The Solar Shadow Angle Calculator determines the minimum distance required between adjacent rows of tilted photovoltaic modules to prevent inter-row shading during low-sun winter conditions.

### Solar Geometry & Inter-Row Spacing Formulas Designing commercial rooftop or ground-mount photovoltaic arrays requires balancing solar collection area against inter-row self-shading losses: - Winter Solstice Solar Declination: Uses the annual solar minimum declination angle $\delta = -23.45^\circ$, representing the lowest solar elevation path of the entire year in the Northern Hemisphere. - Solar Noon Elevation: $$\alpha_{\text{noon}} = 90^\circ - \text{Latitude} + \delta$$ - Morning / Afternoon Target Sun Elevation (9am / 3pm): $$\alpha_{\text{active}} = \max(8^\circ, \alpha_{\text{noon}} \times 0.65)$$ - Panel Projected Height ($H$): Vertical elevation of the tilted panel upper edge above the roof surface: $$H = L \times \sin(\text{Tilt Angle})$$ - Panel Ground Footprint ($B$): Horizontal base projection of the tilted panel: $$B = L \times \cos(\text{Tilt Angle})$$ - Shadow Cast Length ($S$): Horizontal ground shadow projected behind the panel at the selected sun elevation: $$S = \frac{H}{\tan(\alpha_{\text{active}})}$$ - Minimum Inter-Row Pitch ($D$): Total center-to-center row spacing needed to eliminate shading at the selected hour: $$D = B + S = L \times \cos(\text{Tilt Angle}) + \frac{L \times \sin(\text{Tilt Angle})}{\tan(\alpha_{\text{active}})}$$ - Ground Cover Ratio (GCR): The proportion of total ground footprint occupied by solar panel aperture area: $$\text{GCR} = \frac{L}{D} \times 100\%$$

### Worked Row Spacing Example Consider an array at **Latitude 38°N**, with **30° panel tilt**, standard **1.75 m modules (400W)**, evaluated at **9am on the winter solstice**: 1. **Solar Elevation Angles**: - $\alpha_{\text{noon}} = 90 - 38 - 23.45 = 28.55^\circ$ - $\alpha_{\text{9am}} = 28.55 \times 0.65 \approx 18.56^\circ$ 2. **Panel Physical Dimensions**: - Vertical Height: $H = 1.75 \times \sin(30^\circ) = 1.75 \times 0.50 = 0.875\text{ m}$ - Horizontal Base: $B = 1.75 \times \cos(30^\circ) = 1.75 \times 0.866 \approx 1.516\text{ m}$ 3. **Shadow Footprint & Minimum Spacing**: - Shadow Length: $S = \frac{0.875}{\tan(18.56^\circ)} \approx \frac{0.875}{0.3357} \approx 2.61\text{ m}$ - Total Row Pitch: $D = 1.516 + 2.61 = 4.13\text{ m}$ - Ground Cover Ratio: $\text{GCR} = (1.75 / 4.13) \times 100\% \approx 42.4\%$

### Interactive 2D Cross-Section Canvas The tool renders an interactive HTML5 canvas diagram displaying the ground plane, panel tilt angle, projected sun ray vector, and ground shadow footprint in real time. All calculations run strictly in your web browser with zero server data transfer.