Demystifying Solar Position and Sunrise/Sunset Calculations

The accurate determination of solar position is essential in various fields, including solar energy production, architecture, agriculture, and even photography. By understanding the position of the sun at any given time and location, we can optimize the efficiency of solar panels, design buildings with better natural lighting, plan crop planting schedules, and capture stunning sunrises and sunsets in photography.

In this blog post, we will cover the necessary equations and calculations for determining solar positions and predicting sunrise and sunset times. With these tools, you will be well-equipped to apply this knowledge in various applications and make more informed decisions based on the sun’s position.

Solar Position Calculations

Calculating the fractional year (γ)

Definition and Purpose

The fractional year ($\gamma$) represents a specific moment in the year as an angular measurement in radians. This value serves as a critical input for solar position algorithms, enabling accurate calculations of:

• Solar declination angle
• Equation of time
• Solar zenith and azimuth angles
• Solar radiation intensity

The fractional year effectively translates calendar dates into a continuous mathematical value that accounts for Earth’s position in its orbit around the Sun.

Mathematical Formulation

For a non-leap year (365 days), the fractional year is calculated using:

$$\gamma = \frac{2\pi}{365} \times (d-1+\frac{h-12}{24})$$

For a leap year (366 days), the formula is adjusted to:

$$\gamma = \frac{2\pi}{366} \times (d-1+\frac{h-12}{24})$$

Where:

• $d$ = day of year (1-365 or 1-366)
• $h$ = hour of the day in 24-hour format (0-23)
• $\gamma$ = fractional year in radians (0-2π)

Components Explanation

• $\frac{2\pi}{365}$ (or $\frac{2\pi}{366}$): Converts the day count to radians, representing a full orbital cycle
• $(d-1)$: Adjusts the day count to start from 0 (January 1 becomes day 0)
• $\frac{h-12}{24}$: Provides hourly precision by adjusting for the time of day relative to solar noon

Example Calculations

1. January 1 at 12:00 noon (non-leap year):
   • $d = 1$, $h = 12$
   • $\gamma = \frac{2\pi}{365} \times (1-1+\frac{12-12}{24}) = 0$ radians
2. July 2 at 15:00 (non-leap year):
   • $d = 183$, $h = 15$
   • $\gamma = \frac{2\pi}{365} \times (183-1+\frac{15-12}{24}) = \frac{2\pi}{365} \times 182.125 \approx 3.14$ radians
3. December 31 at 23:00 (leap year):
   • $d = 366$, $h = 23$
   • $\gamma = \frac{2\pi}{366} \times (366-1+\frac{23-12}{24}) = \frac{2\pi}{366} \times 365.458 \approx 6.28$ radians

Implementation Considerations

1. When implementing this calculation in code, determine if the current year is a leap year
2. Calculate the day of year from month and day

The fractional year calculation serves as the foundation for subsequent solar position calculations, including the equation of time and solar declination angle, which are essential for precise solar engineering applications.

Equation of Time

The equation of time represents the discrepancy between apparent solar time (measured by sundials) and mean solar time (measured by clocks). This variation occurs due to two primary astronomical phenomena:

1. The Earth’s elliptical orbit around the Sun
2. The 23.44° tilt of Earth’s rotational axis

This results in the Sun appearing to move faster or slower across the sky throughout the year compared to a constant 24-hour solar day. The equation of time is typically expressed in minutes.

Earth’s elliptical orbit

Solar Declination Angle

The solar declination angle ($\delta$) is the angle between the Sun’s rays and the Earth’s equatorial plane. This angle varies seasonally due to Earth’s axial tilt as it orbits the Sun:

• Summer solstice (Northern Hemisphere): $\delta = +23.44°$
• Winter solstice (Northern Hemisphere): $\delta = -23.44°$
• Equinoxes: $\delta = 0°$

Solar declination angle

Mathematical Formulation

For precise calculations, we use the fractional year ($\gamma$) as input, measured in radians:

$$\gamma = \frac{2\pi}{365}(d-1+\frac{h-12}{24})$$

Where:

• $d$ = day of year [1-365]
• $h$ = hour of the day [0-23]

Equation of Time Formula

The equation of time (in minutes) can be calculated using:

$$E_t = 229.18 \times (0.000075 + 0.001868\cos\gamma – 0.032077\sin\gamma – 0.014615\cos2\gamma – 0.040849\sin2\gamma)$$

Solar Declination Angle Formula

The solar declination angle (in radians) can be calculated using:

$$\delta = 0.006918 – 0.399912\cos\gamma + 0.070257\sin\gamma – 0.006758\cos2\gamma + 0.000907\sin2\gamma – 0.002697\cos3\gamma + 0.00148\sin3\gamma$$

Applications

These calculations are fundamental for:

1. Solar energy systems positioning
2. Building energy simulations
3. Solar radiation modeling
4. Agricultural planning
5. Climate studies

Example Calculation

For March 17 (day 76 of the year) at 12:00 noon:

$\gamma = \frac{2\pi}{365}(76-1+\frac{12-12}{24}) = 1.296$ radians

Equation of time:
$E_t = 229.18 \times (0.000075 + 0.001868\cos(1.296) – 0.032077\sin(1.296) – 0.014615\cos(2 \times 1.296) – 0.040849\sin(2 \times 1.296)) = -9.1$ minutes

Solar declination:
$\delta = 0.006918 – 0.399912\cos(1.296) + 0.070257\sin(1.296) – 0.006758\cos(2 \times 1.296) + 0.000907\sin(2 \times 1.296) – 0.002697\cos(3 \times 1.296) + 0.00148\sin(3 \times 1.296) = -0.04$ radians ($\approx -2.3°$)

These values indicate that on March 17, a sundial will be approximately 9.1 minutes behind a clock, and the Sun’s rays make an angle of about -2.3° with the Earth’s equatorial plane.

Calculating true solar time

Definition and Importance

True Solar Time (TST) represents the actual position of the Sun in the sky relative to a specific Earth location. Unlike standard clock time, which divides each day into uniform 24-hour periods, true solar time reflects the Sun’s apparent movement across the local meridian. This measurement is fundamental for:

• Solar energy applications
• Photovoltaic system optimization
• Building energy performance analysis
• Daylighting studies
• Agricultural planning

At true solar noon, the Sun reaches its highest position in the sky, directly above the observer’s local meridian.

Mathematical Formulation

True solar time calculations require several inputs and intermediate steps:

Time Offset Calculation

The time offset (in minutes) between local standard time and true solar time is calculated as:

$$\text{time_offset} = E_t + 4 \times \lambda – 60 \times \text{TZ}$$

Where:

• $E_t$ = equation of time (minutes)
• $\lambda$ = observer’s longitude (degrees, positive eastward)
• $\text{TZ}$ = timezone offset from UTC (hours)

The factor of 4 converts longitude degrees to minutes of time (since Earth rotates at 15° per hour, or 1° per 4 minutes).

True Solar Time Calculation

Once the time offset is determined, true solar time (in minutes since midnight) is calculated as:

$$\text{TST} = h \times 60 + m + \frac{s}{60} + \text{time_offset}$$

Where:

• $h$ = hour in 24-hour format (0-23)
• $m$ = minutes (0-59)
• $s$ = seconds (0-59)

Solar Hour Angle

From true solar time, the solar hour angle ($\omega$) in radians can be derived:

$$\omega = (\text{TST}/4 – 180) \times \frac{\pi}{180}$$

The solar hour angle represents the angular displacement of the Sun east or west of the local meridian due to Earth’s rotation, with morning being negative and afternoon being positive.

Example Calculation

For a location at longitude 2.15° E, in the CET timezone (UTC+1), on March 17 at 14:30:00, with equation of time = -9.1 minutes:

1. Calculate time offset:

$$\text{time_offset} = -9.1 + 4 \times 2.15 – 60 \times 1 = -9.1 + 8.6 – 60 = -60.5 \text{ minutes}$$

2. Calculate true solar time:

$$\text{TST} = 14 \times 60 + 30 + \frac{0}{60} + (-60.5) = 840 + 30 – 60.5 = 809.5 \text{ minutes}$$

This equals 13 hours, 29.5 minutes in true solar time.

3. Calculate solar hour angle:

$$\omega = (809.5/4 – 180) \times \frac{\pi}{180} = (202.375 – 180) \times \frac{\pi}{180} = 22.375 \times \frac{\pi}{180} = 0.39 \text{ radians}$$

Practical Considerations

1. Longitude Correction: The 4-minute correction per degree of longitude accounts for Earth’s rotation rate (15° per hour).
2. Daylight Saving Time: When daylight saving time is in effect, add 60 minutes to the timezone offset.
3. Solar Noon: True solar noon occurs when TST = 12 hours (720 minutes).
4. Diurnal Arc: The period when the Sun is above the horizon can be determined using true solar time in conjunction with the solar declination angle.

True solar time forms the foundation for calculating the precise position of the Sun in the sky, which is essential for optimizing solar energy systems and designing energy-efficient buildings.nd, we can proceed to calculate other essential solar parameters, such as the solar hour angle, and ultimately the solar zenith and azimuth angles.

Finding the solar hour angle

The solar hour angle is a measure of the sun’s position in the sky relative to the observer’s local meridian. It is expressed in degrees and represents the elapsed time (in hours) since solar noon, with each hour corresponding to 15 degrees of rotation. The solar hour angle is an essential parameter for calculating the solar zenith and azimuth angles, which provide the precise position of the sun in the sky at a given location and time.

The equation for the solar hour angle (ha) is as follows:

ha = (tst / 4) – 180

In this equation, tst is the true solar time calculated in the previous section, expressed in minutes. The solar hour angle (ha) ranges from -180 degrees at solar midnight to +180 degrees at solar noon.

By determining the solar hour angle, we can use it along with other parameters such as the solar declination angle and the observer’s latitude to calculate the solar zenith and azimuth angles, providing a complete understanding of the sun’s position in the sky.

Calculating solar zenith and azimuth angles

The solar zenith angle (Φ) is the angle between the sun’s rays and a line perpendicular to the Earth’s surface (the zenith) at a specific location and time. It is measured from directly overhead (0 degrees) to the horizon (90 degrees). The solar zenith angle provides information about the sun’s elevation in the sky and is essential for various applications, including solar energy production and architectural design.

The solar azimuth angle (θ) represents the sun’s position in the sky relative to the observer’s location, measured in degrees clockwise from the north. This angle helps to determine the direction of the sun in the sky and is crucial for accurately positioning solar panels and understanding the sun’s path throughout the day.

Using the solar hour angle (ha), latitude (lat), and solar declination angle (decl) calculated in the previous sections, we can find the solar zenith angle (Φ) using the following equation:

cos(Φ) = sin(lat) * sin(decl) + cos(lat) * cos(decl) * cos(ha)

To calculate the solar azimuth angle (θ), we use the following equation:

cos(180 – 𝜃) = – (sin(lat) * cos(Φ) – sin(decl) * cos(lat) * sin(Φ)) / (cos(decl) * cos(Φ))

By determining both the solar zenith and azimuth angles, we obtain a complete understanding of the sun’s position in the sky at any given location and time, which is essential for various applications, such as optimizing solar panel orientation and predicting sunlight patterns in architectural design.

Sunrise and Sunset Calculations

Setting the zenith for sunrise/sunset

When calculating sunrise and sunset times, it is necessary to adjust the solar zenith angle to account for specific factors that influence the moment the sun appears or disappears below the horizon. The solar zenith angle is set to 90.833° for sunrise and sunset calculations. This value represents a correction that accounts for two main factors:

• Atmospheric refraction: As sunlight passes through the Earth’s atmosphere, its path is bent, causing the sun to appear slightly higher in the sky than its actual position. The bending effect is more significant when the sun is close to the horizon, such as during sunrise and sunset. The 0.833° correction accounts for this refraction effect.
• Solar disk size: The sun appears as a disk in the sky, rather than a single point of light. As a result, sunrise is defined as the moment when the upper edge of the sun’s disk appears above the horizon, and sunset is when the upper edge disappears below the horizon. The 0.000° correction accounts for the sun’s angular diameter, which is approximately 0.53°.

By setting the zenith to 90.833° for sunrise and sunset calculations, we can accurately determine the time at which these events occur, accounting for atmospheric refraction and the sun’s angular diameter.

Calculating the hour angle for sunrise/sunset

To calculate the hour angle for sunrise and sunset, we use the following equation:

ha = ± arccos{ (cos(90.833) / (cos(lat) * cos(decl))) – (tan(lat) * tan(decl)) }

In this equation, lat represents the observer’s latitude, and decl represents the solar declination angle. The positive value of the hour angle (ha) corresponds to sunrise, while the negative value corresponds to sunset.

Determining the UTC time of sunrise and sunset

To determine the UTC time of sunrise (or sunset), we use the following equation:

sunrise = 720 – 4 * (longitude + ha) – eqtime

In this equation, longitude represents the observer’s longitude in degrees (positive to the east of the Prime Meridian), ha is the hour angle calculated in the previous step, and eqtime is the equation of time (in minutes). For calculating the time of sunset, the same equation is used, with the negative value of ha.

The importance of longitude, hour angle, and equation of time in the calculation lies in their contributions to accurately determining the sun’s position relative to the observer’s location on Earth. Longitude helps account for the Earth’s rotation, ensuring the calculation is specific to the observer’s geographic location. The hour angle represents the elapsed time since solar noon, which is crucial for determining the sun’s position in the sky. The equation of time accounts for variations in the Earth’s orbit and axial tilt, which affect the sun’s apparent motion in the sky. By incorporating these factors into the calculation, we can accurately determine the UTC time of sunrise and sunset for any location on Earth.

It is important to note that the result of the equation is given in UTC time, so you may need to convert it to your local time, considering daylight saving time if applicable, to obtain the exact time of sunrise or sunset for your specific location.

In summary, the combination of longitude, hour angle, and equation of time in the sunrise and sunset calculations provides a precise understanding of these events, which is essential for various applications, such as optimizing solar energy production, planning outdoor activities, or predicting daylight hours for agricultural purposes.

Calculating solar noon

Solar noon is the moment when the sun is at its highest point in the sky for a specific location, directly above the local meridian. It is the time when the sun is precisely due north or south (depending on the observer’s latitude) and when shadows are the shortest. Solar noon is essential for various applications, such as solar panel alignment and architectural design, as it represents the sun’s peak daily elevation.

The equation for solar noon (snoon) is as follows:

snoon = 720 – 4 * longitude – eqtime

In this equation, longitude represents the observer’s longitude in degrees (positive to the east of the Prime Meridian), and eqtime is the equation of time (in minutes). The result indicates the time of solar noon in UTC, which can be converted to local time as needed.

Conclusion

Understanding solar position and sunrise/sunset calculations is vital for numerous applications, such as optimizing solar energy production, architectural design, agriculture, and outdoor event planning. The equations and concepts provided in this blog post allow you to accurately determine the sun’s position in the sky and the timing of sunrise, sunset, and solar noon for any location on Earth.

We encourage readers to use these equations and concepts in their own projects or research, as they offer valuable insights into the sun’s behavior and its impact on our daily lives. Accurate solar position and sunrise/sunset calculations can significantly benefit engineering fields related to solar energy, such as photovoltaic system design and solar thermal applications, by enabling the optimization of solar panel orientation, maximizing energy capture, and improving overall system efficiency. Additionally, these calculations can contribute to architectural design, allowing for better use of natural light and passive solar heating, which can lead to energy savings and enhanced occupant comfort.