Final answer:
By applying Kepler's third law adapted with Newton's gravitational law, the mass of a planet can be determined by measuring the orbital period and distance of its orbiting moon.
Step-by-step explanation:
The mass of a planet can be measured by analyzing the velocity of an orbiting moon. We apply Newton's law of gravitation and Kepler’s third law which, when combined with Newton's adaptation, states that the cube of the semimajor axis of a planet's orbit (a³) is proportional to the square of the orbital period (P²) multiplied by the sum of the masses of the planet and the moon (M₁ + M₂). In the case of a moon orbiting a much more massive planet, the mass of the moon can be considered negligible, simplifying the equation to a³ ≈ M₁ × P², where M₁ is the mass of the planet. By knowing the distance and orbital period of the moon, the mass of the planet can be deduced.
Historically, this method was used to calculate the mass of celestial bodies such as the Sun and Jupiter. For objects within our solar system where one body (like the Sun) dominates gravitationally, the mass of the planet can be calculated with high precision. Astronomical data have allowed scientists to calculate the force of gravity and thus the mass of planets through observations of their moons' orbits.
By measuring the orbit and velocity of a moon and substituting these values into the adapted version of Kepler's third law, the mass of the planet is revealed. This understanding is critical for space missions, such as calculating the trajectory of satellites and understanding the orbital dynamics in the solar system.