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What an interesting question, what made you think of it?

Well lets see if I can answer it for you.

The constellation of Orion (the hunter) is a winter constellation (i.e. you can only see it in winter when the earth is on the same side of the sun to where Orion is.

Now if you look at the link I will place below, you will see that the spiral arm of our galaxy in which the Orion constellation is to be found is called the 'local spur' and this is outside the orbit that the sun makes round the galactic center. This means that in winter the sun would be between the earth and the galactic center.

Thus the answer to your question is that the season when Earth is closest to the galactic center is Northern Hemisphere Summer.

The galactic center is to be found in the constellation of Sagittarius (Sagittarius A).

This is the same time the earth is 5 million miles further away from the sun. Is it perhaps the gravity of the galaxy that is pulling on our earths orbit as well, making it slightly elliptic?

One can not argue that the gravity of the galaxy is not keeping the solar system and all the othe stars in orbit around it!

I would like to calculate what that gravitational force must be to balance the solar systems centripetal force acting to fling it outward.

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Q: What season is the earth closest to the galactic center?
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