The equation [tex]T^2=A^3[/tex] shows the relationship between a planet's orbital period, [tex]T[/tex], and the planet's mean distance the sun, [tex]A[/tex], in astronomical units, [tex]AU[/tex]. If factor is
A. [tex]2^{\frac{1}{3}}[/tex]
B. [tex]2^{\frac{1}{2}}[/tex]
C. [tex]2^{\frac{2}{3}}[/tex]
D. [tex]2^{\frac{3}{2}}[/tex]
Answer (1)
The problem states that T 2 = A 3 .
If A is multiplied by 2, then the new distance is 2 A .
The new orbital period T 2 satisfies T 2 2 = ( 2 A ) 3 = 8 A 3 .
Since T 2 = A 3 , we have T 2 2 = 8 T 2 , so $T_2 =
8 T = 2 2 T = 2 2 3 T
. Therefore, the factor is 2 2 3 .
Explanation
Understanding the Problem We are given the equation T 2 = A 3 , which relates a planet's orbital period T to its mean distance from the sun A . We want to find the factor by which T changes when A is multiplied by 2.
Initial Values Let A 1 and T 1 be the initial mean distance and orbital period, respectively. So, we have T 1 2 = A 1 3 .
New Values Now, let A 2 = 2 A 1 be the new mean distance, and let T 2 be the new orbital period. We want to find the factor k such that T 2 = k T 1 . We have T 2 2 = A 2 3 .
Substitution Substituting A 2 = 2 A 1 into the equation T 2 2 = A 2 3 , we get T 2 2 = ( 2 A 1 ) 3 = 8 A 1 3 .
Relating New and Initial Values Since T 1 2 = A 1 3 , we can substitute this into the equation T 2 2 = 8 A 1 3 to get T 2 2 = 8 T 1 2 .
Solving for T2 Taking the square root of both sides, we get T 2 = 8 T 1 = 2 2 T 1 .
Finding the Factor Since 2 2 = 2 1 ⋅ 2 2 1 = 2 2 2 " , w e ha v e T_2 = 2^{\frac{3}{2}}T_1 . T h ere f ore , t h e f a c t or b y w hi c h T$ changes is 2 2 3 .
Final Answer The factor by which T changes is 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 planet twice as far from its star as Earth is from the Sun, we can quickly estimate its orbital period using the relationship T 2 = A 3 . This helps us predict the planet's climate and potential for habitability. The factor 2 2 3 allows us to scale Earth's orbital period (1 year) to estimate the new planet's orbital period, which would be approximately 2.83 years. This kind of estimation is a fundamental tool in exoplanet research.
