If the Earth's core were to mysteriously die and all the water on the planet evaporated, it would have significant effects on the Moon's orbit. However, it's important to note that such a scenario is purely hypothetical and not scientifically feasible. The Earth's core plays a crucial role in generating the planet's magnetic field and maintaining its internal processes, including plate tectonics and volcanic activity. Without a functioning core, Earth would undergo drastic changes, which would likely make the existence of life unsustainable.
Regarding the Moon's orbit, it would be affected by the loss of the Earth's water and the resulting changes in mass distribution. The redistribution of mass caused by the evaporation of all water would alter the Earth's gravitational field, leading to perturbations in the Moon's orbit. These perturbations could manifest as changes in the Moon's distance from the Earth, its orbital speed, and even its orbital inclination.
However, predicting the exact consequences of such a scenario is challenging due to the complex nature of celestial mechanics and the multitude of factors that influence the Moon's orbit. It's safe to say that the Moon's orbit would be significantly affected, but the specific changes would depend on the precise conditions and distribution of mass resulting from the loss of Earth's water.
Again, it's important to emphasize that this hypothetical scenario is highly unlikely to occur naturally, and the Earth's core dying and all water evaporating would have catastrophic consequences for life on the planet.