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Being able to see the Moon is dependant on various factors like sunset timing etc. Considering all these factors, which country or area of the world sees the new moon first? I am asking about the new moon.

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technically, the New Moon is the point when the Moon is between us and the Sun, and we only see the Moon in shadow. The phase of the moon when it becomes visible is the 'waxing crescent'. [Others have already answered that the place to first see the waxing cresent will vary for each lunation]. –  Jeremy Mar 6 at 2:33
    
This kind of crescent moon visibility maps are most easy to get answer to my question- moonsighting.com/visibilitycurves/1435muh_11-4-2013.gif –  Gulshan Mar 6 at 9:46

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The duration of one lunation (the period between one new moon to the next one) isn't neither constant as the Moon rotates around the Earth and it around the Sun (changes between 29.272 and 29.833 days due to the perturbing effects of the Sun's gravity on the Moon's eccentric orbit), nor integer divisible by 24 hours or one Earth's rotation around its axis. So this position of the Moon on the skies where the next new moon as the first of its lunar phases will be first observable constantly changes.

   lunar phases

     Phases of the Moon, as seen looking southward from the Northern Hemisphere. The Southern Hemisphere will see each phase
     rotated through 180°. (Source: Wikipedia on Lunar phase)

Saying it differently, by the time the Moon completes one lunation (or it's synodic period) and starts the next one, it won't be positioned exactly above the same Earth's longitude (East to West) as the one it started at.

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Can the moon-phase vary on the same longitude? –  Gulshan Oct 13 '13 at 16:59
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@Gulshan - Yes, observing the moon phase transitions from a single longitude (i.e. you're stationary on the Earth's surface or only travel South or North) will make it appear as if the moon phase moved to the West in its next cycle. Actually, when the cycle will need nearly 30 days (a bit less), it will appear to have moved East from your vantage point, because 0.833 is closer to 1 that 0 (like 0.272 is). –  Noordung Oct 13 '13 at 17:24
    
Sorry, I misphrased my last question. Can the size(width) of the moon be different on the same longitude at the same time? ie will two places on UTC+6 line always see same size moon at some specific time? –  Gulshan Oct 13 '13 at 17:40
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@Gulshan - Strictly technically no, because 1) it wouldn't be at exactly the same angle at exactly the same time of the day 2) longitude and latitude librations due to its eccentric orbit are synched with its lunation period (it's tidally locked) and 3) on the surface of the Earth, you're not stationed at the Moon-Earth barycenter, which is located within the Earth about 1.000 miles under its surface (i.e. due to the diurnal libration, see the link on #2 for explanation). BTW same goes for observing the Moon when it's at the same angle. –  Noordung Oct 13 '13 at 17:53
    
I should add that to a casual observer, these minute changes in its apparent circumference wouldn't be visible, and are due to its distance to the Earth indeed really tiny even in measurements. As an observer, the more common reason for its apparently larger circumference is the optical illusion of appearing larger when closer to the horizon. –  Noordung Oct 13 '13 at 18:01

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