Can Vera Rubin's findings be explained by a distribution of charge? Vera Rubin found that the rotational velocity of galaxies is much greater than expected at greater distances from the center. Gravity from an invisible mass is assumed to account for this measurement. Can we not account for such a force by a distribution of charge? For example consider the sun and the inner planets positively charged. Perhaps there is even a gradient of decreasing charge the further away from the center. And now consider that the outer planets are oppositely charged. Even perhaps increasing charge as we travel towards the outer edge of the galaxy. Since each solar system is 'insulated' from every other by empty space, the charge separation is maintained. Since solar systems are small compared to galaxies, a uniform charge on all parts of the solar system would not be detectable.
 A: Sure, an overall electric charge distribution in a galaxy in which the core and the outer reaches have opposite charges would explain the faster-than-expected speeds of the outer reaches. The extra force provided by the electric attraction would keep the fast stars in orbit around the galaxy.
There is a large problem with this idea: how is this charge separation maintained?
The electric force is immensely powerful compared to the gravitational force. The electric force between two protons is $10^{34}$ (10 billion trillion trillion) times the strength of the gravitational force between them. To put this in more concrete terms, if you added 600,000 kg (660 tons or 3 blue whales) of pure protons to the Earth and another 600,000 kg of pure protons to the Moon, that would completely negate the gravitational attraction between them. If we used electrons instead, it would only take 300 kg each, or the weight of 4 average adult humans.
Why is this a problem? The Earth is not a closed system. Particles from the sun, stars, and other sources are constantly raining down on Earth. It should be a relatively simple matter to accumulate a large amount of electric charge since so little mass is required. But, we don't see the effects of electric attraction or repulsion. The Moon stays in a very stable orbit around the Earth that can be accurately calculated assuming gravity acts alone.
This is because electric forces are both repulsive and attractive. This leads to most everything in the universe very quickly becoming neutrally charged. For example, solar wind from our Sun is a stream of charged and uncharged particles. If Earth had an overall positive charge, it would repel the positive charges from the Sun and attract the negative charges. The negative charges that the Earth attracts would quickly neutralize the initial positive charge. From then on, Earth would equally absorb positive and negative charges, remaining neutral. Any excess charge gained would be quickly neutralized by attracting the opposite charge.
This goes for galaxies as well. If a galaxy had an overall positive charge in the core and an overall negative charge at the outer reaches, the particles emitted by the stars in the core would have an overall positive electric charge that would stream outwards to neutralize the outer stars. The same would happen with the negative charges in the outer stars.
Because of the strength of the electric force, there is nothing that can cause excess charge to remain bound to a star or any other body. Electric charges will always move about until the overall charge density is uniform.
A: Unfortunately (or fortunately), some back-of-the-envelope calculations show it is unlikely that flat rotation curves are caused by charge separation. The acceleration of the Sun when moving around the center of the galaxy is around $1.6\cdot 10^{-10}\mathrm{m/s^2}$, so, lets say, the acceleration caused by forces from electric field is of the order $a=10^{-10}\mathrm{m/s^2}$. The charge of the Sun is about $Q=77 \textrm{C}$, according to this paper, and therefore the electric field needed to give the Sun needed acceleration would be
$$
E=\frac{M a}{Q}\approx 2\cdot 10^{18} \textrm{V/m},
$$
which is impossibly high (for comparison, an electric field needed to rip an electron from a hydrogen atom is about $10^{11} \textrm{V/m}$)
