What is the semi-major axis of an object that is in 2:1 mean motion resonance with Earth
To determine the semi-major axis of an object that is in a 2:1 mean motion resonance with Earth, we first need to understand what mean motion resonance means. In celestial mechanics, mean motion refers to the average angular speed of an object in its orbit around a celestial body.
In a 2:1 mean motion resonance, the object completes two orbits around its parent body for every one orbit that Earth completes around the same parent body.
Now, let's consider the orbital mechanics. The period of an object's orbit depends on its semi-major axis— a measure of the size of the ellipse that describes its orbit.
To find the semi-major axis of the object in a 2:1 mean motion resonance with Earth, we can use the following equation:
(Period of Earth)^2 / (Period of object)^2 = (Semi-major axis of Earth)^3 / (Semi-major axis of object)^3
Since the object is in a 2:1 resonance, the period of the object will be half of the period of Earth. For convenience, let's represent the semi-major axis of Earth as "a" and the semi-major axis of the object as "x." Also, for this calculation, we will consider the mean orbital periods rather than the sidereal periods.
Using these notations, the equation becomes:
(365.25 days)^2 / [(365.25 days / 2)^2] = a^3 / x^3
Simplifying the equation:
4 = a^3 / x^3
Cross-multiplying and taking the cube root:
x^3 = (a^3)/4
Taking the cube root of both sides:
x = (a / 4)^(1/3)
Therefore, the semi-major axis of the object that is in a 2:1 mean motion resonance with Earth is equal to the cube root of one-fourth of the semi-major axis of Earth.