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:
$F_{\text{SA}} = 4F_{\text{SB}}$
Important Questions from Planetary Bodies
- The correct sequence of planets in order of increasing surface temperature is:
- Atmosphere of planet Mars is almost entirely made up of $\text{CO}_2$. But the surface temperature of Mars is less than that of the Earth because:
- Consider two planets 'A' and 'B' with the following characteristics. The relationship between $T_1$ and $T_2$ is
Distance from the Sun Radius of the planet Incident Solar flux density Equivalent temperature Planet A $d_1$ $r_1$ $F_1$ $T_1$ Planet B $d_2 = 4d_1$ $r_2 = 2r_1$ $F_2$ $T_2$ - Venus is closer to the Sun than Earth, and therefore solar energy incident on Venus is higher than that on Earth. However, the effective radiating temperature of Venus is lower than that of the Earth, because
A planet of radius $r$ has a core of radius $r_c$ and a mantle. It has no crust. Its mean density is $\rho$ and the density of its core is $\rho_c$. What is the density of the mantle?
