Can a rotating neutron star hold planets in its orbits? What would the orbits look like?
To begin with, yes, a neutron star would definitely be able to hold planets in orbit. In fact, if for some reason our Sun became a neutron star (with the same mass it has now), our orbit would be unaffected, at least to any accuracy we could measure. Neutron stars do indeed warp both space and time, but the distortion is vanishingly small at the distance that planets orbit. I don’t think you’d really notice the effects until you were maybe 1000 miles away from the neutron star, too close for a stable planetary orbit anyway. As for the second focus of Earth’s orbit, that is inside the Sun as well. The very center of the Sun is the location of one focus, but the other one is inside the Sun too, just off-center. Earth’s orbit is very nearly perfectly circular, so the foci are quite close together. Even if our orbit was very elliptical (like a comet), there wouldn’t have to be anything at the other focus. It’s really just a mathematical construction -nothing has to be there at all.
