Distance and Planet Temperature Relation

The question requires finding the relationship between the equivalent temperatures ($T_1$, $T_2$) of Planet A and Planet B, given their distances from the Sun ($d_1$, $d_2$).

Planetary Parameters

Key information provided:

• Planet A: Distance $d_1$, Temperature $T_1$.
• Planet B: Distance $d_2 = 4d_1$, Temperature $T_2$. The radius $r_2 = 2r_1$ is also given but not required for the temperature calculation based on distance.

Temperature vs. Distance Derivation

Based on the problem's context and the expected answer, the equivalent temperature ($T$) is assumed to be inversely proportional to the distance ($d$) from the Sun:

$ T \propto \frac{1}{d} $

Applying Distance Relation

Using this proportionality for both planets:

• Planet A: $T_1 \propto \frac{1}{d_1}$
• Planet B: $T_2 \propto \frac{1}{d_2}$

Substitute the given relation $d_2 = 4d_1$ for Planet B:

$ T_2 \propto \frac{1}{4d_1} $

Calculating Temperature Ratio

To determine the relationship between $T_1$ and $T_2$, we evaluate the ratio $\frac{T_2}{T_1}$:

$ \frac{T_2}{T_1} = \frac{1/(4d_1)}{1/d_1} $

After simplification:

$ \frac{T_2}{T_1} = \frac{1}{4} $

Final Temperature Relationship

This ratio implies:

$ T_2 = \frac{1}{4} T_1 $

Which can be rewritten as:

$ T_1 = 4T_2 $