Question

1. suppose, hypothetically, that the earth orbited the sun at half its current distance. (that is, at 1/2 au instead of 1 au). what would be the length of the year? what else would be different?

Explanation:

Step1: Recall Kepler's third law

$T^{2}\propto a^{3}$, where $T$ is the orbital - period (length of the year) and $a$ is the semi - major axis of the orbit. Let $T_1$ be the current period ($T_1 = 1$ year) and $a_1=1$ AU be the current semi - major axis, and $a_2 = 0.5$ AU be the new semi - major axis.

Step2: Set up the ratio

$\frac{T_2^{2}}{T_1^{2}}=\frac{a_2^{3}}{a_1^{3}}$. Substitute $a_1 = 1$ AU, $a_2=0.5$ AU and $T_1 = 1$ year into the equation: $\frac{T_2^{2}}{1^{2}}=\frac{(0.5)^{3}}{1^{3}}$.

Step3: Solve for $T_2$

$T_2^{2}=(0.5)^{3}=0.125$. Then $T_2=\sqrt{0.125}\approx0.354$ years or about 4.25 months.

As for what else would be different: The intensity of sunlight received by the Earth would increase. According to the inverse - square law, the intensity of sunlight $I\propto\frac{1}{r^{2}}$, where $r$ is the distance from the Sun. If $r$ is halved, the intensity of sunlight would be 4 times its current value. This would lead to a much warmer Earth, potentially melting the polar ice caps, changing weather patterns, and having a major impact on the climate and ecosystems.

Answer:

The length of the year would be approximately 0.354 years (or about 4.25 months). The intensity of sunlight would increase to 4 times its current value, leading to significant climate and ecosystem changes.