Talk:Kepler orbit

98.28.209.255 (talk) 03:25, 3 January 2016 (UTC)

Wiki Education Foundation-supported course assignment

This article was the subject of a Wiki Education Foundation-supported course assignment, between 20 August 2018 and 7 December 2018. Further details are available on the course page. Student editor(s): HHarr8001.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 01:43, 17 January 2022 (UTC)

Nonsensical sentence

"The law would change an eccentricity of 0.0. and focus more of an eccentricity of 0.8. which show that Circular and Elliptical orbits have the same period and focus, but different sweeps of area defined by the Sun." I can't figure out what the original intent was, or how to fix it, but something needs done. Chris2crawford (talk) 19:30, 31 October 2020 (UTC)

Constant of integration

I removed the references to Constant of integration as this article is of little use to explain the concept for a differential equation.

Stamcose (talk) 19:42, 16 July 2008 (UTC)

First sentence

This challenged sentence was added by an editor Swpb 22:09 27 July 2008 to obtain the desired Wikipedia style.

Kepler did not know about differential equations but Newton did (one of the main inventors of differential calculus!). It is called "Kepler orbit" because it was the astronomical observations of Kepler that led to the discovery (and proof) of Newton's gravitational theory explaining Keplers laws that Kepler derived from astronomical observations. —Preceding unsigned comment added by Stamcose (talk • contribs) 20:26, 6 August 2008 (UTC)

This is a bit confusing. Is a "Kepler orbit" simply an orbit described by Kepler's laws of planetary motion? These can be derived from Newton's law, but predate it historically. Was this term first used by Newton, or someone else? -- Beland (talk) 05:11, 9 August 2008 (UTC)

Well, I've rewritten the intro to give a more understandable definition, which is hopefully also accurate. Documentation on the historical origin of the term would still be interesting. -- Beland (talk) 14:16, 11 August 2008 (UTC)

This is fine. The terminology is the one that has been used since the 1930s by physicists in describing the 2-body problem solution of the Newton's General Gravitation Law. The logic presented is present in most intermediate and advanced physics mechanics textbooks. This logic using Gravity and Angular Momentum conservation as a solution of the "the Kepler Problem" is given in the graduate textbook, THEORETICAL PHYSICS, by Gerhard A. Blass (direct student of Werner Heisenberg)Appleton Century-Crofts Co.1962 pp.40-51 (all of chapter 3). 69.246.53.93 (talk) 22:49, 22 October 2008 (UTC)

Math, math, and more math...

The level of detail we have for mathematical transformations on this page seems to me like it might go a bit beyond what is appropriate for an encyclopedia. Perhaps it would be a good idea to remove all the intermediate steps and just show important results? -- Beland (talk) 14:18, 11 August 2008 (UTC)

Please do not "improve"

Dear "Beland"

I appreciate that the Wikipedia concept is based on that anybody can do anything to any article. But I hope that the following will convince you not to "improve" this very article.

• This is rock solid mathematical text giving a complete account of a all aspects of Kepler orbits. It is intended as "handbook" for professionals (or future professionals) in the field of "Mathematics of Space Flight"
• Wikipedia is full of articles about "Celestial Mechanics" topics at "layman level" suitable for the part of the "general public" that is uncomfortable with mathematics
• Wikipedia is full of articles about "Celestial Mechanics" topics showing that the authors would need reading "Kepler orbits" (without "improvements"!)

Your introduction really makes no sense!

• How can one "think about an orbit as an ellipse". This has nothing to do with thinking!
• Orbital elements must logically come at the end.
• Orbital elements is certainly no "Main article" that should be references here. Kepler orbits" (without "improvements"!) would in principle make this article (and many more) completely obsolete (except for the part of the publicum that is very "uncomfortable" with mathematics). By the way, many Wikipedia authors do not seem to clearly see that only a Kepler orbit has an "semi-major axis" an "eccentricity" a "true anomaly". For the general orbit these concepts only makes sense as far as the orbit is approximated with a Kepler orbit. These concepts are foggy also in Orbital elements!!!

Stamcose (talk) 10:08, 13 August 2008 (UTC)

Stamcose: Please carefully read the policy Wikipedia:Ownership of articles. You do not have the right to control the content of this or any article, no matter how much work you may have put into it. Blanket reverting of numerous substantial edits by another editor, and actively discouraging other editors from editing, is considered extremely poor behavior on Wikipedia, and will very quickly get you into serious disputes, and even get you blocked. For this reason, I am undoing your revert, and I encourage you to address your concerns about Beland's edits on a change-by-change basis, with small edits and thorough edit summaries, and to use the discussion page for major changes.

It appears that Beland made a number of improvements to the article, especially with regard to prose. I don't mean to insult you, but while you may know a great deal about the mathematics of this topic, your prose could use some improvement. There is no reason that an article which is technically thorough and in-depth cannot also be clear to a general audience – in fact, this is the goal of writing better articles. — Swpb^{τ • c} 00:44, 14 August 2008 (UTC)

Rationale for changes

Stamcose, first, let me thank you for your work getting this article started. I can explain a bit about why I made the changes that I did...

Wikipedia:Make technical articles accessible is a consensus guideline which this article (and all the other articles on celestial mechanics) should follow. This means putting things that a general audience can understand first, and deferring heavy math until later in the article. (A lot of people will stop reading once they start hitting equations.) This does not mean that equations useful to professionals need to be eliminated from the article.

Template:Main, as its description page says, is designed to be used where there is a subarticle which is summarized by a section in a higher-level article. The article orbital elements has more detail on that topic, but it's summarized in one section of this article. It seems to me like a classic situation where this template is appropriate. Otherwise, it becomes more difficult for readers to find out more about this subtopic.

I moved the section on orbital elements to the beginning of the article because 1.) it is accessible to general readers, and 2.) they are a defining aspect of Keplerian orbits, and the most important information in an article goes first. The math which derives or relates these elements is not important or understandable to most people.

I wrote that "The orbit can be thought of either as an ellipse that satisfies Kepler's laws of planetary motion (as Kepler did), or (as was later discovered) as a solution to a differential equation that expresses Newton's law of universal gravitation" to try to improve the previous definition, which was: "Kepler orbit is a solution to a differential equation describing the time variation of a vector". Most people don't know what a differential equation is, and are probably fuzzy on what a vector is. A lot more people do know what an ellipse is. Kepler himself apparently didn't think of these orbits as solutions to differential equations, because they hadn't been invented yet. It's an important fact to mention, and it's relatively accessible to non-mathematicians. Mathematically, the orbits are both ellipses and solutions to a particular differential equation. If the word "think" is objectionable, we could say something like:

A Kelpler orbit in an ellipse (forming a two-dimensional orbital plane in three-dimensional space) defined by Kepler's laws of planetary motion. A Kepler orbit represents a solution to a differential equation that relates the position of the two bodies over time, as they obey Newton's law of universal gravitation. The orbits can be described and manipulated without differential equations; modern calculus had not yet been invented when Johannes Kepler formulated his laws. But Isaac Newton later showed that Kepler's laws could be derived from his own laws.

If the difference between a real orbit and the approximation of a Kepler orbit is unclear in Orbital elements or this article, feel free to suggest or make improvements on the appropriate article or talk pages. I have tried to add clarification in the intro of orbital elements.

-- Beland (talk) 02:12, 14 August 2008 (UTC)

I disagree. Certainly a lot of people will stop reading when hitting equations but the target audience for this article is those who do not! As apparently "Beland" and "Swpb" are of the same opinion this would be a case for a formal "Meditation". And if a "democratic majority" of Wikipedia editors support the "Beland" - "Swpb" opinion then Wikipedia must be the wrong place for this article. Then it should be removed completely (what I think an editor can do!). By the way, the original version was written for and is on the ESA local Wiki made for the project "Knowledge Management". This would then mean that this "knowledge" is not transferred to the world outside the agency!

But in other sectors of Wikipedia (with other editors!) also professional level articles have found a place!

Stamcose (talk) 11:26, 14 August 2008 (UTC)

PS

A compromise solution could be as follows:

Rotation (mathematics) starts with

For a less mathematical treatment of rotations, see rotation.

Here one could say:

This article is an in-depth mathematical treatment. For a descriptive overview, see Orbit

Stamcose (talk) 11:59, 14 August 2008 (UTC)

Kepler orbits seem like a topic which Wikipedia should cover, so deleting the article would make no sense. It is true that sometimes the same topic is treated in different articles with a different level of technical detail, but one must be very careful to avoid Content forking (which is absolutely unacceptable), and this kind of split does not license you to ignore Wikipedia's style guidelines, to ignore the fact that Wikipedia's intended audience is anyone, or to enforce a particular version of the article. This article will likely remain on Wikipedia, and over time it will likely come to resemble nothing like what you initially created or intended. That's ok – that's how Wikipedia works. If you can reconcile yourself with the fact that your work will be changed and sometimes undone, not just by Beland and myself but by any editor who wants to, then I don't think we need to waste anyone's time with mediation. If you can't deal with that fact, then Wikipedia may not be the best place for you. — Swpb^{τ • c} 13:30, 14 August 2008 (UTC)

I changed the hatnote in Rotation (mathematics) to say:

For rotation in fields other than mathematics, see rotation.

which is a bit more accurate. This article is about rotation as understood by the field of mathematics, but the target audience is general, not just mathematicians. There is plenty of prose and there are diagrams to help explain what equations to appear there.

I agree that there's no need to delete this article, but there's also no need for it to be the same as the ESA wiki article or for it to be targeted only at professionals (though I expect it will be useful to them as well). Kepler orbits in particular (not just orbits in general) are of interest to non-mathematicians, like historians of science, and young students (say, in high school, after geometry but before calculus) who want to understand how a "Kepler orbit" is different from other types of orbits. This article is the best place for information accessible to them. -- Beland (talk) 14:56, 14 August 2008 (UTC)

References needed

Citations to reliable sources are needed for verification that the math and other claims in the article are correct. I've tagged the article to indicate this. -- Beland (talk) 02:15, 14 August 2008 (UTC)

Incorrect text in article - intro and orbital elements

The contribution to this article made by "Beland", mainly 11 Aug 2008, must be replaced! The idea of "Beland" is certainly to make a soft "non-mathematical" introduction to the subject. But if this should be done the text must be different!.

Some details:

1.) Beland: In orbital mechanics, a Kepler orbit describes the spatial position of the centre of an orbiting body over time, as approximated by classical mechanics.

Comment: The approximation has nothing to do with "classical mechanics", the approximation has to do with the acting force. For a planet the approximation is that only the gravitational attraction from the Sun is taken into account, not the gravitational attractions from the other planets. For a satellite in LEO the main approximation is that the gravitational attraction from the Earth differs from that of a homogenous sphere due to its oblate shape

2.) Beland: When the central body is so massive that its movement can be neglected, the Kepler orbit is said to be a solution to the Kepler problem. (Examples include a planet orbiting the Sun, or an artificial satellite orbiting the Earth.)

Comment: This has nothing to do with one body being massive. The relative motion will in all cases be a Kepler orbit (This is the two-body problem)

3.) Beland: The orbit can be thought of either as an ellipse that satisfies Kepler's laws of planetary motion (as Kepler did), or (as was later discovered) as a solution to a differential equation that expresses Newton's law of universal gravitation and the initial position and velocity of the objects. (Kepler's laws can be derived from Newton's law; the two are entirely compatible.)

Comment: What Belands wants to say is that "Kepler's laws" were detected by Kepler based on data from astronomical observations and that these laws were used by Isaac Newton to discover and prove the gravitational law. These historical fact are treated by other Wikipedia articles, just make a reference to them.

4.) Beland: By decomposition, Keplerian orbits can also be used to describe systems where the masses of two bodies are fairly similar.

Comment: What Belands wants to say is that the two body problem also results in a perfect Kepler motion, i.e. that his previous "When the central body is so massive that its movement can be neglected, the Kepler orbit is said to be a solution to the Kepler problem." after all is superfluous!

5.) Beland: Introduces a first chapter about "orbital elements".

Comment: If he writes about parameters to describe an ellipse in general it would belong to Ellipse or Conic sections (semi-major axis, eccentricity) respectively Euler angles (right ascension of ascending node, inclination, argument of perigee). If it is about Kepler orbits one must first prove that the body really moves along an ellipse (or parabola/hyperbola)

Stamcose (talk) 10:41, 28 September 2008 (UTC)

---

Now the introduction is at least correct! And with more text (compared to the original version) and with a nice image before the necessary mathematics starts!

Stamcose (talk) 12:37, 30 September 2008 (UTC)

Your points 1-4 are very good, and I think what I wrote was fairly muddled. Wikipedia:Lead section explains that the intro should briefly provide context and summarize the article, but the intro in this article contains a mathematical derivation or motivation, not a summary. The first sentence does not define the title of the article, as the guideline recommends, and the intro is far longer than the four paragraph maximum. Unless someone else wants to work on it, I can try again in perhaps a day or two when I have a bit of time.

As for point 5, Wikipedia:Summary style says that this section should simply be a summary of orbital elements. I've restored an older version which has links to the names of the individual elements, and a diagram which is meant to help readers understand which parameter means what. I don't know what you mean by "having to prove" that the orbit being described is a Kepler orbit; the section begins, "A Kepler orbit is specified by...". Orbital elements are one of the main ideas of Kepler orbits, so they certainly need to be at least mentioned in this article. -- Beland (talk) 18:48, 1 December 2008 (UTC)

Incorrect text in article - Differential equation and simplifying assumptions

A simplifying assumption is that the mass of the central body is very large with respect to the mass of the orbiting body, and that these are the only two gravitationally significant bodies in the system.

Newton's Shell theorem shows that for spherically symmetrical bodies (like a planet or star), the shape and size of the body don't matter. For asymmetrical objects (like artificial satellites), the size and shape matters only at small distances, and so is generally ignored. The gravitational attraction between bodies can be treated as if it were coming from the center of mass of the body, in proportion to the mass of the body.

This results in the differential equation (1) below, describing the time variation of the vector from the (centre of) the central body to (the centre of) the orbiting body.

${\ddot {\bar {r}}}=-\mu \cdot {\frac {\hat {r}}{r^{2}}}\ \ (1)$

where

${\bar {r}}$ is the radius vector from the centre of the central body to the orbiting body

$\mu$ is the product of the general gravitational constant and the mass of the central body

${\hat {r}}$ is the unit vector directed to the centre of the central body

$r$ is the distance to the centre of the central body

The simplifying assumptions are usually reasonable when considering a planet in the solar system, because the mass of the Sun is much greater than all of the other planets put together. So though by Newton's law it experiences the same gravitational pull as the planet under consideration, it is assumed to be stationary relative to an inertial frame of reference. Similarly, because the earth is considerably more massive than artificial satellites, the motion of the planet (and the gravitational influence of other satellites) can be neglected when calculating orbits.

The assumptions made are adequate for most purposes (and greatly simplify the required calculations), but in some cases (interplanetary travel, low orbits, orbits close to a star), the effects of other gravitational bodies, atmospheric drag, or relativity become large enough that they cannot be neglected to achieve the desired level of accuracy.

Belan text:

"Newton's Shell theorem shows that for spherically symmetrical bodies (like a planet or star), the shape and size of the body don't matter."

Comment:

How can one write "shape does not matter" when the theorem applies for the shape "sphere"?

Belan text:

For asymmetrical objects (like artificial satellites), the size and shape matters only at small distances, and so is generally ignored.

Comment:

Strange statement! The body of a spacecraft is very small in comparison to the distance to the centre of the attracting body and it is clear that the gravitational field from the other body is homogeneous in this small volume. The "shell theorem" is non-trivial because it applies also close to the surface of the sphere. At long distance it is trivial! Is Belan thinking about a "black hole" next to the spacecraft?

You're right, I re-worded that explanation to say it's the smallness of the forces generated by the irregularities that can be neglected, especially at great distance. -- Beland (talk) 07:52, 2 December 2008 (UTC)

Additional comment:

There is no need to mention the "Relativity Theory" here, corresponding effects are not really significant for any practical spacecraft mission and can safely be ignored! Physicists keep looking for "gravity waves" etc but do not find much! Stamcose (talk) 08:32, 1 October 2008 (UTC)

See Mercury_(astronomy)#Advance_of_perihelion why relativity cannot be "safely ignored". Bo Jacoby (talk) 18:53, 3 October 2008 (UTC).

Confusing explanation:

I'm slightly confused by these 2 sentences about 3rd assumption: "This assumption is not necessary to solve the simplified two body problem ... " "Under these assumptions the differential equation for the two body case can be completely solved mathematically ..." So, is it necessary or not? Also, I'm generally confused why we need 3rd assumption, it doesn't seem to be explained in the text. If I get it right, the reason is that if m2 would be significant, then the coordinate system with a center at the big mass would not be an inertial frame. Basically, it says that the big mass is stationary. Am I right? — Preceding unsigned comment added by Tevariste (talk • contribs) 07:49, 7 April 2014 (UTC)

Other articles

This article seems to have been created without recognition of the other wikipedia articles on the same subject, such as Kepler's laws of planetary motion, Eccentric anomaly, Two-body problem, and Gravitational two-body problem. The title Kepler orbit is not well chosen, because a Kepler orbit is an ellipse or circle but not a hyperbola.

The article attempts to include all the solutions of the newtonian two-body problem, but one important class of solution is omitted: the one-dimensional ones, which are limiting cases of ellipses and hyperbolae having fixed values of a and p→0. Bo Jacoby (talk) 18:49, 3 October 2008 (UTC).

No, a Kepler orbit is either elliptic (0<= e <1), parabolic (e=1) or hyperbolic (e>1). A circle is the special case of an ellipse having e=0.

Stamcose (talk) 16:48, 22 October 2008 (UTC)

I added mention of the straight-line case to the article, and tried to make links between the articles mentioned. (Some of them need to be merged.) -- Beland (talk) 07:49, 2 December 2008 (UTC)

I agree that the article's title is not well chosen. Something along the lines of "Analysis of orbital motion" would probably be more appropriate (i.e. orbit#Analysis of orbital motion). The article on orbit is already pretty much about Kepler orbits, even though it does not state that explicitly. Jaxcp3 (talk) 17:53, 12 August 2012 (UTC)

Straight line case math

Keplers's first law says:

"The orbit of every planet is an ellipse with the sun at one of the foci."

He certainly did not find any planet in escape orbit heading towards outer space away from the solar system!

Never-the-less the term "Kepler orbit" is nowadays used for elliptic, parabolic and hyperbolic orbits centred at any attracting body, the Sun or a planet or what-so-ever, to honour the discoveries of Johannes Kepler

Stamcose (talk) 07:23, 23 October 2008 (UTC)

Don't forget the particle moving directly towards or away from the sun. The straight line orbit is neither a circle nor an ellipse nor a parabola nor a hyperbola, but it is a solution to the two-body problem. Bo Jacoby (talk) 20:44, 23 October 2008 (UTC).

If this should be written with many words and with full mathematical stricness like in a junior level textbooks one would say:

$r^{2}\cdot {\dot {\theta }}=H\ \ (2)$

${\ddot {r}}-r\cdot {\dot {\theta }}^{2}=-{\frac {\mu }{r^{2}}}\ \ (3)$

the polar coordinate system being selected such that $H>=0$.

If $H=0$ one has that

${\dot {\theta }}=0$

what means that the motion is along a line and that (3) takes the form

${\ddot {r}}=-{\frac {\mu }{r^{2}}}\ \ (3)$

Multiplying both sides of (3) with ${\dot {r}}$ and integrating one gets the "energy equation".

${\frac {{\dot {r}}^{2}}{2}}={\frac {\mu }{r}}+E$

where $E$ is a constant of integration (the total energy).

From this follows that

${\frac {dt}{dr}}={\frac {1}{\sqrt {2\cdot ({\frac {\mu }{r}}+E)}}}$

Time as function of $r$ is therefore the integral of the right side.

If $H>0$ then $\theta$ is an increasing function of the parameter time. With the following relations the differential equation (3) is transformed to the new "independent" variable $\theta$:

etc as in the present text

I do not know (I have not really tried) if one can find a closed form primitive function to

$\operatorname {f(x)} ={\frac {1}{\sqrt {2\cdot ({\frac {\mu }{x}}+E)}}}$

but the "line case" is clearly a limit case of ellipse,parabola and hyperbola depending on if $E<0$, $E=0$ or $E>0$

For this one can use the formulas of "Determination of the Kepler orbit that corresponds to a given initial state" using the actual values for $r>0$ and $V_{r}$ but with a non-zero value for $V_{t}$. Using this Kepler orbit one finds the true anomaly for which the "radius" takes the other value $r>0$ and determines the time from the initial true anomaly to the final true anomaly. Letting $V_{r}\longrightarrow 0$ this time converges to the desired time for the "linear case"

Stamcose (talk) 08:28, 25 October 2008 (UTC)

Sorry, please read:

Letting $V_{t}\longrightarrow 0$ this time converges to the desired time for the "linear case"

Stamcose (talk) 08:33, 25 October 2008 (UTC)

Merge-suggestions opposed

Recently, templates were added to the article "Kepler orbit", to propose that four other articles or sections be merged into "Kepler orbit". The other articles proposed to be affected are "Orbit equation", "Kepler problem", "Gravitational two-body problem", and the section "Position as a function of time" currently contained in "Kepler's laws of planetary motion".

I'm posting this message to oppose that suggestion. (a) No reasons in support appear to have been offered by the editor who added the tags.

(b) It does look as if there may be an undue proliferation of separate articles related to two-body dynamics right now. But the proposed merger would likely leave the encyclopedia less clear than its current state, and does not like a good way to deal with the proliferation.

(c) It is suggested that to evaluate or make a merger proposal of the kind proposed, three questions (at least) ought to be addressed: 1 How many articles are needed to cover the subject matter? 2 What are the most appropriate names (and contents) for them? and 3 What alternative names should be linked by redirection?

I can't right now give full answers, but suggest that the following elements are relevant:

On the 'how many articles' point, it looks as if there may be three distinguishable aspects here, the basic physics/celestial mechanics, the application to astrodynamical practice, and the history.

On the 'what names' point, the chosen names should most usefully be either, a name that a user of the encyclopedia would most likely look for, or a name that is in current frequent use by relevant experts as a distinctive special technical term.

It's not clear that the term 'Kepler orbit' fits either of these bills -- are there even reliable sources to show that it is a distinctive technical term for identifying the subject matter? (As far as I know, it is only just one, and perhaps not even a widely-preferred one, among many possible descriptive/jargon phrases, other being 'unperturbed two-body orbit', 'Keplerian orbit', 'Keplerian ellipse', etc. etc. etc.)

Of the articles proposed to be merged, "Gravitational two-body problem" looks as if it would be much better and more accessible if merged with "Two-body problem" -- which also might even be a better 'home' for the material currently covered under 'Kepler orbit' itself.

Terry0051 (talk) 23:58, 28 November 2009 (UTC)

Oppose: An IP, with no discussion or explanation, proposed merging 3 articles and 1 section from Kepler's laws of planetary motion into this article. The proposer did not state any reasons, did not set up a discussion on this talk page, did not date the merge templates as required, and did not inform any of the editors of the pages involved. Kepler's laws of planetary motion is a pretty good article, and I would not want to remove an integral part of it to move it here. I don't see any justification for merging the other articles into this one. —Finell 01:24, 29 November 2009 (UTC)

Further comment by way of oppose: A serious merge proposal on a technical subject, if it is to offer a prospect of improving the encyclopedia, probably needs a a lot more than just a hand-waving tag insertion. For example, some outline or the beginnings of a concrete idea about the detail, i.e. an idea of how the merger would be practically implemented, would not come amiss. In the absence of any arguments at all supporting merger, after more than a month for consideration, and in the absence of any discussion offered by the editor who added the general 'duplication' tag to explain what actually is alleged to be duplicated, I've removed these tags. Terry0051 (talk) 14:53, 3 January 2010 (UTC)

Removing the merge tags. Bo Jacoby (talk) 08:14, 2 June 2010 (UTC).

Incorrect sign

Equation (1) should have a negative sign. To anyone who's thinking of arguing that the vector is pointing inward: no matterr which way it's pointing, its magnitude tends to decrease (more precisely, increase its rate of decrease) because the force is attractive! So the sign is wrong. I'm sure enough to be bold but I don't know how to change it. Somebody please do it. Thanks. --Dhatsavan (talk) 03:24, 13 September 2010 (UTC)

This incorrect sign was caused by the edits of 129.78.64.101 3 August 2010. But that was not all, the math stayed incorrect also with the sign corrected! Now I have corrected the rest of the damage caused by 129.78.64.101 (from Sydney University)

Stamcose (talk) 19:06, 12 June 2011 (UTC)

Equations (2) and (3)

The vector equation (1) does not resolve into equations (2) and (3). Equation (3) is indeed the radial component but the theta component just tells us that the rate of change of specific angular momentum is zero (as expected for a central force). Equation (2) in fact comes from the definition of the cross product. I found this a bit confusing and wasted about an hour on it working everything out myself. — Preceding unsigned comment added by 194.176.105.151 (talk) 12:49, 29 December 2011 (UTC)

Major Fixes Needed

I feel that this article requires an overhaul. The article is very confusing putting elliptical equations and hyperbolic equations next to each other without any clear separation between the two. I also can't say I've ever seen eccentric anomaly used to define hyperbolic orbits. Generally hyperbolic anomaly is used to describe these (unfortunately it generally has the symbol H, which has been used for angular momentum here). I will do some research and hopefully be able to clear some of this confusion up. At first glance, the article seems to be overloaded with equations in general. Some work could probably be done to slim this down and make it look less daunting.

There are also a few places that the prose could be cleaned up and made a little more professional (but I am an engineer, so writing eloquently isn't exactly my specialty). I also feel like additional source material should be added.

Based off this talk page, it seems that no major edits have been made since 2008 or so, which seems like quite a while, given the state of the article. I apologize if my criticism offends, but I feel rather strongly about the subject matter and truly believe that the article is in sore need of improvement

I have started a sandbox page (User:Jaxcp3/Kepler orbit) where I am working on the article. I would appreciate any input that anyone has to offer on the work in progress. I would prefer that comments be made in the talk page rather than the sandbox.

Jaxcp3 (talk) 00:59, 9 August 2012 (UTC)

Updates:

I have split the first section into an introduction and a section about the simplified two body problem. I have added a significant amount of information to the introduction. My writing is not the best, but its better than what was there previously. Any factual and grammatical corrections are more than welcome.

Jaxcp3 (talk) 00:44, 10 August 2012 (UTC)

Edited the second section that I created last time. It needed elaboration on "standard assumptions" that are mentioned in various articles but never seem to be outlined in detail. I also felt that some of the equations about vector magnitude and unit vectors were a little extraneous. I also moved the orbital elements section up to there. This may or may not be the best place for it, but part of me feels that the orbital elements should be mentioned early on in the article, if for no other reason that to put a,e and nu in the correct context

Jaxcp3 (talk) 17:45, 12 August 2012 (UTC)

I fully support any and all attempts to shorten this article, since currently it's unreadable. As you have mentioned on your sandbox page, there are currently at least five articles on orbital mechanics that are largely restatements of the same information (or at least contain sections that are): orbital equation, orbital mechanics, Kepler's laws of planetary motion, Kepler problem, and Kepler orbit (this article), not to mention the individual articles for elliptic orbit, parabolic trajectory, and hyperbolic trajectory. If it came down to it, I think this article, Kepler problem, and orbital equation could all be merged, and information duplicated in the other two articles (Kepler's laws of planetary motion and orbital mechanics) could be severely truncated. If you look at what's in the three articles I mentioned, it's all just variations on:

• Solving for the orbit equation starting from the two-body problem
• Discussing the taxonomy of the orbit equation according to the value of the eccentricity (with links to Kepler's laws of planetary motion, elliptic orbit, parabolic trajectory, and hyperbolic trajectory)
• Discussing the interrelations between e, a, E, h, etc.

These three things can fit easily into a single article. Zueignung (talk) 22:06, 1 September 2012 (UTC)

Positively hyperbolic trajectory treatment

a) This reader appreciates the presentation of the hyperbolic trajectory. He appreciates the mathematical presentation of the hyperbolic anomaly E as the "missing link" between time and position. Because this is nowhere else, not in any other language on Wikipedia. Maybe the math is not completely complete (I did not go thru it completely, but equations (40)ff are here), but it gives enough clues, and this for the first time. I just need it today... The term hyperbolic anomaly E is not in the text, but I picked it from this talk page

What is the source of equations (24) and (32)? I do not remember it seen anywhere else, it looks like original work. Equations (24) and (32) are convincing and easy. 77.59.144.68 (talk) 00:18, 4 September 2015 (UTC)

Somewhat ironic

Kepler studied elliptical orbits only. As far as I know, he knew nothing of parabolic or hyperbolic orbits, although we use similar orbital elements for all of them. Maybe we should make a note of it. Tfr000 (talk) 02:30, 27 November 2015 (UTC)

Question equation (20)

Your equation (20) relationship of x to E "eccentric anomaly" does not agree with the definition in the link above "eccentric anomaly". They do not include the little e eccentricity term. p.s. Thanks for all your work. I like all the equations when they flow.

I checked the definition of the eccentric anomaly and confirmed that the eccentricity does not belong in equation (20). I have corrected the equation. Argument of Periapsis (talk) 12:40, 3 December 2020 (UTC)

Notation incoherence

In the beginning, when the formulas are given, nu is used as the true anomaly, whereas in all the other drawings and in demonstrations, it is referred to as theta. I edited the notation, and I interverted figures showing types of orbit, for needs of clarity, because the latter one had theta represented on the figure. — Preceding unsigned comment added by Badidzetai (talk • contribs) 09:00, 18 June 2016 (UTC)

Simplified 2 body

I've a couple of minor quibbles with this. The first concerns the sentence "The difference between an irregular shape and a perfect sphere also diminishes with distances..." Logically, the difference is what it is, it doesn't "change" with distance. The difference in the gravitational effects of an irregular compared to a spherical object do indeed diminish with distance. Second, I see nowhere in the explanation of the math the admission that it assumes instantaneous force while in our Universe we know that the force takes time (based on distance and the speed of light) to affect other bodies, this is negligible for electric forces, but not genearlly negligible when objects several AU apart are concerned. Newton's Law of Universal Gravitation assumes, erroneously, that action at a distance happens instantaneously. Third, the sentence:"Even Jupiter's mass is less than the sun's by a factor of 1047,[3] which would constitute an error of 0.096% in the value of μ." is not only quite confusing, it is wrong in general. It is confusing because in the "simplified" treatment there is no way to accommodate a third body - how do we calculate a "0.096%" error? Not only that, but according to my calculations for the year 2038 A.D. the force component due to Jupiter constitutes almost 2.5% of the force Mars experiences on April 4th of that year. Suggesting that the other planets can be neglected and two body solutions are "good" to 0.1% is misleading at best, possibly cherry picking at worst (I admit, I picked the April date to maximize the Jovian contribution, but had only 2 years, 2037-2038, of JPL-Horizons data to work with). What's worse, there's virtually no discussion that none of the Solar System's planet's orbits are stable, orbit to orbit, they all precess and exhibit other non-osculatory (non-ideal) behavior, behavior which accumulates over time, making the 2-body approach of quite limited value. It might, imho, also be worth noting that due to the emission of energy via gravitational waves, no orbit is infinitely stable, all orbits will eventually decay.98.21.212.86 (talk) 22:34, 20 April 2017 (UTC)

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