<p>I want to prove Kepler's Third Law via Vector Analysis of motion. I know of other derivations, but want to know this as I think it will be interesting. I have been unable to prove it. But I know of proofs of 1st and 2nd law via Vector Analysis. Can anyone please tell me how to proceed on it, please? One equation can be
<span class="math-container">\begin{equation}
\frac{1}{2} \times \vec r × \frac{d \vec r}{dt}  = constant. 
\end{equation}</span>
Please, can anyone help me go further?</p>
<p>I tried using <span class="math-container">$\vec r= acosωt\hat i +bsinωt\hat j$</span>,</p>
<p>So the areal velocity came <span class="math-container">$\frac{1}{2} abω \hat k$</span>. But then I tried integrating areal velocity over time thinking it would give total area, but the relation of 3rd law didn't come. So can anyone please help?</p>

CC BY-SA 4.0
2021-01-13T10:59:24.760Z
Any help please..
Guji2203
CC BY-SA 4.0
2021-01-13T12:04:35.330Z
You may find this helpful: http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/KeplersLaws.htm
PM 2Ring
CC BY-SA 4.0
2021-01-13T12:36:43.790Z
I'm not sure what you mean, because unless you assume an inverse square law, Kepler's third law isn't correct.  IIRC you can deduce an inverse square law from assumptions of "closed orbits" and "centripetal accelleration which tends to zero at infinite distance".  But it is easy to show that (for example) a Hookes Law orbit doesn't obey Kepler's third law.
James K
CC BY-SA 4.0
2021-01-13T13:07:12.967Z
I have taken the inverse square law to be true, then only could one  derive the 1st and 2nd laws. I have seen proofs of Kepler's laws via vector in some books like Stewart and Murray Spiegel schaum, but only the first two were present . It says that third can also be done in vectorial way. So I started to try on it but the correct result wasn't coming so i wanted to know how to do it. I know about the other derivation but the vectorial way fascinated me so i wanted to know the proof of 3rd law also in that way.
Guji2203
CC BY-SA 4.0
2021-01-13T13:08:53.037Z
Sorry for my illogical comment i just wanted to mean that i wanted the derivation which does a bit vector analysis of the orbit with the help of inverse square law. Thanks for pointing the mistake.
Guji2203
CC BY-SA 4.0
2021-01-13T14:25:32.620Z
I have edited the question also and given my attempt in it. Can anyone help please.
Guji2203
CC BY-SA 4.0
2021-01-13T19:37:11.023Z
Your parametrization of $\vec{r}$ is not the orbit of a planet. It gives an ellipse all right, but the center of that ellipse is at the point of symmetry rather than at one of the focal points.
Jyrki Lahtonen
CC BY-SA 4.0
2021-01-14T00:03:41.150Z
According to Wikipedia, “A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example, pages 161–164 of Meriam, J.L. (1971) [1966]. Dynamics, 2nd ed. New York: John Wiley. ISBN 978-0-471-59601-1.” https://archive.org/details/dynamics00meri
Pierre Paquette
<mathematics><kepler>
Vector analysis of Kepler's Third Law
Guji2203
# Vector analysis of Kepler's Third Law

(Asked by: username_0 on 2021-01-13)

I want to prove Kepler's Third Law via Vector Analysis of motion. I know of other derivations, but want to know this as I think it will be interesting. I have been unable to prove it. But I know of proofs of 1st and 2nd law via Vector Analysis. Can anyone please tell me how to proceed on it, please? One equation can be
\begin{equation}
\frac{1}{2} \times \vec r × \frac{d \vec r}{dt} = constant.
\end{equation}
Please, can anyone help me go further?

I tried using $\vec r= acosωt\hat i +bsinωt\hat j$,

So the areal velocity came $\frac{1}{2} abω \hat k$. But then I tried integrating areal velocity over time thinking it would give total area, but the relation of 3rd law didn't come. So can anyone please help?

**Tags:** mathematics, kepler

> **Comments:**
> **username_0**: Any help please..  
> **username_1**: You may find this helpful: http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/KeplersLaws.htm  
> **username_2**: I'm not sure what you mean, because unless you assume an inverse square law, Kepler's third law isn't correct.  IIRC you can deduce an inverse square law from assumptions of "closed orbits" and "centripetal accelleration which tends to zero at infinite distance".  But it is easy to show that (for example) a Hookes Law orbit doesn't obey Kepler's third law.  
> **username_0**: I have taken the inverse square law to be true, then only could one  derive the 1st and 2nd laws. I have seen proofs of Kepler's laws via vector in some books like Stewart and Murray Spiegel schaum, but only the first two were present . It says that third can also be done in vectorial way. So I started to try on it but the correct result wasn't coming so i wanted to know how to do it. I know about the other derivation but the vectorial way fascinated me so i wanted to know the proof of 3rd law also in that way.  
> **username_0**: Sorry for my illogical comment i just wanted to mean that i wanted the derivation which does a bit vector analysis of the orbit with the help of inverse square law. Thanks for pointing the mistake.  
> **username_0**: I have edited the question also and given my attempt in it. Can anyone help please.  
> **username_3**: Your parametrization of $\vec{r}$ is not the orbit of a planet. It gives an ellipse all right, but the center of that ellipse is at the point of symmetry rather than at one of the focal points.  
> **username_4**: According to Wikipedia, “A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example, pages 161–164 of Meriam, J.L. (1971) [1966]. Dynamics, 2nd ed. New York: John Wiley. ISBN 978-0-471-59601-1.” https://archive.org/details/dynamics00meri  

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## 0 answers

