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I am trying to figure out how often Neptune and Uranus align. Internet sources says that Neptune's orbit around the sun is 165 years, and that Uranus' is 84 years. It also claims that the two planets align every 170 years. Is 170 years really accurate? It sounds too long.

I am therefore searching for a planet alignment calculator. One that is non-astrological. Are they to be found somewhere on the net, or in an app?

Alternatively, is there a simple formula one can use to calculate this with paper and pen? Or, alternatively a brain-twist way of getting to the right answer.

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  • $\begingroup$ See en.wikipedia.org/wiki/Orbital_period#Synodic_period There's a formula in that article. $\endgroup$
    – PM 2Ring
    Commented Nov 7, 2021 at 1:38
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    $\begingroup$ You are asking multiple questions at once. Try to stick with one question per post. $\endgroup$
    – Nilay Ghosh
    Commented Nov 7, 2021 at 2:25
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    $\begingroup$ @PM 2Ring. That formula is too much for my limited math knowledge. I think I prefer a calculator. The internet is full of various calculators. Why wouldn’t there be one for this too. A straight answer to my problem would otherwise be helpful. $\endgroup$
    – Constantthin
    Commented Nov 7, 2021 at 2:38
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    $\begingroup$ Sorry. That's one of the simplest formulas in celestial mechanics. It can also be written as $\frac{p_1p_2}{p_2-p_1}$. Here's a synodic period calculator $\endgroup$
    – PM 2Ring
    Commented Nov 7, 2021 at 9:07
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    $\begingroup$ It's mathematically identical to Nilay's formula. Try the calculator I just wrote, which basically use the same formula (adapted slightly so that it doesn't matter whether $p_1$ is larger or smaller than $p_2$). $\endgroup$
    – PM 2Ring
    Commented Nov 7, 2021 at 9:16

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If you go to Wikipedia article of conjunction, there is a formula given for the average time between two conjunctions between a planet pair in siderial years.

$$\mathrm{p_{conjunction} = \frac{1}{\frac{1}{p_{1}}-\frac{1}{p_{2}}}}$$

where p1 and p2 are the orbital time periods of two planets respectively.

The orbital time periods of Uranus and Neptune are 84.012 and 164.782 years respectively. Plugging the numbers, you get the value of 171.396 years.

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    $\begingroup$ Thx. Put that formula into a scientific calculator just now and got the same answer as you did. $\endgroup$
    – Constantthin
    Commented Nov 7, 2021 at 3:42

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