Radiation Balance Practice Problems
For each problem, make a sketch of the geometry, showing the shape of the geometric solid and indicating
which direction the solar radiation is coming from and the face it is hitting. Then, derive the equation for the
effective radiating temperature of the planet. Solve your equation using the following values:
S = Solar Flux = 1361 W/m2 ; A= albedo=0.3
1. The Earth has been transformed into a cylinder with a radius of 1000 km and height of 2000 km. The
sun shines on one of the circular faces at all times. What is the effective radiating temperature?
2. The Earth has been transformed into a cylinder with a radius of 1000 km and height of 2000 km. The
sun shines directly on the side of the cylinder at all times. What is the effective radiating temperature?
3. The Earth has been transformed into a hemisphere with a radius of 1000 km. The sun shines directly
on the flat side of the hemisphere at all times. What is the effective radiating temperature?
4. The Earth has been transformed into a rectangular prism with sides l=700 km, w=500 km, and h=1000
km. The sun shines directly on the side with the smallest area (l x w) at all times. What is the effective
radiating temperature?
5. The Earth has been transformed into a pyramid with a square base with side length l=1000 km. The
sun shines directly on the same triangular side at all times. What is the effective radiating temperature?
6. The Earth has been transformed into a pyramid with a square base with side length l=1000 km. The
sun shines directly on the square base at all times. What is the effective radiating temperature?
7. The Earth has been transformed into a triangular prism with height h=1000 km and triangular bases
that are equilateral triangles with side length s=500 km. The sun shines directly on one of the
equilateral triangle ends at all times. What is the effective radiating temperature?
8. The Earth has been transformed into a triangular prism with height h=1000 km and triangular bases
that are equilateral triangles with side length s=500 km. The sun shines directly on the same
rectangular side at all times. What is the effective radiating temperature?
9. The Earth has been transformed into an equilateral tetrahedron (all four sides are equilateral triangles)
with side length s=500 km. The sun shines directly on the same triangular side at all times. What is the
effective radiating temperature?
10. The Earth has been transformed into a thin sheet with l=2000 km and w=500 km and negligible
thickness. The sun shines directly on the same side of the sheet at all times. What is the effective
radiating temperature?
1.
2.
3.
4.
𝜎𝑇𝑒4 ∗ (2𝜋𝑟 2 + 2𝜋𝑟ℎ) = 𝑆(1 − 𝐴) ∗ 𝜋𝑟 2 ; 230 𝐾
𝜎𝑇𝑒4 ∗ (2𝜋𝑟 2 + 2𝜋𝑟ℎ) = 𝑆(1 − 𝐴) ∗ 2𝑟ℎ; 244 𝐾
𝜎𝑇𝑒4 ∗ 3𝜋𝑟 2 = 𝑆(1 − 𝐴) ∗ 𝜋𝑟 2 ; 274 𝐾
𝜎𝑇𝑒4 ∗ (2𝑙𝑤 + 2𝑙ℎ + 2𝑤ℎ) = 𝑆(1 − 𝐴) ∗ 𝑙𝑤; 209 𝐾
5.
𝜎𝑇𝑒4 ∗ (4 (
6.
𝜎𝑇𝑒4 ∗ (4 (
7.
8.
9.
√3 2
√3
𝑙 ) + 𝑙2 ) = 𝑆(1 − 𝐴) ∗ 𝑙2 ; 227 𝐾
4
4
√3 2
2
2
𝑙 ) + 𝑙 ) = 𝑆(1 − 𝐴) ∗ 𝑙 ; 280 𝐾
4
√3
√3
𝜎𝑇𝑒4 ∗ (2 ( 𝑠 2 ) + 3𝑠𝑙) = 𝑆(1 − 𝐴) ∗ 𝑠 2 ; 180 𝐾
4
4
√3
𝜎𝑇𝑒4 ∗ (2 ( 𝑠 2 ) + 3𝑠𝑙) = 𝑆(1 − 𝐴) ∗ 𝑠𝑙; 264 𝐾
4
𝜎𝑇𝑒4 ∗ 4 (
√3 2
√3
𝑠 ) = 𝑆(1 − 𝐴) ∗ 𝑠 2 ; 255 𝐾
4
4
10. 𝜎𝑇𝑒4 ∗ 2𝑙𝑤 = 𝑆(1 − 𝐴) ∗ 𝑙𝑤; 303 𝐾