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The formula for calculating the center of mass is

$$ r_{center} = \frac{m_1 \cdot r_1 + m_2 \cdot r_2}{m_1+m_2} $$

Why can't I use it to calculate the barycentre of two planets?

I understand how to use ratios to approach this problem.

$$ \frac{r_1}{r_2} = \frac{m_1}{m_2} $$ $$ r_1 = \frac{m_2}{m_1+m_2}\cdot (d_{Sun->Earth}) $$

However, I do not understand why I can not use the $r_{center}$ formula.

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  • $\begingroup$ You can use it. $\endgroup$
    – mikhailcazi
    Commented Aug 7, 2013 at 12:46

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You can use the center of mass formula.

Set the origin of your coordinate system at the center of the Earth, then $\vec{r}_1 = \vec{0}$ and $\vec{r}_2 = d$ and $$r_{center} = \frac{m_1r_1+m_2r_2}{m_1+m_2} = \frac{m_2}{m_1+m_2} \cdot d$$ as you have as well.

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  • $\begingroup$ That makes sense - thank you a lot! Could you also help me with this question: If I now want to calculate the velocity of the sun orbiting arount r_center. Why did they use the circumference of the earth * pi / circulation period of the earth? Is it the same for the sun? I don't get it. $\endgroup$
    – libjup
    Commented Aug 7, 2013 at 8:09
  • $\begingroup$ That's an application of $v=\frac{s}{t}$, but that's the velocity on the surface of a rotating sphere not around $r_{center}$. You might want to consider asking a new question. $\endgroup$
    – pfnuesel
    Commented Aug 7, 2013 at 8:28
  • $\begingroup$ Thank you very much again, pfnuesel; +1 and checked as answer $\endgroup$
    – libjup
    Commented Aug 7, 2013 at 13:55

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