To determine which planet is closer to the sun, we can use Kepler's third law of planetary motion. This law states that the square of the orbital period [tex]\(T\)[/tex] of a planet is directly proportional to the cube of the semi-major axis (average distance from the sun) of its orbit, which can be approximated by the radial distance [tex]\(R\)[/tex] from the sun for circular or nearly circular orbits.
Mathematically, Kepler's third law can be expressed as:
[tex]\[ T^2 \propto R^3 \][/tex]
Or more explicitly:
[tex]\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \][/tex]
Given the durations of the years for planets [tex]\(Y\)[/tex] and [tex]\(Z\)[/tex]:
- Planet [tex]\(Y\)[/tex]: [tex]\(T_{Y} = 224\)[/tex] days
- Planet [tex]\(Z\)[/tex]: [tex]\(T_{Z} = 320\)[/tex] days
First, we observe that [tex]\(224\)[/tex] days is less than [tex]\(320\)[/tex] days ([tex]\(T_Y < T_Z\)[/tex]). According to Kepler's third law, the shorter the orbital period, the closer the planet is to the sun.
Therefore, since [tex]\(T_Y < T_Z\)[/tex], planet [tex]\(Y\)[/tex] must be closer to the sun than planet [tex]\(Z\)[/tex]. This happens because a closer planet experiences a stronger gravitational pull from the sun, resulting in a higher orbital speed and consequently a shorter orbital period.
Thus, the correct explanation is:
Planet [tex]\(Y\)[/tex], because the sun exerts a greater gravitational force on it than on planet [tex]\(Z\)[/tex].
So the answer is:
- Planet [tex]\(Y\)[/tex], because the sun exerts a greater gravitational force on it than on planet [tex]\(Z\)[/tex].
