"Now we need to know where the sun rises at dawn on the summer solstice. Sarenites are the most likely to have those records." Luckily Sarenrae is one of the more popular deities in Osirion. Bhornkild asks around town to find the local Sarenites.
"I don't know anything about where the sun would rise on the solstice, but I'm pretty sure where we can find out. The local Sarenrae temple is this way" he points off in the general direction of where he believes the temple to be.
Ki'dhir looks around at the ruins and considers what he knows of the sun's yearly path.
Clueless about the position of the sun at midsummer, Zahira follows along.
Ki'dhir does not quite remember the exact position of the sun. In fact, his best guess, he believes is off by a few degrees. However, Marius does recall that there is an observatory in Tephu called the Eyes of Heaven and the astronomers may be able to provide you with the information needed, as they have recorded the position of the sun daily since the founding of the city. A quick trip over to the observatory and a 10 gp donation to their research fund and you have the position of the sun.
Now to calculate where the shadow of the tower would point involves a Knowledge (Engineering) check. I am open to what other ways you think you may discern the information.
Marius begins looking over the schematics of the tower. Taking a page out of his journal he begins scribbling notes on the paper.
"It should not be difficult to calculate..." The smartest barbarian in the world says.