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Would dark matter have a similar effect on the Solar System as it does on the Milky Way? I'd imagine it could help hold together some of the Solar System bodies or at least affect their motion, right? Maybe to a lesser extent, but would it be significant?

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The dark matter model that is used to explain the "missing mass" problem relating to our Galactic rotation curve, consists of a pseudo-spherical distribution that is much more extended than the visible stars and gas. Even though this "halo" contains more than ten times the mass of the visible matter, when you work out what it's density should be in the solar neighbourhood, it turns out to be around 100 times less than the "normal" interplanetary medium.

This means, that inside the Earth's orbit, there could be as much as $10^{10}$ kg of dark matter. This sounds a lot and in principle, yes, this would speed up the orbits of planets in a way that depended on their distance from the Sun. However, the predicted effects of this are still about 6 orders of magnitude below the current precision of measurement.

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