Solar Angles – Fundamentals of Solar Positioning

In today’s rapidly evolving renewable energy landscape, optimizing solar panel installation is critical for maximizing energy production. Two fundamental parameters determine how effectively your solar panels capture sunlight: orientation and tilt. These factors, when properly calculated, can significantly increase your system’s efficiency and return on investment.

Orientation refers to the directional facing of your panels relative to the sun’s path (typically measured as azimuth angle), while tilt represents the angular position relative to the horizontal ground. Together, these parameters determine how directly sunlight strikes your panels throughout the day and across seasons.

The Science Behind Solar Angles

Solar energy optimization relies on understanding several interconnected angular relationships:

Solar Declination (δ)

The Earth’s axial tilt of 23.45° relative to its orbital plane creates a daily variation in the angle between the Earth-Sun line and Earth’s equatorial plane. This angle, known as solar declination (δ), can be calculated using:

δ = 23.45° × sin[360° × (284 + N)/365]

Where N represents the day of year (with January 1 = 1).

Solar Position Coordinates

The sun’s position is defined by:

• Solar altitude (β): Angle between the sun and the horizontal plane
• Solar azimuth (ϕ): Angular displacement from true south measured in the horizontal plane

Surface-Solar Relationships

For optimal energy capture, we must determine the angle of incidence (θ) between direct solar radiation and the panel surface normal. This depends on:

• Surface tilt angle (Σ): Measured from horizontal
• Surface azimuth (ψ): Direction the surface faces relative to true south
• Surface-solar azimuth (γ): Angular difference between solar azimuth and surface azimuth

Calculating Optimal Panel Positioning

The angle of incidence (θ) for any surface with tilt angle Σ can be determined by:

cos θ = cos β × cos γ × sin Σ + sin β × cos Σ

For specific surface types:

Vertical Surfaces (Σ = 90°)

cos θ = cos β × cos γ

Horizontal Surfaces (Σ = 0°)

cos θ = sin β

Advanced Considerations for Solar Tracking

The sun’s position at any hour (τ) is expressed through the hour angle (ω):

ω = 15° × (τ – 12)

Where τ represents the solar time in hours, with morning hours negative and afternoon hours positive.

Solar altitude can be determined with:

sin β = sin δ × sin φ + cos δ × cos φ × cos ω

Where φ represents latitude.

More deep on solar angles?

The axis about which the earth rotates is tilted at an angle of 23.45 degrees to the plane of the earth’s orbital plane and the sun’s equator. The earth’s axis results in a day-by-day variation of the angle between the earth–sun line and the earth’s equatorial plane called the solar declination δ. This angle may be estimated by the following equation:

$$\delta = 23.45 \sin\left[\frac{360}{365}(284 + N)\right]$$

where N = year day, with January 1 + 1

To determine the angle of incidence θ between a direct solar beam and the normal to the surface, the surface azimuth ψ and the surface-solar azimuth γ must be known. The surface-solar azimuth is designated by γ and is the angular difference between the solar azimuth ϕ and the surface azimuth ψ. For a surface facing the east of south, γ = ϕ − ψ in the morning, and γ = ϕ + ψ in the afternoon. For surfaces facing the west of south, γ = ϕ + ψ in the morning and γ = ϕ − ψ in the afternoon. For south-facing surfaces, ψ = 0 degree, so γ = ϕ for all conditions. The angles δ, β, and ϕ are always positive.

Table 1: Solar Declination Values Throughout the Year (2025)

Month	Representative Day	Day Number (N)	Declination (δ)
January	15th	15	-21.27°
February	15th	46	-13.28°
March	15th	74	-2.82°
April	15th	105	9.41°
May	15th	135	18.79°
June	15th	166	23.31°
July	15th	196	21.52°
August	15th	227	13.78°
September	15th	258	2.22°
October	15th	288	-9.97°
November	15th	319	-19.15°
December	15th	349	-23.34°

Table 2: Optimal Fixed Tilt Angles by Latitude

Latitude (°N)	Optimal Year-Round Tilt (°)	Winter Optimal Tilt (°)	Summer Optimal Tilt (°)
0 (Equator)	0	15	15
10	10	25	5
20	20	35	5
30	30	45	15
40	40	55	25
50	50	65	35
60	60	75	45

Table 3: Surface-Solar Azimuth (γ) Calculation Reference

Surface Orientation	Morning Calculation	Afternoon Calculation
East of South	γ = ϕ – ψ	γ = ϕ + ψ
West of South	γ = ϕ + ψ	γ = ϕ – ψ
Directly South	γ = ϕ (ψ = 0°)	γ = ϕ (ψ = 0°)

Table 4: Solar Position Angles at Different Hours (Example for 40°N Latitude on May 15, 2025)

Solar Time (hr)	Hour Angle (ω)	Solar Altitude (β)	Solar Azimuth (ϕ)
6:00	-90°	8.7°	-110.8°
8:00	-60°	28.7°	-88.7°
10:00	-30°	48.0°	-53.8°
12:00 (noon)	0°	58.8°	0.0°
14:00	30°	48.0°	53.8°
16:00	60°	28.7°	88.7°
18:00	90°	8.7°	110.8°

Table 5: Estimated Energy Production Ratio by Orientation and Tilt (Normalized to Optimal)

Panel Orientation	Panel Tilt	Annual Energy Ratio	Summer Energy Ratio	Winter Energy Ratio
South	Latitude	1.00	0.98	1.00
South	Latitude-15°	0.98	1.00	0.93
South	Latitude+15°	0.97	0.93	1.00
East	Latitude	0.85	0.87	0.81
West	Latitude	0.85	0.87	0.81
SE/SW	Latitude	0.95	0.95	0.94
Horizontal	0°	0.89	0.95	0.76

Table 6: Solar Tracking System Comparison

Tracking System Type	Energy Gain vs. Fixed	Complexity	Maintenance Requirements	Relative Cost
Fixed (optimized)	Baseline	Low	Minimal	Low
Single-axis (E-W)	+25-35%	Medium	Moderate	Medium
Single-axis (N-S)	+15-20%	Medium	Moderate	Medium
Dual-axis	+35-45%	High	Significant	High
Seasonal manual adjustment	+4-8%	Low	Low (quarterly)	Very Low

These tables provide critical reference data for solar system design, allowing for quick evaluation of optimal configurations based on geographic location, installation constraints, and seasonal performance requirements.

Practical Applications

The position of the sun can be defined in terms of its altitude β above the horizon and its azimuth ϕ measured in horizontal plane.

Understanding these relationships allows us to:

1. Determine optimal fixed panel positions based on location-specific parameters
2. Calculate seasonal adjustments to maximize year-round production
3. Evaluate potential benefits of tracking systems versus fixed installations
4. Estimate energy production throughout different times of day and year