Jerusalem in 1187: Fight or surrender?

It takes Saladin nine years to conquer Aleppo after surviving an assassination plot. Thus, in 1183 he controls the valleys of Egypt, Tigris and Euphrates and is about 600 miles from Jerusalem. The Templar Reynald angers him by attempting to destroy a Muslim shrine in Egypt and sacking a caravan that, legend say, included Saladin’s sister. Fed-up and encouraged by the death of the young leper King of Jerusalem in 1184, he sets to war and defeats the Christians at Hattin on July 4th, 1187. This gets him about 100 miles from Jerusalem. On September 20 he is right under the walls and asking for unconditional surrender. The clerics refuse and threaten to kill themselves, the Muslims and destroy the Holy places on top of it. Saladin does not want that harm and a fierce siege and desperate defense starts. On October 2 Saladin breaks one of the Jerusalem’s walls and gets in and what he asked for beforehand: unconditional surrender.
In Europe, the Pope is profoundly affected and The Holy Roman Emperor Frederick Barbarossa, France’s King Filip II and England’s King Henry the Lion set to retake Jerusalem. That’s the Third Crusade. On a personal basis, Frederick sends a letter to Saladin, threatening that he’ll have to deal with him now, on the battlefield. Unfortunately, Frederick dies by drowning in the Saleph river. Umberto Eco story is that Frederick went for a swim while historical sources say that he fell from his horse in the river. I choose to stay with Umberto in my movie adaptation.