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Yesterday I was reviewing some calculations using Kepler's laws and when working the third law that relates orbital periods and distances, it is very normal that, given the parameters of some object and some parameter of the other, through this law, we can find the missing parameter. For example, given the period and distance of the earth from the sun, we could find any of these two parameters of another object with respect to the sun according to the relationship:

$T^2 /a^3 = C$

Which has always defined that, for any planet, the square of its orbital period is directly proportional to the cube of the length of the semimajor axis of its elliptical orbit. We have always been completely sure of this. Where, $T$ is the orbital period (time it takes to go around the Sun), $a$ the average distance of the planet with the Sun and $C$ the constant of proportionality. These laws apply to other astronomical bodies that are in mutual gravitational influence, such as the system formed by the Earth and the sun. My question is, if the relationship between period and distance is a constant, why do not I get that same quotient for the earth when I do it with jupiter, for example?

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closed as off-topic by Bill N, rob Apr 18 at 20:04

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    $\begingroup$ I'm voting to close this question as off-topic because it is based on incorrect arithmetic, not a real conceptual difficulty. $\endgroup$ – Bill N Apr 18 at 17:42
  • $\begingroup$ I agree. It was an error to post this $\endgroup$ – jormansandoval Apr 18 at 20:20
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Are you sure you've calculated and used the correct values?

I'm going to use units for the semi-major axis and the orbital period that are very convenient for Earth: Astronomical Units (AU) and years.

Earth's semi-major axis is 1 AU and its orbital period is one year (by definition!) $$c=T^2/a^3 = (1 year)^2/(1 AU)^3 = 1 year^2/AU^3$$ Jupiter's semi-major axis is 5.2044 AU, and its orbital period is 11.862 years. $$c=T^2/a^3 = (11.862 year)^2/(5.2044 AU)^3 = 0.99817 year^2/AU^3 \approx 1 year^2/AU^3$$

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  • $\begingroup$ thank you!!!!! Attentions like these are what help us to be better every day $\endgroup$ – jormansandoval Apr 18 at 16:35

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