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 $d_2 = 4d_1$ $r_2 = 2r_1$ $F_2$ $T_2$
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 $
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 radii $2r$ and $r$, respectively. Let their distances from the Sun be $d$ and $2d$, respectively. The solar constants for A ($F_{\text{SA}}$) and B ($F_{\text{SB}}$) are related by
- 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?
