ou recently read that a new possibly habitable planet was discovered. The planet, Burt, receives 2100 W/m2 from it’s companion star. If planet Burt has a a radius of 7500 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 know you use the Pradiation equation
P=AesT^4
e=epsilon
s=sigma
T= temp in Kelvin
To find the average surface temperature of planet Burt, we can use the Stefan-Boltzmann law, which states that the power radiated by a blackbody is proportional to the fourth power of its temperature. The equation is given as:
P = A * e * σ * T^4
Where:
P is the power (energy per unit time) radiated by the planet
A is the surface area of the planet
e is the emissivity of the planet (in this case, 1 for a blackbody)
σ is the Stefan-Boltzmann constant (approximately equal to 5.67 x 10^-8 Watts per meter squared per Kelvin to the fourth power)
T is the temperature of the planet in Kelvin
In order to find the temperature of planet Burt, we need to rearrange the equation to solve for T:
T^4 = P / (A * e * σ)
First, let's find the power radiated by planet Burt. We know that it receives 2100 W/m^2 from its companion star. To calculate the power, we multiply this value by the surface area of the planet.
The surface area of a sphere is given by the equation:
A = 4 * π * r^2
Where:
A is the surface area
π is a mathematical constant (approximately equal to 3.14)
r is the radius of the planet
Substituting the given values:
A = 4 * π * (7500 km)^2
Make sure to convert the radius to meters to match the units used in the equation. 1 kilometer is equal to 1000 meters.
Now, we have the surface area A. Multiply it by the power received from the star:
P = 2100 W/m^2 * A
With the power P and the surface area A, we can now calculate the temperature T using the rearranged equation:
T^4 = P / (A * e * σ)
Finally, take the fourth root of both sides to find the average surface temperature of planet Burt.