### Reading Assignment

*SECS*,**Chapter 6**: Review both sections of "Moments, Hours, and Days" as well as Sun-Observer Angles."

In reviewing these sections, you should notice that three common angular symbols keep popping up: the declination ($\delta $), the local latitude ($\varphi $), and the hour angle ($\omega $). As we shall again see in the next section, these are three of our key *Earth-Sun angles*.

We additionally include the use of longitude ($\lambda $) in our calculation of time, and in particular, converting time to an angle: the hour angle.

### Now, let's convert "time" into an angle, for our future trigonometric relations:

When we convert time to an angular value, we can no longer use a 24 hr format. We need to convert hourly time into a useful angle based on the properties of a sphere, again using spherical trigonometry.

#### Video: Hour Angles (5:03)

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**Finishing Step for Time Conversions!** The Hour Angle ($\omega $) in decimal degrees.

We represent the apparent displacement of the sun away from solar noon, either as a negative or positive angle. An $\omega $ of zero indicates that the sun is at its highest point for that given day.

- The sign of the hour angle before noon is negative, because we have to count backward from the "zero hour" of solar noon.
- The sign of the hour angle after noon is positive, because we are counting forward from the "zero hour" of solar noon.

Another way of looking at it is the angular difference between the local meridian of the observer/collector and the meridian that the beam of the sun is intersecting at a given moment.

$$\omega =\frac{{360}^{\circ}}{24\text{hr}}({t}_{sol}-12)=\frac{{15}^{\circ}}{\text{hr}}({t}_{sol}-12)$$ $$\omega =\{\begin{array}{c}-0-180\text{degrees},\text{if before noon (morning)}\\ +0-180\text{degrees, if after noon (evening)}\end{array}$$

Now, the day won't really begin at -180 degrees anywhere between the arctic circles. However, I wanted to emphasize that the day only begins at -90° on *two days* a year. Those days occur during the Equinox moments in the orbit of the Earth about the Sun, and the length of those days is actually 12 hours long. All other days are either shorter than 12 hours, or longer than 12 hours. As such, they end either less than 90° or greater than 90°.

The images below demonstrate singular arcs of the Sun for each hour in solar time, at five different latitudes on Earth (no analemma correction necessary). The peak hour (or the hour with the highest solar altitude angle) is defined as solar noon. You will notice that the solar equinox has twelve sun spots for latitudes below the arctic circle, and that the sun rises due East, while setting due West. During the solar solstices, you see multiple arcs: one for winter and one for summer. Notice that there are more hours of the day in the summer, and the sun rises farther from the equator in the Summer (sun rise in the northeast for the Northern Hemisphere).

What else do you notice in comparing the four critical times of year at different latitudes?

#### Solar Equinox

**Solar Solstice**

Now, we are ready to use hour angle to find out positions in our solar design projects!

#### Self-check questions:

Using the information and equations you learned above, calculate the solar angle for the following times.

1. What is the solar angle for 10 am?

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^{o}

2. What is the solar angle for 3 pm?

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^{o}

3. What is the solar angle for noon?

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4. What is the solar angle for 7:30 pm?

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