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There is no question the earth's gravity is the highest in early January.
That is, the earth is closest to the sun in January (seasons have little to do with this though since the tilt of the earth, not the distance from the sun, explains why we have seasons)
Earth is farthest from the sun in July
Closer to the sun would mean more intense force of gravity.
However, also the earth rotation on its axis is slowest in January as it reaches minimum distance from the sun. The slower earth's rotation, except on the poles, the more you weigh.
Earth rotates fastest when it is exactly halfway between each January and July (so October and April) but when approaching closer to the sun is when the speedup and slowdowns are way stronger (so October)
Question is then when is gravity less? July, October, or sometime in between?
Wait I messed that up. The answer is July. Earth spin is fastest when farthest distance from the sun in July.
October just happens to be the time of year when combining axial tilt impact with earth's rotation speed, solar time gets the most minutes ahead of clock time "mean solar time".
This is partly because in July and August, the axial tilt related to June solstice would have pushed the clock time much farther ahead of solar time, but the fast earth rotation speed offset a lot of this. This is why in September equinox the affect of clock time losing minutes to solar time accumulates as far as it does to the point that by November solar noon occurs 16 minutes earlier than average.
I think Earth's gravity is a function of its mass in accordance to general relativity; to a first approximation at the scales of mass around Earth Newton's gravitational laws work just as well. Since the Earth's mass is a constant (over the timescale of a human life) then the Earth's gravity is also constant.
Now if you want to talk about the gravitational field near a hypothetical object close to the Earth's surface then it's a little more complicated as that depends not just on the Earth but also on the nature of that hypothetical object itself and other "nearby" objects--where nearby is also relative--such as the moon, the Sun and other planets. However, that net gravitational field is not the Earth's gravity: it's "everything's" gravity.
The Earth's spin does not effect gravity AFAIK. It simply effects how a hypothetical scale on the surface of the Earth would measure the net force on an object. However, the gravity itself does not change.
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