Does the Earth circle the sun?

[N1LAF]

Now for a better question. Does the earth move under my feet ?

You have to define a reference point now

;)

[N1LAF]

You are missing the entire point that I am making. What I am trying to point out is is that your definition of a circular path is not entirely correct since there are numerous simple closed curves (other than circles) that fit the criteria you described. The equation for area and circumference of a circle are not entirely relevant to this discussion since to describe planetary motion we would be more interested in examining position vectors expressed in parametric form (i.e. vector equations).. In the plane we would be looking at equations that approximate an elliptical orbit and, if we are to consider the motion of the plane through 3-space we would have to add additional parameters

In answer to your question no, the formulas don't apply if the orbital path in the plane is non-circular (elliptical). But assuming it were actually circular since you can have a circular path in 3-space and any circle in 3-space can be defined in terms of a 2-dimensional subspace (i.e a projection) . Think of it in terms of the earth orbiting the sun in a plane and envision the plane in motion through space.In terms of vectors the path of the planet in the plane would be a component of the vector equations for it's position in space.

Said differently, for an observer standing somewhere out in space the position of a given planet at some time "t" would be given by a set of parametric (vector) equations. The 2-dimensional motion in the plane would be a component of those equations i.e. the projection of that motion onto some plane subspace of 3-space.

At the end of the day, the planetary orbits are actually a corkscrew motion, not circular, in an absolute galactic coordinate system

[KC2UGV]

At the end of the day, the planetary orbits are actually a corkscrew motion, not circular, in an absolute galactic coordinate system

Well, as seen from 0,0,0 Milky Way; it's not a corkscrew. It'd be a speeding forwards, then slowing down, speeding forwards, then slowing down, with a slight waver in it's path :)

[N2RJ]

At the end of the day, the planetary orbits are actually a corkscrew motion, not circular, in an absolute galactic coordinate system

I would like to see the reasoning behind that.

[KC2UGV]

At the end of the day, the planetary orbits are actually a corkscrew motion, not circular, in an absolute galactic coordinate system

And, which Galactic Coordinate System? The standard one, uses Sol as it's 0,0,0

[n2ize]

At the end of the day, the planetary orbits are actually a corkscrew motion, not circular, in an absolute galactic coordinate system

I am fully aware of that. That is what I wrote above. The only thing I was attempting to do was to better define "circle" or "circular path" in terms of closed curves. Regardless of whether we are talking about astronomy it still boils down to Mathematics and the principles of analytical geometry and coordinate systems and transformations. A component of the "corkscrew motion" you are describing is the elliptical motion of planets in the plane. To an observer in the plane the planet moves in an elliptical orbit Now if we take that plane of planetary motion and move it through space (lets say in a straight line perpendicular to the plane of motion for simplicity) an observer at some reference point (origin) in space would observe (say off to the side somewhere would observe a spiral (corkscrew path). Now if that observer were to be smart enough to write down the equations of that corkscrew path the planar path would be a component of that motion.

An analogy would be I am riding a bicycle at night in a straight line path while twirling a flashlight in a plane perpendicular to the direction I am riding. To me sitting on the bicycle I am moving with the flashlights plane of motion and to me it traces out a circular path in a plane. But if you are sitting on a bench observing me riding past the path of the light to your frame of reference is a corkscrew path. the planar motion of the light as I twirl it is a component of that corkscrew.

I am not trying to criticize your example. Matter of fact I like this example because when we think of planetary motion we often only think only of the elliptical orbit in the plane (as if the plane were stationary) as opposed to the frame of reference of a point in space in which it is moving in the plane as the plane of motion is hurtling through space.