Let's solve this problem step by step to identify by what factor the orbital period increases if planet [tex]\( Y \)[/tex] is twice the mean distance from the sun as planet [tex]\( X \)[/tex].
### Step 1: Understanding the Relationship
The equation [tex]\( T^2 = A^3 \)[/tex] describes the relationship between the orbital period [tex]\( T \)[/tex] of a planet and its mean distance [tex]\( A \)[/tex] from the sun.
### Step 2: Relating the Distances
Given:
- Planet [tex]\( X \)[/tex] has a mean distance [tex]\( A_X \)[/tex] from the sun.
- Planet [tex]\( Y \)[/tex] has a mean distance [tex]\( A_Y \)[/tex] from the sun, where [tex]\( A_Y = 2A_X \)[/tex].
### Step 3: Determining the Orbital Periods
We need to find the orbital periods [tex]\( T_X \)[/tex] and [tex]\( T_Y \)[/tex] in terms of their mean distances:
- For planet [tex]\( X \)[/tex], we have:
[tex]\[
T_X^2 = A_X^3
\][/tex]
- For planet [tex]\( Y \)[/tex], since [tex]\( A_Y = 2A_X \)[/tex], the relationship becomes:
[tex]\[
T_Y^2 = (2A_X)^3 = 8A_X^3
\][/tex]
### Step 4: Solving for the Orbital Period of Planet [tex]\( Y \)[/tex]
Using the equation above:
[tex]\[
T_Y^2 = 8A_X^3
\][/tex]
To find [tex]\( T_Y \)[/tex], take the square root of both sides:
[tex]\[
T_Y = \sqrt{8A_X^3}
\][/tex]
Rewrite [tex]\( 8 \)[/tex] as [tex]\( 2^3 \)[/tex]:
[tex]\[
T_Y = \sqrt{2^3 \cdot A_X^3}
\][/tex]
Separating the terms under the square root:
[tex]\[
T_Y = \sqrt{2^3} \cdot \sqrt{A_X^3}
\][/tex]
Since [tex]\( \sqrt{2^3} = (2^3)^{\frac{1}{2}} \)[/tex]:
[tex]\[
\sqrt{2^3} = 2^{\frac{3}{2}}
\][/tex]
And [tex]\( \sqrt{A_X^3} = A_X^{3/2} \)[/tex]:
[tex]\[
T_Y = 2^{\frac{3}{2}} \cdot A_X^{\frac{3}{2}}
\][/tex]
### Step 5: Comparing [tex]\( T_Y \)[/tex] to [tex]\( T_X \)[/tex]
Recall that [tex]\( T_X = A_X^{\frac{3}{2}} \)[/tex]. To express [tex]\( T_Y \)[/tex] in terms of [tex]\( T_X \)[/tex], we get:
[tex]\[
T_Y = 2^{\frac{3}{2}} \cdot T_X
\][/tex]
### Step 6: Conclusion
The orbital period [tex]\( T_Y \)[/tex] is increased by a factor of [tex]\( 2^{\frac{3}{2}} \)[/tex] compared to [tex]\( T_X \)[/tex]. Thus, the correct answer is:
[tex]\[
2^{\frac{3}{2}}
\][/tex]
So, by what factor is the orbital period increased?
[tex]\[
\boxed{2^{\frac{3}{2}}}
\][/tex]
