If Kepler 1606b has no atmosphere and an albedo similar to the Earth’s (albedo of 0.3), what would the equilibrium average temperature of the planet be? Give your answer in °C.
The Correct Answer and Explanation is:
To find the equilibrium temperature of Kepler 1606b, we can use the Stefan-Boltzmann Law, which relates the energy received by a planet to its temperature. The equilibrium temperature is determined by balancing the energy absorbed from the star with the energy radiated by the planet. The formula is: Teq=((1−A)S4σ)1/4T_{eq} = \left( \frac{(1 – A) S}{4 \sigma} \right)^{1/4}Teq=(4σ(1−A)S)1/4
Where:
- TeqT_{eq}Teq is the equilibrium temperature of the planet,
- AAA is the albedo (reflectivity),
- SSS is the solar flux (energy per unit area received from the star),
- σ\sigmaσ is the Stefan-Boltzmann constant, 5.67×10−8 W/m2 K45.67 \times 10^{-8} \, \text{W/m}^2 \, \text{K}^45.67×10−8W/m2K4,
- The factor of 4 accounts for the fact that only one side of the planet receives sunlight at any given time.
The solar flux SSS at Earth’s distance from the Sun (which we assume is similar for Kepler 1606b) is approximately: S=1361 W/m2S = 1361 \, \text{W/m}^2S=1361W/m2
We can now plug in the values:
- Albedo A=0.3A = 0.3A=0.3,
- S=1361 W/m2S = 1361 \, \text{W/m}^2S=1361W/m2,
- σ=5.67×10−8 W/m2 K4\sigma = 5.67 \times 10^{-8} \, \text{W/m}^2 \, \text{K}^4σ=5.67×10−8W/m2K4.
First, calculate the term inside the parentheses: (1−A)=1−0.3=0.7(1 – A) = 1 – 0.3 = 0.7(1−A)=1−0.3=0.7 (1−A)S4σ=0.7×13614×5.67×10−8≈2.68×109 W/m2\frac{(1 – A) S}{4 \sigma} = \frac{0.7 \times 1361}{4 \times 5.67 \times 10^{-8}} \approx 2.68 \times 10^9 \, \text{W/m}^24σ(1−A)S=4×5.67×10−80.7×1361≈2.68×109W/m2
Now, take the fourth root to find TeqT_{eq}Teq: Teq=(2.68×109)1/4≈283.1 KT_{eq} = \left( 2.68 \times 10^9 \right)^{1/4} \approx 283.1 \, \text{K}Teq=(2.68×109)1/4≈283.1K
Finally, convert from Kelvin to Celsius: Teq=283.1−273.15≈9.96 °CT_{eq} = 283.1 – 273.15 \approx 9.96 \, \text{°C}Teq=283.1−273.15≈9.96°C
Thus, the equilibrium average temperature of Kepler 1606b would be approximately 10°C.
Explanation:
- The albedo of 0.3 indicates that the planet reflects 30% of the incoming solar radiation and absorbs the remaining 70%.
- The equilibrium temperature is calculated by considering the balance between the absorbed radiation and the energy the planet radiates back into space.
- The result of around 10°C suggests that without an atmosphere, Kepler 1606b would have a temperate climate, similar to Earth’s average temperature, assuming it has an albedo close to Earth’s and the same solar flux.
