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 the orbital period of planet $Y$ is twice the orbital period of planet $X$, by what factor is the mean distance 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 periods and mean distances of planets X and Y.
Use the given relationship T Y = 2 T X and the equation T 2 = A 3 to set up equations.
Substitute and simplify to find the ratio of mean distances: ( A X A Y ) 3 = 4 .
Solve for the factor by which the mean distance is increased: A X A Y = 2 3 2 .
2 3 2
Explanation
Define 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 T Y = 2 T X . We want to find the factor by which the mean distance is increased, which is A X A Y .
Set up equations We have the equations T X 2 = A X 3 and T Y 2 = A Y 3 . Substituting T Y = 2 T X into the second equation, we get ( 2 T X ) 2 = A Y 3 , which simplifies to 4 T X 2 = A Y 3 .
Substitute Since T X 2 = A X 3 , we can substitute this into the previous equation to get 4 A X 3 = A Y 3 .
Simplify Dividing both sides by A X 3 , we have 4 = A X 3 A Y 3 = ( A X A Y ) 3 .
Solve for the factor Taking the cube root of both sides, we find the factor by which the mean distance is increased: A X A Y = 3 4 = 4 3 1 = ( 2 2 ) 3 1 = 2 3 2 .
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 whose orbital period is known, we can estimate its distance from its star using the equation T 2 = A 3 . This helps us determine if the planet lies within the habitable zone, where liquid water could exist, making it a potential candidate for life. This principle also applies to satellite orbits around Earth; knowing the orbital period allows us to calculate the satellite's altitude.
