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Earth and moon rotate around a center together which is near to the surface of earth.
If they were rotating in an orbit more near to the sun they would rotate around it much faster. But wouldn't another effect of this closeness be a wobbling in that center due to tidal effect of sun's gravity.

When earth or moon in their orbit would pass near the sun it would pull that body down and the other one on the far side would follow with some lag. So that the orbit of earth around sun would look like concave and convex bumpy indent at the frequency of every 14 day going around the sun?

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I think the obvious answer to your question is yes, at least there would be increased wobble, though I don't know what a dented 104 node ellipse is. The wobble would be quite small in comparison to the Orbit, and as the big orbit would be measurably faster, I'm not sure the wobble would be more pronounced, even with the Earth-Moon orbit closer.

The answers to this question goes into detail about the Moon and I'll borrow the picture of what the Moon's orbit actually looks like in relation to the sun. It's quite close to an ellipse. This visible wobble is primarily the Moon's orbit around the Earth (not tidal). The tidal effect that the sun has on the Earth-Moon system is just a tiny fraction of the visible wobble in these pictures.

enter image description here

and

https://i.sstatic.net/ROFmP.gif

and that's the Moon's wobble, the Earth's is 81 times smaller.

If the Earth-Moon system was moved closer to the sun those wobbles would visibly increase in size compared to the big orbit, by the ratio to the distance (and the smaller tidal effects by the cube of the ratio), but there would also be fewer of them across one orbital cycle so the visible different wouldn't be all that much. Something else would happen before you got much closer too. The Earth's stable zone of it's hill sphere would shrink inside the Moon's orbit and the Moon would escape the Earth. You can't perturb an orbit all that much more than the Sun already perturbs the Moon without shaking the moon lose from the Earth. Venus distance, the Moon would still be stable, I think, and just barely, 5 or 10 million miles closer than Venus, it begins to enter the danger zone where the Earth likely loses the Moon and it becomes a very menacing near earth object.

Hope that's clear.

If someone else wants to answer with more detail, feel free.

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  • $\begingroup$ Thanks. I am really setting up the 3 body motion in a hypothetical configuration where the tidal action would be large. Where it will definitely drag the earth or moon drastically closer. A case where the distance between moon and earth would change by 10% - 15% each 28/2=14 days hence creating major wobbling on their orbit around the sun. in that case the center of common orbit between moon and earth would even migrate to a point outside the mass of earth when moon is dragged far away to sun and its orbit has stretched wide out. Same happens at a less intensity when earth is in and moon out. $\endgroup$
    – kamran
    Commented Sep 7, 2016 at 15:24

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