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Does the direction we are tilted change in relation to the background stars? And the points where our ellipse is pointiest? (most elongated?) Does this change the timing of our aphelion and perihelion? In other words, were they always one month after each solstice?

Also, are all of the planets' ellipses aligned? So that the semi-major axis is the same?

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Yes, yes and yes - these changes are called axial precession and apsidal precession.

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