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Let's say I have an imaginary meridian line over new york going from north to south. I know that the sun will not always be exactly on top of my head here in New York, but every day it will have to cross somewhere on top of this imaginary line. Does it cross this at the exact same time every day?

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No, it will not. I suspect that this motion of the Sun simulator is more informative than the explanation I can give. In the simulator, check the boxes for "show analemma" in the lower left, and choose "step by day" in the animation controls box in the bottom middle. Then, click "start animation", and you can see how the Sun's position at a given time will vary from day to day.

The reason the Sun is not in the exact same position along a meridian line is a combination of the non-circular orbit of the Earth around the Sun, and the axial tilt of the Earth. If you are interested in reading more about this, analemma is the name of this phenomenon, so you can check out wikipedia, the library, etc., for more details.

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The sun crosses the southern meridian at an average of 24 hours... This 'mean solar time' gives us the basis of civilian time. However due to changes introduced by the orbit of Earth around the sun the actual amount of time taken varies.

To calculate the early/late arrival of the Sun due South we use the 'Equation of Time'. This allows you to calculate the offset correctly. It can vary by around 15 minutes over the year.

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The short answer is "no".

The longer answer is the following:

In current life we use solar time, which is measured in solar days, which measures the time between two sun cross over at the same meridian. This defines the True Solar Day. It is in fact the sun's angular time, it is 0hour when the Sun crosses over the local meridian.

Trouble is, the angular time does not vary uniformly. Indeed, the Earth orbit being an elipse, the speed of the earth around the sun varies: it is higher during winter, and slower during the summer (I'm talking from the northern hemisphere :)) This means that for the sun to appear at exactly the same local meridian, the earth will have to revolve a little bit more during the winter, and a little bit less during the summer. This means that the duration of the True Solar Day varies during the year.

For practical reasons, in order to obtain a constant time unit, we use in our everyday life the Mean Solar Time, which is the average of the durations of all the true days during the years. It is different from the True Solar Time.

So to answer your question, no, the sun does not cross over the meridian at the same time every day.

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