91Ó°ÊÓ

• The semi-major axis of Pluto is $5.91×{10}^{12}\text{″¾}$.
• The eccentricity of Pluto is $0.249$.

The law states that the square of the orbital period of a planet is mainly proportional to the cube of the "semi-major axes" of the orbit of the particular planet.

The product of the cube of the orbital radius of Pluto and the Kepler’s constant gives the orbital period of Pluto.

${\mathrm{T}}^2=\left(\frac{4{\mathrm{Ï€}}^2}{\mathrm{GM}}\right){\mathrm{r}}^3$

Where the value $\mathrm{Ï€}$ is3.14 andT is the orbital period. Apart from that, M is the mass of the sun which is $1.99×{10}^{30}\text{ k²µ}$. Furthermore, G is the gravitational constant $6.673×{10}^{−11}\text{ }\frac{{\text{N.m}}^{\text{2}}}{{\text{kg}}^{\text{2}}}$, and r is Pluto's orbital radius $5.91×{10}^{12}\text{″¾}$.

Substituting the values in the above equation, we get,

$\begin{array}{c}{\mathrm{T}}^2=\frac{4×{(3.14)}^2×{(5.91×{10}^{12}\text{″¾})}^3}{6.673×{10}^{−11}\text{ }\frac{{\text{N.m}}^{\text{2}}}{{\text{kg}}^{\text{2}}}×1.99×{10}^{30}\text{ k²µ}}\\ {\mathrm{T}}^2=6.13×{10}^{19}\\ \mathrm{T}=7829431652\text{ s}\end{array}$

Thus, the orbital period of Pluto is about $7829431652\text{ s}$.

b)

From the Kepler’s third law, the equation of the closest distance of the Pluto from the sun is-$R=a(1−e)$

Here, R is the closest distance, a is the semi-major axis and e is the eccentricity.

Substituting the values in the above equation, we get-

$\begin{array}{c}\mathrm{R}=5.91×{10}^{12}\text{″¾}×(1−0.249)\\ =4.4×{10}^{12}\text{″¾}\end{array}$

From the Kepler’s third law, the equation of the farthest distance of the Pluto from the sun is-

$r=a(1+e)$

Here, r is the closest distance, a is the semi-major axis and e is the eccentricity.

Substituting the values in the above equation, we get-

$\begin{array}{c}\mathrm{R}=5.91×{10}^{12}\text{″¾}×(1+0.249)\\ =7.38×{10}^{12}\text{″¾}\end{array}$

Thus, the closest distance of Pluto from the sun is $4.4×{10}^{12}\text{″¾}$and the farthest distance of Pluto from the sun is $7.38×{10}^{12}\text{″¾}$.