'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.'
It appears that Durin's Day in 2010 will be in early December, the threshold of Winter.
Winter itself starts on December 21st, 2010 at 1747 UTC, with the Full Moon occuring at 3:14 am EST (0814 UTC) that day.
The last moon of Autumn starts on this threshold (Sunday afternoon, December 5th at 12:36 pm EST or 1636 UTC).
On that day, the moon sets at 4:51 pm (2151 UTC) while sunset for my location (near N 41° 30' 36.4212" W 81° 34' 44.1588") is 4:56 pm (2156 UTC), five minutes later. In The Hobbit, we read that on Durin's Day, "the sun sank, the moon was gone", implying that sunset takes place first, so December 5th isn't Durin's Day.
Not only is the order of setting wrong, but the amount of moon showing is so slight (due to the short apparent angular distance between the moon's and the sun's centers) that it's doubtful it would be seen.
On December 6th, the sun sets at 4:56 pm (2156 UTC) and the moon sets at 5:50 pm (2250 UTC), nearly an hour later. At that point, it will be nearly 0.03% illuminated, just a thin crescent.
I think this day (December 6th) is the best candidate for Durin's Day in 2010.
The order of setting is correct and there's likely to be enough moon to see (between 0.016% and 0.03% visible).
The United States Naval Observatory indicates that the record for an early sighting of a lunar crescent is 15.5 hours from New Moon for a naked-eye sighting and December 6th falls outside that mark.
The computed phase of the moon for a few hours later on the 6th is shown nearby.
The website crescentmoonwatch.org seems to agree with us. While they indicate (pdf and jpg) that it will be difficult unless the atmospheric conditions are correct, they do think that northeast Ohio should be within the visible range. The likelihood of seeing it on December 7th is even greater (pdf).
So there you have it. Durin's Day is the first day after the last New Moon before Winter, December 6th, 2010. Head out to the west at sunset and look for the tiny crescent.
Durin's Day for 2007, 2008 and 2009.
The moon/sun position images were taken from Stellarium.
