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 for orbital period and mean distance for both planets.
Substitute the given relationship A Y = 2 A X into the equation T Y 2 = A Y 3 .
Use the relationship T X 2 = A X 3 to simplify the equation.
Solve for T Y in terms of T X to find the factor of increase: 2 2 3 .
Explanation
Setting up the problem Let T X and A X be the orbital period and mean distance of planet X , respectively, and 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 . We have T X 2 = A X 3 and T Y 2 = A Y 3 .
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 factor Take the square root of both sides: T Y = 8 T X 2 = 8 T X = 2 2 T X = 2 2 3 T X .
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 know that a newly discovered exoplanet is three times farther from its star than 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 timing for communication and observation purposes.
