SME
Historical Geology · Problem type
The faint young Sun
Use this for “4 billion years ago the Sun was dimmer. With today’s albedo of 0.3 and no greenhouse effect, how cold was Earth?”
Diagram
Averaged
over
the
globe,
1,008
4
=
252
W/
m
2
arrives.
With
α
=
0.3,
76
is
reflected
and
F
=
1,008
×
1 − 0.3
4
=
176
W/
m
2
is
absorbed.
In balance the ground sends F back out: σTₑ⁴ = 176, so Tₑ = 236 K (−37 °C).
This is an example. Type your own values to start a new problem.
t
Time ago
example
L
Sun’s brightness then
calculated
S
Sunlight
calculated · W/m²
α
Albedo
example
F
Sunlight absorbed
calculated · W/m²
Tₑ
Balance temperature
calculated · K
Formulas
L
=
1
÷
(1
+
0.4
×
(1
−
4.57 −
t
4.57
))
0.7407
=
1
÷
(1
+
0.4
×
(1
−
4.57 − 4
4.57
))
S
=
1,361
×
L
1,008.0665 = 1,361 × 0.7407
F
=
S
×
1 −
α
4
176.4116
=
1,008.0665
×
1 − 0.3
4
Tₑ
=
∜(
F
5.67 ×
10
-8
)
236.1761
=
∜(
176.4116
5.67 ×
10
-8
)
Assumptions
•
The Sun formed 4.57 Ga and has brightened about 30% since: L = 1 ÷ (1 + 0.4(1 − age ÷ 4.57)), its age then being 4.57 − t.
•
Sunlight then is today’s solar constant, 1,361 W/m², times L; σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
•
Tₑ has no greenhouse effect, and the albedo is kept at today’s 0.3 unless you change it.
•
Today’s Tₑ is 255 K; a much colder Tₑ while liquid water flowed is the faint young Sun paradox.
Step-by-step
WE KNOW
t = 4 Ga, α = 0.3
FIND
Sun’s brightness then (L), sunlight (S), sunlight absorbed (F), balance temperature (Tₑ)
Step 1 · Find Sun’s brightness then (L)
Use L = 1 ÷ (1 + 0.4 × (1 − (4.57 − t) ÷ 4.57))
The
Sun
was
4.57
−
t
old
then;
the
share
of
its
life
still
to
come
sets
how
much
dimmer
it
was.
L
=
1
÷
(1
+
0.4
×
(1
−
4.57 − 4
4.57
))
4.57 − 4 = 0.57 Ga
1
−
0.57
4.57
=
0.8753
1 + 0.4 × 0.8753 = 1.35
L
=
0.7406807
Step 2 · Find sunlight (S)
Use S = 1,361 × L
Today’s
1,361
W/
m
2
,
scaled
by
how
bright
the
Sun
was.
S
=
1,361
×
0.7406807
S
=
1,008.066
W/
m
2
Step 3 · Find sunlight absorbed (F)
Use F = S × (1 − α) ÷ 4
The share not reflected, spread over the whole globe: 4 times the disk that catches it.
F
=
1,008.066
×
1 − 0.3
4
F
=
1,008.066
×
0.7
4
F
=
705.6462
4
F
=
176.4116
W/
m
2
Step 4 · Find balance temperature (Tₑ)
Use Tₑ = ∜(F ÷ (5.67 × 10⁻⁸))
In balance the ground sends out as infrared, σTₑ⁴, what it absorbs.
Tₑ
=
∜(
176.4116
5.67 ×
10
-8
)
Tₑ
=
∜(3.11
×
10
9
)
Tₑ = 236.1761 K
Check
Put the numbers back in. Both sides should match.
0.7406807
=
1
÷
(1
+
0.4
×
(1
−
4.57 − 4
4.57
))
✓
1,008.066 = 1,361 × 0.7406807 ✓
176.4116
=
1,008.066
×
1 − 0.3
4
✓
236.1761
=
∜(
176.4116
5.67 ×
10
-8
)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.