The famous astronomical essay people keep asking for. Quoting from "The Silmarillion", one can prove that the sun in Middle-earth was static. It was located, mostly, over the realm of Hithlum.
Introduction
Truth can be hidden deeply in Tolkien's words. In a sentence he sometimes says more than other authors in whole books. The following article is just one example for that.
This essay is a transcript of a speech I first held at the Conference on Cosmogony and Astronomy in First Age Middle-earth, which my colleagues at FATS (Fredonian Academy of Tolkien Studies) hosted in 2006 [1]. Unfortunately, the transcript was lost at some time in 2007 due to a malevolent plot of my many enemies who are trying to erase my name from history whenever I look the other way. I would like to thank Count N. Vagorovich, who kept a printout of the original text and sent it to me, so truth could be restored.
The Static Sun Theory
"But Lúthien was silent, and from that hour she sang not again in Doriath. A brooding silence fell upon the woods, and the shadows lengthened in the kingdom of Thingol." (The Silmarillion, Of Beren and Lúthien)
From these simple sentences we can draw the surprising conclusion that the sun in Tolkien's universe was static. If the sun was moving the way it did today, rising in the morning and setting in the evening, then the shadows would be long every morning, then shorten and reach minimum length every noon, and lengthen again every afternoon. This is basic physics and can be observed every day. There simply would be no need to explicitly state that "the shadows lengthened" if the shadow changed every day anyhow [2]! No, the sun must have been in the same place all the time, so that a relatively minor change in position was noteworthy to the chronist.
The concept of static lights is not alien to Tolkien's world - the major light source before sun and moon were two glowing trees, which were undoubtedly immobile [3].
How far was the static sun from Doriath?
If we operate with a static sun, the next question inevitably is where over Middle-earth it was located. It is unlikely that it was positioned directly over Doriath, for then its rays would have hit Doriath exactly vertically, and there would have been hardly any shadows to begin with. In this case, after the change, Tolkien would have written something like "...and shadows appeared in the kingdom of Thingol". I think we can safely say that the sun in this mythological setup was not too far away from the ground. A few hundred miles perhaps, but not a few million miles like today. So a small distance from Doriath's borders would already have produced an angle of considerably less than 90°, and thus considerable shadows. I argue that the sun was located northwest of Doriath, somewhere over Hithlum. Reasons for this will be given below.
Though not explicitely stated in the quoted sentences, there must have been a relation between the fact that Lúthien stopped singing and the lengthening of the shadows, i.e. the moving of the sun to a more acute angle. Now Lúthien was the daughter of Melian, an "angelic" being, so I guess she must have had an impressive voice. Let me state once again that the sun in this world was not "a mass of incandescent gas, a gigantic nuclear furnace" [4], but Arien, a hot girl (pardon the pun) in a flying boat. So shouting up to her would not really be a big thing for a demi-angel. If Lúthien stood at the edge of Doriath and sang really loudly, the sound might well have reached Arien.
My theory is that there were strong atmospheric winds in this region, continuously pushing Arien's boat away, and only Lúthiens fair voice motivated her to steer into the opposite direction and stay close to Doriath. Without the voice, Arien was bored and just drifted off, which in turn lengthened the shadows in Doriath. That Lúthien's voice could attract people is a proven fact, if we look at what it did to Beren. The graphics illustrate the mechanics behind this process.
Where exactly was the static sun located?
If Lúthiens voice was so loud, it must have been heard in far-away countries which lay in the general direction of her singing. If we assume that Lúthien faced the sun when she sang - to make it easier for her voice to reach Arien - then we simply have to look for hints of her voice on the ground to learn where the static sun was located. Interestingly, to the northwest of Doriath we find a continuous trail of place names which have the words "sound" or "echo" in it: Dor-Lómin (Land of Echo), Ered Lómin (Echoing Mountains), Lammoth (The Great Echo). So I think that the sun was somewhere over this area, directly over Dor-lómin at first and then slowly drifting away over Lammoth.
The weather can give further proof for that. Directly underneath Arien's ship, her radiation heats up the air, while the air on the ground, being farther away, remains relatively cool. Such a situation - cold air on the ground, warm air above it - is called an inversion, and this is a weather pattern which frequently causes smog in today's cities. The lands under the sun must have been troubled by this effect too. And indeed Tolkien explains that Hithlum, the realm of which Dor-lómin is a province, comes from hísi-lóme (Land of Mist).
I think all of this provides enough proof for the Static Sun Theory so that we can safely label any disagreeing scholar as a heretic and exert merciless punishment upon him.
Footnotes
[1]: A German translation of the speech was published under the title "Die Lúthiengesangssonnenstatiktheorie" in "Neueste wohlfeile Wahrhaftigkeythen betreffend die Balrogspropellierung und dero weithere übelst streithige Punkte in des Tollkyns gar löblichem Werke", University of Zwornitz (Oberlausitz) 1792, so this is where the speech was heard first in absolute chronology. However, I think I remember I fell into that particular time rift after the FATS conference, so in my timeline, I first held it there.
[2]: Bqggz (2006) pointed out that this could mean the average length of shadows on a day, perhaps calculated over the course of an entire year, which would still allow us to work with a moving sun. He speculated that the silence of Lúthien may have somehow tilted Earth's axis, so that Doriath was nudged from a more equatorial towards a more polar position. However, Earth is a pretty big thing to tilt, whereas the sun in Tolkien's world is a burning woman in a boat, infinitely easier to halt or move around. So I think Occam's Razor is on my side here, and besides, by definition everything Bqggz says is a lie.
[3]: Except if they were Entwives, but I have never heard anyone seriously claim that. Not even Bqggz.
[4]: They Might Be Giants (1993).