SME
Physical Geography
Earth–sun relationships
Use this for “How high is the noon sun at 40° N on the June solstice, and how long is the day?”
Diagram
The Sun is overhead at δ = 23.4°; the place is at φ = 40°.
Noon sun angle = 90° − |φ − δ| = 90° − 16.6° = 73.4°.
cos
H
=
−tan
φ
tan
δ,
so
H
=
111°:
61.9%
of
the
parallel
is
lit,
day
=
2H
15
=
14.8
h.
This is an example. Type your own values to start a new problem.
α
°
=
90°
−
|
φ
°
−
δ
°|
H
Sunrise hour angle
calculated · °
D
Day length
calculated · h
Formulas
α
=
90
−
|
φ
−
δ
|
73.44 = 90 − |40 − 23.44|
cos(
H
°)
=
−tan(
φ
°)
×
tan(
δ
°)
cos(111.3342°) = −tan(40°) × tan(23.44°)
D
=
2
×
H
15
14.8446
=
2
×
111.3342
15
Assumptions
•
The declination δ is the latitude where the Sun is overhead at noon (±23.44° at the solstices).
•
Earth turns 15° an hour; refraction and the Sun’s width (a few minutes) are ignored.
•
Past a polar circle the day can be 24 h (midnight sun) or 0 h (polar night).
•
A negative noon angle means the Sun stays below the horizon all day.
Step-by-step
WE KNOW
φ = 40°, δ = 23.44°
FIND
noon sun angle (α), sunrise hour angle (H), day length (D)
Step 1 · Find noon sun angle (α)
Use α = 90 − |φ − δ|
The
Sun
is
overhead
at
δ
;
each
degree
of
latitude
away
from
it
lowers
the
noon
sun
a
degree.
α
=
90
−
|40
−
23.44|
α
=
90
−
|16.56|
α
=
90
−
16.56
α
=
73.44°
Step 2 · Find sunrise hour angle (H)
Use cos(H°) = −tan(φ°) × tan(δ°)
The sunrise hour angle: how far Earth turns from sunrise to noon, where the parallel meets the circle of illumination.
−1 < tan(40°) × tan(23.44°) < 1: the Sun rises and sets
H
=
arccos(−tan(
φ
°)
×
tan(
δ
°))
H
=
arccos(−tan(40°)
×
tan(23.44°))
tan(40°) × tan(23.44°) = 0.3638
H
=
111.3342°
Step 3 · Find day length (D)
Use D = 2 × H ÷ 15
Sunrise
to
sunset
is
twice
H
of
turning,
and
Earth
turns
15°
an
hour.
D
=
2
×
111.3342
15
D
=
222.6684
15
D
=
14.8446
h
Check
Put the numbers back in. Both sides should match.
73.44 = 90 − |40 − 23.44| ✓
cos(111.3342°) = −tan(40°) × tan(23.44°) ✓
14.8446
=
2
×
111.3342
15
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.