Talk:Kepler's laws of planetary motion

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Figure 1[edit]

The size of the lettering within Figure 1 could be increased, to make it readable.

Neither the figure nor its caption seems to indicate where the ends of the distances a1 & a2 are.

82.163.24.100 (talk) 13:44, 27 August 2009 (UTC)

Shouldn't this say "the star" not the sun, as this diagram does not show our own solar system, but one with exaggerated elliptical motion? Richard LaBorde (talk) 23:19, 24 May 2010 (UTC)

I see no reason to say "surface area" rather than just "area". No surface is involved. Jeff Root (talk) 13:55, 27 January 2011 (UTC)

Stable orbits[edit]

According to the third law, the period of a planetary motion is propotional to the 3/2 th power of radius ( or semi major axis. ) But this is an idealised case. In reality, there is a very diffuse gas in the Solar system and the motion of the planets around Sun may be retarded by this gas. To be sure, the retarding effect is negligable. But the cumulative effect over millions of years may be considerable, which means that the delicate balance between the gravitational and cetrifugal forces is in danger. But luckily, we haven't observed any spirally inward motion of any planet. I think that the article must have a section to explain how the planets maintain their orbits over billions of years. Nedim Ardoğa (talk) 06:33, 15 October 2009 (UTC)

I think that this subject belongs in another article, perhaps secular variations of the planetary orbits. This article is about Kepler's laws.

The third law ratio a13/2 would be fine if a1 or a2 labeled the semimajor axis in the diagram, but it doesn't. The semimajor axis is half the major axis, ie a1/2 and a2/2. --Laertes Oleander (talk) 15:54, 19 May 2012 (UTC)

Yes, however there is nothing that is incorrect about the caption because the ratio of two semimajor axes raised to some power is the same as the ratio of the two major axes raised to the same power. Johnuniq (talk) 01:45, 20 May 2012 (UTC)

NPOV[edit]

Stating these laws in terms of the sun is heliocentric. The intro should either place the laws firmly into their historic context, or should avoid any biased language. -Craig Pemberton (talk) 21:15, 25 October 2009 (UTC)

this argument is absurd because the historical context is heliocentric. "Kepler's laws are concerned with the motion of the planets around the sun. Newton's laws of motion in general are concerned with the motion of objects subject to impressed forces." if you have evidence contradicting this, please offer it. -- 99.233.186.4 (talk) 02:32, 29 October 2009 (UTC)
I never claimed the laws were not heliocentric, only that the article is being heliocentric because the intro gives no such context. Even what you've said above would improve the article immensely in this regard. As it stands there is no such context. When I find the time I'll take a stab reworking the intro. -Craig Pemberton (talk) 03:53, 29 October 2009 (UTC)
As all wikipedia readers live in this solar system, being heliocentric does not violate WP:NPOV. Bo Jacoby (talk) 22:48, 5 November 2009 (UTC).

Merge proposal opposed[edit]

This is to oppose the recent merge proposal in respect of the section 'Position as a function of time': Discussion is offered here. Terry0051 (talk) 00:08, 29 November 2009 (UTC)

Current text[edit]

The current text says, "... Kepler stated these laws as they apply to the Sun and the planets". The four brightest moons of Jupiter were coming to the notice of the public in Europe in 1610. I am not sure who was the first to note that the three laws applied to the four moons. —Preceding unsigned comment added by 86.157.177.188 (talk) 13:44, 29 December 2009 (UTC)

Kepler used the observations of Tycho Brahe who did not use a telescope and consequently did not see the moons of Jupiter. Bo Jacoby (talk) 13:16, 4 February 2010 (UTC).
The four brightest moons of Jupiter are visible without a telescope. I agree that Tycho
probably over-looked them in point of fact. Kepler himself had bad eye-sight
and probably could not see any moons of Jupiter. Some one other than Kepler was probably the first to say Kepler's laws applied to the four moons. —Preceding unsigned comment added by 86.177.254.83 (talk) 09:22, 19 May 2010 (UTC)
"The four brightest moons of Jupiter are visible without a telescope." I really doubt that. Truly astounding seeing and eyesight would be required. It is geometrically plausible, Callisto getting as far 10 minutes of arc away from Jupiter at opposition (the naked eye's angular resolution is about one minute of arc). The problem is glare: the apparent magnitudes are -2.94 vs. 5.65, a factor of 2700 ratio (note also how close Callisto's apparent magnitude is to the naked eye's limit of about magnitude 6). Urhixidur (talk) 15:14, 23 November 2011 (UTC)
See Godefroy Wendelin. Wendelin noted that Kepler's third law applied to the moons of Jupiter. He wrote in 1643. Glare can be eliminated by getting Jupiter behind an opaque object, such as some brick-work. — Preceding unsigned comment added by 86.171.253.163 (talk) 10:36, 28 January 2012 (UTC)

Relations now known as Kepler's laws: problems over accuracy[edit]

The text of recent edits says: "Almost a century later, Isaac Newton was able to derive Kepler's relationships from Newton's own laws of motion and law of universal gravitation, using classical Euclidean geometry."

The corresponding earlier text said: "Later, Newton's work showed mathematically how the three relationships resulted from a central gravitational attraction according to an inverse-square law, as an approximation that would become more exact as the relevant planetary masses could be assumed smaller in relation to the mass of the sun and as the planetary mutual perturbations could be thus ignored."

Maybe the older text was not of the best, but there is a problem about the new text. What Newton derived was different from Kepler's relationships (at least as Kepler stated them). He showed that relationships of that form would prevail only in certain idealized circumstances that don't occur naturally (e.g. 2 bodies only, and/or orbiting bodies of vanishingly small mass), and specifically, that in a multi-planet system they only approach exactness as the planetary masses tend towards zero. He stated correctly that the relationships are not exactly fulfilled in the real solar system. (As approximate observations, the "laws" are of course approximately true enough to the levels of accuracy achievable in Kepler's time.)

What is likely to happen, if the distinction between approximation and exactness is not carefully maintained here according to the sources, is that readers will come away with the (mis)understanding that Kepler found and stated the laws that apply to the planets' motions and Newton then proved what Kepler had stated. This is both untrue to the reliable sources, and likely to lead to all sorts of faulty conclusions if relied on by readers.

Part of the problem of faithfulness to the sources here is that while the sources do show that "Kepler's laws" have been called "laws" for about 270 years (though not in the time of Kepler or Newton), their status as "laws" is something very peculiar, seeing that the sources also show that they are not (exactly) true as Kepler stated them, and that what Newton proved was not (exactly) what Kepler stated.

The article text does therefore seem to need some more work including in the lead section. This looks like a case where the inevitable and necessary process of summarizing is specially sensitive, because it can easily change the sense of what is summarized (and inadvertently falsify it). Terry0051 (talk) 21:55, 29 December 2009 (UTC)

Copernicus was not exactly right because Kepler's laws are more precise. Kepler's laws are not exact because Newton's laws are better. Newton's laws are not exact because general relativity is better. General relativity is not exact because it doesn't take quantum mechanics into account. Probably no natural law is exactly true. Bo Jacoby (talk) 09:57, 4 February 2010 (UTC).

Thank you for your response.
(1) Before your edit, the intro section was, in an important respect, in accordance with the reliable sources here. The sources clearly identify Kepler's 'laws' as approximate (and also show that they were not called 'laws' in Kepler's time or for a century afterwards.) Because this is (to some people) a surprising idea (see the G E Smith source cited in the article for support and explanation of that point), it seems important that the intro also tries to offer the reader a 'handle' on the true position, with citations. Your edit took away from the intro section the essential qualification that the 'laws' are not exact, took away the inline references to the citations, and left incorrect and unsupported statements, untrue to the cited reliable sources, that Kepler's laws apply (without the qualification 'approximately') to the planetary system and other planetary systems etc.
(2) Your post seems to try to brush aside the approximate character of the Kepler relationships, and to suggest that this is the same as, say, the approximate character of Newton's laws when seen in the light of relativity. Where do you find support for that? I can only suggest that you look at the already-cited sources on this matter (especially the cited items by G E Smith and Curtis Wilson.) And where is there support in sources for the 'because' statements in your post? It seems unlikely that such sources are available; all of those 'because' statements appear to be false. As a specific example, sources show that the exactness/inexactness of the relationships or laws that you mention has to do with their comparison to physical reality, in other words it's about how well they 'model' physical reality -- it's not "because" of the existence of later or other theories.
(3) I have no problem with a history section -- as long as the result is supported by the reliable sources and clearly conveys the content to the reader. But the introductory section still needs to be left in a state that is consistent with the sources, and properly foreshadows the citation-supported content, not to contradict it. Terry0051 (talk) 11:44, 4 February 2010 (UTC)

  1. I merely moved the historical information away from the introduction and into a special section.
  2. The Perihelion_precession_of_Mercury is a physical reality accounted for by general relativity, but not by classical mechanics.
  3. The introductory section should be a summary. References to sources are found later in the article.

Do you object against the use of the word law for a statement that is only approximately true? Well, a lot of approximate rules are called law in science, see Physical law#Laws_as_approximations. Bo Jacoby (talk) 12:33, 4 February 2010 (UTC).

About your point 1: Agreed your edit was a move, but it was not a 'mere' move; it took away from the introductory section the previous mention of characteristics essential to the correct meaning according to the sources.
About your point 2: Yes, I agree the facts about the perihelion precession of Mercury -- but that doesn't seem to say anything about the current topic. General relativity is still standing up well (last I heard) to tests of its accuracy. Newton's theory also stood up accurately to tests for a time, and it was only after about a century and a half (towards the end of the 19th century), that it became clear that the Mercury phenomenon was beyond its explanation. But in contrast to both of those, there never was any time when the Kepler 'laws' looked accurate without observations being known that were contradictory to their accuracy. Even Kepler himself noticed some of the discrepancies, as per the cited sources.
About your point 3: Agreed the introductory section should be a summary: but of course it also should not be a misleading summary. (I believe there is no rule against references to citations in an introductory section. They are also important in a subject like this.)
About "Laws as approximations", the article-section to which you refer lacks support from any citation. I think it will be found that the position shown by reliable sources is rather different from the oversimplified version that the unsupported article sets out. Answering your question "Do you object against the use of the word law for a statement that is only approximately true?", I have no problem with anything that clearly accords with the sources, and is not misleading. The present case is possibly unusual (see earlier posts in this section). If the characteristics discussed go without mention, then the resulting oversimplified statements are shown by the sources to be incorrect and misleading. (Just to try and avoid misunderstanding: it's worth reminding that the 'misleading' enters so far as the 'laws' are said to apply to the solar system bodies without qualification as to their approximate character -- and Kepler's statements were indeed about the planets in the solar system. By contrast, relationships with the same form as Kepler's laws were shown by Newton to apply exactly in idealized systems, e.g. the limit case of planetary masses tending to zero. But the limit cases of accuracy are not the real cases found in nature.) User:Terry0051|Terry0051]] (talk) 20:33, 4 February 2010 (UTC)

I suggest that you reestablish my edit and improve the article from there on, rather than just having it reverted, according to the rules of Wikipedia:Reverting.

It seems that we agree that a section on history is appropriate.

That physical laws are only approximately true is the rule rather than the exception, e.g. Boyles law. Newton was aware that his theory of gravitation was unlikely to be the final truth because instantaneous action at a distance is unexplicable. Einstein was aware that his general theory of relativity was unlikely to be the final truth because is does not take quantum mechanics into account. So that Kepler also noticed some discrepancies is not in contrast to Newton and Einstein.

Bo Jacoby (talk) 10:03, 5 February 2010 (UTC).

Thank you for your further comment. Yes, we are agreed that a history section is appropriate. I also agree that some shortening of the introductory paragraphs is appropriate too. Hopefully, consensus can be reached at least on the central points, and because of your suggestion I have just now made an edit that implements both the intro-shortening and the history section along a generally similar format to the one you chose, and I believe it is also consistent with the desiderata about accuracy.
It seems that you may be suggesting that my revert of your former edit was contrary to WP:Reverting, but if necessary, I would certainly point to details of the nature of the edit, and the history that went before it, and defend both the substance of the revert and the procedure adopted as in accord with the guideline. It seems desirable, and helpful to the reliability of the encyclopedia, that new edits respect any accuracy points that are already on the recent edit record of an article and its talk page. Perhaps there is no need for me to go further at present, it may be that there is sufficient agreement on a few main editing issues. Terry0051 (talk) 20:19, 5 February 2010 (UTC)
  1. Why write "The relationships now known as Kepler's laws" instead of simply "Kepler's laws"? It adds nothing to the meaning or understanding.
  2. Why highlight the approximate nature of Kepler's laws? They are not special in this respect.
  3. The rule for reverting is: Don't revert a good faith edit.

Bo Jacoby (talk) 10:14, 6 February 2010 (UTC).

1 "The relationships now known as Kepler's laws": This originates in the sources, see for example page 73 in the article on Johannes Kepler by O Gingerich in Planetary Astronomy from the Renaissance to the Rise of Astrophysics, Part A : Tycho Brahe to Newton, Volume 2, Part 1; eds. René Taton, Curtis Wilson, Michael Hoskin (Cambridge, 1995, CUP), for a reference to "the three relationships now called Kepler's laws". (I also refer again to the other already-cited sources, especially the G E Smith and C Wilson references -- as well as to the posts here that preceded your edit -- for the reasons why this is significant, and why for the general reader it adds to the meaning and understanding.)
2 About the approximate nature of the laws: Again I refer you to the already-cited sources, esp. the G E Smith (2007) citation, for a better explanation than I have succeeded in giving, about why this is specially significant in connection with Kepler's laws.
3 Reverting: I refer to the article (wp:reverting) itself.

Terry0051 (talk) 11:12, 6 February 2010 (UTC)

  1. Your link page 73 says: "the Epitome astronomiae Copernicanae gave a systematic tretment of all of heliocentric astronomy including the three relationships now called Kepler's laws." This means that the relationships were not called Kepler's laws in the Epitome astronomiae Copernicanae, but now they are called Kepler's laws, so we should call them that.
  2. Surely Kepler's laws are approximate, but so are the other planetary theories of Ptolemaios, Copernicus, Newton and Einstein, so it is not specific for Kepler's laws.
  3. wp: reverting says: "Revert vandalism on sight, but revert a good faith edit only as a last resort".

Bo Jacoby (talk) 14:03, 6 February 2010 (UTC).

1: Please do not forget the effects of all of the other references referred to in the discussion and cited in the article.
2: The cited sources explain how the degree of approximation was different and rougher here than in the later examples, and actually known to be so by reference to observations known and available in Kepler's & Newton's time.
3: I am sorry that I could not see any alternative to reverting your edit, but please would you also bear in mind: that the error reintroduced had recently been mentioned on the talk page and in the recent edit history; the text for which you altered the meaning was clearly supported by the cited reliable sources; you did not attempt to seek any consensus for your changes; and the nature of your edit seems to indicate that you did not make any attempt to take into account the matters that had previously been discussed in thh edit summaries and talk page. As against all of that, having the introduction stand with wrong information in spite of all of the above points is a substantial negative. I appeal to your sense of professionalism to respect the need for the encyclopedia articles to rely on support in reliable sources. I have taken the trouble to reintroduce at your suggestion the presentational changes that you wanted to bring to the opening sections. I note that all of the arguments about content that you have brought to the present discussion lack any reference to reliable sources.

Terry0051 (talk) 15:18, 6 February 2010 (UTC)

  1. The reference you relied upon did not support your point of view. Of course Kepler's laws are called Kepler's laws.
  2. A discussion on the accuracy of planetary theories would be nice, but not within the scope of this article.
  3. Apology accepted. You are not precise in what you call "wrong information". I refer to C.D.Murray and S.F.Dermott, Solar System Dynamics. See [1].

Bo Jacoby (talk) 02:06, 7 February 2010 (UTC).

It might be worth mentioning that Kepler's second law was originally that the speed of a planet varies in inverse proportion to distance from the sun (i.e., not correct), and Kepler derived the correct form from that by means of an erroneous argument (as described in both Thomas Kuhn's The Copernican Revolution and Richard S. Westfall's The Construction of Modern Science). Lippard (talk) 22:14, 24 February 2010 (UTC)

And yet it does work out that the inverse of the square of the velocity of a planet is directly proportional to its radius. Uglysses —Preceding undated comment added 19:59, 14 March 2011 (UTC).

Oh no!! TASDELEN´s recent changes (May 5th) are in bad form, and seem inflammatory, overly opinionated, and likely to confuse and mislead the new reader. Will somebody please edit?

77.249.214.182 (talk) 15:08, 5 May 2010 (UTC)

As[edit]

Under the heading "Generality", the phrase "as to know" appears. I am not sure what it means. It is true that Kepler's laws apply approximately out-side the Solar System. —Preceding unsigned comment added by 81.154.6.133 (talk) 08:50, 25 May 2010 (UTC)

I rephrased it.--Patrick (talk) 09:05, 25 May 2010 (UTC)

Variables[edit]

Throughout the article there is heavy use of variables but some of these are never defined. A section in its own right which defined each of this would be very helpful. The biggest issue is with M. It is only ever defined as the mean anomaly but later it is used as the mass of the sun, though that it is being used as such is never stated. Que? (talk) 02:46, 2 June 2010 (UTC)

The M is used first as the mass of a star with an exoplanet around it. The formula on exoplanets is neither necessary nor sufficient in this context and I am going to remove it. Bo Jacoby (talk) 08:12, 2 June 2010 (UTC).
I also inserted explanations for the variables in Newton's laws. It is not good that the mean anomaly and the sun mass share the same symbol M. What change do you suggest? Bo Jacoby (talk) 08:40, 2 June 2010 (UTC).

Delete "Estimating the eccentricity of earth orbit"?[edit]

The section "Estimating the eccentricity of earth orbit" seems to be out of place in the article. It's not really relevant to the rest of the article, is inaccurate and is unreferenced. Its presence does not enhance the rest of the article in any way. I do not see any compelling reason why it should be retained. -- B.D.Mills  (T, C) 04:17, 13 July 2010 (UTC)

It is commonly complained that wikipedia math articles are too theoretical to be comprehensible to newcomers. A tiny bit of explanation bringing the theory down to earth may be helpful to the reader. Bo Jacoby (talk) 05:00, 13 July 2010 (UTC).
This really doesn't address any of the issues I raised.
  • Relevance: It may be more because it is misplaced within the article that makes the relevance questionable. If we need an example on the computation of the eccentricity, why isn't it with the other discussions on eccentricity such as the First Law section, instead of several sections before the First Law is defined? This placement is very sloppy.
  • Accuracy: The example is clearly inaccurate. This on its own would be sufficient grounds to remove it from the article even if the other issues did not exist. Firstly, it splits the Earth's orbit on the equinox to equinox line which as a way of splitting the Earth's orbit is incorrect because the perihelion point is not at a solstice but is about 14 days from the solstice; only a split along the semiminor axis would be correct. Secondly, it uses a 365-day year which introduces another inaccuracy.
  • Unreferenced: The example lacks references. Again, this on its own would be sufficient grounds to remove it if references cannot be found.
This brings me to an additional issue that I did not raise earlier:
  • Probable original research: If the method in the example was made up by a contributor rather than being derived from an independent source, it is in violation of the Wikipedia guidelines on the prohibition of original research. If it uses an independently-published method to derive the example, this is fine. However, the sloppy accuracy and lack of citations makes it likely that this section does contain original research. Again, original research on its own is sufficient grounds to delete material from an article.
To fix this section, the following must be done:
  1. Move it to the section on the First Law. This improves the flow of the article and addresses the relevance issue (which is really more a matter of bad placement within the article than a matter of relevance per se).
  2. Do not use the equinox to equinox split of the Earth's orbit. This is inaccurate. Instead, use referenced sources to derive the length of time that the Earth takes to travel from one semiminor point to the other.
  3. Use the anomalistic year (perihelion to perihelion), not the tropical year (equinox to equinox: does not account for polar precession) or sidereal year (fixed star to fixed star: does not account for orbital precession).
  4. Use accuracy better than 1 day. I recommend 0.001 day because this demonstrates the difference between the tropical year of 365.242 days, the sidereal year of 365.256 days and anomalistic year of 365.260 days.
  5. Provide references.
In short, the section has issues but it can be salvaged if the above issues are remedied. This demonstration can be helpful but such demonstrations need to be accurate and referenced just like any other Wikipedia content. -- B.D.Mills  (T, C) 00:26, 22 July 2010 (UTC)
You are course welcome to improve the article. The message of the subsection "Estimating the eccentricity of earth orbit" is merely that the eccentricity is nonzero. This is important because it distinguishes Kepler's laws from simpler models of the solar system. Perhaps the title should be changed to reflect this, and perhaps the contents should be reduced to remark that nonzero eccentricity explains the unequal 'halfyears'. I don't think that this article should contain a detailed description on how to determine orbital elements observationally. The difference between the siderial year and the anomalistic year violate Kepler's law and should perhaps be explained in another article. Bo Jacoby (talk) 07:07, 22 July 2010 (UTC).

Relation Of The Period with Areal Velocity[edit]

You can relate the period with the areal velocity by,

T=\frac{p^2 (\log (1-\epsilon )-\log (\epsilon -1))}{\text{K1} \left(\epsilon ^2-1\right)^{3/2}}

were K1 is the areal velocity. This means that it will be inversely proportional to the angular velocity at a given point, were the proportionally constants will depend only on \epsilon. For \epsilon=0 (circular motion) this will simply be,

\frac{p^2 \pi }{\text{K1}}

as expected. This can be derived by solving the keepler's differential equation, resulting in,

t=\frac{p^2 \left(\epsilon  \sqrt{\epsilon ^2-1} \sin (\theta )-2 \tanh ^{-1}\left(\frac{(\epsilon -1) \tan
   \left(\frac{\theta }{2}\right)}{\sqrt{\epsilon ^2-1}}\right) (\epsilon  \cos (\theta )+1)\right)}{2 \left(\epsilon
   ^2-1\right)^{3/2} (\epsilon  \cos (\theta ) \text{K1}+\text{K1})}

--Paclopes (talk) 23:22, 1 September 2010 (UTC)

--Could you please derive these equations? or provide a reference where I can better understand them? —Preceding unsigned comment added by 157.193.10.67 (talk) 13:16, 21 October 2010 (UTC)

I am sory, i don't have much time, but i think i made some error in the derivation, my current result is:

T=\frac{\pi  p^2}{\text{K1} \left(1-\epsilon ^2\right)^{3/2}}

I calculated it using T equal to 2 times the value of t at \theta=\pi/2. Actually i took the limit. All the calculations were done using a symbolic calculation package. The equation for t was done by solving the equation, (1/2 r[t]^2 \[Theta]'[t] == K1) with (r[t] -> p/(1 + \[Epsilon] Cos[\[Theta][t]]). —Preceding unsigned comment added by Paclopes (talkcontribs) 21:37, 21 October 2010 (UTC)

Badly placed foci in Figure 2[edit]

In Figure 2 in the section "First law", the foci are drawn way too far apart from each other for the ellipse shown. This can easily be seen by looking at the point at the top of the minor axis: the sum of its distances to the two alleged foci is obviously much greater than twice the semi-major axis. Can someone with graphics skills fix this figure? Duoduoduo (talk) 17:57, 22 April 2011 (UTC)

misplaced new section[edit]

The recently introduced subsection on Scale Invariance does not belong here, in my opinion. Bo Jacoby (talk) 07:22, 18 November 2011 (UTC).

Maybe this section should be moved to the scale invariance page. But this property is an (may be the most) important aspect of Kepler's third law, and it is the key to deriving Newton's law of gravitation. So I think it deserves mention at least. Cstalg (talk) 14:37, 21 November 2011 (UTC)

No, the inverse square law for acceleration is already derived in the article. It is strictly kinematic, involving only geometry and time. Your new subsection uses dynamical concepts, (force, mass, angular momentum etc.), which are alien to Kepler's laws. So please move the subsection somewhere else. Bo Jacoby (talk) 15:13, 21 November 2011 (UTC).

Though Kepler's laws are kinetic, they leads to the law of gravitation, which is dynamical. In the section about Newton's gravitional law, we do make use of Newton's dynamics. Without dynamics, Kepler's laws will be of less interest. I have to emphasise that, as I've mentioned in the section about scale invariance, Kepler's third law dose have something to do with mass: the orbit of the planet is independent of its mass! This makes the essential difference between the motion of particles in a gravitational field and the motion of particles in an electric field. Cstalg (talk) 14:38, 23 November 2011 (UTC)

We have other articles on gravitation and dynamics (mechanics) and electricity, but this article is about Kepler's laws of planetary motion. The historical and logical relation between Kepler's laws and dynamics is of cause important, and it is treated in the article. But the full-grown arsenal of dynamical concepts belongs somewhere else. Bo Jacoby (talk) 21:28, 23 November 2011 (UTC).

Kepler's efforts to explain the underlying reasons for such motions are no longer accepted[edit]

It doesn't mention what those reasons are. (I also can't find them anywhere with internet searches.) Should the reasons Kepler came up with be given? — Preceding unsigned comment added by Bazbsg (talkcontribs) 04:47, 28 September 2012 (UTC)

Without reference to detailed explanation to what Kepler's efforts were about, the sentence is not helpful to the reader. I will remove it. Bo Jacoby (talk) 17:38, 19 November 2012 (UTC).

Kepler's second law not a law according to Kepler[edit]

I accidentally posted this on the biography of Johannes Kepler page. I guess I will leave it but it is more relevant here:

Forgive the typos as it got processed by an OCR. I don't want to add it but people should be aware. Also this book would make a good primary source.


Since,
therefore, in addition, the real diurnal arcs which are in proximity
are greater still on account of the greater velocity, and the real
arcs in the remote aphelion are smaller still on account of the
retardation, it results, as l have shown in my Commentaries on
Mars, that tbe apparent diurnal arcs of one eccentric are almost
exactly inversely proportional to tbe square of tbeir distances from
tbe Sun.‘ As, for instance, if a planet in one of its days when it is
in aphelion is distant from the Sun 10 units, in any measure
whatsoever, and in its opposite day, when it is in perihelion, is
distant 9 units of exactly the same kind, it is certain that, as seen
from the Sun, its apparent progress in aphelion will be to its
apparent progress in perihelion as 81 is to 100.
Now this is true with these reservations: first, that the arcs of
the eccentric be not large, that they may not have different dis-
tances varying greatly, that is, that they may not cause a sensible
variation in the distances of their ends from the apsides; secondly,
that the eccentricity be not very great, for the greater the eccen-
‘ Or “the ratio of the apparent diurnal arcs of one eccentric is almost exactly
twice the inverse of the ratio of their distances from the Sun." M X % =
twiCe ‘/2. (Note by translator.)
tricity, that is the greater the arc, the greater is the increase of the
angle of that appearance in comparison with its own advance
toward the Sun, according to Theorem 8 of the “Optics” of
Euclid. But there is another reason why I give this warning.
The arcs of the eccentric about the middle of the anomalies are
observed obliquely from the center of the Sun, and this obliquity
diminishes the size of their appearance, while, on the other hand,
the arcs around the apsides are presented to the sight, which is
supPosed to be on the Sun, from directly in front. When, there-
fore, the eccentricity is very great, the relation of the motions is
sensibly disarranged if we apply the mean diurnal motion without
diminution to the mean distance, as if it appeared from the mean
distance as large as it is; and this will appear below in the case of
Mercury. All this matter is treated at greater length in “Epitome
Astronomiae Copernicae," Book V, but it had to be given here
because it concerns the very terms themselves of the celestial
harmonies, when considered apart each by itself. 

A source book in astronomy, by Harlow Shapley and Helen E. Howarth New York ètc. McGraw-Hill book co. 1929. 1St ed. pages 36-37 (From “Harmonice Mundi," Opera Omnia, Volumen Quintum; Edidit Dr- Ch. Frisch, 1864; translation by Dr. John H. Walden, 1928.) — Preceding unsigned comment added by 207.229.179.97 (talk) 05:02, 19 November 2012 (UTC)

Orbital period from Kepler/Newton laws[edit]

Should we add that the orbital period of the secondary in any two-body system is given by P = 2\pi\cdot
\sqrt{
\dfrac {r^3} {GM}
} , where P is the period in seconds, r is length of the semi-major axis length in metres, G is Newton's constant and M is the mass of the primary, in kilograms? Or we could point the readers to Orbital_period#Small_body_orbiting_a_central_body, which has this equation. CS Miller (talk) 21:50, 19 February 2013 (UTC)

No, it had nthing to do with it. It belongs in an article on celestial mechanics.77Mike77 (talk) 14:57, 20 February 2013 (UTC)

Following CS miller's equation, the main article saying that "n squared a cubed should have the same value" is incorrect. It should say "n squared / a cubed should have the same value".Jburdettelinn (talk) 18:16, 20 January 2014 (UTC)

This is an encyclopedia, not a physics handbook.[edit]

This article is ridiculous. It states the laws in the most awkward way imaginable. They are extremely simple laws, but this article makes them seem complicated. Also, Newton's proof of the laws came much later, and is a separate topic.77Mike77 (talk) 14:55, 20 February 2013 (UTC)

Is the whole article rediculous or just a few formula's? It would help a lot if the description of the problem as you see it would be more exact. Also, could you give one example, or a few, and provide us with your ideas about a solution? How would you have written down the formula's? And if you have a source that supports your statement on the date of Newton's proof, can you cite it here? Thanks for your feedback. Wikiklaas (talk) 20:59, 21 February 2013 (UTC)
There isn't anything wrong with it as a summary of chapter of a university textbook on celestial mechanics, but it is far beyond the target readership of wikipedia, as in "too technical". For example, probably 999 out of a thousand wikipedia readers would not know that the two dots over the letter refers to the second derivative with respect to time. (In fact, probably most have never taken calculus, and don't even know what a derivative is, let alone be familiar with the quirky Newtonian notation. I'm used to it, but even a first year university math student would probably write the first time derivative as dr/dt, rather than an r with a dot.) Also, writing r in bold-face indicates that it is a vector, not a scalar, and nobody but a university physics student would know this. Simply put, the level at which it is written is ridiculously beyond what is required of an encyclopedia article. The talk of "Mean anomaly", etc., is also severely inappropriate.
I don't suggest deleting all of that, but maybe have the article in two parts, where the first part is readable to those who have no more than high school algebra, and the second part under a broad separate category, Technical Discussion. This isn't a criticism of the person who did all that work, it's just that it doesn't belong in the body of an encyclopedia article targeting the lay public.
Re my claim that Newton proved Kepler's laws much later, I offer the fact that Kepler died in 1630, and Newton wasn't born until 1642.
I was going to suggest moving all of the technical stuff to an article about Celestial Mechanics, but I see that the entry (http://en.wikipedia.org/wiki/Celestial_mechanics ) has no math, and it suggests seeing this article for the math details. Hmm. In that case, perhaps there could be a note (sentence) after the opening history section, saying, "See Celestial Mechanics (linked) to see the broader context in which Kepler's Laws appear," or something like that.
The following site I found is at a more appropriate level, for example. http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html
I hope my comments will be of assistance in making the article better.77Mike77 (talk) 00:10, 22 February 2013 (UTC)
You seem to have a rather precise notion of the target readership of Wikipedia. I don't know where you got that but I don't have it. At least I feel that this encyclopedia is not just for the lay public. If it were, I would stop contributing right now. Second you seem to adhere to one special way of notation (for a vector for example), while there are nearly always several (I'm used to indicate a vector by putting a right-headed arrow over it). I was hoping you would give an example of a simple formula to replace one you criticise. By not doing so, you place the burden of improving the article entirely upon someone else's shoulders. Furthermore, I don't think every user would visit this page, only the ones who are interested in the subject, which makes the chances of attracting readers with at least some basic understanding of math slightly better. Last, if this article would fail to give the appropriate equations, it would also disappoint quite a number of readers. Wikipedia is not meant as a children's book either. The nice thing about this project is that we can use links to articles where basic principles are explained. Otherwise, every article that's not about basic principles but relies on them, would have to start by repeating the same principles, resulting in having more or less the same text dozens of times in different articles. "Mean anomaly" and "true anomaly" for example, are both linked to the articles where the reader can find explanations on these topics.
I do not entirely disagree with you though. If possible, a simple explanation in words could be given at the start of the treatment of each of the three laws. I don't feel however the whole article should be split in two. The way it is build up now leaves plenty of opportunity to add some simple explanation at the appropriate points. The part you seem to criticise most, on computing accelerations, is already at the end. Also the relation with Newton's laws is only discussed after Kepler's laws have been treated. I see I misinterpreted your first remark on Newton's proof coming much later. You're right of course about the time gap. But as we are able to look at the topic from a 21st century viewpoint, I think it is relevant to make the connection, not just in an article on Newton or his laws, but also in this article. It shows, with hindsight, how relevant Kepler's laws were, I guess.
You are invited of course, to add your improvements to this article yourself. Wikiklaas (talk) 02:56, 22 February 2013 (UTC)

I thought that the "-pedia" part of "Wikipedia" implied that it is an encyclopedia, e.g. Encyclopedia Britannica, where the articles are not targeting post-graduate students, but rather the general reading public. I think that anybody who would understand the math university level math would already have a big textbook all about it, so wouldn't be looking at wikipedia. Re vectors, most people use a little arrow, as I do, which is my whole point, i.e. that it is mostly in physics textbooks that you see bold-face representing a vector, and the average person would miss it. So, between the "bold-face = vectors" and the "dot notation" for time derivatives, the article is already using two conventions that only university-level physics students would know about. You may be "disappointed" if a wikipedia article doesn't cater to your area of expertise, but that is not what an encyclopedia is for. I will have to check the wikipedia mission statement, because if it is supposed to be a collection of arcane details targeting specialists only, and over the head of everyone not already trained in the specialty, then it is not an encyclopedia at all, and I am wasting my time here. This article is an interesting summary, though, I have to admit, but not for the general public.77Mike77 (talk) 01:24, 23 February 2013 (UTC)

I was just reading this: http://en.wikipedia.org/wiki/Wikipedia:Make_technical_articles_understandable This article is borderline "too technical", but could be fixed with a few more explanations re what the math is about. I think it is a cop-out to have an article containing a lot of linked jargon which is not understandable without having to jump constantly to the links. For example, if I hadn't already studied Kepler's Laws, parts of this article would be impossible to understand. I may try to put in a few explanatory statements when I get the time.77Mike77 (talk) 01:42, 23 February 2013 (UTC)

In the section on Zero Eccentricity, in this sentence "but the equator cuts the orbit into two parts with area...", the word "equator" is linked to an article on the earth's equator, which makes no obvious sense. Does the author mean to say, "the line through the focus perpendicular to the major axis (latus rectum)" rather than "equator"?77Mike77 (talk) 02:29, 23 February 2013 (UTC)

Is there a proposal under discussion? This is not the right place to discuss the purpose of Wikipedia but I'll mention that there are thousands of highly specialized articles where an average citizen would not understand more than a few fragments. The lead seems to do a pretty good job of stating the laws, and anything more understandable by the general public is likely to be less precise. Regarding links: many articles have dubious links because people love adding them. I haven't formed an opinion about that one, but there are likely to be several in any article which should be removed per WP:OVERLINK. Johnuniq (talk) 03:07, 23 February 2013 (UTC)
I agree with the point that all articles should be understandable to as wide an audience as possible otherwise wikipedia is failing in its mission. The policy 77Mike77 pointed to at Wikipedia:Make technical articles understandable says it all. Obviously some articles, such as this one, will get very technical with formulae that many won't understand but the text should be as readable as possible. As an example the first sentence I get to that, as a layman, I'm not too sure about is "Because of the nonzero planetary masses and resulting perturbations, Kepler's laws apply only approximately and not exactly to the motions in the solar system". I'm sure that could be much more easily explained without using the term "nonzero planetary masses", or at least explaining what it means. One problem with the article is that there are notes and references all mixed up together so the average reader probably wouldn't realise that there are explanations there, as in the text they all look like references. It's quite easy to sort this out (if a little tedious initially) by splitting up the notes and references into seperate sections as I did with the Magnetic field article. Then when you get to a bit of text followed by superscript with something like [nb1] you know it's note of explanation and not just a reference. When I have a bit of time I'll try and sort that out. I'll also have another read through the article and try and find other bits I don't follow and ask here for an explanation. Richerman (talk) 10:43, 23 February 2013 (UTC)
I just wanted to say that I think this article is great. It really helped me understand basic orbital mechanics. By all means, let's try to make it more accessible where we can, but equally let's make sure we don't throw away good material. Martijn Meijering (talk) 10:56, 23 February 2013 (UTC)
Ok, I've split up the notes and references so it should be more obvious which is which now. Note 1 and 4 are identical though, perhaps they should be more specific to the section where they are used. Richerman (talk) 13:32, 23 February 2013 (UTC)

Kinda skimmed over the later portions of this section (tldr), but there are a couple good points. This article does have a lot of exessive mathematical formulae which can be found in other articles about OM/astrodynamics. For example, there are other articles regarding position as a function of time, and mean/eccentric/true anomaly do not have huge relevance to the topic of keplers laws. The section on planetary acceleration is even more of a deviation from the main topic. Jaxcp3 (talk) 15:54, 19 July 2013 (UTC)

re "often" versus "preferable"[edit]

The comment for changing what I wrote, from "preferable" to "often" is the idea that "preferable" indicates an opinion. I suppose if I were to write, "It is preferable for non-swimmers to use the bridge to cross the river," it woud be changed to, "Non-swimmers often use the bridge to cross the river". I guess the fact that it is extremely messy to treat Kepler's Laws with Cartesian coordinates, and most people prefer the much easier treatment (polar coordinates) doesn't enter into it, in this person's mind. Also, re the edit of an earlier statement, ellipses are NEVER introduced for the first time using polar coordinates, so the word "often" is flatly wrong. Those edits are borderline vandalism, but fine, I'm not going to fight it. I'm out of here. Good luck to the serious people in your attempts to improve it.77Mike77 (talk) 17:27, 24 February 2013 (UTC)

Wikipedia doesn't give advises, not even to non-swimmers, and if someone here would advise non-swimmers to take the bridge, that statement would almost certainly be deleted. What's preferable and what's not is for the reader to decide. I'm convinced you did your very best to improve this article and you were well on your way to do so and shake things up for the good. It's a pity you don't see that others have the same motive when they make adjustments to your contributions. Adding opinions to articles is a beginners mistake that is easily spotted but also easily corrected. No offence was meant there. It's the way Wikipedia works. It's more than a pity you value as vandalism an edit that was meant as an improvement. Wikiklaas (talk) 20:16, 24 February 2013 (UTC)

Okay, I accept you were trying to improve it, but your changes eliminated the meaning. In mathematics, if Method A is simpler and more elegant than Method B, it is automatically "preferable" - that is not an "opinion" nor a personal judgment call. Part of explaining math and physics involves explaining why things are done a certain way; if wikipedia forbids this, then it is pointless to try. You are saying that wikipedia is not an encyclopedia, so what is it? It is, in fact, preferable to use polar coordinates here (which is why the original author used them). I was explaining why polar coordinates were preferable. It is worthless to say polar coordinates are "often" used; they are always used here, and I am not allowed to explain why, which is ridiculous. It makes no sense to carefully choose my words, then have them changed to something that doesn't convey the meaning. That's why I can't contribute further to this.77Mike77 (talk) 00:01, 25 February 2013 (UTC)

Please stop arguing about something that's so simple. Try to learn from it and just apply it when you make further edits. The article you are editing here is about Kepler's laws, not on ellipses. Furthermore, in most cases there's no necessity to explain why things are done. It is often enough to explain how they are done. Wikiklaas (talk) 03:59, 25 February 2013 (UTC)

Fine, then you do it. That's all from me.77Mike77 (talk) 05:32, 25 February 2013 (UTC)

Kepler's Third Law, stated more simply[edit]

The square of the orbital period of a planet is directly proportional to the cube of the major axis of its orbit.

(If it's proportional to the cube of the semi-major axis, then by definition it's also proportional to the cube of the major axis, which will be 8 times the cube of the semi-major axis.) HarmonicSphere (talkcontribs) 18:09, 29 March 2013 (UTC)

That is correct, only in describing the elliptical orbit, it is customary to refer to the semi-major axis. Wikipedia could decide to choose its own course, but that would be a course leading away from the main body of literature. I feel we should not point the way but follow. Wikiklaas (talk) 23:38, 16 April 2013 (UTC)

I agree with HarmonicSphere. An encyclopedia may deviate from the main body of literature by improving the clarity of exposition. The simpler version should be preferred. Bo Jacoby (talk) 13:52, 17 June 2013 (UTC).

Actually, I believe the original stating of the law refers to the cube of mean distance from the sun (which can be shown geometrically to be the semi-major axis). I agree with Wikiklaas in regards to preference for SMA over MA, pretty much all literature regarding orbital mechanics discusses the semi-major axis exclusively and Wikipedia should be descriptive, not prescriptive. However, if a simpler stating of the is desirable, I would suggest mean distance over either of the other two options. Jaxcp3 (talk) 15:33, 19 July 2013 (UTC)

Hyperbolic[edit]

A link apparently targeting http://www.nasa.gov/mission_pages/stereo/news/SECCHI_P2003.html was used as a citation to support the claim "Heavenly bodies such as comets with parabolic or even hyperbolic orbits are possible under the Newtonian theory and have been observed." (The original link is archived here http://web.archive.org/web/20100420005926/http://erc.ivv.nasa.gov/mission_pages/stereo/news/SECCHI_P2003.html and appears substantially identical to the new link.)

The citation does not appear to support observation of a hyperbolic orbit. In fact, the observations specifically confirmed the object was on a periodic (i.e. elliptical orbit):

"This latest STEREO/SECCHI discovery ("recovery") is noteworthy for a couple of reasons. First, it means that we now know much more accurately where the comet is in space, so now ground-based observers can find it when it moves away from the Sun in January as a fading 10th magnitude object. Secondly, it means the comet can be declared "officially" periodic, and redesignated as such, instead of bearing just the provisional "P/2003 K2" designation."

http://adsabs.harvard.edu/full/1991JBAA..101..119H "On Hyperbolic Comets" by David W. Hughes (1991) discusses the possibility of hyperbolic comet observations and claims "Out of the 120 well-observed comets with periods longer than 61000 years, a total of 46...were seen to leave the solar system on hyperbolic orbits." But these were slingshotted out of the solar system: their orbits were disturbed and not on a purely hyperbolic trajectory that would be described by Kepler's laws.

1738[edit]

We are told that Voltaire was the first to call Kepler's laws laws, in 1738. I am not sure why this fatuous information has been put into the article. — Preceding unsigned comment added by 86.143.239.144 (talk) 13:15, 13 August 2013 (UTC)

Spoken version[edit]

The pictures don't match the text[edit]

In the pictures the sun is placed in the left-hand focal point, but the text assumes that the perihelion is to the right of the sun, assuming that the axis θ=0 is pointing to the right. Couldn't we have pictures with the sun in the right-hand focal point? And the figures are overloaded with information. I would like a figure showing an ellipse, a figure showing an ellipse with focal points, a figure showing an ellipse with perihelion distance rmin, a figure showing an ellipse with semi latus rectum p, a figure showing an ellipse with aphelion distance rmax, a figure showing an ellipse with semi major axis a, a figure showing an ellipse with semi minor axis b, and a figure showing an ellipse with a planet and the distance r and the angle θ. Bo Jacoby (talk) 08:04, 9 December 2013 (UTC).

The mathematics is wrong[edit]

This article contains significant errors, even the statement of Kepler's laws is mathematically incorrect. Thank you, Wikipedia, you have motivated me to better understand this subject. ilan (talk) 08:38, 23 January 2014 (UTC)

Please provide a clue concerning which statement you believe to be incorrect, and why. Johnuniq (talk) 09:25, 23 January 2014 (UTC)
I did provided a clue in my previous statement. ilan (talk) 10:20, 23 January 2014 (UTC)
ilan, perhaps you misunderstand how Wikipedia works. Unlike a journal, there is no board of editors to make edits for you. Wikipedia is the encyclopedia anyone can edit so if you see an error you can correct it yourself. If other people disagree with your edit then the disagreement can be discussed here. Martin Hogbin (talk) 12:11, 23 January 2014 (UTC)
Thanks, but I understand only too well how Wikipedia works: it is mostly edited by anonymous contributors with no demonstrable expertise in the subject (since you don't even know who they are). In particular the above comment was made by such an anonymous user. In my opinion, pages regarding scientific correctness should not be modified by anonymous users. I used to make Wikipedia modifications, but most of the time, they were modified back to wrong by anonymous users, with resulting modification war, so I'm not going to go through that again. P.S. I just noticed historical errors, due to a naive understanding of the subject. This page is pretty much a mess. ilan (talk) 13:00, 23 January 2014 (UTC)
You are wasting your time posting here if you think that someone is going to divine exactly what what changes you would like and then make them for you. If you want to contribute by all means so so. If your edits are challenged then defend them. There is no other way to work here.
As it happens I do post under my real name and I do recognise WP as a unique experiment. In some ways I believe it is failing but, for the moment at least, I continue to edit here. Martin Hogbin (talk) 13:18, 23 January 2014 (UTC)
Check out the third law. The error was introduced when it was modified on January 20, 2014 by User:Jburdettelinn. It should be either P^2/a^3 or n^2 a^3. Since the text talks about period I prefer the first variant. Rolbit (talk) 13:59, 23 January 2014 (UTC)
I prefer n2a3 which is used later. Bo Jacoby (talk) 14:09, 24 January 2014 (UTC).
The point is that P^2/a^3 is the literal mathematical formulation of Kepler's third law, which is what this part of the text says it is going to do. ilan (talk) 15:30, 24 January 2014 (UTC)

The history is wrong[edit]

In the article it's stated that Kepler's laws are based on Copernicus. However, one of Copernicus' main motivations was his dislike of Ptolemy's equant, because he wanted to reestablish uniform circular motion. Now Kepler started off with Copernicus, but then realised that Ptolemy's equant was the way to go, since it's a first approximation to the ellipse model. So, saying that Kepler's laws are based on Copernicus is pretty much the worst historical comment you could come up with. At least, if my understanding of the history is correct.

That's just the main point, the rest of the historical summary is equally wrong. For example, it's not the geocentric model of Aristotle, it's the one by Eudoxus, since Aristotle was just reporting existing astronomical knowledge of his contemporaries.

Then there are completely ridiculous statements such as: "proving that the planets' speeds varied." That statement is wrong on so many levels it's hard to list them all. Everyone always knew that planets' speeds varied, that's why they are called "planets", which in Greek means "wanderer", i.e., a celestial object that does not move regularly with respect to the fixed stars. Secondly, Kepler didn't prove anything, his laws are hypotheses based on experimental evidence. It was Isaac Newton who proved these laws based on his laws of motion.

This page confirms that the Wikipedia concept isn't ready for historically accurate scientific articles. ilan (talk) 23:21, 27 January 2014 (UTC)

Second Law query[edit]

When talking about "equal areas swept" within a planet's orbit, does that also extend to mean that ALL planets within a particular system cover the same amount of area as one another? What I mean is: does Mercury cover the same amount of area in 1 second (a comparatively fat and short sector) as Pluto covers in that same second (a really long and thin sector)? If so, could a very simple comparative example be provided, please (with, say Mercury and Pluto)? And if not, how does the area swept decrease/increase with distance? (evenly, exponentially...?)

Actually, examples would be great in the other two Laws' sections, too - all 3 sections contain almost exclusively mathematical formulae, whereas just a couple of real-world examples in each would do wonders! BigSteve (talk) 21:17, 20 February 2014 (UTC)

I guess that doesn't work, even with circular orbits of radius r. It would say P was proportional to the square of r? Which is not what the third law gives in this special case. Charles Matthews (talk) 19:34, 10 March 2014 (UTC)

Anachronism[edit]

I agree with the general idea that something should be done about the history in the article.

I have added {{anachronism}} not because I think a textbook treatment of the three laws is not useful to have. Rather it indicates that the history of what people are reading should be clearer.

I have divided the existing section History so that the terminological questions come first. Another thing that needs doing is finding a way by which what Kepler actually did can coexist with the textbook modern treatment. I think one obvious way is to improve what is said in Astronomia nova about the laws, and refer to that.

I'm actually here because a Kepler scholar asked me to intervene, at the editathon at the Royal Society last week. Please comment. Charles Matthews (talk) 16:06, 10 March 2014 (UTC)

I'm sure you're (both) right, fwiw, Johnbod (talk) 22:06, 10 March 2014 (UTC)

Thanks. It would look useful to have an article also about the Epitome Astronomiae Copernicanae, Kepler's later work. Charles Matthews (talk) 06:55, 12 March 2014 (UTC)

So Epitome Astronomiae Copernicanae now exists. This is apparently where the "third law" was published. That article needs plenty more work. There are certainly indications that Kepler did not use the working concept "natural law", suggesting that a more careful explanation of his actual approach is needed here. Charles Matthews (talk) 16:06, 20 June 2014 (UTC)

Original statements of Kepler's laws in his own writings[edit]

I have been trying to locate the original statements of Kepler's laws in his own writings. This undertaking has been difficult for several reasons:

(1) I thought that, given the importance of Kepler's laws, there would be many sources listing quotes from his original works, in which he states his laws. Unfortunately, so far I've not found any such source. Instead, the information is scattered among many scholarly works, which are oftentimes difficult to access. Even these works often cite only translations of Kepler's work, not the original texts.

(2) A further problem is created by Kepler's presentation of his laws:

  • He did not present the first law in 1609; rather, he merely showed that Mars' orbit is an ellipse. Only subsequently, in his Epitome, did he confirm that all of the planets travel in elliptical orbits.
  • He presented his second law in 1609 in two different forms: one is called the "distance law" whereas the other is called the "area law". The former is wrong, but it was the one that Kepler favored because it was based on his theory of gravitation. Furthermore, the area law is expressed not explicitly but implicitly; that is, it is not stated clearly in its modern form. It was only later, in his Epitome, that he stated the law clearly and explicitly.

Therefore, please be patient with my numerous edits while I continue my investigations.
Cwkmail (talk) 19:46, 2 June 2014 (UTC)

Please carry on. I have just reorganised the article somewhat, to separate the basic exposition from the heavier mathematics, which deals with another issue that has been raised with me. Charles Matthews (talk) 16:37, 20 June 2014 (UTC)