You recently read that a new possibly habitable planet was discovered. The planet, Burt, receives 1700 W/m2 from it’s companion star. If planet Burt has a a radius of 6500 km and acts like a blackbody ( emissivity =1 ) and thus radiates same amount of energy it absorbs, what is its average suraface temperature. For comparison the earth has a surface temperaure of 278 K.
I guess we assume the temp is uniform over the surface as is the incoming power (obviously untrue)
5.67*10^-8 T^4 = 1700
T^4 = 300 * 10^8
T = 4.16 * 10^2 = 416 K
416 - 273 = 143 C
Hotter than boiling water. Habitable?
To calculate the average surface temperature of planet Burt, you can use the Stefan-Boltzmann law, which states:
E = σ * A * T^4
where:
E is the energy radiated by the planet,
σ is the Stefan-Boltzmann constant (equal to 5.67 × 10^(-8) W/(m^2 K^4)),
A is the surface area of the planet,
T is the temperature of the planet's surface.
In this case, since the planet is a blackbody, it radiates the same amount of energy it absorbs. Therefore, the energy radiated E is equal to the energy received from the companion star, which is 1700 W/m^2.
The surface area A of a sphere can be calculated using the formula:
A = 4πr^2
where r is the radius of the planet.
Now, let's plug in the values and calculate the average surface temperature of planet Burt step by step:
1. Calculate the surface area A:
A = 4π * (6500 km)^2
A = 4 * 3.14159 * (6500 km)^2
A = 4 * 3.14159 * (6500 km * 1000 m/km)^2
A = 4 * 3.14159 * (6500000 m)^2
A ≈ 5.305 × 10^14 m^2
2. Write down the energy equation:
E = σ * A * T^4
3. Substitute the known values:
1700 W/m^2 = 5.67 × 10^(-8) W/(m^2 K^4) * 5.305 × 10^14 m^2 * T^4
4. Rearrange the equation to solve for T:
T^4 = (1700 W/m^2) / (5.67 × 10^(-8) W/(m^2 K^4) * 5.305 × 10^14 m^2)
T^4 ≈ 5.405 × 10^3 K
T ≈ (5.405 × 10^3 K)^(1/4)
T ≈ 18.4 K
Therefore, the average surface temperature of planet Burt is approximately 18.4 K.
To find the average surface temperature of planet Burt, we need to use the Stefan-Boltzmann Law, which relates the power radiated by a blackbody to its surface area and temperature. The formula for the power radiated by a blackbody is given by:
P = σ * A * T^4
Where:
P is the power radiated
σ is the Stefan-Boltzmann constant (approximately equal to 5.67 * 10^-8 W m^-2 K^-4)
A is the surface area of the blackbody
T is the temperature of the blackbody in Kelvin
In this case, we know the power received by planet Burt from its companion star, which is 1700 W/m^2. Since planet Burt is assumed to be a blackbody, it radiates the same amount of energy it absorbs, meaning the power radiated is also 1700 W/m^2.
We can calculate the surface area of planet Burt using the formula for the surface area of a sphere:
A = 4πr^2
Where:
A is the surface area
r is the radius of the sphere
Plugging in the values, we have:
A = 4π(6500 km)^2
Now, we can rearrange the Stefan-Boltzmann Law to solve for the temperature T:
T = (P / (σ * A))^0.25
Plugging in the values we know, we have:
T = (1700 W/m^2) / (5.67 * 10^-8 W m^-2 K^-4) * (4π(6500 km)^2))^0.25
To get the temperature in Kelvin, we need to convert the radius from kilometers to meters.
Now we can calculate the average surface temperature of planet Burt.