What time was this photo taken?

As I read through my facebook feed, I was struck with following picture:

Immediately -- shows how much of a dork I am -- I thought "I wonder what time this picture was taken?!"

You see, as the earth rotates around the sun, shadows rotate around the objects that form them. In the northern hemisphere, these shadows rotate clockwise -- which is why clockwise is clockwise. The first clocks ever made were sundials, made by people living in the North, and then clocks were built later.

I figured I should be able to figure out the angle of the shadow of the arch and use it to figure out what time of day the picture was taken.  I could also figure out the date the picture was taken by looking at the length of the shadow. You see, everyday the angle of the sun at a given time changes. Right now, during the spring, the sun is higher in the sky every day at a specific time, which makes shadows shorter. Measure your shadow at 11:00am today and measure it again tomorrow and it will be smaller!

So I found a map of St. Louis, and used Geogebra to figure out the angle of the shadow of the sun, and the length of the shadow.  After about five minutes, I had placed a point on the map that represented where I thought the top of the shadow was, and had drawn a vector from that point to the point that represented the top of the arch. I compared that with the scale of the map, and estimated the length of the shadow to be about 1,000 ft.  After looking on wikipedia, I knew the height of the arch, and a little trig revealed the altitude of the sun to be about 32 degrees.

In a few more minutes I had estimated the angle of the the vector and converted that into a compass heading, which gives me the azimuth of the sun of approximately 111 degrees.  

I knew there is only two times a year where the sun has that exact altitude and azimuth, once in spring and again sometime in the fall -- and I took a chance that this picture was taken on spring break (reasonable enough right?). So I looked up the altitude and azimuth for the sun on the days during spring break:

Since the photo was tagged as uploaded on April 1* I started with that date, and found the following data in the table:

The first column is the time (AM), the second column is the altitude of the sun, and the third column is the azimuth of the sun.  I was disappointed that I didn't see my exact values in the table -- but I didn't expect to either, for two reasons:
  1. I didn't know if this was the correct date -- the picture might have been uploaded that day but taken several days (or even a half a year?!) earlier.
  2. There is some degree of uncertainty in my measurements. As I moved around the point where I thought the top of the shadow was, the angles varied somewhat. To be specific, they varied less than a degree more or less than my values, but that's significant enough to make my answers have to be estimates.

Let me treat each of these reasons separately.  Assuming the picture was actually taken on April 1, and my measurements were slightly off, I would estimate that the picture was taken around 8:34 am local time (I could be off by an hour if the website doesn't account for daylight-savings time, but I'm going to assume they were smart enough for that).

If I don't assume to know the date the picture was taken, and trust my measurements, I would argue that the picture wasn't actually taken on the 1st.  Looking at similar tables for other days, I get much closer altitude/azimuth combinations for a few days later:

 If I had nothing else to go on, I would estimate the date/time of the picture was April 3, 8:33am.

Perhaps the photo takers will provide the true answer in the comments below?

*There was some discrepancy between my wife and I as to when the picture was actually uploaded onto Facebook. It was posted April 5th, "tagged" April 1, but I have reason to doubt the "tagged" date. Only time will tell who wins our little "argument" -- although regardless of who wins, I will probably lose -- right guys?  I love you honey!