Earth-Sun Geometry For example, on April 20, N = 30 (number of days from the closest equinox, March 21), and Subsolar point = 23.5 \cdot sin (30°) = 23.5 \cdot (0.5) = 11.75° or 11° 45′ N On December 9, N = -78 (number of days from September 22, negative since it is between the September and March equinoxes), and Subsolar point = 23.5 \cdot sin (-78°) = -22.90° or 22° 53′ S Notice that we use negative values for the South- ern Hemisphere and positive ones for the Northern Hemisphere–again by convention. 4. Calculate the subsolar point on: a. March 21 b. June 21 c. September 22 d. December 22 e. Today’s date 5. Calculate the noon Sun angles for New Orleans, USA (30° N), and Helsinki, Finland (60° N), on each of the following dates: a. March 21 b. June 21 c. September 22 d. December 22 e. Today’s date New Orleans Helsinki Compare the declination values for the three additional dates you calculated in question 4 with those on the analemma. 6. What are the zenith and solar elevation angles for your city today? Optional Exercise: Measuring Solar Elevation Angle (Sun Angle) d. Using a calculator or table of tangent values,
The Correct Answer and Explanation is:
Analyzed
pythonAlways show detailsCopyfrom datetime import datetime
import math
# Constants
latitude_new_orleans = 30 # degrees North
latitude_helsinki = 60 # degrees North
declination_max = 23.5 # maximum declination in degrees
# Date info
today = datetime.today()
march_21 = datetime(today.year, 3, 21)
june_21 = datetime(today.year, 6, 21)
sept_22 = datetime(today.year, 9, 22)
dec_22 = datetime(today.year, 12, 22)
# Days from equinoxes
def days_from_march_21(date):
return (date - march_21).days
# Subsolar point calculation
def subsolar_point(N):
# Convert N to degrees and find declination using sine
declination = declination_max * math.sin(math.radians(N))
return declination
# Noon Sun angle = 90° - |latitude - declination|
def sun_angle(latitude, declination):
return 90 - abs(latitude - declination)
# Dates to evaluate
dates = {
"March 21": march_21,
"June 21": june_21,
"September 22": sept_22,
"December 22": dec_22,
"Today": today
}
# Prepare results
results = {}
for label, date in dates.items():
N = days_from_march_21(date)
decl = subsolar_point(N)
angle_nola = sun_angle(latitude_new_orleans, decl)
angle_helsinki = sun_angle(latitude_helsinki, decl)
results[label] = {
"N": N,
"Declination (°)": round(decl, 2),
"New Orleans Noon Sun Angle (°)": round(angle_nola, 2),
"Helsinki Noon Sun Angle (°)": round(angle_helsinki, 2)
}
results
Result
{'March 21': {'N': 0,
'Declination (°)': 0.0,
'New Orleans Noon Sun Angle (°)': 60.0,
'Helsinki Noon Sun Angle (°)': 30.0},
'June 21': {'N': 92,
'Declination (°)': 23.49,
'New Orleans Noon Sun Angle (°)': 83.49,
'Helsinki Noon Sun Angle (°)': 53.49},
'September 22': {'N': 185,
'Declination (°)': -2.05,
'New Orleans Noon Sun Angle (°)': 57.95,
'Helsinki Noon Sun Angle (°)': 27.95},
'December 22': {'N': 276,
'Declination (°)': -23.37,
'New Orleans Noon Sun Angle (°)': 36.63,
'Helsinki Noon Sun Angle (°)': 6.63},
'Today': {'N': 82,
'Declination (°)': 23.27,
'New Orleans Noon Sun Angle (°)': 83.27,
'Helsinki Noon Sun Angle (°)': 53.27}}
Here are the calculated results for the subsolar point and noon Sun angles on key dates, including today:
4. Subsolar Point (Solar Declination)
Calculated using the formula:
Subsolar Point (°) = 23.5 × sin(N°)
where N is the number of days from March 21:
| Date | N (days from Mar 21) | Declination (°) |
|---|---|---|
| March 21 | 0 | 0.00° |
| June 21 | 92 | 23.49° |
| September 22 | 185 | -2.05° |
| December 22 | 276 | -23.37° |
| Today (June 11) | 82 | 23.27° |
5. Noon Sun Angles
Formula:
Noon Sun Angle = 90° – |latitude − declination|
| Date | New Orleans (30°N) | Helsinki (60°N) |
|---|---|---|
| March 21 | 60.00° | 30.00° |
| June 21 | 83.49° | 53.49° |
| September 22 | 57.95° | 27.95° |
| December 22 | 36.63° | 6.63° |
| Today (June 11) | 83.27° | 53.27° |
6. Zenith and Solar Elevation Angles (Today)
- Zenith angle = angle between the Sun and the vertical direction =
Zenith = 90° − Solar Elevation - From today’s results:
- New Orleans: Zenith = 90° − 83.27° = 6.73°
- Helsinki: Zenith = 90° − 53.27° = 36.73°
📘 300-word Explanation (Textbook Style)
The Earth-Sun geometry determines the solar declination—the latitude where the Sun is directly overhead at solar noon, also called the subsolar point. This point shifts between 23.5°N and 23.5°S due to the Earth’s axial tilt. The declination angle follows a sinusoidal pattern throughout the year, and is calculated as:
δ = 23.5 × sin(N°),
where N is the number of days from the spring equinox (March 21), and the angle is measured in degrees.
On equinoxes (March 21 and September 22), the Sun is directly overhead at the equator, so the declination is 0°. On solstices, declination reaches its extreme: about +23.5° on June 21 (Tropic of Cancer) and −23.5° on December 22 (Tropic of Capricorn).
The solar elevation angle at local noon is the angle of the Sun above the horizon. It depends on both the observer’s latitude and the solar declination and is calculated using:
Noon Sun Angle = 90° − |Latitude − Declination|
For example, in New Orleans (30°N), the Sun reaches 83.3° above the horizon near the June solstice, meaning the Sun is nearly overhead. In contrast, Helsinki (60°N) experiences a much lower noon Sun even on June 21, peaking at 53.5°, and drops dramatically to only 6.6° by December 22, contributing to short, dim winter days.
The zenith angle is the complement of the solar elevation and describes how far the Sun is from being directly overhead. It is smallest when the Sun is highest in the sky. Today, June 11, the subsolar point is near the Tropic of Cancer, so Northern Hemisphere cities like New Orleans and Helsinki receive strong midday sunlight.
