Talk:Circular orbit

WTF

I don't know if I'm just retarded and I'm missing something complete obvious, but it seems to me that the speed is given by $v={\sqrt {2\mu \over {r}}}$ and not $v={\sqrt {\mu \over {r}}}$. I mean, the kinetic energy is $mv^{2} \over {2}$ and the potential energy is :$-GMm \over {r}$. On the orbit, total energy is minimized (E=0).

Therefore :$v={\sqrt {2\mu \over {r}}}$ .

This error carries on to the period. $T=\pi {\sqrt {2r^{3} \over {\mu }}}$ not $T=2\pi {\sqrt {r^{3} \over {\mu }}}$

And also, if we're talking about circular orbits, might as well give the real orbital energy conservation equation ${v^{2} \over {2}}-{\mu \over {r}}=0$ . I have no clue why it was equal to something other than 0 before.

It's as if someone went through this and purposely screwed things up. Anyway I cleaned things up.

Headbomb 03:22, 17 May 2007 (UTC)

The energy is not zero. The formulas were correct.--Patrick 12:53, 22 June 2007 (UTC)

Equation of Motion, delta V and Virial theorem

The equation of motion became redundant, and I really don't see what mentionning the Virial theorem or the delta V brought to the topic. Headbomb 03:33, 17 May 2007 (UTC)

I restored it.--Patrick 12:58, 22 June 2007 (UTC)

Orbital velocity in general relativity

Is the orbital velocity of circular orbits in general relativity exactly the same as in the Newtonian case or are there any subtle differences? Agge1000 12:57, 11 November 2007 (UTC)

I'm aware this is an old question, but it looks bad unanswered. Yes, there are differences and I attempted to describe how to calculate the speed in GR. Ebvalaim (talk) 23:34, 21 February 2012 (UTC)

Why the gamma factor has been introduced? And how? I mean, as far I know the gamma factor is a construct of and for special relativity. I don't understand why the scalar product of two different velocity vectors in any metric is equal to the gamma factor. In other words, why the orbital velocity has been defined through the gamma factor and, for example in spherical coordinates, not simply as v(r) = r d\phi/dr? In that case I obtain a different result. What is the difference? — Preceding unsigned comment added by Dundanox (talk • contribs) 18:54, 16 February 2019 (UTC)

I think you mean v = r d\phi/dt? The problem here is that dt doesn't actually measure time relative to the local observer. It's just an arbitrary coordinate. The actual time difference is ${\sqrt {g_{tt}}}dt$ in Schwarzschild metric. If you take this into account, ${\frac {1}{\sqrt {g_{tt}}}}r{\frac {d\phi }{dt}}$ is actually the same result. Ebvalaim (talk) 12:41, 13 November 2020 (UTC)

I don't know what textbook was employed here, but this formula for the orbital speed doesn't make a whole lot of sense to me. When you compute the angular momentum for the circular orbit and minimize this angular momentum, you should get the ISCO radius of 3r_s, no? The formula v=(GM/(r-r_s))^(1/2) doesn't appear to give the correct ISCO radius when you follow this method. You do get the correct ISCO radius if you use the standard formula v=(GM/r)^(1/2), however. — Preceding unsigned comment added by 213.10.18.209 (talk) 19:15, 17 December 2019 (UTC)

How do you calculate the angular momentum? The conserved quantity is $r^{2}{\dot {\phi }}$, which isn't equivalent to $rv$ (because of the issue with time mentioned above in my answer to Dundanox). Ebvalaim (talk) 12:41, 13 November 2020 (UTC)

More details please

This is the first time I'm looking at these equations, can someone clarify what the two different R's are in the equation: $\mathbf {a} =-{\frac {v^{2}}{r}}{\frac {\mathbf {r} }{r}}=-\omega ^{2}\mathbf {r}$ — Preceding unsigned comment added by Nhilton (talk • contribs) 20:07, 27 September 2012 (UTC)

As far as I can tell, $\mathbf {r}$ in all cases is radius. I do not understand why it is presented as it is, and further, the article Circular_motion#Uniform_circular_motion presents the formula as $a\,={\frac {v^{2}}{r}}\,={\omega ^{2}}{r}$, and for the sake of clarity, I am going to replace the former with the later. T.Randall.Scales (talk) 19:43, 2 January 2015 (UTC)

Late to the party, but the bold r is the position relative to the center of the massive body as a vector. The regular r is the magnitude of this vector, which is also the distance from the center, or the radius. Ebvalaim (talk) 13:54, 13 November 2020 (UTC)

standard gravitational parameter does not agree with link.

The definition of the standard gravitational parameter from here:

http://en.wikipedia.org/wiki/Standard_gravitational_parameter   

does not agree with the text from here:

http://en.wikipedia.org/wiki/Circular_orbit#Velocity  — Preceding unsigned comment added by 98.125.78.211 (talk) 09:05, 19 February 2015 (UTC)