I am re-reading The Hobbit at the moment. Work has been, well, work, recently and I wanted something that I could enjoy without expending too much effort. And then, I found myself thinking about this passage:
'Then what is Durin's Day?" asked Elrond.
'The first day of the dwarves' New Year,' said Thorin, 'is as all should know the first day of the last moon of Autumn on the threshold of Winter. We still call it Durin's Day when the last moon of Autumn and the sun are in the sky together. But this will not help us much, I fear, for it passes our skill in these days to guess when such a time will come again.'
This is from The Hobbit, Chapter III, "A Short Rest", pg 56, emphasis added. Winter begins at the Winter Solstice, on or around December 22nd each year (in my hemisphere, anyway). The sun and moon can appear in the sky together in a number of different ways, but if we assume that a "moon" starts at the new moon (i.e., when there is no moon), then the first chance that this new moon and the sun have at being in the sky together is at the sunset after the new moon, when the sliver of new moon trails the sun just a little. Tolkien reinforces these details with another passage, later in the book, in Chapter XI, "On the Doorstep", pg 179.
'Tomorrow begins the last week of autumn,' said Thorin one day.
'And winter comes after autumn,' said Bifor.
So we know that our theory about Durin's day being in late autumn is correct.
All day Bilbo sat gloomily in the grassy bay gazing at the stone, or out west through the narrow opening. He had a queer feeling that he was waiting for something.
Looking west gives us the clue that we're thinking about sunsets here.
If he lifted his head he could see a glimpse of the distant forest. As the sun turned west there was a gleam of yellow upon its far roof, as if the light caught the last pale leaves. Soon he saw the orange ball of the sun sinking towards the level of his eyes. He went to the opening and there pale and faint was a thin new moon above the rim of the Earth.
So we've confirmed so far that it is late autumn, the new moon has just taken place and we're looking west toward the sunset at a sliver of a moon chasing the sun toward the horizon.
Quickly Bilbo explained. They all fell silent: the hobbit standing by the grey stone, and the dwarves with wagging beards watching impatiently. The sun sank lower and lower, and their hopes fell. It sank into a belt of reddened cloud and disappeared. The dwarves groaned, but still Bilbo stood almost without moving. The little moon was dipping to the horizon. Evening was coming on. The suddenly when their hope was lowest a red ray of the sun escaped like a finger through a rent in the cloud. A gleam of light came straight through the opening into the bay and fell on the smooth rock-face. . . . There was a loud crack. A flake of rock split from the wall and fell. A hole appeared suddenly about three feet from the ground.
. . . .
Then Thorin stepped up and drew the key on its chain from round his neck. He put it to the hole. It fitted and it turned! Snap! The gleam went out, the sun sank, the moon was gone, and evening sprang into the sky.
The clues we can get from these passages are many. We confirm that we're looking for the last new moon in late autumn. The clues about looking west confirm for us that this is a sunset event and the sequence ("the sun sank, the moon was gone") helps us place things precisely. We're to watch the sunset on the evening of the new moon or the evening after the new moon. Why the imprecision in our precision? Because, depending on your longitude and the time of day for the new moon, not everything may appear on the first sunset.
We are currently on the threshold of winter (begins about 12/22/2007). The last moon (of Autumn) starts this weekend (Sunday afternoon, December 9th at 1241). On that day the moon sets at 1634, while sunset for my location (near N 41° 30' 36.4212" W 81° 34' 44.1588") is twenty minutes later, at 1657. I'd like to see that moon if it were possible. On Monday, the sun sets at 1657 and the moon sets at 1726, So as I read it, we just might be able to see the moon and the sun at the same time at sundown Monday, December 10th, 2007.
That would be Durin's Day.
I'll be watching.