From my home on the west bank of the Columbia River in Richland, Washington (46.3°N
latitude), I have an unobstructed view of the sunrise along the east bank during an entire
year. As a retirement project after a career of teaching and research in physics, I
decided to take photographs of the sunrise for a year and, and at the same time, carry out
calculations to determine * 1)* the hours of daylight on a
given day and

*the elevation angle of the Sun at noon. My goal is to use geometrical diagrams along with a simplified model so that results are very easy to understand. In this model, the spherical Earth revolves around the Sun in a circular orbit, and the parallel sunbeam striking Earth is directed along a line connecting their centers. However, Earth’s axis of rotation maintains the same direction throughout its motion around the Sun, which causes the seasons and the variations that we observe.*

**2)**From my home on the west bank of the Columbia River in Richland, Washington (46.3°N
latitude), I have an unobstructed view of the sunrise along the east bank during an entire
year. As a retirement project after a career of teaching and research in physics, I decided to
take photographs of the sunrise for a year and, and at the same time, carry out calculations
to determine * 1)* the hours of daylight on a given day and

*the elevation angle of the Sun at noon. My goal is to use geometrical diagrams along with a simplified model so that results are very easy to understand. In this model, the spherical Earth revolves around the Sun in a circular orbit, and the parallel sunbeam striking Earth is directed along a line connecting their centers. However, Earth’s axis of rotation maintains the same direction throughout its motion around the Sun, which causes the seasons and the variations that we observe.*

**2)**My article entitled “95 Sunrises along the Columbia Riverbank” has recently been published on *Sky and Telescope’*s website Stargazers Corner,^{1} where collages of the sunrise photos show the path of the
sunrise along the east bank. The distance of a sunrise on a given date from that on December
21 shows a sine wave behavior that is expected for such a periodic occurrence.

My calculations have been completed using the simplified model. While the direction of the Sun’s rays on Earth does change gradually by 0.986° (360°/365 d) during 24 h, the model further assumes that the direction of the Sun’s rays on Earth remains constant during a single day, but changes by 0.986° each day. This paper will show how information from two geometrical diagrams can be used to determine the number of daylight hours (defined as the time between sunrise and sunset) and the elevation angle of the Sun at noon. These calculations show a small difference of about 0.2% to 2%, when compared with precisely calculated values available on the internet that include many small corrections to the basic theory. Trigonometry is the highest mathematical level in this paper.

A literature search was carried out to find related articles and texts. Many websites on the
internet provide a great deal of information about the times of sunrise and sunset for a given
location and about the solar elevation angle at noon at any time and location. The algorithms
used to determine these values are explained by Meeus.^{2} Refs. 2–4 also describe
algorithms for very precise calculations should the reader wish a comparison.

## Some basics

Figure 1 shows the traditional view of Earth’s
counterclockwise path around the Sun with a top view of the Northern Hemisphere, but with
some embellishments. To indicate the tilt of Earth’s axis of rotation, the North Pole is
indicated off-center with a solid black circle, while the South Pole (on the underside) is
inferred with a prominent black X. Note that the axis of rotation, passing through the two
poles and Earth’s center, does not change direction as Earth revolves around the Sun.^{5} This has some important consequences, such as
the seasons of the year and the differing number of hours between sunrise and sunset.
Immediately, you see that the North Pole is tilted away from the Sun on December 21 and
toward it on June 21.

Realistically, the size of Earth relative to the Sun in Fig. 1 should be about the size of a pinpoint or a period since the radius of the Sun
is 109 times the radius of Earth. Therefore, the rays that strike Earth, indicated by vector ** S**, must be along the line connecting the center of Earth
and Sun because the rays in any other direction will simply not strike Earth! These rays
form a broad parallel beam, as indicated by a rectangular patch of light approaching Earth
on December 21. As seen from Earth, this patch has a large size and engulfs one hemisphere
of Earth, with the other one in darkness.

^{6}The circular boundary between the hemispheres is indicated by asterisks in figures throughout this paper, but only two asterisks are used here. Note that the sunlight strikes a slightly different part of Earth each day and causes the seasons. For example, equinoxes on March 21 and September 21 have 12 h of daylight because the sunlight illuminates both poles.

As discussed in the introduction, the model for the calculations assumes that ** S** is constant during one day and changes by 0.986° each
day. This means that the patch of sunlight in Fig. 1 moves forward but does not move “upward” (along the +

*y*

_{OP}axis) or “downward” during one day. Thus, we view the sunlight and darkness as remaining fixed in space for one day with Earth rotating

*through*it—sometimes in sunlight and sometimes in darkness. Our goal is to determine the fraction of the time in sunlight as Earth rotates about its axis 360° in one day.

Conceptually, the calculations are easier to develop if we view Earth as static with the
Sun revolving around it, as shown in Fig. 2. Note that,
on any given day, the ** S** vector has the same direction as in
Fig. 1. The direction of vector

**is designated by the angle kappa (Κ) with respect to the negative x**

*S*_{OP}axis. Since Κ = 45° on February 4, this date occurs 45°/(0.986° per day), or 46 d, after December 21.

## Calculation of the daylight hours in Richland on December 21

*Realizing that the sunlight and darkness in Fig. 3 remain stationary during 24 h (for the simplified model), while Earth rotates
once about its axis during that time, is essential to understanding the varying hours of
daylight.* You can easily see in Fig. 3(a) that, due to the tilt of 23.5°, the fraction of the time in the sunlight depends on the
observer’s latitude, with the hours of sunlight increasing at more southern latitudes. On
December 21, the North Pole is in complete darkness, the equator has 12 h of daylight, and
the South Pole has 24 h of sunlight. When it is winter in the Northern Hemisphere, it is
summer in the Southern Hemisphere, and vice versa. These facts are quite familiar to us.

Next, we want to determine from Fig. 3(b) the fraction of 24 h that an observer in Richland, Washington (46.3°N latitude), spends in daylight on December 21. Viewing a model of Earth, as shown in Fig. 4, is a helpful first step in that direction.

*D*, and establishes the location of the asterisks. The larger circle indicates regions of sunlight and darkness that strike Earth. Although these regions are infinite in expanse, they are terminated by the larger circle in Fig. 3(b) for convenience. As Earth rotates counterclockwise through the stationary sunlight and darkness, the observer reaches the first asterisk (with +

*y*

_{E}) and sees the sunrise. As the day continues, the sunlight seems to change direction, due to Earth’s rotation, even though the vector

**has not changed direction! The Sun rises in the sky until it reaches a maximum elevation when the observer is directly across from point P. It is noon—lunch time. After that, the elevation of the Sun decreases until the observer reaches the second asterisk (with –**

*S**y*

_{E}) and views the sunset. Local noon occurs midway between sunrise and sunset. Therefore, the observer sees sunlight as it rotates through the angle 2B, shown in Fig. 3(b). Since the observer rotates 360° during 24 h, the hours of daylight are given by the following proportionality:

*B*, we first find the length of the thick black line

*D*in two right triangles and equate them. In Fig. 3(a), the length of the dashed line in the right triangle is equal to height of the latitude circle above the equator, which is given by

*R*

_{E}sin 46.3°, where

*R*

_{E}is the radius of Earth. Because the angle between axis

*z*

_{OP}and

*z*

_{E}is 23.5°, the distance

*D*is given by

This value compares very well with the precise value of 8.6019 h, obtained from Ref. 7. The difference between the values obtained in Eq. (5) and the precise value is about 2.5%, which is about the uncertainty stated in the introduction for the basic theory.

At 18°N latitude on December 21, there are 10.9 h of daylight, using an obvious adjustment to Eq. (4). On June 21, the direction of the sunlight is opposite that on December 21, as shown in Fig. 3(a). Therefore, in a similar figure for June 21, the regions of sunlight and darkness are switched, and so there are 24−8.39 h, or 15.6 h, of daylight at 46.3°N latitude. The regions of sunlight and darkness are also switched in a new version of Fig. 3(b).

We know, of course, that the number of daylight hours at a given latitude changes during the year, which is our next subject. Because the direction of the sunlight striking Earth changes each day, the LDC and the hours of daylight also change.

If a model of the sphere, like that shown in Fig. 4, is available, the length of the illuminated section of the latitude circle and the circumference of the latitude circle can be measured using a flexible tape measure. Then the fraction of the circumference illuminated multiplied by 24 h is equal to the number of daylight hours.

## Predicting the daylight hours for any day of the year

*A*is the amplitude of the sine wave and, for a given date, DN is the number of days after March 21. Using this formula, the average value of the day length over an entire year is 12 h, because the average of a sine function is zero over one cycle. The value of the amplitude

*A*is obtained from the difference between 12 h and the 15.6 h of daylight, which is 3.6 h. Also, we have pairs of dates. The day lengths on June 21 and December 21 have an average value of (15.6+8.4)/2 = 12 h, and so on, for two other dates where the

**vectors are oppositely directed. Because one hemisphere is always illuminated, this is very reasonable. It is well known from observations of the sunrise and sunsets that the Sun moves more slowly near the solstices and much more swiftly near the equinoxes. The author’s photographs of the sunrise**

*S*^{1}also show these effects. These observations also suggest a sine wave function.

In Fig. 5, the black circles show the hours of
daylight for four days of the year, while the solid line is the sine wave fitted to them.
The open circles show the precise values obtained from websites.^{7} The open circles are, on average, about 0.2 h above that
predicted by the sine wave, leading to an uncertainty of 2.3% for December 21 and 1.3% for
June 21. This is about the uncertainty for the basic theory as stated in the introduction.
Considering the simplicity of the model, this is a very small uncertainty.

Refraction of the sunlight occurs as it enters Earth’s atmosphere due to the increasing density of the atmosphere, causing its path to curve. As discussed in Ref. 8, this causes the time between sunrise and sunset to increase, compared to what it would have been without the atmosphere. Because the simplified model does not include refraction, this is one reason why the precise values in Fig. 5 are larger than those predicted by the simplified model.

## Elevation angle of the Sun at noon on December 21

We are well aware that, at noon, the Sun rises high in the sky in summer, but not nearly as
high in winter.^{9,10} To gain some
understanding of this, we consider how five observers view the Sun at noon on December
21 in.Fig. 6. The goal is to determine the Sun’s
elevation angle at noon on December 21, defined as the angle between the horizontal (for the
observer) dashed line and the vector –** S**, for each observer.
Only the observer at 23.5°S latitude sees the Sun directly overhead—an elevation angle of
90°. The Tropic of Capricorn is the special designation for 23.5°S latitude and Tropic of
Cancer, for 23.5°N latitude, where the Sun is directly overhead in summer at each location.
Table I is helpful in determining the elevation angle
for the other observers.

Observer . | Latitude of Observer . | Angle between Another Observer and Observer #2 . | Sun Elevation Angle at Noon . |
---|---|---|---|

#1 | 46.3°S | 22.8° | 67.2°, N |

#2 | 23.5°S | — | 90°, overhead |

#3 | 0°, equator | 23.5° | 66.5°, S |

#4 | 23.5°N | 47.0° | 43.0°, S |

#5 | 46.3°N | 69.8° | 20.2°, S |

Observer . | Latitude of Observer . | Angle between Another Observer and Observer #2 . | Sun Elevation Angle at Noon . |
---|---|---|---|

#1 | 46.3°S | 22.8° | 67.2°, N |

#2 | 23.5°S | — | 90°, overhead |

#3 | 0°, equator | 23.5° | 66.5°, S |

#4 | 23.5°N | 47.0° | 43.0°, S |

#5 | 46.3°N | 69.8° | 20.2°, S |

Columns 1 and 2 in.Table I describe the observer’s location. The term “angle” in column 3 can be visualized by “drawing” a radius from the center of Earth to its surface for each observer. The angle between these two radii is the value in column 3. Because the Sun is overhead at noon for #2, the elevation angle in column 4 is equal to 90° minus the angle between the two observers. The north and south directions are also indicated in column 4. At noon on December 21, observer #5 looks due south first and then raises his/her eyes 20.2° above the horizon, while observer #1 looks due north and then 67.2° above the horizon. The elevation angles are also shown in Fig. 6. Therefore, at 46.3°N latitude in Richland, Washington, the elevation angle of the Sun at noon ranges from 20.2° on December 21 to 67.2° on June 21.

## Elevation angle of the Sun at noon on March 21 at 46.3°N latitude

In Fig. 7 for noon on March 21, the ** S** vector is directed along the
+

*y*

_{E}axis and illuminates the North and South Poles, yielding 12 h of daylight, but we see in Fig. 2 that the

**vector is along the +**

*S**y*

_{OP}axis for March 21. However, nothing is amiss because the +

*y*

_{E}and +

*y*

_{OP}axes are parallel, as described in the caption to Fig. 3(a). The thick black lines and the dashed line have the same definitions as in Fig. 5. Inspection of Fig. 7 shows that the angle between the –

**vector and the observer’s vertical direction is 46.3°. Therefore, the elevation angle—the angle between the –**

*S***vector and the dashed line—is 43.7°. For the equinoxes, the elevation angle of the Sun is in general given by 90° minus the latitude angle.**

*S*## Predicting the elevation angle of the Sun at noon for any day of the year

*B*=23.5° and DN is the number of days after March 21. The open circles in Fig. 8 show the precise values obtained from the National Oceanic and Atmospheric Administration website,

^{12}which are in excellent agreement with Eq. (8). The errors for June 21 and December 21 are only 0.2%.

## Conclusions

Two geometrical diagrams, Fig. 3 and Fig. 6, enhance our understanding of why the number of daylight hours changes during the year and why the Sun is so low in the sky at noon in winter. The use of geometry and trigonometry leads directly to the fraction of the latitude circle illuminated by sunlight, and thus, the number of daylight hours. In Fig. 6, geometry obtains the solar elevation angle at noon for five observers at different latitudes, and those angles are labeled directly on the diagram. The examples are concrete and not abstract. The straightforward calculations are easy to understand, thus satisfying one of the author’s goals.

The analyses of the data for the hours of daylight and the elevation angle show that both can be fitted with a sine wave function. In addition, 13 precise values, obtained from the internet, are also plotted, and these results show that the error for the simplified model is in the range of 0.2–2%. This seems like a very acceptable amount of error for a model that provides so much understanding.

## References

*Practical Astronomy with your Calculator or Spreadsheet*

*Celestial Calculations: A Gentle Introduction to Computational Astronomy*

*Practical Astronomy with Your Calculator or Spreadsheet*

*Phys. Teach.*

*Phys. Teach.*

**Dr. Margaret Greenwood** *taught physics and introductory astronomy at DePaul University in Chicago and
carried out research in nuclear physics at Argonne National Laboratory. When teaching
astronomy, she emphasized the seasonal changes we see every day, and that interest led to
further investigations and to writing this article in retirement. In 1990 she joined
Pacific National Laboratory and designed ultrasonic instruments.*