91影视

• The semi-major axis of Pluto is $5.91脳{10}^{12}\text{鈥尘}$.
• 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鈥檚 constant gives the orbital period of Pluto.

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

Where the value $\mathrm{蟺}$ is3.14 andT is the orbital period. Apart from that, M is the mass of the sun which is $1.99脳{10}^{30}\text{鈥塳驳}$. Furthermore, G is the gravitational constant $6.673脳{10}^{鈭�11}\text{鈥�}\frac{{\text{N.m}}^{\text{2}}}{{\text{kg}}^{\text{2}}}$, and r is Pluto's orbital radius $5.91脳{10}^{12}\text{鈥尘}$.

Substituting the values in the above equation, we get,

$\begin{array}{c}{\mathrm{T}}^2=\frac{4脳{(3.14)}^2脳{(5.91脳{10}^{12}\text{鈥尘})}^3}{6.673脳{10}^{鈭�11}\text{鈥�}\frac{{\text{N.m}}^{\text{2}}}{{\text{kg}}^{\text{2}}}脳1.99脳{10}^{30}\text{鈥塳驳}}\\ {\mathrm{T}}^2=6.13脳{10}^{19}\\ \mathrm{T}=7829431652\text{鈥塻}\end{array}$

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

b)

From the Kepler鈥檚 third law, the equation of the closest distance of the Pluto from the sun is-$R=a(1鈭�e)$

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-

$\begin{array}{c}\mathrm{R}=5.91脳{10}^{12}\text{鈥尘}脳(1鈭�0.249)\\ =4.4脳{10}^{12}\text{鈥尘}\end{array}$

From the Kepler鈥檚 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-

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

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