If it just went away very cleanly with nothing left behind, a great science fiction plot by the way, the Earth would detect this loss 8.5 minutes later as the loss of the solar gravitational field ( which travels at the speed of light). The Earth would then behave as a free body unfettered by the solar gravitational field. Its speed would remain fixed at its tangential orbital velocity at the moment the Sun's field went away, and its trajectory would be in a roughly straight line in the direction of the sky it was moving at that instant. Only the Milky Way's gravitational field would now have much affect. What you would see is that daylight would never come and you would have a perpetual nighttime everywhere. The stars would still move in their 24 hour cycle, and the precession of the equinoxes would still occur. Over thousands of years Earth astronomers would see familiar constellations being left behind and the bright stars shift in the sky.