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Chapter 8: Problem 45

The asteroid Pasachoff orbits the Sun with period 1417 days. Find the semimajor axis of its orbit from Kepler's third law. Use Earth's orbital radius and period, respectively, as your units of distance and time.

Short Answer

Expert verified
The semimajor axis of asteroid Pasachoff's orbit, in terms of the Earth's orbital radius, is approximately 2.62 Astronomical Units.

Step by step solution

01

Understanding Kepler's Third Law

Kepler's Third Law states that the square of the period of a planet is directly proportional to the cube of the semimajor axis of its orbit. Mathematically, it is represented as \(T^2 = k \cdot a^3\), where T is the period of revolution, a is the semimajor axis of the orbit, and k is the constant of proportionality.
02

Reference Units Conversion

In this problem, use the Earth's orbital period and radius as the units of time and distance, respectively. Therefore, \(T_0 = 1\) year, \(a_0 = 1\) Astronomical Unit (AU). For asteroid Pasachoff, \(T = 1417\) days or \(T = \frac{1417}{365}\) years.
03

Application of Kepler's Third Law

Given the reference units and applying Kepler's Law, we can derive the following equation: \( (\frac{T}{T_0})^2 = (\frac{a}{a_0})^3\), where \(T_0\) and \(a_0\) represent Earth’s period and semimajor axis respectively. Replace \(T\), \(T_0\), \(a_0\) with their respective values and solve for \(a\).
04

Calculation

Substitute \(T = \frac{1417}{365}\) years, \(T_0 = 1\) year, \(a_0 = 1\) AU into the equation, to get \( (\frac{\frac{1417}{365}}{1})^2 = (\frac{a}{1})^3\). Simplify and solve for a with a = \( \sqrt[3]{(\frac{1417}{365})^2} \).
05

Final Result

After calculating the cubic root, you will find the semimajor axis of its orbit in terms of Earth's orbital radius, which is the answer.

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