Sun Altitude Calculator & Explanation | Your Trusted Source

Sun Altitude Calculator

Precisely determine the sun’s height in the sky for any location and time.

Sun Altitude Calculator

Results

–.-°

Solar Declination: –.-°

Hour Angle: –.-°

Zenith Angle: –.-°

Altitude is calculated using the formula: Altitude = 90° – Zenith Angle.

Key Assumptions:

Location: Lat –.–°, Lon –.–°

Date: –/–/—-, Time (UTC): –:–

Atmospheric Refraction (standard): ~0.5° at horizon

Sun Altitude Over the Day (UTC)

Sun Altitude
Solar Declination

Sun Altitude Data Table

Time (UTC)	Hour Angle	Solar Declination	Zenith Angle	Sun Altitude

Approximate data for the selected day. Values adjust dynamically.

What is Sun Altitude?

Sun altitude, also known as solar elevation angle, is a fundamental astronomical and geographical measurement representing the angle between the horizon and the center of the Sun’s disk. It is measured vertically upwards from the horizon. An altitude of 0° corresponds to the Sun being exactly on the horizon, while an altitude of 90° means the Sun is directly overhead (at the zenith).

Understanding sun altitude is crucial for a wide array of applications, from optimizing solar panel efficiency and architectural design to conducting astronomical observations and even planning outdoor activities. The sun altitude varies continuously throughout the day and year, depending on your geographical location (latitude and longitude), the date, and the time.

Who should use it:

Solar energy system designers and installers
Architects and urban planners
Astronomers and skywatchers
Farmers and gardeners (for crop placement)
Surveyors and civil engineers
Anyone interested in celestial mechanics or the position of the sun.

Common misconceptions:

Misconception: The sun is always highest at noon. Reality: Solar noon (when the sun is highest) occurs when the sun is on the local meridian, which may not precisely align with clock noon due to time zones and daylight saving. The calculated altitude is based on UTC and longitude, not local clock time directly.
Misconception: Sun altitude is the same everywhere on Earth at a given time. Reality: Sun altitude is highly dependent on latitude. At the equator, the sun can be directly overhead (90° altitude), while at the poles, it may remain below the horizon for months.
Misconception: Solar declination is constant throughout the year. Reality: Solar declination changes daily, ranging from approximately +23.5° in the Northern Hemisphere summer solstice to -23.5° in the Southern Hemisphere summer solstice.

Sun Altitude Formula and Mathematical Explanation

Calculating the sun altitude involves several steps, using astronomical formulas to determine the sun’s position relative to an observer on Earth. The primary formula relates the sun’s altitude (α) to its zenith angle (z):

Altitude (α) = 90° – Zenith Angle (z)

The zenith angle is the angle between the observer’s local vertical (zenith) and the sun. To find the zenith angle, we typically use the following formula, which incorporates latitude, declination, and hour angle:

cos(z) = sin(φ)sin(δ) + cos(φ)cos(δ)cos(H)

Where:

z is the zenith angle
φ (phi) is the observer’s latitude
δ (delta) is the Sun’s declination
H is the hour angle

Let’s break down the components:

1. Solar Declination (δ)

The Sun’s declination is its angular distance, north or south, from the celestial equator. It varies throughout the year due to the Earth’s axial tilt (about 23.5°). A common approximation for declination is:

δ ≈ 23.5° * sin( (360/365.25) * (N – 81) )

Where N is the day number of the year (1 for January 1st).

2. Hour Angle (H)

The hour angle represents the angular displacement of the Sun east or west of the local meridian. It is 0° at solar noon and increases by 15° for every hour away from solar noon (since the Earth rotates 360° in 24 hours, or 15° per hour). It’s calculated relative to UTC and longitude:

H = (Local Solar Time – 12) * 15°

To get Local Solar Time from UTC:

Local Solar Time (hours) = UTC (hours) + (Longitude / 15)

The hour angle formula can be expressed more precisely using UTC:

H (degrees) = ( (UTC Hour + UTC Minute/60 + UTC Second/3600) – 12 ) * 15 + Longitude

(Note: The longitude term is added to account for the difference between UTC and local solar time. A positive longitude is East, negative is West.)

3. Zenith Angle (z) and Altitude (α)

Once we have latitude (φ), declination (δ), and hour angle (H), we can plug them into the cosine rule for spherical trigonometry:

cos(z) = sin(φ)sin(δ) + cos(φ)cos(δ)cos(H)

Then, calculate z by taking the arccosine (inverse cosine) of the result. Finally, the altitude is:

α = 90° – z

Atmospheric Refraction: For practical purposes, especially near the horizon, atmospheric refraction bends sunlight, making the Sun appear slightly higher than it geometrically is. A common correction is to add about 0.5° to the calculated altitude when the Sun is near the horizon (altitude < 10°).

Variables Table:

Variable	Meaning	Unit	Typical Range
α (Altitude)	Angle of the Sun above the horizon	Degrees (°)	-90° to 90° (practically 0° to 90° for visible sun)
z (Zenith Angle)	Angle of the Sun from the overhead point (zenith)	Degrees (°)	0° to 180° (practically 0° to 90° for visible sun)
φ (Latitude)	Observer’s position north or south of the equator	Degrees (°)	-90° (South Pole) to 90° (North Pole)
δ (Declination)	Sun’s angular position relative to the celestial equator	Degrees (°)	~ -23.5° to ~ 23.5°
H (Hour Angle)	Sun’s angular distance from the local meridian	Degrees (°)	-180° to 180° (often considered -90° to 90° for a single day)
N (Day Number)	Ordinal day of the year	Integer	1 to 366

Practical Examples (Real-World Use Cases)

Example 1: Solar Panel Optimization in California

Scenario: A homeowner in Los Angeles, California wants to determine the sun’s peak altitude to optimize the angle of their new solar panels.

Inputs:

Latitude: 34.0522° N
Longitude: -118.2437° W
Date: Summer Solstice (June 21st, 2024)
Time (UTC): 19:00 (which corresponds to 12:00 PM Pacific Daylight Time)

Calculation Steps & Results:

Day Number (N) for June 21st is 173.
Declination (δ) ≈ 23.5° * sin( (360/365.25) * (173 – 81) ) ≈ 23.5° * sin( 92 * 0.9856 ) ≈ 23.5° * sin(90.67°) ≈ 23.45°.
Hour Angle (H) = ( (19 + 0/60) – 12 ) * 15 + (-118.2437) = (7 * 15) – 118.2437 = 105° – 118.2437° = -13.2437°.
cos(z) = sin(34.05°)sin(23.45°) + cos(34.05°)cos(23.45°)cos(-13.2437°)
cos(z) ≈ (0.5597 * 0.3979) + (0.8289 * 0.9174 * 0.9736) ≈ 0.2227 + 0.7392 ≈ 0.9619
Zenith Angle (z) = arccos(0.9619) ≈ 15.8°.
Sun Altitude (α) = 90° – 15.8° = 74.2°.

Interpretation: On the summer solstice around solar noon, the sun reaches a high altitude of approximately 74.2° in Los Angeles. This indicates that solar panels would receive direct sunlight for most of the day, and a relatively steep tilt angle (close to 90° – 74.2° = 15.8°, or adjusted for optimal annual performance) would be beneficial.

Example 2: Shadows for Architectural Design in London

Scenario: An architect needs to know the lowest sun altitude during winter to assess potential shadowing from a neighboring building onto a proposed development in London.

Inputs:

Latitude: 51.5074° N
Longitude: -0.1278° W
Date: Winter Solstice (December 21st, 2024)
Time (UTC): 12:00 (Local solar noon approximation)

Calculation Steps & Results:

Day Number (N) for December 21st is 356.
Declination (δ) ≈ 23.5° * sin( (360/365.25) * (356 – 81) ) ≈ 23.5° * sin( 275 * 0.9856 ) ≈ 23.5° * sin(271°) ≈ -23.45°.
Hour Angle (H) = ( (12 + 0/60) – 12 ) * 15 + (-0.1278) = (0 * 15) – 0.1278 = -0.1278°. (Very close to 0°)
cos(z) = sin(51.51°)sin(-23.45°) + cos(51.51°)cos(-23.45°)cos(-0.1278°)
cos(z) ≈ (0.7826 * -0.3979) + (0.6224 * 0.9174 * 0.9999) ≈ -0.3115 + 0.5712 ≈ 0.2597
Zenith Angle (z) = arccos(0.2597) ≈ 74.9°.
Sun Altitude (α) = 90° – 74.9° = 15.1°.

Interpretation: On the winter solstice around solar noon, the sun reaches a low altitude of only about 15.1° in London. This low angle means the sun will cast long shadows, and any obstructions to the south (like a tall building) could significantly block sunlight, especially during the winter months. Architects must account for this low sun angle and potential overshadowing.

How to Use This Sun Altitude Calculator

Our Sun Altitude Calculator is designed for simplicity and accuracy. Follow these steps to get your results:

Enter Latitude: Input your geographical latitude in decimal degrees. Northern latitudes are positive (e.g., 40.71), and Southern latitudes are negative (e.g., -33.87).
Enter Longitude: Input your geographical longitude in decimal degrees. East longitudes are typically positive (e.g., 139.69) and West longitudes are negative (e.g., -74.01).
Select Date: Choose the specific date for which you need the sun altitude calculation using the date picker.
Enter Time (UTC): Input the time in Coordinated Universal Time (UTC). This is crucial for accurate astronomical calculations. If you know your local time, you’ll need to convert it to UTC.
Click Calculate: Press the ‘Calculate’ button.

Reading Your Results:

Main Result (Sun Altitude): This is the primary output, displayed prominently in degrees (°). It represents the angle of the sun above your horizon at the specified time and location.
Intermediate Values:
- Solar Declination: The sun’s angle relative to the Earth’s equator on that day.
- Hour Angle: How far the sun is from its highest point (local meridian) in angular degrees.
- Zenith Angle: The angle between the sun and the point directly overhead (zenith).
Chart and Table: Visualize how the sun’s altitude (and related angles) change throughout the selected day. The table provides precise data points, while the chart offers a graphical overview.
Assumptions: Note the location, date, time used, and the standard assumption for atmospheric refraction.

Decision-Making Guidance:

Solar Energy: A higher sun altitude generally means more direct sunlight. Use this data to orient solar panels optimally.
Architecture & Landscaping: Understand shadow patterns by knowing the sun’s path. Low winter sun angles are important for passive heating and shadow analysis.
Astronomy: Plan observations based on when celestial objects (like the Sun) are at favorable altitudes.

Key Factors That Affect Sun Altitude Results

Several factors interact to determine the sun’s altitude at any given moment. Understanding these is key to interpreting the calculator’s output:

Latitude: This is arguably the most significant factor. Higher latitudes experience much greater seasonal variations in sun altitude, with the sun being lower in the sky during winter and potentially never setting in summer above the Arctic/Antarctic Circle. Lower latitudes experience less variation and can have the sun directly overhead.
Time of Year (Declination): The Earth’s axial tilt causes the sun’s apparent path across the sky to shift north or south throughout the year. This results in the solar declination angle, which is highest around the summer solstice (+23.5°) and lowest around the winter solstice (-23.5°). This directly impacts the sun’s maximum daily altitude.
Time of Day (Hour Angle): The Earth’s rotation causes the sun’s position to change relative to the local meridian. The hour angle measures this rotation. The sun is highest (lowest zenith angle, highest altitude) when the hour angle is zero (solar noon). As time moves away from solar noon, the hour angle increases, and the sun’s altitude decreases.
Longitude: While latitude determines the general seasonal pattern, longitude primarily affects the *timing* of solar events like sunrise, sunset, and solar noon relative to UTC. It’s crucial for calculating the correct hour angle for a specific moment in UTC.
Atmospheric Refraction: Earth’s atmosphere bends light rays, especially near the horizon. This makes celestial objects appear slightly higher than they geometrically are. For the sun, this effect is about 0.5° when it’s right on the horizon, diminishing as altitude increases. Our calculator may include a standard correction.
Elevation: While not directly used in the standard spherical trigonometry formulas for altitude, the observer’s elevation above sea level can slightly increase the effective horizon and thus the apparent sun altitude, especially in mountainous terrain. This effect is generally minor compared to the other factors.
Seasons: This is a combination of Latitude and Declination. The interaction dictates the overall sun path throughout the year. For instance, at the equator, the sun can be high year-round, while at mid-latitudes, the difference between summer and winter sun angles is dramatic.

Frequently Asked Questions (FAQ)

Q1: What is the difference between Sun Altitude and Sun Azimuth?: Sun Altitude is the vertical angle above the horizon. Sun Azimuth is the horizontal angle along the horizon, typically measured clockwise from North (0° North, 90° East, 180° South, 270° West). They together define the Sun’s position in the sky.
Q2: Why does my calculator give a different result than another source?: Differences can arise from the exact formulas used (e.g., precision of declination calculation), how atmospheric refraction is handled, the exact time standard (UTC vs. local mean time), and slight variations in input values (e.g., precise coordinates for a location).
Q3: Can the sun altitude be negative?: Geometrically, yes. A negative altitude means the sun is below the horizon. This occurs during twilight and nighttime. Our calculator focuses on the visible sun, typically showing altitudes from 0° upwards.
Q4: How do I find my exact latitude and longitude?: You can find your latitude and longitude using online mapping services (like Google Maps), GPS devices, or smartphone map applications. Many allow you to right-click or long-press on a location to get its coordinates.
Q5: Is the time input in Local Time or UTC?: This calculator specifically requires the time in Coordinated Universal Time (UTC). You may need to convert your local time (considering your time zone and any daylight saving adjustments) to UTC before entering it.
Q6: How accurate is the declination formula used?: The formula used (δ ≈ 23.5° * sin( (360/365.25) * (N – 81) )) is a good approximation for general use. More complex astronomical algorithms provide higher precision, accounting for orbital eccentricities and other factors, but this approximation is sufficient for most practical applications like solar panel angling or shadow studies.
Q7: What does it mean if the sun altitude is 90°?: An altitude of 90° means the sun is directly overhead at the zenith. This only occurs at specific locations (between the Tropics of Cancer and Capricorn) on certain dates when the sun’s declination matches the observer’s latitude.
Q8: How does the calculator handle leap years?: The simple declination formula uses 365.25 days per year, which implicitly averages out leap years. For highly precise calculations spanning many years, specific leap year adjustments might be needed, but for a single day’s calculation, this approximation is generally adequate.