Formula & Calculator
Solar Altitude Angle
Gives the Sun's angle above the horizon at a given latitude, solar declination, and hour angle from solar noon.
Interpretation
sin(a) = sin(φ) sin(δ) + cos(φ) cos(δ) cos(H). Calculates the Sun's altitude angle at a given latitude, declination, and hour angle. Used in solar energy and navigation.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| a | Solar altitude angle | deg |
| φ | Observer latitude | deg |
| δ | Solar declination | deg |
| H | Hour angle from solar noon | deg |
What it means
The solar altitude angle (a) is the angle of the Sun above the horizon. This formula gives its sine in terms of latitude φ, solar declination δ, and hour angle H (time of day). It is fundamental for solar energy applications (estimating insolation), for shadow calculations, and for navigation (finding direction from the Sun). It also appears in astronomical navigation (sight reduction). Understanding this formula allows engineers, architects, and sailors to predict solar position for optimal design and for wayfinding.
Worked example
Solar Altitude Angle – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| φ (deg) | 40 |
| δ (deg) | 23.44 |
| H (deg) | 0 |
| Parameter | Value |
|---|---|
| φ | 51.5 |
| δ | 10 |
| H | 30 |
Common mistakes
- Solar altitude angle a: The angle of the Sun above the horizon.
- φ: Latitude, δ: declination, H: hour angle – all in degrees.
- sin(a): The result is the sine of the altitude – use arcsin to get the angle.
- Hour angle H: Measured from local noon – negative in the morning, positive in the afternoon.
- Units: Use degrees for all angles – ensure your trig functions are in degrees.
Applications
The solar altitude angle, given by sin(a) = sin(φ)sin(δ) + cos(φ)cos(δ)cos(H), relates the Sun's elevation above the horizon (a) to latitude (φ), solar declination (δ), and hour angle (H). This formula is fundamental for solar energy engineering, architecture, and climatology. It allows the calculation of the Sun's position at any time and location, enabling the design of solar collectors, window shading, and building energy performance. By computing the altitude angle, engineers can estimate the intensity of sunlight and its heat gain. This formula is also used in astronomy for measuring stellar altitudes. Understanding solar altitude is essential for harnessing solar energy and for understanding the Earth's climate.
- Design of solar thermal and photovoltaic systems
- Building design for passive solar heating and daylighting
- Climate modelling and solar radiation estimation
- Astronomical navigation and sextant use
- Agricultural planning for sunlight exposure
Frequently Asked Questions
sin(a) = sin(φ)sin(δ) + cos(φ)cos(δ)cos(H). It gives the altitude (a) of the Sun above the horizon for a given latitude φ, solar declination δ, and hour angle H (measured from solar noon).
- φ – latitude of the observer (degrees).
- δ – solar declination (degrees).
- H – hour angle (degrees), where 0° is solar noon, positive in the afternoon.
Hour angle in degrees = 15° × (local solar time in hours − 12). For example, at 10:00 solar time, H = 15 × (10−12) = −30°. Negative values are before noon.
The altitude is highest at noon (H=0) and decreases as the Sun moves away from the meridian. At sunrise and sunset, a = 0° (the Sun is on the horizon). The formula gives a negative altitude when the Sun is below the horizon.
The maximum altitude at noon (H=0) is a_max = 90° − |φ − δ|. This occurs at the summer solstice for the northern hemisphere when δ is maximal.
When sin(a) < 0, the Sun is below the horizon. The formula still gives a negative altitude. For practical purposes, you can check if a > 0 to determine daytime.
- Forgetting to convert degrees to radians in the trigonometric functions.
- Using clock time instead of local solar time without correcting for longitude and equation of time.
- Not accounting for atmospheric refraction, which raises the apparent Sun by about 0.5°.
It determines the angle of incidence of sunlight on a solar panel. For maximum output, the panel should be oriented perpendicular to the Sun, so the altitude helps in adjusting tilt.
By setting a = 0° and solving for H, you get cos(H) = −tan(φ)tan(δ), which is the hour angle of sunrise/sunset. This is the basis for the day length formula.
The shadow length of an object is inversely proportional to tan(a). When the Sun is low (small altitude), shadows are long; when it is high, shadows are short. This is important in architecture and solar design.