If the Moon were to suddenly turn into a black hole, its size would determine whether it would engulf the Earth or not. To avoid engulfing the Earth, the black hole's event horizon—the boundary beyond which nothing can escape its gravitational pull—would need to be smaller than the Moon's original radius.
The event horizon of a black hole is determined by its mass. The radius of the event horizon, known as the Schwarzschild radius, can be calculated using the formula:
r = 2GM/c^2,
where r is the Schwarzschild radius, G is the gravitational constant, M is the mass of the black hole, and c is the speed of light.
The Moon has a mass of approximately 7.34 × 10^22 kilograms. If we assume that the Moon's entire mass is compressed into a black hole, we can calculate the corresponding Schwarzschild radius. Plugging the values into the formula:
r = 2 × (6.674 × 10^(-11) m^3 kg^(-1) s^(-2)) × (7.34 × 10^22 kg) / (3 × 10^8 m/s)^2,
r ≈ 1.09 × 10^-2 meters or 10.9 millimeters.
Therefore, for the Moon not to engulf the Earth as a black hole, it would need to have a Schwarzschild radius smaller than 10.9 millimeters. In other words, it would need to be smaller than this size to prevent its gravitational influence from extending beyond the Moon's original radius.