3.1 Solar Geometry, Sun Angles & Sun Path Diagrams

Fundamentals of Solar Geometry

Solar geometry forms the foundation of climate-responsive schematic design in the ARE 5.0 Project Planning & Design (PPD) division. To control passive solar heat gain, optimize daylighting autonomy, and reduce mechanical heating and cooling loads, an architect must precisely track the sun's position relative to the building envelope across all hours of the day and seasons of the year.

Celestial Coordinates: Altitude and Azimuth

The position of the sun in the sky vault at any given solar time is defined by two fundamental celestial angles:

1. Solar Altitude ($\beta$ or $h$): The vertical angle measured from the horizontal ground plane up to the sun's position. Altitude ranges from $0^\circ$ when the sun is on the horizon (sunrise and sunset) to a theoretical maximum of $90^\circ$ at the zenith directly overhead.

2. Solar Azimuth ($\phi$ or $A_s$): The horizontal angle representing the sun's compass bearing. In solar architectural engineering, azimuth is conventionalized with True South as $0^\circ$, where morning angles to the east are negative (e.g., $90^\circ\text{ East} = -90^\circ$) and afternoon angles to the west are positive (e.g., $90^\circ\text{ West} = +90^\circ$). In standard civil navigation, azimuth is measured clockwise from True North as $0^\circ$ (North $= 0^\circ$, East $= 90^\circ$, South $= 180^\circ$, West $= 270^\circ$). Architects must verify the coordinate convention in exam prompts.

Earth-Sun Orbital Mechanics and Declination

Earth orbits the sun on an elliptical plane with its rotational axis tilted at an angle of $23.45^\circ$ ($23.5^\circ$) relative to the perpendicular of the orbital plane. As Earth revolves around the sun, the angle between the equatorial plane and the sun's rays—known as the solar declination ($\delta$)—varies continuously throughout the year between $+23.45^\circ$ and $-23.45^\circ$:

• Summer Solstice (June 21 in Northern Hemisphere): Declination reaches its maximum positive value, $\delta = +23.45^\circ$. The Northern Hemisphere tilts toward the sun, resulting in the longest daylight duration and the highest solar noon altitude of the year.

• Winter Solstice (December 21 in Northern Hemisphere): Declination reaches its minimum negative value, $\delta = -23.45^\circ$. The Northern Hemisphere tilts away from the sun, resulting in the shortest daylight duration and the lowest solar noon altitude.

• Equinoxes occur around March 20–21 and September 22–23. Solar declination is near zero. Geometric day and night are approximately equal away from polar conditions, but atmospheric refraction and the finite solar disk make observed sunrise-to-sunset day length differ from exactly 12 hours.

Calculating Solar Noon Altitude

At solar noon the sun crosses the local meridian and ordinarily reaches its highest daily altitude; this is not necessarily the zenith. The general geometric relation is $\beta=90^\circ-|L-\delta|$, where latitude and declination use consistent signed angles. For the northern mid-latitudes in the following table, the familiar seasonal expressions reduce to the forms below. They should not be extended unchanged into the tropics or polar-night conditions.

$\beta_{\text{equinox}} = 90^\circ - L$ $\beta_{\text{summer solstice}} = 90^\circ - L + 23.45^\circ$ $\beta_{\text{winter solstice}} = 90^\circ - L - 23.45^\circ$

---

Reading Sun Path Diagrams

Architects evaluate seasonal solar exposure and site overshadowing using graphical sun path diagrams (solar charts). Two standard cartographic projections are utilized in architectural practice:

1. Stereographic Sun Path Charts (Polar Projections)

A stereographic sun path diagram represents the hemispherical sky dome projected onto a flat circular horizon plane, viewed from directly above:

• Concentric Circles: Represent lines of constant solar altitude, progressing from $0^\circ$ at the outer perimeter (the horizon) inward to $90^\circ$ at the exact center point (the zenith).

• Radial Spokes: Represent solar azimuth lines radiating outward from the center point to compass bearings along the perimeter.

• Curved East-West Lines: Represent the sun's path across the sky on specific dates of the year, with June 21 occupying the path nearest the center (highest altitude) and December 21 occupying the outermost path (lowest altitude).

• Intersecting Hour Loops (Analemmas): Represent hours of the day (e.g., 8:00 AM, 12:00 PM, 4:00 PM). The intersection between a specific date line and hour line identifies the exact altitude and azimuth of the sun at that moment.

2. Cylindrical Sun Path Charts (Rectilinear / Cartesian Projections)

A cylindrical sun path diagram plots the sky dome onto a rectangular Cartesian coordinate grid:

• Vertical Y-Axis: Represents solar altitude ($0^\circ$ to $90^\circ$).

• Horizontal X-Axis: Represents solar azimuth (typically $-180^\circ$ through $0^\circ$ South to $+180^\circ$, or $0^\circ$ North through $180^\circ$ South to $360^\circ$).

• Site Obstruction Masks: Rectilinear charts simplify plotting surrounding site obstructions. An architect records the angular height (altitude) and bearing (azimuth) of adjacent tall buildings, retaining walls, or mature trees using an inclinometer/compass survey. Plotting these obstruction contours onto the rectilinear chart visually outlines the exact hours and months when the proposed building or solar collectors will be in shadow.

---

Solar Profile Angles and Shading Geometry

Designing effective horizontal overhangs requires understanding that the sun rarely strikes a building façade perpendicularly. As a result, horizontal shading devices do not respond to the true solar altitude angle; they respond to the Solar Profile Angle.

True Altitude vs. Solar Profile Angle

• True Solar Altitude ($\beta$): The vertical angle between the sun and the horizontal ground plane.

• Surface-Solar Azimuth Angle ($\gamma$): The angular difference between the solar azimuth ($\phi$) and the orientation of the wall normal ($\psi_{\text{wall}}$): $\gamma = |\phi - \psi_{\text{wall}}|$. When the sun is directly aligned with the wall normal, $\gamma = 0^\circ$.

• Solar Profile Angle ($\Omega$ or POA): The vertical angle of the sun projected onto a vertical plane perpendicular to the window. It represents the apparent angle of the sun as viewed in a section drawing cut through the wall.

The mathematical relationship between true altitude, surface azimuth, and profile angle is expressed by:

$\tan(\Omega) = \frac{\tan(\beta)}{\cos(\gamma)}$

When the sun is perpendicular to the wall ($\gamma = 0^\circ$, $\cos(0^\circ) = 1.0$), the profile angle equals the true altitude ($\Omega = \beta$). However, when the sun strikes the façade obliquely (e.g., $\gamma = 60^\circ$, $\cos(60^\circ) = 0.50$), $\tan(\Omega) = \tan(\beta) / 0.50 = 2 \times \tan(\beta)$, making the profile angle significantly steeper than the true solar altitude.

Horizontal Overhang Sizing for South Glazing

Horizontal overhangs are exceptionally effective on south-facing elevations (within $15^\circ$ of True South) because midday summer sun strikes at high profile angles, while winter sun strikes at low profile angles.

To size a continuous horizontal overhang:

1. Determine the Cutoff Profile Angle ($\Omega_{\text{cutoff}}$): Identify the critical date when 100% shading is required (e.g., late summer, August 21, when cooling degree days peak) and calculate the profile angle at solar noon.

2. Identify the Glazing Height ($H$): Measure the vertical distance from the bottom of the window sill to the underside of the overhang.

3. Calculate Required Overhang Projection Depth ($D$):

$D = \frac{H}{\tan(\Omega_{\text{cutoff}})}$

If the overhang is mounted flush at the window head, direct sun begins striking the glass as soon as the sun dips below the profile angle. In high-performance design, architects introduce an overhang gap (mounting the projection slightly above the window head) to allow warm air trapped under the soffit to escape while preventing the overhang from cutting off high-angle diffuse daylight.

Shading Limitations on East and West Orientations

Horizontal overhangs alone often perform poorly against low-angle east and west sun. Evaluate operable exterior screens, louvers, vegetation and reduced glazing. Vertical fins can help at oblique sun angles but may also be ineffective against sunlight nearly normal to the facade; test the actual sun geometry.

• East/West Shading Solution: Vertical fins, operable louvered shutters, automated exterior fabric drop screens, or minimal fenestration area.

• Vertical Fin Geometry: Vertical fins respond to the surface-solar azimuth angle ($\gamma$). As the sun moves past the fin, the fin's projection casts a horizontal shadow across adjacent glass, cutting off direct beam radiation.

---

Fenestration Performance Metrics & Orientation Strategy

Glazing specification directly governs building envelope heat balance and daylight distribution.

Core Glazing Metrics

• Solar Heat Gain Coefficient (SHGC): The fraction of incident solar radiation admitted through a window assembly, both via direct transmission and through absorption followed by inward release. SHGC is expressed as a dimensionless number between 0.00 and 1.00. Lower values ($0.20 - 0.25$) indicate high solar heat rejection (vital for cooling climates); higher values ($0.45 - 0.60$) indicate high passive solar heat admission (vital for south glass in cold heating climates).

• Visible Transmittance (VT): The fraction of visible spectrum light ($380 - 740\text{ nm}$) transmitted through the glazing, expressed between 0.00 and 1.00. High VT ($>0.50 - 0.65$) enhances daylight autonomy and interior visual quality.

• U-Factor: The overall rate of non-solar conductive/convective heat transfer through the fenestration assembly (expressed in $\text{Btu}/(\text{hr}\cdot\text{ft}^2\cdot^\circ\text{F})$ or $\text{W}/(\text{m}^2\cdot\text{K})$). Lower U-factors indicate superior insulating capability.

• Light-to-Solar-Gain (LSG) Ratio: The ratio of visible transmittance to solar heat gain coefficient:

$\text{LSG} = \frac{\text{VT}}{\text{SHGC}}$

Spectrally selective low-emissivity (low-e) coatings maximize the LSG ratio ($\text{LSG} \ge 1.50 - 2.00$). They transmit visible light while selectively reflecting shortwave infrared (heat) energy, allowing architects to provide naturally daylit spaces without incurring excessive summer chiller cooling penalties.

Façade Orientation Response Matrix

---

Shading Strategies: Fixed, Operable & Vegetative Systems

Exterior vs. Interior Shading

From a thermodynamic perspective, exterior shading devices are vastly superior to interior blinds:

• Exterior shading intercepts sunlight before it enters the glazing and often controls solar load more effectively than an interior shade. The reduction depends on geometry, surface properties, glazing and sun angle; a universal 80–90% rejection factor is not a design input.

• Interior Shading (Blinds, Curtains, Roller Shades): Solar radiation passes through the glass, strikes the interior shading surface, and is absorbed. The blind heats up and re-radiates thermal energy as longwave infrared radiation. Because glass is largely opaque to longwave infrared, this thermal energy is trapped inside the room (the greenhouse effect), imposing a direct convective cooling load on the HVAC system.

Vegetative Microclimate Shading: Deciduous vs. Coniferous

Strategically positioned landscape vegetation provides seasonal microclimatic control:

• Deciduous trees can shade summer exposures and permit more winter sun after leaf fall. Branches still cast shade. Model mature canopy geometry, seasonal density, root zones, water demand and the desired heating/cooling balance instead of applying fixed transmission percentages.

• Evergreen planting can provide year-round shelter or shade where those are project goals. Preserve winter solar access where beneficial and locate windbreaks using the actual winter wind direction. A fixed north/northwest planting instruction is not appropriate for every site or hemisphere.

---

Worked Sizing Scenario: South-Facing Classroom Overhang

Scenario Parameters

An architect is designing an elementary school in Denver, Colorado (Latitude: $40^\circ\text{N}$). The design features south-facing classroom windows that are $6\text{ ft}$ high ($1.83\text{ m}$), with the window head located directly below the structural roof fascia. The design objective is to provide 100% shade at solar noon on the Summer Solstice (June 21) to prevent interior overheating, while permitting substantial winter solar penetration across the lower glass at solar noon on the Winter Solstice (December 21) for passive solar heating.

Step 1: Calculate Solar Noon Altitudes

• Summer Solstice Noon Altitude: $\beta_{\text{summer}} = 90^\circ - 40^\circ + 23.5^\circ = 73.5^\circ$

• Winter Solstice Noon Altitude: $\beta_{\text{winter}} = 90^\circ - 40^\circ - 23.5^\circ = 26.5^\circ$

Because the windows face due South, the surface-solar azimuth angle at solar noon is $\gamma = 0^\circ$. Therefore, the profile angles equal the true solar altitudes: $\Omega_{\text{summer}} = 73.5^\circ$ and $\Omega_{\text{winter}} = 26.5^\circ$.

Step 2: Calculate Required Overhang Projection Depth for Summer Shading

To provide 100% shading across the full window height ($H = 6.0\text{ ft}$) from the window head down to the sill:

$D = \frac{H}{\tan(\Omega_{\text{summer}})} = \frac{6.0}{\tan(73.5^\circ)} = \frac{6.0}{3.376} \approx 1.78\text{ ft} \approx 1\text{ ft } 10\text{ in}$

Step 3: Verify Winter Solstice Solar Penetration

With an overhang depth $D = 1.78\text{ ft}$, determine how far the shadow line extends down from the window head on the winter solstice at solar noon:

$H_{\text{shadow, winter}} = D \times \tan(\Omega_{\text{winter}}) = 1.78 \times \tan(26.5^\circ) = 1.78 \times 0.4986 \approx 0.89\text{ ft} \approx 10.7\text{ in}$

The shadow line extends less than $11\text{ inches}$ down from the head of the $72\text{-inch}$ window, leaving more than $85\%$ of the glass completely bathed in direct passive solar radiation at midday in midwinter.