From your intergalactic survey base, you observe a moon in a circular orbit about a faraway planet. You know the distance to the planet/moon system, and you determine the maximum angle of separation between the two and the period of the moon's orbit. Assuming that the moon is much less massive than the planet, explain how you can determine the mass of the planet.
G•M•m/r² = m•v²/r
v =2•π•r/T
G•M•m•/r²= 4• π²•m•r/T²
M = 4•π²•r³/G•T²
Since r and R are very large r = R•θ,
where R is the distance to the system and the θ is an angle separation
To determine the mass of the planet, you can use the combination of the distance to the planet/moon system, the maximum angle of separation between the two, and the period of the moon's orbit. Here's how you can go about it:
1. Measure the distance to the planet/moon system: This can be done using various methods, such as parallax measurements or radar ranging. By knowing the distance, you have a crucial parameter for calculating the mass of the planet.
2. Measure the maximum angle of separation: From your intergalactic survey base, you can observe the moon's orbit around the planet and measure the maximum angle of separation between the two. This angle is typically known as the angular diameter or angular separation.
3. Determine the period of the moon's orbit: By tracking the moon's motion over a period of time, you can determine the time it takes for the moon to complete one orbit around the planet. This is known as the period of the moon's orbit and is usually measured in days, months, or years.
Now, with these measurements, you can use the following formula, derived from Kepler's Third Law, to calculate the mass of the planet:
M = (4π² × R³) / (G × T²)
Where:
- M is the mass of the planet
- R is the distance between the center of the planet and the center of the moon's orbit
- G is the gravitational constant (6.67430 × 10^-11 m³ kg^-1 s^-2)
- T is the period of the moon's orbit
By plugging in the values of R and T that you've measured, along with the known value of G, you can compute the mass of the planet.
It's important to note that this method assumes that the moon is much less massive than the planet, which allows us to neglect the moon's gravitational effect on the planet's motion. If the moon's mass is significant, it would need to be considered in the calculations using more complex equations.