Solar Flux and Distance Relation

The solar constant ($F$) represents the solar radiation flux received per unit area at a specific distance ($D$) from the Sun. According to the inverse square law, the intensity of radiation from a point source like the Sun decreases proportionally to the square of the distance from the source.

Mathematically, this relationship is expressed as:

$F \propto \frac{1}{D^2}$

The radii of the planets do not affect the value of the solar constant itself, as it's defined as flux per unit area at that distance.

Planet A and B Solar Constant Calculation

We are given:

• Planet A: Radius $ = 2r$, Distance from Sun $ = d$
• Planet B: Radius $ = r$, Distance from Sun $ = 2d$

Using the inverse square law:

• For Planet A, the solar constant $F_{\text{SA}}$ is proportional to $1/d^2$:

  $F_{\text{SA}} \propto \frac{1}{d^2}$

• For Planet B, the solar constant $F_{\text{SB}}$ is proportional to $1/(2d)^2$:

  $F_{\text{SB}} \propto \frac{1}{(2d)^2} = \frac{1}{4d^2}$

Derived Solar Constant Relationship

To find the relationship between $F_{\text{SA}}$ and $F_{\text{SB}}$, we can take the ratio:

$\frac{F_{\text{SA}}}{F_{\text{SB}}} = \frac{1/d^2}{1/(4d^2)}$

Simplifying the expression:

$\frac{F_{\text{SA}}}{F_{\text{SB}}} = \frac{1}{d^2} \times \frac{4d^2}{1} = 4$

Therefore, the relationship is:

$F_{\text{SA}} = 4F_{\text{SB}}$