To find the orbital period of a planet, we can use the equation:
T = 2 * pi * (a^3 / (G * M))^(1/2)
where T is the orbital period, a is the semi-major axis (average distance) of the orbit, G is the gravitational constant, and M is the mass of the star.
First, we'll convert the average distance from kilometers to meters:
a = 112 million km = 112 x 10^6 m
Next, we'll find the semi-minor axis using the formula:
b = a * sqrt(1 - e^2)
where e is the eccentricity.
b = 112 x 10^6 m * sqrt(1 - 0.3^2) = 112 x 10^6 m * sqrt(0.91) = 112 x 10^6 m * 0.954
Next, we'll find the semi-major axis:
a = 112 x 10^6 m
Finally, we'll substitute these values into the formula for the orbital period:
T = 2 * pi * (a^3 / (G * M))^(1/2)
T = 2 * pi * (112 x 10^6 m)^3 / (G * M)^(1/2)
Since M is the mass of the sun, we'll use the mass of the sun in kilograms:
M = 1.989 x 10^30 kg
G = 6.67430 x 10^-11 m^3 kg^-1 s^-2
T = 2 * pi * (112 x 10^6 m)^3 / (6.67430 x 10^-11 m^3 kg^-1 s^-2 * 1.989 x 10^30 kg)^(1/2)
T = 2 * pi * (112 x 10^6 m)^3 / (6.67430 x 10^-11 * 1.989 x 10^30)^(1/2)
T = 2 * pi * (112 x 10^6 m)^3 / (1.327 x 10^-20)^(1/2)
T = 2 * pi * (112 x 10^6 m)^3 / (1.153 x 10^-10)
T = 2 * pi * (112 x 10^6 m)^3 / 1.153 x 10^-10
T = 2 * pi * (112 x 10^6 m)^3 / 1.153 x 10^-10
T = 2 * pi * (112 x 10^6 m)^3 / 1.153 x 10^-10
T = 5.201 x 10^7 s
So, the orbital period of the planet is approximately 5.201 x 10^7 seconds, or about 1.6 years.
The nearest and farthest orbital distances from the star are equal to the semi-minor and semi-major axes, respectively.
Nearest distance = b = 112 x 10^6 m * 0.954 = 106.6 x 10^6 m
Farthest distance = a = 112 x 10^6 m