Chapter 4: Problem 10

Consider a hypothetical trans-Neptunian object located 100 AU from the Sun. What would be the orbital period (in years) of this object?

Short Answer

Expert verified

Answer: The orbital period of the hypothetical trans-Neptunian object located 100 AU from the Sun is approximately 1000 years.

Step by step solution

Kepler's Third Law

According to Kepler's Third Law of Planetary Motion, the square of the orbital period of an object (T) is directly proportional to the cube of the semi-major axis (a) of its orbit. Mathematically, this is represented as: T^2 ∝ a^3 To find the constant of proportionality, we must first determine the relationship between the Earth's orbital period and the Earth's distance from the Sun.

Earth's orbital period and distance from the Sun

The Earth's orbital period (T_Earth) is 1 year, and its distance from the Sun (a_Earth) is 1 Astronomical Unit (AU). Plugging these values into Kepler's Third Law equation, we can derive the constant of proportionality (k): T_Earth^2 = k * a_Earth^3 1^2 = k * 1^3 k = 1 Now, we can write Kepler's Third Law equation with the constant of proportionality: T^2 = a^3

Calculate the orbital period of the trans-Neptunian object

The trans-Neptunian object is located 100 AU from the Sun, so its semi-major axis (a) is 100 AU. We can plug this value into the Kepler's Third Law equation to find the orbital period (T): T^2 = a^3 T^2 = (100)^3 Now, solve for T: T = sqrt(100^3) T ≈ 100 * sqrt(100) T ≈ 100 * 10

Find the orbital period in years

The orbital period T is expressed in years: T ≈ 100 * 10 T ≈ 1000 years Hence, the orbital period of the hypothetical trans-Neptunian object located 100 AU from the Sun is approximately 1000 years.

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Consider a hypothetical trans-Neptunian object located 100 AU from the Sun. What would be the orbital period (in years) of this object?

Short Answer

Step by step solution

Kepler's Third Law

Earth's orbital period and distance from the Sun

Calculate the orbital period of the trans-Neptunian object

Find the orbital period in years

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