Talk:Two-body problem

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After the section "Change of variables", the solving of the equation is not pursued. A detailed discussion on the subject may be found at Eric Weisstein's world of physics

1 Huh?
2 Yuh huh! :D
3 Deletion of "Newtonian gravity" section
4 Deletion of "General Relativistic Gravity" section

[edit] Huh?

I followed this link from another page, and from the first sentence, I was lost. I'm not a physicist or a scientist of any sort. Can someone who understands this stuff please write a layman's description for the first sentence? (I've noticed this happens a lot with physics-related articles, but this is the first time I've mentioned it.)

Yeah, I know :( I've been meaning to fix this article, but I've been busy with a few others lately. Why don't you check back in a few days? Thanks for being patient. :) WillowW 16:44, 13 June 2006 (UTC)

P.S. If you find other physics articles that are too inscrutable, would you please let me know on my talk page? You shouldn't be shy about asking for more intelligibility. Sometimes it can't be helped; some topics are just hard to understand because they're so foreign to everyday life and also rely on lots of careful little definitions. But I'm sure that there are many others that need only a little help to make them clear. You could try some of the Physics articles mentioned on my User page; how about centripetal force? Thanks for your help! WillowW 16:44, 13 June 2006 (UTC)

[edit] Yuh huh! :D

Hi, rather than waiting, I decided to try my best at fixing the article now. Please let me know if you like it as is, or if there are still confusing parts. I didn't tinker with the stuff at the very end; maybe someone else can decide whether it's worth keeping? WillowW 19:43, 13 June 2006 (UTC)

[edit] Deletion of "Newtonian gravity" section

I apologize for having deleted the section on "Newtonian gravity", but I hope that the following will explain my reasoning, and that everyone will agree that the article is improved.

The author seems to have intended this section mainly as a collection of rules of thumb for working with two-body problems. However, given that the general formulae are given in the earlier text, such rules of thumb do not seem to be needed. Moreover, it is not clear how these particular rules of thumb are useful or explanatory, and they do not seem to be strongly organized. Finally, the topic of "two-body problem" is independent of the particular interaction potential between the two bodies; hence, rules of thumb specific to gravitation or to the Earth-Sun (or a hypothetical Sun-Sun) system seem not general enough for this article. Perhaps they would do better under Kepler's laws of planetary motion?

Therefore, since the main points of this section were covered elsewhere (e.g., in the general formulae preceding this section), it seemed to help the flow of the article to eliminate this section and go straight to the explanatory examples. Thanks for your patience with me, Willow 07:22, 6 August 2006 (UTC)

Some text from the first half of what you deleted is replicated at standard gravitational parameter#Two bodies orbiting each other, so I added the a^3/T^2 = M formula there, and changed the link at Alpha Centauri#System components. -- Paddu 16:16, 27 August 2006 (UTC)

I reproduce the deleted section here, in case anyone would like to glean from it

Applying the gravitational formula we get that the position of the first body with respect to the second is governed by the same differential equation as the position of a very small body orbiting a body with a mass equal to the sum of the two masses, because

m 1 m 2 / μ = m 1 + m 2

Assume:

the vector r is the position of one body relative to the other (above called x)
r, v, the semi-major axis a, and the specific relative angular momentum h are defined accordingly (hence r is the distance)
$μ = G (m 1 + m 2)$ , the standard gravitational parameter (the sum of those for each mass)

where:

$m 1$ and $m 2$ are the masses of the two bodies.

Then:

the orbit equation applies; recalling that the positions of the bodies are $m 2 / (m 1 + m 2)$ and $- m 1 / (m 1 + m 2)$ times r, respectively, we see that the two bodies' orbits are similar conic sections; the same ratios apply for the velocities, and, without the minus, for the angular momentum with respect to the barycenter and for the kinetic energies
for circular orbits $r v 2 = r 3 ω 2 = 4π 2 r 3 / T 2 = μ$
for elliptic orbits: $4π 2 a 3 / T 2 = μ$ (with a expressed in AU and T in years, and with M the total mass relative to that of the Sun, we get $a 3 / T 2 = M$ )
for parabolic trajectories $r v 2$ is constant and equal to $2μ$
h is the total angular momentum divided by the reduced mass
the specific orbital energy formulas apply, with specific potential and kinetic energy and their sum taken as the totals for the system, divided by the reduced mass; the kinetic energy of the smaller body is larger; the potential energy of the whole system is equal to the potential energy of one body with respect to the other, i.e. minus the energy needed to escape the other if the other is kept in a fixed position; this should not be confused with the smaller amount of energy one body needs to escape, if the other body moves away also, in the opposite direction: in that case the total energy the two need to escape each other is the same as the aforementioned amount; the conservation of energy for each mass means that an increase of kinetic energy is accompanied by a decrease of potential energy, which is for each mass the inner product of the force and the change in position relative to the barycenter, not relative to the other mass
for elliptic and hyperbolic orbits $μ$ is twice the semi-major axis times the absolute value of the specific orbital energy

For example, consider two bodies like the Sun orbiting each other:

the reduced mass is one half of the mass of one Sun (one quarter of the total mass)
at a distance of 1 AU: the orbital period is ${1\over 2} \sqrt{2}$ year, the same as the orbital period of the Earth would be if the Sun would have twice its actual mass; the total energy per kg reduced mass (90 MJ/kg) is twice that of the Earth-Sun system (45 MJ/kg); the total energy per kg total mass (22.5 MJ/kg) is one half of the total energy per kg Earth mass in the Earth-Sun system (45 MJ/kg)
at a distance of 2 AU (each following an orbit like that of the Earth around the Sun): the orbital period is 2 years, the same as the orbital period of the Earth would be if the Sun would have one quarter of its actual mass
at a distance of $\sqrt[3]{2} \approx 1.26$ AU: the orbital period is 1 year, the same as the orbital period of the Earth around the Sun

Similarly, a second Earth at a distance from the Earth equal to $\sqrt[3]{2}$ times the usual distance of geosynchronous orbits would be geosynchronous.

[edit] Deletion of "General Relativistic Gravity" section

My reasons for deleting this section are slightly different. Here, the author points out an interesting correction of general relativity to the planetary motion expected from solving a classical two-body problem of gravity.

This section, although covering a fascinating topic, does not seem at home in this article, since it is specific for the relativistic gravitational two-body problem, and not the general two-body problem. Therefore, it would seem to belong rather under the general theory of relativity, or under a more specific article about the motion of planets. It would be possible to cover the classical and GR gravitational two-body problems as illustrative examples in this article, but that would significantly increase the article's length. What does everyone think -- is it worth it? Willow 07:38, 6 August 2006 (UTC)

Here's the deleted text again, in case anyone wishes to glean from it

In the general theory of relativity gravity behaves somewhat differently, but, to a first approximation for weak fields, the effect is to slightly strengthen the gravity force at small separations. Kepler's First Law is modified so that the orbit is a precessing ellipse, its major and minor axes rotating slowly in the same sense as the oribital motion. The law of conservation of angular momentum still applies (Kepler's Second Law). Kepler's Third Law would in principle be altered slightly, but in practice, the only way to measure the sum of the masses is by applying that Law as it stands, so there is effectively no change. These results were first obtained approximately by Einstein, and the rigorous two body problem was later solved by Howard Percy Robertson.

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