Shadow length and depth on Dec 21 3PM for any latitude?

I posted a question yesterday about the shadow length (see link below) and it was fully answered.

http://answers.yahoo.com/question/index;_ylt=ArvbP573dkyTVLdet4Gin5_ty6IX;_ylv=3?qid=20111008122802AAFuRnB





PART 1: To test that i built in excel the right formula I built a simple 3D model with Google Sketchup (a 3D tool that simulates shadows for specific latitudes among other things). What Sketchup drew does not match the result of formulas provided in my previous questions. (the link below includes a screenshot of the sketchup model showing latitude, day and time with all dimensions of the object and the shadow)

https://premierpower.box.net/shared/m2544miqhrm8t9276agn

I am not sure if i didn't use the formula correctly or if Sketchup has a bug. Any help is appreciated.



PART 2: In addition, i need to calculate the depth of the shadow. In other words, the rod or object will project a shadow of length L at an angle (given that is 3PM DEC21 i assume the angle of the shadow is 45 degrees). If i draw a line from the rod/object going north until if forms a right triangle with end of the shadow that is what i call the shadow depth (6.05m in the sketchup model). Can i calculate the depth with simple trigonometry? Is 45 degrees correct?Shadow length and depth on Dec 21 3PM for any latitude?
I answered your question yesterday. I'll answer Part 1 first, and come back shortly with Part 2.

Part 1:
In Google Sketchup, you used a location of N40.445, E17.415944, with a time zone of UTC+2 and a time of 3 PM on 12/21.

My personal astronomy software shows that the sun then has an altitude of 19.147 degrees; so if you have a stick of height H, the shadow length should be
L = H / tan(19.147) = 2.88 H

In your example with Google Sketchup, the height is H=2.46 m, so I would say the length should be
L = 2.46 x 2.88 m = 7.08 m

This agrees within 1% with the length given by Sketchup (7.14 m), so my calculations are consistent with Sketchup.

The formula in yesterday's answer, however, says
L = 4.11 H

Why are these two formulae for L different (4.11 vs 2.88)? The answer is that you're using the wrong time. My answer yesterday assumed a local solar time of 3 PM -- that is, three hours after the sun is due south. For the coordinates and time zone you gave today, the sun is due south not at noon but at 12:48 PM. If you set the time to 3:48 PM, Google Sketchup should agree closely with my result.

(Note that from an astronomical point of view, UT+2 is not the proper time zone for a longitude of 17 degrees east; but time zones are determined by politics as much as astronomy.)

---

Part 2:
No, the sun's azimuth from south at 3 PM local solar time is not 45 degrees (unless you happen to be at the north or south pole). It's more complicated than that, and once again we have to use spherical trigonometry.

Let's define the following:
Z = azimuth of sun measured from south (0 if sun is due south, 45 if sun is in southwest, 90 if sun is in west, etc.)
D = "depth" of shadow as you define it -- that is, the length of the north/south component of the shadow

Like yesterday, we can split this problem into two parts. The easy part is one of simple trigonometry:

D = L cos(Z)

For instance, if the sun is due south, the depth equals the length. If the sun is due west, the depth is 0 (because the shadow runs along the east/west line).

The more difficult part is calculating Z. I again refer you to the spherical triangle defined in yesterday's answer. At this point, we know all three sides (since we have calculated h, the azimuth of the sun) and angle A (45 degrees). Given this information, it's easy to use the law of sines to calculate the unknown angles.

The angle of interest is B, which is related to Z as follows:
Z = 180 - B

The law of sines for spherical trigonometry says that
sin(A)/sin(a) = sin(B)/sin(b) = sin(C)/sin(c)
where A, B, and C are the angles of the vertices, and a, b, and c are the lengths of the opposite sides.

We make use of the following:
sin(A)/sin(a) = sin(B)/sin(b)

Using the triangle defined yesterday, this becomes
sin(45)/cos(h) = sin(180-Z) / sin(113.4)
= sin(Z) / sin(113.4)

Therefore,
sin(Z) = sin(113.4) sin(45) / cos(h)
Z = arcsin (0.64895 / cos(h))

example:
Let's use the coordinates mentioned above at 3:48 PM (three hours after the sun is due south). The altitude of the sun at that time is 13.66 degrees. Using the above formula, we get
Z = arcsin (0.64895 / cos(13.66))
= 41.9 degrees

(My astronomy software says that the sun at that time is 41.8 degrees west of south, which is nearly identical to the above answer. The difference could be caused by rounding or by tiny discrepancies in the definition of time, since solar days are of slightly variable length.)

Now we put this all together:

Step 1: Yesterday's answer showed how to calculate h. Because h=13.66,
L = H/tan(13.66)
= 4.11 H

Step 2: Today's answer shows how to calculate Z. The depth is given by
D = L cos(Z)
= L cos(41.9)
= 0.744 L
= 0.744 (4.11 H)
= 3.06 H

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