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How massive does a satellite have to be in order to shift the earth's orbit around the sun? I am wondering how massive a satellite has to be in order to be able to shift the orbit of the earth by rotating around it. I heard it would be possible, but it would be a difficult task, but how massive does it have to be in order to be able to do that? Could you provide real-life example of something massive enough to do that, or perhaps fictional examples from science-fiction movies?

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    $\begingroup$ There isn’t some magic mass above which the orbit is affected. For a small mass there is a small effect. For a large mass there is a large effect. $\endgroup$ – G. Smith Sep 5 at 18:08
  • $\begingroup$ The center of mass of the Earth-Moon system (and everything in orbit) follows a smooth elliptical orbit around the sun. Nothing we do within the system will change that. $\endgroup$ – R.W. Bird Sep 5 at 18:36
  • $\begingroup$ The moon is an example. But also would any satellite if you allow for small enough effect. $\endgroup$ – Brick Sep 5 at 18:58
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How much the earth will displaced by the satellite depends not only on the mass of the satellite, but also on its distance from the earth.

According to Wikipedia "Barycenter - Two-body problem" the shift of the earth can be calculated by $$ r_1 = a \cdot \frac{m_2}{m_1 + m_2} = \frac{a}{1+\frac{m_1}{m_2}}$$ where $r_1$ is the shift of the earth,
$a$ is the distance between the centers of earth and satellite,
$m_1$ and $m_2$ are the masses of the earth and satellite.

Because of the large mass of the earth ($m_1 = 6\cdot 10^{24}\ \text{kg}$) the shift of the earth by a man-made satellite is unmeasurable small ($r_1 \approx 10^{-13}\ \text{m}$), even if its mass $m_2$ is many tons and its distance $a$ is many thousand km. You should insert some numbers into the formula above and do the calculation by yourself.

Another real-life example is the moon which is also a satellite orbiting the earth. According to the same Wikipedia article "Barycenter - Primary secondary examples" the shift of the earth due to the moon is $4670\ \text{km}$.

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