The equation $T^2=A^3$ shows the relationship between a planet's orbital period, $T$, and the planet's mean distance from the sun, $A$, in astronomical units, $A U$. If planet $Y$ is twice the mean distance from the sun as planet $X$, by what factor is the orbital period increased?
A. $2^{\frac{1}{3}}$
B. $2^{\frac{1}{2}}$
C. $2^{\frac{2}{3}}$
D. $2^{\frac{3}{2}}$
Answer (1)
Define variables: Let T X , A X be the period and distance of planet X , and T Y , A Y be the period and distance of planet Y .
Use the given relationship: A Y = 2 A X .
Substitute into the equation: T Y 2 = ( 2 A X ) 3 = 8 A X 3 .
Solve for the factor: T X T Y = 2 2 3 .
The orbital period is increased by a factor of 2 2 3 .
Explanation
Understanding the Problem We are given the equation T 2 = A 3 , which relates a planet's orbital period T to its mean distance from the sun A . We are also told that planet Y is twice the mean distance from the sun as planet X . We need to find the factor by which the orbital period is increased.
Defining Variables Let T X and A X be the orbital period and mean distance of planet X , respectively. Similarly, let T Y and A Y be the orbital period and mean distance of planet Y , respectively. We are given that A Y = 2 A X .
Setting up Equations We have the equations T X 2 = A X 3 and T Y 2 = A Y 3 . We want to find the ratio T X T Y .
Substitution Substitute A Y = 2 A X into the equation for planet Y : T Y 2 = ( 2 A X ) 3 = 8 A X 3 .
Relating the Equations Since T X 2 = A X 3 , we can write T Y 2 = 8 T X 2 .
Solving for the Ratio Taking the square root of both sides, we get T Y = 8 T X 2 = 8 T X = 2 2 T X = 2 2 3 T X .
Finding the Factor Therefore, the factor by which the orbital period is increased is T X T Y = 2 2 3 .
Final Answer The orbital period is increased by a factor of 2 2 3 .
Examples
Understanding the relationship between a planet's orbital period and its distance from the sun is crucial in astronomy. For instance, if we discover a new exoplanet twice as far from its star as Earth is from our sun, we can estimate its orbital period using the equation T 2 = A 3 . This helps us understand the planet's climate and potential for habitability. This concept is also used to calculate the orbital periods of satellites around Earth, ensuring proper positioning and timing for communication and observation purposes. By applying this relationship, we can predict and manage the movements of celestial bodies, enhancing our understanding of the universe and improving technological applications.
