Calculation of Solar Motion for Localities in the USA

Even though the longest day occurs on the June solstice everywhere in the Northern Hemisphere, this is NOT the day of earliest sunrise and latest sunset. Similarly, the shortest day at the December solstice in not the day of latest sunrise and earliest sunset. An analysis combines the vertical change of the position of the Sun due to the tilt of Earth’s axis with the horizontal change which depends on the two factors of an elliptical orbit and the axial tilt. The result is an analemma which shows the position of the noon Sun in the sky. This position is changed into a time at the meridian before or after noon, and this is referred to as the equation of time. Next, a way of determining the time between a rising Sun and its passage across the meridian (equivalent to the meridian to the setting Sun) is shown for a particular latitude. This is then applied to calculate how many days before or after the solstices does the earliest and latest sunrise as well as the latest and earliest sunset occur. These figures are derived for 60 cities in the USA. The selection was initially based on the most populous urban areas but was extended to ensure that each of the 50 states has a representative city.


Introduction
The June solstice is the longest day of the year in the USA and the December solstice is the shortest. However, calculations in this paper show that: The earliest sunrise happens before the June solstice and the latest sunset after; The latest sunrise happens after the December solstice and the earliest sunset before; The effect at the December solstice is more pronounced than for the June one and The number of days departure for earliest/latest sunrise/sunset from the longest and shortest days is latitudinally dependent and increases towards the equator.

Dates Used in this Paper
The solstices, equinoxes, perihelion (closest distance to Sun) and aphelion (furthest point from Sun) occur at instances and these are given in Universal Time, the time at Greenwich, England. The USA has a wide spread of time zones from 5-10 hours behind Greenwich. A solstice, for example, happens at an instant and may occur on different days in different time zones. Calculations in this paper are based on Central Standard Time of -6 hours difference, with Chicago being the representative locality. 2022 is the year selected. One advantage to this is that it avoids a leap year. So, relevant orbital and seasonal dates chronologically for the USA in 2022 Astropixels [1] are shown in Table 1.

Outline of Paper
1. The elevation of the Sun at noon over a year is related to the tilt of the spin axis of Earth which is inclined at 23°.44 Seidelman_2006 [2] to the vertical of its orbital plane (obliquity). This is determined graphically.
2. The east-west variation of the Sun from the meridian at noon depends on two factors. These are an elliptical rather than a circular orbit for Earth and the obliquity. These factors are independent of each other. This derivation is analyzed in steps (as in Yeow_2002 [3]) with first taking only the elliptical orbit and setting the obliquity as 0°.
3. Second, the influence of the obliquity under a circular orbit is found.
4. Third, the two factors from 3.2 and 3.3 are combined. 5. The result from 3.4 is joined with that from 3.1 to display the angle of the Sun and its east-west variation from the meridian over a year. 6. 3.5 is converted from an angle to time in minutes to produce the Equation of Time.
7. Rising and setting times may be calculated for one's latitude.

Variation of the Position of the Sun over the Year: Elevation
The basic unit for the calendar is the time between successive March equinoxes and its duration is 365.242 19 days [2]. From the point of view of Earth, the average angular movement of the Sun is 360°/365.242 19 days=0°.986 per day. (1) Declination is the angle from the celestial equator moving away from it at right angles with + to the north and -to the south. Over the year declination of Sun=23.44 sin (0.986 (286+D)) (2) where 286 is the number of days from the March equinox of the previous year (here March 20 2021 [1]) to 31 December, and D is the number of days from 01 January in the current year (2022).
In Excel, highlight column A, Format Cells to Custom and d-mmm. In cell A1 place 1/01/2022 and in A2 5/01/2022. These will change to 1-Jan and 5-Jan respectively. In A3 place=A2+5 and copy to 31-Dec. In B1 and B2 respectively place 1 and 5. In B3 put=B2+5 and copy to 365. In C1,=23.44*SIN(RADIANS(0.986*(286+B1))). Copy will be taken for granted from now. Graph C against B to give Figure 1.   The upper blue line is parallel to the lower one. On this scale, the Sun is far enough away that the upper parallel line also points to the Sun. By convention, declinations south of the equator are negative. Hence, from geometry for the Sun in either hemisphere elevation=90° -(latitude -declination). ( For a fixed latitude, the elevation of the Sun at noon varies due to the declination. In

Deviation from Meridian for Elliptical Orbit Only
The derivation following [3] fixes the tilt angle at 0°. If N is the number of days after perihelion, m=0. 986N (1) gives the angular change from perihelion of a uniformly moving Sun on a circle centered on Earth. The position of the real Sun uses Earth at one focus of an ellipse. The following equation gives the angular change v of the actual Sun in an elliptical orbit Duffet-Smith _1992 [5] where the eccentricity e of Earth's orbit is 0.016 71 [2].  The elliptical orbit is based on the first law of planetary motion of Kepler. His second law is demonstrated in Figure 4 by the maximum speed at perihelion and the minimum at aphelion.
Equation (4) may be rearranged so that is the angular difference between a circular and elliptical orbit for the Sun. A positive value gives a faster moving Sun so that at noon it is west of the meridian. Conversely, a negative value gives a slower moving Sun, which at noon is east of the meridian. Thus, Z 1 represents the position of the Sun relative to the meridian based on the elliptical orbit. In H1,=-1.915*SIN(RADIANS(0.986*E1)). Graph H against B for Figure 5.

Deviation from Meridian for Obliquity Only
A circular orbit is selected. The celestial equator and ecliptic are inclined at 23°.44 to each other in Figure 6. Movement on the celestial equator is the basis for clock time, but the real Sun moves at an angle to this. From Figure  6, cos 23°.44=0.917 so that for a 1° movement of the Sun along the ecliptic, its component on the celestial equator is 0°.917. The "real Sun" and the "clock Sun" meet at the equinoxes. The real Sun and the clock Sun are moving parallel at the solstices, so they are again in step. The maximum difference will be halfway between them, that is, at 45° along the orbit. The difference in angular movement at 45° gives the amplitude of a graph due to this factor only. Spherical geometry Kaler [6] 45° -42°.52=2°.48.
While the period for the elliptical factor is one year, it is 6 months (182.5 days) for the obliquity factor. In 2022 there are 75 days between perihelion and the March equinox. One quarter of a year is 91.25 days. If N is taken as previously, the effect here is [3]

Deviation from Meridian for Elliptical Orbit and Obliquity Combined
The two effects of elliptical orbit and tilt angle are now combined from (5) and (8).

Elevation and Deviation from the Meridian Combined
The union of the elevation for a specific locality ( Figure 3) and deviation from the meridian (Figure 8) for the Sun results in the construction of an analemma, which is in the shape of a figure eight. The word is derived from a pedestal for a sundial. This is what would be observed for the noon Sun over a year at a specific place. However, a general analemma is constructed by having the vertical axis the declination (Figure 1) and the horizontal axis as the angle of the noon Sun from the meridian (Figure 8) with positive values west and negative ones east.
Plot C against J for Figure 9. The actual position in the sky is obtained by raising or lowering the entire curve to match the elevation at one's latitude.
As the Sun is not on the meridian at noon at the solstices, the analemma is skewed and does not line up with the vertical axis.

Equation of Time
The sidereal rotation of the Earth is 23 hours 56 minutes 04 seconds=0.997 269 days. 0.997 269 days x 24 x 60 / 360°=3.989 minutes per degree. (11) Thus, from the position of the Sun given as an angle, one may determine the time earlier or later than noon that it crosses the meridian (Figure 8 into Figure 10). For the Sun west of the meridian at noon, it is faster than a clock and will be shown here as a positive time.
In K1,=3.989*J1. Graph K against B for Figure 10. The relative sizes of the crests show the effect in December is more pronounced than in June.

Rising and Setting Times for One's Latitude
The time from sunrise to its crossing of the meridian (and from the meridian to sunset) depends on both the declination of the Sun and the latitude of the observer. Even though the longest and shortest days are generally the same date for all places in the USA, earliest and latest sunrise and sunset are latitudinally dependent. Houston Texas, being close to the equator, is used as an example as this gives a large difference in days. However, adopt your own latitude. Then, the number of days before or after the solstices for earliest/latest sunrise and sunset is calculated for your locality.
The algorithm for the time interval between the rising of a star (here the Sun) and its transit over the observer's meridian (or equivalently from the meridian to setting) for particular declinations is derived firstly by determining an angle from spherical geometry [6] and is Ridpath [8] cos (semi-diurnal arc)=-tan (declination) x tan (latitude) (12) where the semi-diurnal arc is the angle that a star makes from either horizon to the meridian. Then, a general formula for converting this angle to length of time is  The following procedure Wagon _1990 [9] is based on the semi-diurnal time for Houston, Texas, but use your own latitude.
Addition of the minutes before or after noon in Figure 10 to the minutes of half daylight in Figure 11 gives the actual minutes before noon of sunrise.
In M1,=K1+L1 for time of sunrise in minutes before noon. Graph M against B for Figure 12.
Subtraction of the minutes before or after noon in Figure  10 from the minutes of half daylight in Figure 11 gives the actual minutes after noon of sunset.
In N1,=L1-K1 for time of sunset in minutes after noon. Graph N against B for Figure 13 where the vertical axis is reversed to have a sense that the spacing between the plots in Figures 12 and 13 is the length of daylight.
Inspection of Figures 12 and 13 shows that the earliest sunrise precedes the June solstice and the latest sunset trails it, and the latest sunrise trails the December solstice, and the earliest sunset precedes it.  Addition of the minutes before noon of sunrise ( Figure 12) and minutes after noon of sunset ( Figure 13) gives the length of daylight for each day.
The data in Excel have been calculated for each 5 days. To ascertain the number of days difference from the solstices for the shortest/longest sunrise/sunset, the information can be set daily, but it is only necessary for June to early July 2022 and late November 2022 to early January 2023. As the effect is less pronounced further from the equator, these spans of the calendar could be even smaller. There is a slower rate of change at these maxima and minima, and it may be necessary to go to 3 decimal places to isolate one particular day. Also, these equations have some rounding effect for simplicity, and may differ ever so slightly from results put out from say, the United States Naval Office. Also, the clock time is for the longitude of a certain time zone. A difference in one's locality from this longitude introduces a minor adjustment.
From O in these daily calculations, one should see that the figures do correspond to the longest and shortest days, given the caveats above. From O also, one should determine the number of days of earliest sunrise before the June solstice, latest sunset after the June solstice, latest sunrise after the December solstice and earliest sunset before the December solstice.
The offset effect should be more pronounced for December than for June.

Extra for the Mathematically Inclined
The graphs in Figures 12 and 13 (for the daily calculations) could be lined one above the other. Let one be a function of p against time t and the other q versus t. Let the vertical difference q -p=r. At the maximum and minimum values of r, its differential dr/dt=0. Hence, dq/dt -dp/dt=0, and this will occur when these differentials are equal. This corresponds to the days when the slopes of these two curves are equal. Draw these slopes on the daily section of the graphs to verify this for the June solstice and then for the December solstice. They are shown for Houston, Texas at the June solstice for sunrise in figure 14 and sunset in figure 15. The days are aligned if figure 14 is placed above figure 15.

Data for 60 Cities for Calculation of Solar Motion for Localities in the USA
60 cities of the USA were selected for analysis. The 25 most populous urban areas from the estimated population at the end of 2016 were in the first group included. Then, for any state not represented in this collection, its most populous city was chosen. These data were actually calculated for 1999, and while there is a slight variation from year to year, the pattern should be similar to the table.

Conclusion
The longest and shortest days are the same throughout the USA but these do NOT correspond to earliest/latest sunrise/sunset. This resulted in the following observations from calculations.
The earliest sunrise occurs before the June solstice and the latest sunset after.
The latest sunrise occurs after the December solstice and the earliest sunset before.
The effect at the December solstice is more pronounced than for the June one.
The effect increases with decreasing latitude.