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How Do They Get the Solar Zenith Angle?
Ever wonder how they obtain those nasty equations for the Solar Zenith Angle? At
l...
∠ = −
∠HAS is the hour angle h. It is determined by Local Apparent Time(LAT); by
definition, the Solar Zenith Angle is gre...
rotational axis with ∠HAS. The normal vector should then be parallel to axis OS.
To avoid any confusions, we shall do the ...
=
( ) ( )
0
( ) ( )
{ } − ( ∙ ̂ ){ ̂ } =
( − )
0
( − )
−
( ) ( )
0
( ) ( )
=
− ( ) ( )
0
( ) ( )
{ × ̂ } =
̂ ̂
( − ) 0 ( −...
Appendix: How Do They Get the Equation of Time
The Equation of Time tells you how the LAT varies with local mean solar tim...
noon occurs (phase 3). The angular velocity of optical solar movement on the
celestial globe is an equivalent depiction of...
=
−
Where Omega_Earth is the angular velocity of Earth’s rotation and T_Earth is the
period of one rotation. This means th...
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How do they get the solar zenith angle?

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This article shows how to derive the solar zenith angle, an important parameter in solar energy engineering and building energy analysis. It also shows how the "equation of time" is derived, and why the angular velocity of optical solar movement on the celestial globe differs over the entire year.

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How do they get the solar zenith angle?

  1. 1. How Do They Get the Solar Zenith Angle? Ever wonder how they obtain those nasty equations for the Solar Zenith Angle? At least that was what I pondered whenever this subject was brought up at course. Renewable engineers and energy efficiency experts often lack the natural science knowledge to know the premise behind these equations. On the other hand, the astronomy scientists won’t bother to explain something that seems so trivial to them. As an amateur of both fields, I will give my best shot to explain those equations in this article. At this very moment I don’t know the real equation and I’m not sure if my methods are right, but we’ll see if it works out in a moment. So now image you are standing alone on a wide, flat field spreading beyond the horizon. Image the sky is a hemisphere, half of the entire celestial globe. Now this is how the movement of the sun would look like to you: By definition, the Solar Zenith Angle is the ∠ZOS. To obtain it we must know the value of the other two angles: ∠ZOH and ∠HAS. ∠ZOH is determined by the latitude φ you are at and the current solar declination δ. To see why, let us draw a complete celestial globe. We can see from the celestial globe below that
  2. 2. ∠ = − ∠HAS is the hour angle h. It is determined by Local Apparent Time(LAT); by definition, the Solar Zenith Angle is greatest when LAT is 12:00. At that very moment, ∠HAS is zero and the Solar Zenith Angle is simply ∠ZOH. For any given LAT and solar day, we can calculate h by the following equation. ℎ( , ) = ( , ) : Where omega is the angular velocity of the optical solar movement. It varies throughout the year, mostly because 1. the earth does not revolt the sun with constant distance and 2. the inclination between ecliptic circle and earth’s equator. Consequently, every solar day in a year has various length. See the appendix at the very end of this article to learn more about this messy omega issue. But to make life easier, at this moment we can neglect those variations and assume that angular velocity to be uniform. That’s 15 degrees per hour. So we can simplify the equation into: ℎ( ) = 15° × (12 − ) The determination of angle between axis OZ and axis OS can then be done with rotation matrices. Suppose we define a normal vector n parallel to axis OZ, rotate it about y axis with ∠ZOH, and then rotate it about a new x axis parallel to the Earth
  3. 3. rotational axis with ∠HAS. The normal vector should then be parallel to axis OS. To avoid any confusions, we shall do the above calculation step by step. First is the rotation about y axis, yielding the normal vector n’: { } = ( − ) 0 ( − ) 0 1 0 − ( − ) 0 ( − ) { } = ( − ) 0 ( − ) (To work out the above matrix, you might one to think of the 2-D example. Treat the z-axis as the first basis and the x-axis as the second when applying the 2-D rotational matrix.) But we will also have to rotate the original x-axis. The new x-axis will be parallel to the Earth’s rotational axis, so it would be: { ̂ } = ( ) 0 ( ) (If you feel uncomfortable with this result, think this way: The Earth’s rotational axis would be parallel to the horizon when you are on the equator, and perpendicular to that when you at the poles.) Now we shall rotate n’ about the new x-axis. To do that, we split n’ into two components. The first is parallel to the new x-axis, and the second is perpendicular to it. Only the second component of n’ will be affected by the rotation. Thus we shall have { } = ( ∙ ̂ ){ ̂ } + ({ } − ( ∙ ̂ ){ ̂ }) (ℎ) + { × ̂ } ‖ × ̂ ‖ ‖{ } − ( ∙ ̂ ){ ̂ }‖ (ℎ) In which ( ∙ ̂ ){ ̂ } = ( ( − ) ( ) + ( − ) ( )) ( ) 0 ( ) = ( ( ) ( ) − ( ) ( ) ( ) + ( ) ( ) ( ) + ( ) ( )) ( ) 0 ( )
  4. 4. = ( ) ( ) 0 ( ) ( ) { } − ( ∙ ̂ ){ ̂ } = ( − ) 0 ( − ) − ( ) ( ) 0 ( ) ( ) = − ( ) ( ) 0 ( ) ( ) { × ̂ } = ̂ ̂ ( − ) 0 ( − ) ( ) 0 ( ) = ‖ × ̂ ‖ ̂ Thus { } = ( ) ( ) − ( ) ( ) (ℎ) ( ) (ℎ) ( ) ( ) + ( ) ( ) (ℎ) Then the cosine of the Solar Zenith Angle Z is just the cross product of n’’ and n. ( ) = ∙ = ( ) ( ) + ( ) ( ) (ℎ) This result is very useful for solar engineering and building energy analysis. For example, if you want to know when is the sunrise/sunset time, simply plug in the equation and take Z as pi/2: ( ) ( ) + ( ) ( ) (ℎ ) = 0 ℎ = (− ( ) ( ))
  5. 5. Appendix: How Do They Get the Equation of Time The Equation of Time tells you how the LAT varies with local mean solar time (an annual average of LAT). As mentioned earlier, there are two major factors that cause LAT to vary: 1. The earth does not revolt the sun with constant distance. 2. The inclination between ecliptic circle and earth’s equator. These two factors affect the angular velocity of optical solar movement on the celestial globe. To understand why this will also affect LAT, consider the following plot from Wikipedia: The Earth rotates one full round during the time duration between phase one and phase two. However, because the Earth also revolts around the sun when it rotates, it will actually take the Earth to rotate slightly more than one round when the next
  6. 6. noon occurs (phase 3). The angular velocity of optical solar movement on the celestial globe is an equivalent depiction of this. The first factor that affects this angular velocity omega is a consequence of angular momentum conservation, or you might recall it as the second Kepler’s Law from high school physics. It is just saying that ̇ = . The second factor is a little bit more counter intuitive. Since the celestial longitude at which the sun is positioned is what that actually matters, the angular velocity of the optical solar movement must be projected on the equatorial plane. To best comprehend this concept, let us first write down a parameter equation for a circular motion: { ̂} = ( ( )) ( ( )) 0 Now we rotate the circular motion about the y-axis with the inclination between ecliptic circle and earth’s equator, alpha: { ̂′} = ( ) 0 − ( ) 0 1 0 ( ) 0 ( ) { ̂} = ( ) ( ( )) ( ( )) ( ) ( ( )) The velocity of motion, by definition, is { ̂′} = − ( ) ( ) ̇( ) ( ( )) ̇( ) − ( ) ( ) ̇( ) But only the x and y components contribute to longitude changes, thus = − ( ) ( ) ( ) 0 ̇( ) = 1 − ( ) ( ) ̇( ) If we combine the effects of factor one and two, we can obtain the general form of equation of time: = 1 − ( ) ( ) ( ) Where K is just some constant. A solar day is then
  7. 7. = − Where Omega_Earth is the angular velocity of Earth’s rotation and T_Earth is the period of one rotation. This means that the greater omega_sun, the great a solar day is. The effects of sine theta and inverse square distance is drawn below. Their combined effect is the graph we have shown in the beginning. A more rigorous angular velocity of the optical motion of sun on the sky can also be obtained: ( , ) = − This gives the most general form to calculate hour angle: ℎ( , ) = ( , ) :

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