SME
Climatology
Energy balance
Use this for “Sunlight 1,361 W/m², albedo 0.30, and one layer absorbing 78 % of the infrared: how warm is the ground?”
Diagram
F
=
S(1 − α)
4
=
1,361
×
1 − 0.3
4
=
238.2
W/
m
2
,
so
Tₑ
=
(
F
σ
)
1
4
=
254.6
K.
The
layer
absorbs
ε
=
0.78
of
the
ground’s
infrared
and
sends
half
back
down,
so
Tₛ
=
Tₑ
(
2
2 − ε
)
1
4
=
254.6
×
1.132
=
288.1
K
(15
°C).
At the top 85.9 + 152.3 = 238.2 W/m² goes out, as much as comes in.
This is an example. Type your own values to start a new problem.
S
Sunlight
example · W/m²
α
Albedo
example
F
Absorbed sunlight
calculated · W/m²
Tₑ
Balance temperature
calculated · K
ε
Layer emissivity
example
Tₛ
Surface temperature
calculated · K
Formulas
F
=
S
×
1 −
α
4
238.175
=
1,361
×
1 − 0.3
4
Tₑ
=
(
F
5.67 ×
10
-8
)
1 ÷ 4
254.5823
=
(
238.175
5.67 ×
10
-8
)
1 ÷ 4
Tₛ
=
Tₑ
×
(
2
2 −
ε
)
1 ÷ 4
288.0685
=
254.5823
×
(
2
2 − 0.78
)
1 ÷ 4
Assumptions
•
One atmospheric layer, transparent to sunlight, absorbs a share ε of the ground’s infrared and sends half of it back down.
•
The sunlight on Earth’s disk is spread over the whole sphere, 4 times the disk’s area.
•
σ = 5.67 × 10⁻⁸ W/(m²·K⁴); ε = 1 gives Tₛ = 21/4 Tₑ, about 1.19 Tₑ.
Step-by-step
WE KNOW
S = 1,361 W/m², α = 0.3, ε = 0.78
FIND
absorbed sunlight (F), balance temperature (Tₑ), surface temperature (Tₛ)
Step 1 · Find absorbed sunlight (F)
Use F = S × (1 − α) ÷ 4
Keep the share not reflected, then spread it over the whole globe, 4 times its disk.
F
=
1,361
×
1 − 0.3
4
F
=
1,361
×
0.7
4
F
=
952.7
4
F
=
238.175
W/
m
2
Step 2 · Find balance temperature (Tₑ)
Use Tₑ = (F ÷ (5.67 × 10⁻⁸))1 ÷ 4
In
balance
the
planet
sends
out
σTₑ⁴
=
F
,
so
take
the
fourth
root
of
F
σ
.
Tₑ
=
(
238.175
5.67 ×
10
-8
)
1 ÷ 4
238.175
5.67 ×
10
-8
=
4.2006 ×
10
9
∜(4.2006
×
10
9
)
=
254.58
Tₑ = 254.5823 K
Step 3 · Find surface temperature (Tₛ)
Use Tₛ = Tₑ × (2 ÷ (2 − ε))1 ÷ 4
The
layer
sends
εG
2
back
down,
so
the
ground
must
send
out
G
=
2
F
2 −
ε
.
Tₛ
=
254.5823
×
(
2
2 − 0.78
)
1 ÷ 4
2
2 − 0.78
=
1.639
∜1.639 = 1.132
Tₛ = 288.0685 K
Check
Put the numbers back in. Both sides should match.
238.175
=
1,361
×
1 − 0.3
4
✓
254.5823
=
(
238.175
5.67 ×
10
-8
)
1 ÷ 4
✓
288.0685
=
254.5823
×
(
2
2 − 0.78
)
1 ÷ 4
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.