According to Kepler's third law, the orbital period T of a planet is related to the radius R of its orbit by

Question

According to Kepler's third law, the orbital period T of a planet is related to the radius R of its orbit by

T2 R3.
A planet's orbit is larger than Earth's by a factor if 3.39. What is the planet's orbital period?

Answer 1

Well, according to Kepler's third law, if a planet's orbit is larger than Earth's by a factor of 3.39, we can assume that its orbital radius is also larger by the same factor. So, if we substitute this into the equation, we get:

(T^2) = (3.39^3) * (R^3)

Now, we know that the orbital period of Earth is about 365.25 days, so let's call the planet's orbital period T_planet. We can solve for T_planet as follows:

(T_planet^2) = (3.39^3) * (365.25^3)

Now, I'll let you do the math to find the orbital period of the planet. Just remember to bring a calculator and a good sense of humor!

Answer 2

To find the planet's orbital period, we can use Kepler's third law, which states that the square of the orbital period is directly proportional to the cube of the radius of the orbit.

Let's say T represents the planet's orbital period and R represents the radius of its orbit. We can express Kepler's third law as an equation:

T^2 = k * R^3

From the given information, we know that the planet's orbit is larger than Earth's orbit by a factor of 3.39. This means that the radius of the planet's orbit (R) is 3.39 times larger than the radius of Earth's orbit (R_E).

R = 3.39 * R_E

Substituting this value of R in the equation, we have:

T^2 = k * (3.39 * R_E)^3

Now, since Kepler's third law is a proportionality relationship, the constant of proportionality (k) remains the same for all planets orbiting around the same star. So, we can write:

T^2 = k * 3.39^3 * R_E^3

Now, if we divide both sides of the equation by k * R_E^3, we get:

(T^2) / (R_E^3) = 3.39^3

Simplifying further, we have:

T^2 = 3.39^3 * R_E^3

To find the planet's orbital period (T), we need to take the square root of both sides of the equation:

T = sqrt(3.39^3 * R_E^3)

Now, if we know the orbital period of Earth (T_E), we can substitute its value. The orbital period of Earth is approximately 365.25 days or 1 year.

T_E = 1 year

Using this information, we can calculate the planet's orbital period (T):

T = sqrt(3.39^3 * (1 year)^3)

Now, we can substitute the values and calculate the result:

T = sqrt(3.39^3 * 1^3)

T = sqrt(3.39^3)

T ≈ 3.39^(3/2)

Using a calculator, we find:

T ≈ 7.289 years

Therefore, the planet's orbital period is approximately 7.289 years.