Final answer:
A frame fixed to the Earth's center but not rotating with it is not an inertial frame because it rotates and accelerates relative to a frame at rest with distant stars. A frame at the center of the Sun is nearly inertial, yet not perfect, due to the Sun's own motion and minimal gravitational influences from other celestial bodies.
Step-by-step explanation:
If you are in a frame which is fixed to the Earth's center but not rotating with the Earth, it is not an inertial frame. The reason is that any frame that accelerates or rotates relative to a known inertial frame, such as a frame at rest relative to the distant stars, is not considered inertial. In detail, the Earth's rotation on its axis and revolution around the Sun causes acceleration relative to these distant stars. Furthermore, a non-rotating frame of reference placed at the center of the Sun is very nearly inertial but not exactly because the Sun itself is in motion within our galaxy and is also affected, albeit very slightly, by the gravitational pull of other bodies in the universe. This influences the motion of the Sun, and as a result, the frame would also be subject to this non-inertial influence, making it not a perfect inertial frame.