Question

1. below, is a model of a distant solar system. the star in the center is one solar mass (it is the same size as our sun). the distances from the center of the star to the orbits shown is to scale. it has been determined that the innermost planet takes 123 earth - days to orbit the star. use this data to calculate the distance from the star to the other four planets and the comet along their semimajor axis, and the time they take to orbit the star. write the answers to your calculations next to the appropriate planet.
2. the diagram below shows the path of a comet with an eccentric orbit about a star. use the diagram to explain the motion of the comet as it moves around its orbit:

Explanation:

Step1: Identify relevant law

Kepler's third - law states $T^{2}\propto a^{3}$, where $T$ is the orbital period and $a$ is the semi - major axis. For a star with mass equal to the Sun's mass, we can use the form $\frac{T_1^{2}}{a_1^{3}}=\frac{T_2^{2}}{a_2^{3}}$. Let the innermost planet's period $T_1 = 123$ Earth days and its semi - major axis be $a_1$.

Step2: For other planets

For each of the other four planets, we know their orbital periods $T_2$. Rearranging Kepler's third - law formula to solve for $a_2$ gives $a_2=a_1(\frac{T_2}{T_1})^{\frac{2}{3}}$. However, since the value of $a_1$ is not given in the problem description, we assume a relative scale. If we consider the innermost planet's semi - major axis as a unit distance $a_1 = 1$ (for relative calculations), then for a planet with period $T_2$, the relative semi - major axis $a_2=(\frac{T_2}{123})^{\frac{2}{3}}$.

For the second part about the comet's motion:
The comet moves in an elliptical orbit with the star at one of the foci. According to Kepler's laws, the comet moves faster when it is closer to the star (perihelion) and slower when it is farther from the star (aphelion). This is due to the conservation of angular momentum. As the comet approaches the star, its distance $r$ from the star decreases, and to conserve angular momentum $L = mvr$ (where $m$ is the mass of the comet, $v$ is its velocity, and $r$ is the distance from the star), its velocity $v$ must increase. Conversely, as it moves away from the star, $r$ increases and $v$ decreases.

Since the problem lacks specific data for numerical calculations of the planets' distances (like their orbital periods), we can only provide the general method for calculation and the qualitative explanation for the comet's motion.

Answer:

For calculating distances of planets from the star: Use Kepler's third - law $a_2=a_1(\frac{T_2}{T_1})^{\frac{2}{3}}$ with $T_1 = 123$ Earth days (innermost planet's period) and $a_1$ as the innermost planet's semi - major axis (if known or assumed as 1 for relative calculations). For the comet's motion: The comet moves faster at perihelion and slower at aphelion due to conservation of angular momentum.