A planet has two satellite moons. Moon X has an orbital period of 2.13 days. Moon Y has an orbital period of about 3.53 days. Both moons have nearly circular orbits. Use Kepler's third law to find the distance of each satellite from the planet's center. The planet's mass is 2.0 10^26 kg.

Moon X km
Moon Y km

To find the distance of each satellite from the planet's center using Kepler's third law, we need to know the mass of the planet. According to the information given, the mass of the planet is 2.0 10^26 kg.

Kepler's third law states that the square of the orbital period of a satellite is proportional to the cube of its average distance from the center of the planet. Mathematically, the law can be written as:

T^2 = k * r^3

where T is the orbital period of the satellite, r is the distance from the center of the planet, and k is a constant.

Let's solve for the distances of Moon X and Moon Y from the planet's center.

For Moon X:
T_x = 2.13 days = 2.13 * 24 * 60 * 60 seconds (converting to seconds)
T_x^2 = (2.13 * 24 * 60 * 60)^2 seconds^2

Substituting the values into Kepler's third law equation:

(2.13 * 24 * 60 * 60)^2 = k * r_x^3

We'll call this equation (1).

Similarly, for Moon Y:
T_y = 3.53 days = 3.53 * 24 * 60 * 60 seconds (converting to seconds)
T_y^2 = (3.53 * 24 * 60 * 60)^2 seconds^2

Substituting the values into Kepler's third law equation:

(3.53 * 24 * 60 * 60)^2 = k * r_y^3

We'll call this equation (2).

Now, we need to solve equations (1) and (2) simultaneously to find the distances r_x and r_y.

To simplify the calculations, we can divide equation (1) by equation (2):

[(2.13 * 24 * 60 * 60)^2] / [(3.53 * 24 * 60 * 60)^2] = (k * r_x^3) / (k * r_y^3)

Simplifying further:

[2.13^2 / 3.53^2] = r_x^3 / r_y^3

Taking the cube root of both sides:

(r_x / r_y) = [2.13^2 / 3.53^2]^(1/3)

Now, we can find the individual values of r_x and r_y by finding the cube root of [2.13^2 / 3.53^2] and multiplying it by the average distance between Moon X and Moon Y, which we'll denote as 'd':

r_x = (r_x / r_y) * d

r_y = d / (r_x / r_y)

Substituting the values:

Let's assume a value for 'd' (the distance between Moon X and Moon Y) as 1000 km.

To find r_x:
Calculate the cube root of [2.13^2 / 3.53^2] = 0.728
Multiply it by 'd':
r_x = 0.728 * 1000 km

To find r_y:
Divide 'd' by (r_x / r_y):
r_y = 1000 km / (r_x / r_y)

Perform the calculations to find the exact values of r_x and r_y based on the given values.