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Let us say that Earth and Venus are both moved 10 million additional miles from the sun. How would the synodic period of Venus appear to change for an observer on Earth? If 584 days is Venus' current synodic period, how many days would elapse if the planets were adjusted as mentioned?

Thank you so much for your help! Math to help me do it myself is most welcome!

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    $\begingroup$ Any thought you spend on the problem or researching it yourself? $\endgroup$
    – planetmaker
    Aug 18, 2022 at 20:01
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    $\begingroup$ planetmaker is correct. When you post a question you should say what you have already tried. That is so we don't repeat what you already know. Suppose you already knew the 1/p=1/e+1/s law and Kepler's law, that would make my answer completely useless. So when you post another question be sure to tell us what thoughts you have spent on the problem. See How to Ask $\endgroup$
    – James K
    Aug 19, 2022 at 16:18
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    $\begingroup$ James K., thank you for taking the time to play the mediator here. Planetmaker, if I misinterpreted the tone and intent of your comment, I apologize for getting hot. $\endgroup$
    – JM Yaden
    Aug 19, 2022 at 16:27
  • $\begingroup$ meta.stackexchange.com/help/how-to-ask says what I miss here "Have you thoroughly searched for an answer before asking your question? Sharing your research helps everyone. Tell us what you found and why it didn’t meet your needs. This demonstrates that you’ve taken the time to try to help yourself, it saves us from reiterating obvious answers, and above all, it helps you get a more specific and relevant answer!". Your question makes me believe you know kepler's laws and some math, so something is missing somewhere. But where exactly is your problem and where do you stand? $\endgroup$
    – planetmaker
    Aug 19, 2022 at 16:37
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    $\begingroup$ Planetmaker, I appreciate your response here, and both your and James K.'s willingness to give me a little education into how this forum operates. As per your question, James K. has provided me with exactly what I need to continue below, so I think I am good. I am here primarily to get assistance with the mathematics necessary to work out my inquiries. I assure you, I am not here to simply get answers, but to learn more math and astrophysics so that I can help myself in the future. Again, thank you for your patience with my "outburst" above, and I hope to see your comments on my future posts. $\endgroup$
    – JM Yaden
    Aug 19, 2022 at 17:27

1 Answer 1

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The formula relating the synodic period of Venus to its sidereal period is

$$\frac1P = \frac1E + \frac1S$$

Where $P$ is the sidereal period, $S$ is the synodic period, and $E$ is the orbital period of Earth (1 year).

Now both E and P can be calculated from Kepler's law, explicitly

$$P = 2\pi\sqrt{\frac{r_P^3}{\mu}}$$

Where $r_P$ is the semi-major-axis of Venus (distance to sun), and $\mu$ is the sun's gravitational parameter $\mu=1.33\times 10^{20}$ in SI units (m,kg,s). The same formula works for $E$

So now you can use this law to work out $P$ and $E$ for Venus and Earth in their new positions, and with that find $S$. Take care with units you can't use metres and seconds in one part of the formula and km - years in another.

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    $\begingroup$ Answered as a procedure for find the answer, rather than an answer. $\endgroup$
    – James K
    Aug 18, 2022 at 20:24
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    $\begingroup$ Thank you so much for taking the time to post this helpful response and including the math! Much appreciated! $\endgroup$
    – JM Yaden
    Aug 19, 2022 at 15:46
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    $\begingroup$ Make sure you use the distance in metres ie 125,000,000,000 then the period will be in seconds. Convert to days by dividing by 86400. Here is the calculation for the real venus $\endgroup$
    – James K
    Sep 1, 2022 at 19:23
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    $\begingroup$ No, the left hand side is positive! 1/225 is larger than 1/366 But you are making another mistake, You also said that you were going to "move earth out 10%" So you will need to calculate the new orbital period of Earth. It isn't going to be 366. $\endgroup$
    – James K
    Sep 1, 2022 at 20:29
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    $\begingroup$ Looks plausible. Feel free to write up an answer to your own question. $\endgroup$
    – James K
    Sep 1, 2022 at 21:32

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