan (0) = 0

and

tan (S) = sin (-13.5) x 0 =  0

so we aren't getting anywhere fast.

The only way to escape the conunudrum is actually not to work with hour angles at all, but to use azimuth angle (A)  and calculate the extreme azimuth dates for the setting (or rising) Sun say, and then the *median* azimuth date between them. Again, this is to obtain a generic answer to your question. (Since you aren't asking for a specific shadow length for a specific original length)

I am not about to go through all the azimuth calculations (since I do not know what your math background is) but the basic relation for a given azimuth angle A is:

A = arcos [ sin (D)/ cos (lat)]

And using this one obtains extremum values for the azimuth of the sun (e.g. on the western horizon) for it setting times, for the two solstice dates. The median azimuth (90 deg) is computed for either of the two equinox dates. By a process of simple geometry and reasoning from this, one can then deduce that the date you want is on either of the solstice dates at noon, local time.

Now, if I have misapprehended what you actually want, and it isn't what I have provided - please let me know.