Analytical Solutions and a Clock for Orbital Progress Based on Symmetry of the Ellipse
Round 1
Reviewer 1 Report
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Comments.pdf
Author Response
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Reviewer 2 Report
The authors visit the classical Keplerian problem, and presents an interesting observation that some of the Keplerian relations might look simpler in the image focus polar coordinates. Their main finding is the more uniform angular speed of the polar angle of the image focus, hence provides a good educational aide. This works for Mercury at eccentricity of 0.2, however, wasn't demonstrated to work for very eccentric orbits, e.g., comet Halley, so the misconception might overweigh the gain unless the author can show otherwise. The fluctuation is due to the non-uniformness of the factor 1/cos(delta)^2, which ranges from 1 to 1/(1-e^2) (see Table 1 or Eq 14), hence it breaks down for a nearly radial orbit, but works well for e<0.3 (at 10%), but is unacceptably wrong at e>0.8.
It's also not clear if the image focus works any better for a hyperbolic orbit with large e~2 than a real focus. Unless the feasibility of the image focus view is demonstrated for a wide range of situations, it's not sufficient to be of interest.
The Runge-Lenz vector in the image focus view is also worth a discussion.
All real binary Keplerian systems involve a fixed centre of mass, but a non-inertially-moving real focus and image focus. The image view is of very limited interest to visualise binary motion.
Author Response
Reviewer 2 (numbered pp in italics) plus Response
- The authors visit the classical Keplerian problem, and presents an interesting observation that some of the Keplerian relations might look simpler in the image focus polar coordinates. Their main finding is the more uniform angular speed of the polar angle of the image focus, hence provides a good educational aide. This works for Mercury at eccentricity of 0.2, however, wasn't demonstrated to work for very eccentric orbits, e.g., comet Halley, so the misconception might overweigh the gain unless the author can show otherwise. The fluctuation is due to the non-uniformness of the factor 1/cos(delta)^2, which ranges from 1 to 1/(1-e^2) (see Table 1 or Eq 14), hence it breaks down for a nearly radial orbit, but works well for e<0.3 (at 10%), but is unacceptably wrong at e>0.8.
Response. The reviewer misunderstands that only our equation 39 (the last equation) is an approximation. The first 38 equations are exact, many are purely geometrical, and several are new. These exact equations apply to any ellipse of any eccentricity. So, we had demonstrated the generality of many new results. The reviewer discusses a single aspect of our paper, our “orbital clock” approximation, then complains that it is not perfect, and ignores all other key findings.
In detail, our abstract had stated: “….yields new geometric formulae and conservation rules. These results simplify computation of exact orbital time. Additionally, a simple approximation appears: the radial hand….” Therefore, the approximation we discuss is a secondary finding. But because the reviewer misunderstood, we rewrote the abstract to emphasize that our key equations are exact and hold for ellipses of all eccentricities.
The last sentence of the reviewer’s comment repeats our statements regarding the conditions under which our "clock" approximation is good, namely low eccentricity. Importantly, as stated in the paper, and as is well known, Mercury is the most eccentric planet. Consequently, we demonstrated and clearly stated that the “orbital clock” is useful for all planets and most moons in the Solar system. This is far more important than addressing eccentricity > 0.8, which pertain only to comets.
To emphasize the existence of our exact equations, valid for all eccentricity, and to quantify when the approximation is useful, we added Figure 3b that compares the “orbital clock” approximation to our exact formula, using an eccentricity that is more than twice that of Mercury. The errors for any planet and most moons in the Solar system are far lower than that shown in this new figure.
- It's also not clear if the image focus works any better for a hyperbolic orbit with large e~2 than a real focus. Unless the feasibility of the image focus view is demonstrated for a wide range of situations, it's not sufficient to be of interest.
Objects following hyperbolic and parabolic trajectories are not bounded, and so these objects are not part of the solar system. Their positions cannot repeat, so they have been useless to the long historic quest to understand time.
Again, the reviewer missed that we provide exact equations for ellipses having any eccentricity, so we actually cover a wide range of situations.
The topic of the paper is the geometry of the ellipse: The suggestion to explore hyperbolae is another subject.
- The Runge-Lenz vector in the image focus view is also worth a discussion.
Response. As we clearly stated in the introduction, our derivations neither use nor require vectors. Our approach is easily understood, and thus has significant pedagogic value. We further emphasize pedagogy in our revised discussion section (8.1).
Moreover, the Runge-Lenz vector is not mentioned in most presentations of Kepler's works. Discussing the Runge-Lenz vector would be peripheral to our presentation, and is also beyond the scope of the present report.
- All real binary Keplerian systems involve a fixed centre of mass, but a non-inertially-moving real focus and image focus. The image view is of very limited interest to visualise binary motion.
Response. An equivalent statement would be that Kepler's deductions of planetary orbits are of "limited interest"; which is unjustifiable.
It appears that the reviewer missed our statement of purpose in the introduction, “to improve understanding of the solar system” which is why our paper focuses on bound orbits about a massive central object, which case is reasonably approximated as stationary. This also was stated in the introduction. The same assumption is made in textbook presentations of Keplerian orbits. We added emphasis in the revision.
The topic of the paper is not binary stars with somewhat similar mass, which would require a different set of equations. Again, this suggestion is beyond the scope of the present report, which concerns the geometry of the ellipse and behavior of the vast majority of bodies in solar system
Ellipses find application to numerous problems in mathematics, engineering, and science, most of which do not involve orbits. Because our paper yields new geometrical formulae that are true and exact for any ellipse of any eccentricity, it is of general interest, not just to applications intrinsic to astronomy and planetary science. Mathematically-oriented readers of Symmetry should find many of our new geometrical constants and equations of interest.
Round 2
Reviewer 1 Report
I believe that the updated paper may be accepted for publication.
