Final answer:
To maintain its current orbital radius after the planet's mass is doubled, the moon's orbital speed must increase by a factor of the square root of 2, due to the gravitational force being directly proportional to the mass of the planet.
Step-by-step explanation:
If the mass of a planet were suddenly doubled, the moon's orbital speed would need to increase to maintain its current orbital radius. This is because the gravitational force holding the moon in orbit around the planet is directly proportional to the mass of the planet.
According to Newton's law of universal gravitation, the force of gravity between two masses is equal to the gravitational constant times the product of the two masses divided by the square of the distance between their centers. If the mass of the planet doubles, the gravitational force would double as well, requiring the orbital speed of the moon to increase by a factor of the square root of 2 to keep the same orbital radius.
For example, if the moon's orbital radius is initially 100,000 km its speed is 10,000 m/s, and the mass of the planet is then doubled, the moon's new orbital radius would be 200,000 km. To maintain the same orbital period, the moon's speed would need to be increased to approximately 14,142 m/s.