# How massive does a satellite have to be in order to shift the earth's orbit around the sun?

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?

• 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. – G. Smith Sep 5 at 18:08
• 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. – R.W. Bird Sep 5 at 18:36
• The moon is an example. But also would any satellite if you allow for small enough effect. – Brick Sep 5 at 18:58

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 $$4670\ \text{km}$$.