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Palos Verdes

Theory and data

Insolation

Sun position

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Sun position

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The position of the sun is crucial in determining the amount of energy the house receives in the form of radiation. Therefore, it is essential to understand how it can be modeled based on factors such as latitude, longitude, time of day, and season of the year.

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Points of reference on the surface of the planet

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There are two key points for calculating the position of the sun in the sky. Firstly, there's the geographic position (OP) of a location, which can be defined by its latitude ($\theta_0$, represented by a LEGO® piece in purple) and longitude ($\varphi_0$, represented by a LEGO® piece in dark pink). When representing this graphically, it's important to remember that the axis defining the coordinates is rotated by the angle of the Earth's axis inclination ($\vartheta$, represented by a LEGO® piece in light pink):

The second point is the position on Earth where the sun is directly overhead, known as the subsolar point (SP). It can also be described by a latitude ($\theta_s$, represented by a LEGO® piece in dark blue) and longitude ($\varphi_s$, represented by a LEGO® piece in azure blue) that change with time ($t$, represented by a LEGO® piece in white):

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Calculation of the latitude of the subsolar point

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The subsolar point corresponds to the point on the Earth's surface that is closest to the sun. Due to the planet's axis being inclined at an angle of 23.44 degrees (0.409 radians), the latitude of the subsolar point varies throughout the year.

To represent this graphically, we can imagine a sphere with an axis tilted at the same angle ($\vartheta$, represented by a light pink LEGO® piece) that rotates over the course of the year ($t$, represented by a white LEGO® piece), assuming values for the latitude ($\theta_s$, represented by a light blue LEGO® piece) that can reach a maximum equal to the inclination but not less than the negative value of the inclination:

Since the point of closest approach to the sun occurs at the December solstice, which is approximately 10 days before the start of the year, and the progression is at a constant angular velocity, the latitude of the subsolar point can be calculated using the formula:

$ \theta_s = -0.409 \cos\left(2\pi\displaystyle\frac{ n +10}{365}\right)$

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Effect of the tilt of the earth's axis

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Being able to model how declination affects each location on Earth throughout the year is crucial.

The equation, simply represented by a bright light blue brick and a white one (the latter representing time), provides us with the zenith angle extremes. In summer, when our hemisphere is closer to the Sun,

the maximum zenith angle is equal to the sum of the latitude and inclination. In Palos Verdes, this corresponds to 63.37°.

On the other hand, in winter, when the hemisphere is farther from the Sun,

the maximum zenith angle is obtained by subtracting the inclination angle from the latitude. In Palos Verdes, this corresponds to 16.49°.

In this way, the movement of the Sun corresponds to circles around the Earth's axis that vary in zenith angle throughout the year. Similarly, the times when the Sun rises and sets shift across the horizon, as shown in the following diagram:

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Calculation of the hour angle

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To calculate the longitude of the subsolar point $SP$ ($\varphi_s$, represented by a light blue LEGO® piece), we can examine the difference between this longitude and that of the geographical location ($\varphi_o$, represented by a dark pink LEGO® piece). More than calculating the longitude of the subsolar point itself, what matters is the difference in the position of this point concerning longitude, and, therefore, the angle that is either missing or exceeding compared to the situation where the sun is at its highest point depending on the time of day or time ($t$, represented by a white LEGO® piece):

Therefore, the angle can be calculated using the following formula:

$ \varphi_s - \varphi_o = 2\pi \displaystyle\frac{ t }{24\cdot 60\cdot 60}$

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Calculation of solar azimuth angle

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The solar azimuth angle can be estimated using the following approximate equation:

$ \sin \varphi_a = - \displaystyle\frac{\sin( \varphi_s - \varphi_o )\cos \vartheta }{\sin \theta_z }$

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Calculation of zenith angle

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The latitude and longitude of a location allow us to describe the direction of the zenith with respect to the center of the Earth.

Meanwhile, using the angles of inclination and hour angle, we can find the direction of the subsolar point, which can be moved parallel to the location under study.

With these two directions, we can easily calculate the angle between them, giving us an equation that depends on the four angles mentioned earlier. However, since the angles describing the position of the subsolar point depend on time, we can express the zenith angle in terms of latitude, longitude, and time.

In mathematical terms, the direction of the subsolar point ($SP$) has a latitude $\theta_s$ and a longitude $\varphi_s$. The plane of the equator is inclined relative to the plane of the ecliptic (the plane of Earth's orbit around the Sun) by an angle $\vartheta$, which corresponds to the latitude of the point $OP$ $\theta_o$.

By considering both directions, we can calculate the cosine of the zenith angle as the dot product between the two directions:

$\cos \theta_z = \sin \theta_s \sin \vartheta + \cos \theta_s \cos \vartheta \cos( \varphi_s - \varphi_o )$

The vector indicating the position of $OP$ is given by:

$\vec{OP} = \cos\theta_s\cos\varphi_s\hat{x}+\cos\theta_s\sin\varphi_s\hat{y}+\sin\theta_s\hat{z}$

While the point $SP$ is located at:

$\vec{SP} = \cos\theta_o\cos\varphi_o\hat{x}+\cos\theta_o\sin\varphi_o\hat{y}+\sin\theta_o\hat{z}$

The cosine of the zenith angle is obtained as the dot product of these two vectors:

$\cos\theta_z=\vec{OP}\cdot\vec{SP}$

Therefore, we have:

$\cos(\varphi_s-\varphi_o)=\cos\varphi_s\cos\varphi_o+\sin\varphi_s\sin\varphi_o$

and

$\theta_o = \vartheta$

Resulting in:

$\cos \theta_z = \sin \theta_s \sin \vartheta + \cos \theta_s \cos \vartheta \cos( \varphi_s - \varphi_o )$

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Instructions for calculating the zenith angle and azimuth

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From the latitude and longitude of a location on Earth and the position of the subsolar point, you can calculate the direction in which the sun will be seen. To do this, we first calculate how the latitude of the subsolar point varies and the solar hour angle, which indicates how the subsolar point moves in longitude relative to the location on Earth being considered. With this information, we can calculate the zenith and azimuth angles, allowing us to formulate a vector pointing towards the sun:

In this case, the following calculations can be identified:

1. Latitude of the subsolar point

2. Solar hour angle

3. Zenith angle

4. Azimuth angle

5. Vector pointing in the direction of the sun.

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