Dating the Trojan War: Odyssey, Book XX: 350-357
“And now the voice of the godlike prophet Theoclymenus was heard. ‘Unhappy men’, he cried, ‘what horror is this that has descended on you? Your heads, your faces and your knees are veiled in night. The air is ablaze with lamentations; cheeks are streaming with tears, their hearts with forebodings of grief. The walls and lovely alcoves are splashed with blood. The porch is filled with ghosts. So is the court - ghosts hurrying down to darkness and to the Underworld. The sun is blotted out from heaven and a foul mist has crept upon the world.’” Translation by: E.V. Rieu, Penguin Classics
Greek Text: "ἠέλιος δὲ οὐρανοῦ ἐξαπόλωλε, κακὴ δ᾽ ἐπιδέδρομεν ἀχλύς.” (Line midway 356 and 357)
My translation applying the verb rules of Latin Grammar1: "and the sun utterly perished from heaven and an evil darkness had covered all." (Because the verb ἐξαπόλωλε is in the 3rd person singular of the Ind. Perfect tense and the other verb ἐπιδέδρομεν is in the pluperfect tense)
“…And the sun has perished out of heaven and an evil mist covers all.” Translation by: A.T. Murray, Loeb Classical Edition
Putting this quotation in its context, it refers to the time when Odysseus (Ulysses) returns to his home in Ithaca as a beggar and, after an elaborate scheme to single out the members of his household who had remained faithful from those that had become unfaithful, he moves into action to exterminate his wife's, Penelope, suitors. The expression in italic is interpreted as referring to a solar eclipse. Now we know that a solar eclipse can be of three types: partial, total or annular. The following chart shows what a partial and total eclipse would look like:
What evidence, of a solar eclipse occurring in or around the period of the Trojan War, do we have?
Thanks to knowledge of celestial mechanics that we have today, we are able to make past and forward calculations and plots of solar eclipses. Based on this knowledge we have access to graphic charts made available in the public domain by NASA which plot the course of an eclipse across the ages, past and future. The image shown on the right-hand-side shows a total eclipse that occurred in the area of the Mediterranean over 
Sicily and Corfu on the exact date of 16 April 1178 B.C. (-1177 in the image); see the blue line which shows the trajectory of this eclipse and the width over which it would have been seen as a total eclipse, "utterly perished," according to Homer's text.
Our area of interest is highlighted by the golden oval shape. From the position of Ithaca, not very far from Corfu, it would have been seen as an almost total solar eclipse. The chart also shows another total eclipse that occurred on 6 April 1169 B.C., (-1168 in the image). But this eclipse was seen primarily in Northern Europe, the English Channel, and visible as a strong partial eclipse in the Midlands and Northern England and south down to about Northern Switzerland. Hence, since the recount given by Odysseus (Ulysses) in Book XX: 350-357 took place 10 years after the Trojan War, the date of the Trojan War and the specific event of the "utterly perished from heaven" must be placed on 16 April 1188 B.C. give or take a few months due to the uncertainty of dating the exact day and month when Ulysses actually executed the Suitors. However, it is reasonable to assume that it may have been the early part of the year because, we read, that he needed to wear a coat and a fleece to sleep during the night. We know that in the heat of summer, at those latitudes, you can't even sleep under a cotton sheet, unless you happen to be on top of a hill around 500 metres above sea level. This total eclipse lasted 4 minutes and 32.5 seconds, whereas the duration of the partial eclipse, from beginning to end, lasted 2 hours 35 minutes and 17 seconds.
Another total eclipse took place on 19 August 1157 B.C., again over the area of Sicily and Ithaca, Zante and Kefallonia, see the highlighted gold oval. By the same calculation this would place the Trojan War 10 years earlier, hence on 19 August 1167 B.C. However in the middle of August you certainly would not need any fleece to sleep at night and not even a cotton linen sheet over you. For this reason I would take the earlier eclipse as more probable. This second eclipse is shown by the narrowish blue curve in the following illustration taken from the NASA website.
1: Some Classicist or Linguists might well object to this approach, but before passing judgment it should be noted that there does not exist a Consecutio Temporum between Classical Greek and English. It does not exist, even, between Greek and Latin. The object here is scientific: did the writer mean a total eclipse or a partial eclipse? From the choice of verb tenses my conclusion is that it referred to a total eclipse, and this is verified by the evidence that I have presented. Q.E.D.