Where Is the Sun? Find the altitude angle and azimuth angle for the sun at 3:00 P.M. solar time in Boulder, Colorado (latitude 40º) on the summer solstice.

Question

Where Is the Sun? Find the altitude angle and azimuth angle for the sun at 3:00 P.M. solar time in Boulder, Colorado (latitude 40º) on the summer solstice.

Accepted Answer

Since it is the solstice we know, without computing, that the solar declination δ is 23.45º. Since 3:00 P.M. is three hours after solar noon, from (7.10) we obtain

[latex]H \ = \ \left(\frac{15º}{h} ight) \ \cdot \ \left( ext{hours before solar noon} ight) \ = \ \frac{15º}{h} \ \cdot \ \left(-3 \ h ight) \ = \ −45º[/latex]

Using (7.8), the altitude angle is

[latex]\begin{matrix} \sin \ \beta & = \ \cos \ L \ \cos \ \delta \ \cos \ H \ + \ \sin \ L \ \sin \ \delta \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \ \ \ & = \ \cos \ 40º \ \cos \ 23.45º \ \cos \left(-45º ight) \ + \ \sin \ 40º \ \sin \ 23.45º \ = \ 0.7527 \ \quad \ \ \beta & = \ \sin^{-1} \left(0.7527 ight) \ = \ 48.8º \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \ \ \end{matrix}[/latex]

From (7.9) the sine of the azimuth angle is

[latex]\begin{matrix} \sin \ \phi _{S} & = \ \frac{\cos \ \delta \ \sin \ H}{\cos \ \beta} \quad \quad \quad \quad \quad \quad \quad \quad \ & = \ \frac{\cos \ 23.45º \ \sin \ \left(-45º ight)}{\cos \ 48.8º} \ = \ −0.9848 \end{matrix}[/latex]

But the arcsine is ambiguous and two possibilities exist:

[latex]\begin{matrix} \phi_{S} & = \ \sin^{-1} \left(−0.9848 ight) \ = \ −80º \quad \left(80º ext{west of south} ight) \ \ \ \ \ \ \ ext{or} & \phi_{S} \ = \ 180 \ - \ \left(-80 ight) \ = \ 260º \quad \quad \left(100º ext{west of south} ight) \end{matrix}[/latex]

To decide which of these two options is correct, we apply (7.11):

[latex] ext{if} \quad \cos \ H \ \geq \ \frac{ an \ \delta}{ an \ L}, \quad ext{then} \ \left|\phi_{S} ight| \ \leq \ 90º; \quad ext{otherwise} \ \left|\phi_{S} ight| \ \gt \ 90º[/latex] (7.11)

[latex]\cos \ H \ = \ \cos \left(-45º ight) \ = \ 0.707 \quad ext{and} \quad \frac{ an \ \delta}{ an \ L} \ = \ \frac{ an \ 23.45º}{ an \ 40º} \ = \ 0.517[/latex]

Since [latex]\cos \ H \ \geq \ \frac{ an \ \delta}{ an \ L}[/latex] we conclude that the azimuth angle is

[latex]\phi_{S} \ = \ -80º \quad \left(80º ext{west of south} ight)[/latex]

Question 7.3: Where Is the Sun? Find the altitude angle and azimuth angle ......

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