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As there is no influence or gravity in intergalactic space, e.g. like pull force from the Sun, can a small force make a massive stationary object like earth move?

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The first part is wrong because there is a gravitational field from all the matter, dark matter and dark energy everywhere. This doesn't really matter for the purposes of your question, which is really best dealt with in a more simplified Newtonian model.

In the case of two objects Newtonian physics tells you there is an equal and opposite force. So any force one body exerts on another, it gets the same back, so the net force on the system as a whole is zero.

So if you have an apple and a large planet, both experience the same force in opposite directions. The apple is relatively small so it's acceleration is relatively large, while the planet is relatively big so it's acceleration in very small.

All of this is because acceleration is force divided by mass. So even though they experience the same magnitude of force, the difference in mass decides how much acceleration they experience.

Another way of seeing this is the notion of Barycenter, which is explained on the Wikipedia page I linked to. Briefly, it means show that in a two body system, both bodies move around a common center, but the larger body moves less because the force divided by it's mass is smaller than that for the smaller body.

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