Say for example that at the center of our solar system you replaced the sun, with something that wasn't a star, lets say that it was electrified and still produced the same amount of heat, would that planetary system be stable. If so that what else could a sun be replaced with, a water planet? a Gas giant? An Ice planet? A planet made of Uranium? what else could the center of a planetary system be replaced with?
If we assume all of the rest of the planets are neutral in charge, then only the mass of the sun matters for the Solar system. The gravity between the star and planets and other objects forms the main force that maintains the rotation of smaller objects around the sun and be stable for a long time. No matter what you fill in the sun, the mass of the sun should be far larger than other objects. Otherwise, objects will be rotating around some mass center outside of the sun which may cause collapse under some complicated movements. On the other hand, if the mass of the sun is very large as the current sun, then giant cloud of gas and ice may clash into a denser body because of the gravity. In the end, the sun will be a body with high pressure and high temperature like the usual stars you can talk.
If we assume all the rest of planets can be charged and ionized to orbit a electrified sun mainly due to the electrical force among them, then the planets themselves will have to compensate the repulsing force due to the charges of particles forming the planets. The most likely way to overcome this force is the gravity which is formed by the mass of the planet. We know that gravity is a relatively weak force compared with electrical forces, and it is unlikely to hold the planet balanced with gravity and electrical force -- unless its so densed as a black hole. However, it is possible to replace the sun with a charged black hole which has a large gravity field to maintain a large "solar" system with planets for a relatively long time no matter whether the planets are charged or not.
orbiting, graviting, means only mass maters. It could be gaz, diamon, a bag of cockroach, or even a black hole, it would not change gravitational behaviors that orbits are.
( NB: yes, black hole too. Nothing different occurs in gravity as well when a star turn into blackhole. And it was attracting distant matter before exactly as well.)