Analemma: The Sun\’s Journey In The Sky

Abstract.

The Sun\’s position in the sky changes when looked at during the same time each day from the same location throughout the year. The diagram showing how the Sun changes its position in the sky throughout the year is known as an Analemma. There are two independent factors that affect the change in sun’s position. First, the elliptical orbit of Earth around the Sun and second, the axial tilt of Earth\’s rotational axis with its orbital axis around the Sun. Since both of these factors are independent, the absence of any one of them affects the shape of the Analemma. When there is an elliptical orbit factor present but no axial tilt then the Analemma is a horizontal line, but when there is axial tilt and the orbit has no elliptical character, the Analemma has a figure of eight structure with both loops equal in size.

The Second result is a bit strange as one wonders why axial tilt will cause the Sun to shift east and west in addition to shifting it north and south from its mean position, thus causing this figure of eight patterns. This paper will provide a possible explanation as to why the axial tilt of the Earth causes the Sun to shift sidewards.

In order to explain it we will picture that instead of the Earth revolving around the Sun, the Sun revolving around the Earth. From there we will picture the motion of the Sun around the Earth. We will then use this to explain the solution to the question of why axial tilt causes the Sun to shift sidewards in the sky from its mean position.

After reading this paper you will understand why the axial tilt of Earth\’s rotational axis causes the Sun to shift sidewards in the sky from its mean position. Also, this paper will provide insight into the factors that are responsible for the formation of the Analemma which will help us to understand the motion of the sun relative to other planets in our solar system, leading to a better understanding of our solar system.

Introduction

– An Analemma is the diagram showing the position of the Sun in the sky, as seen from a fixed location at the same time, throughout the year. Its structure looks similar to that of the figure: “8”. But unlike the figure of 8, the two loops of the Analemma are unequal in size with one of them larger than the other. In astronomy, the Analemma is considered one of the most difficult and demanding phenomena to image because it is never present all at once. It requires a virtual image made at the same time of day on 30 to 50 days throughout the year.^[1] Figure 1 shows how an Analemma on Earth looks like. You can see the difference in the size of the two loops.

Figure – 1: Photo was taken in 1998–99 of Analemma from the office window of Bell Labs, Murray Hill, NJ.

Analemma has its significance in astronomy as the study of it gives us a better understanding of the Earth\’s revolution around the Sun and about our solar system.

There are two factors that contribute to the formation of the Analemma pattern and they are completely independent of each other.^[2]

^{The two factors are as follows:}

1) Earth revolves around the Sun in an elliptical orbit.

2) The Earth is tilted on its axis 23.5° in relation to the plane of its orbit around the Sun.

Both of these factors contribute to the formation of Analemma independently so Let us see how these factors affect the shape of the Analemma.

Elliptical Orbit

– The Earth revolves around the Sun in an elliptical orbit^[A]. Whilst revolving in an elliptical orbit around the Sun, the distance between the Earth and the Sun changes throughout the orbit. The Earth gets closest to the Sun around January 3 and this point in the orbit is known as perihelion while it is farthest from the Sun around July 4 at a point in the orbit known as aphelion. This is shown by Figure 2.

Fig-2: The Earth revolving around the Sun in an elliptical orbit.

According to Kepler\’s second law of planetary motion, a planet will have a faster orbital speed when it is closer to the Sun, and a slower orbital speed when farther away from the Sun^[3]. Following this law, the Earth revolves with different orbital speed around the Sun, depending on the time of year. Earth’s orbital speed is slower near aphelion and faster near perihelion.

Due to this varying orbital speed of the Earth, the apparent solar time^[G] will be unequal to the mean solar time^[I] since mean solar time is constant while apparent solar time changes throughout the year.

This difference between mean solar time and apparent solar time is represented by the Equation of Time^[K]. Due to this difference, the apparent Sun position will lead or lag its mean (or expected) position, represented by the mean Sun, as it traverses the sky. The Equation of Time takes account for the sideways shift which is an eastwards or westwards shift of the true Sun in relation to the mean Sun. If the apparent Sun is west of the mean Sun then the equation of time is positive, but if the apparent Sun is east of the mean Sun then the equation of time is negative^[4].

Figure 3 depicts how much time difference there is between the actual solar time and the mean solar time.

Fig-3: Graph depicting how much the apparent Sun lags and lead to mean Sun. The Y-axis represents the time difference (between the apparent and the mean Sun) and the X-axis represents the day of the year (1 unit = 10 days).

This lag or lead of the Sun in the sky is seen as an eastward or westward shift of Sun\’s position in the sky with respect to the mean position of the Sun. The mean position of the Sun is a theoretical position of the sun in the sky where the Sun would be expected to be seen without the effect of axial tilt or changes in the orbital speed. When the Sun lags, it is eastward with respect to the mean Sun and when it leads, it is westward with respect to the mean Sun. This is because of the fact that Earth spins on its axis towards the east.

This is how the elliptical orbit of Earth around the Sun contributes to the formation of the Analemma.

Axial Tilt

– In astronomy, axial tilt is the angle between a celestial object\’s rotational axis^[L] and its orbital axis^[M]. And for Earth, right now, this axial tilt is about 23.5 degrees^[5]. This axial tilt of 23.5 degrees also contributes in the formation of an Analemma.

As the Earth moves through its orbit around the Sun with an axial tilt of 23.5 degrees, the Sun\’s overhead position changes. At a time when the Earth is at summer solstice, the northern hemisphere is tilted 23.5 degrees towards the Sun. This means that the Sun is directly above the tropic of cancer as shown by Figure 4. At this time, the Sun is at its northernmost point in the sky.

Fig-4: The Earth at summer solstice.

Similarly, when Earth is at winter solstice, the northern hemisphere is tilted 23.5 degrees away from the Sun, and the Sun is directly above the tropic of capricorn. This is when the Sun is at its southernmost point in the sky.

In between come two points: the vernal equinox and the autumnal equinox where the Sun is directly overhead at equator. Figure 5 shows the Sun at two solstices^[N] and equinoxes^[O].

Fig-5: Diagram showing Earth throughout the year.

So this tilt of the rotational axis of the Earth with respect to its orbital axis causes the Sun to move from north to south and then from south to north in the sky throughout the year. This north-south movement of the Sun in the sky is represented by Sun\’s declination.

Declination^[P] is positive towards the north pole, which has declination of 90 degree, and negative towards south pole, which has declination of -90 degree^[6].

The Analemma is formed due to the cumulative effect of both of these factors. Figure 6 shows the structure of the Analemma.

Fig-6: Diagram showing how the Analemma is formed by combining the effects of the axial tilt of Earth\’s rotational axis and the Earth\’s elliptical orbit around the Sun.

The asymmetry in the two loops of the Analemma is because of the fact that the two points – perihelion and aphelion – lie at different distances from the equinoxes.

Now as both these factors are independent of each other, what would happen to the shape of the Analemma if one of them was missing?

Elliptical orbit but no axial tilt.

If the Earth was to revolve around the Sun in an elliptical orbit but without axial tilt, then the shape of the Analemma would be a straight horizontal line from east to west. The reason for this is that due to absence of axial tilt, the declination of the Sun would be zero and thus the Sun would always be overhead of the equator. However, due to the elliptical orbit, the apparent Sun will still shift eastwards or westwards from the mean Sun depending upon Earth\’s position in the orbit around the Sun. So there is only east-west motion of the Sun in the sky and due to this, the Analemma will have a figure of a straight line extending from east to west.

The shape of the Analemma in this case is shown by Figure 7. The yellow horizontal line represents the shape of the Analemma. The blue sphere shows the Earth with zero axial tilt and below it is the Earth’s orbit around the Sun with an eccentricity of 0.0167.

Fig – 7 : Diagram showing how the Analemma would look if the Earth revolved in an elliptical orbit around the Sun but with zero axial tilt of its rotational axis.

With axial tilt but a circular orbit.

When the Earth has axial tilt but revolves around the Sun in a circular orbit, then the shape of the Analemma will be that of figure eight with both the loops equal in size^[7].

The shape of the Analemma, in this case, is shown in Fig-7. The shape of the figure of eight in yellow is what the Analemma looks like in this case. Here also, the blue sphere represents the Earth but this time it has an axial tilt of 23.5 degrees, and below it is the Earth’s orbit around the Sun with an eccentricity of 0.

Fig-8 : Diagram showing how the Analemma would look if Earth revolves around the Sun in a circular orbit with the same axial tilt of 23.5 degrees. The blue sphere shows the Earth having an axial tilt of 23.5 degrees and below is its orbit around the Sun which is perfectly circular.

The above result is a bit strange because if the Analemma has a shape of a figure of eight, this means that in addition to the north-south motion of the Sun in the sky there is also some east-west motion of the Sun in the sky too due to the axial tilt of the Earth.

Now why is axial tilt causing the Sun to shift sideways in addition to making it move north-south in the sky? Why do we have a component of the Equation of Time due to axial tilt?

This paper will give a detailed explanation as to why the axial tilt of Earth causes the sun to shift sidewards from its mean position.

Aim

To know why the axial tilt of the Earth makes the Sun shift sidewards from its mean position in the sky.

Methodology.

In order to explain why the axial tilt of the Earth makes the Sun shift sidewards in the sky from its mean position, we assume that instead of Earth revolving around the Sun, the Sun revolves around the Earth in circular orbit^[8]. This assumption is important because of the fact that since Earth exhibits two kinds of motion, rotation and revolution, it makes the overall explanation a bit hard and confusing. So in order to make it simple and easier to understand and avoid confusion this assumption is taken.

Explanation.

Earth exhibits two kinds of motions. The first one is rotation on its own axis and the second one is a revolution around the Sun. In this explanation, it is assumed that instead of the Earth revolving around the Sun, the Sun revolves around the Earth in a circular orbit.

Figure 9 represents our assumption. Here one can see the Sun revolving around the Earth in a circular orbit.

Fig:9 Diagram showing the Sun revolving around the Earth in a circular orbit. The small red sphere is the true Sun and the small blue sphere is the mean Sun.

In figure 9, the large sphere is the Earth and the small red sphere is the true Sun^[D] with the small blue sphere representing the mean Sun^[E].

Now as the mean Sun revolves around the equator and has zero axial tilt, its orbit is not tilted and is in the plane of Earth’s equator. This compares to the apparent Sun that has an axial tilt of 23.5 degrees and so its orbit is tilted as shown by Figure 9.

Also, both the mean Sun and the true Sun meet twice in a year and these points are the two different equinoxes. This is when both the true Sun and the mean Sun are directly above Earth’s equator.

As Earth\’s axis of rotation is tilted with respect to its orbital axis by 23.5 degrees, the orbit of the true Sun is tilted with respect to Earth\’s axis of rotation, as shown in figure 9. However, the orbit of the mean Sun has no axial tilt.

We start this explanation assuming both the mean Sun and the true Sun are located at the vernal equinox.

Earth completes one rotation around its own axis in 23 hours 56 minutes 4 seconds,^[9] meaning that to bring the Sun into the same position in the sky the next day, the Earth has to rotate a little more than once around its axis. This is because of the fact that during this period the Sun also moves a little in its orbit around the Earth.

Since the mean Sun is revolving around the Earth along the equator with constant speed it will move a constant distance towards the east each day during that period. This means that mean solar day^[J] is equal to 24 hours or it takes 24 hours for the mean Sun to return to the same position in the sky.

The true Sun also moves around the Earth with constant speed but it also has axial tilt due to which instead of moving completely towards east, it instead moves obliquely towards the north pole. In this way, the true Sun has both eastward and northward components in its motion around the Earth. Consequently, due to this oblique motion, the net eastward shift of the true Sun will be less than that of the mean Sun^[10].

Figure 10 shows the motion of the true sun and the mean sun around the earth. The true sun has less net eastward shift than the mean sun.

Fig-10: Diagram showing motion of mean Sun and true Sun. The true Sun has less net eastward shift than mean Sun.

According to mean solar time, which our clocks follow, the Earth has to rotate for an extra 4 minutes after one complete rotation around its own axis in order to bring the Sun (mean Sun) into the same position in the sky. However, since the true Sun has less eastwards shift than the mean Sun it will come to the same spot in the sky before 4 minutes or the true Sun will reach the same spot in the sky before the mean Sun and as result after the end of 4 minutes true Sun will be west of where it would be if there was no axial tilt of the Earth’s rotational axis or west of its mean position in the sky with more declination than the mean Sun since the true Sun also has a northward component in its motion, but this declination doesn\’t matter because the equation of time only takes account for the sidewards shift.

As the true Sun continues to move towards the north, its declination increases in the sky until summer solstice, meaning that its westward shift from its mean position increases up to this point. After the summer solstice, the Sun\’s declination starts decreasing until the autumnal equinox. This causes the westward shift from the mean position to reduce. At the autumnal equinox, both the mean and the true Sun are at the same position in the sky. Again, from the autumnal equinox to winter solstice, the Sun starts moving towards the south and the true Sun shifts eastwards from its mean position. Maximum eastward shift from mean Sun is at the winter solstice. When the Sun returns from the south to the equator, between the winter solstice and the vernal equinox, its eastward shift reduces and the mean Sun and the true Sun are once again at the same position in the Sky at the vernal equinox.

Figure 11 Shows the graph of the Equation of time due to the axial tilt of the Earth.

Fig-11: Graph of the Equation of Time due to axial tilt of the Earth, on y axis, and the day of the year on the x axis.

The axial tilt of Earth causes a sidewards shift of the true Sun in the sky from its mean position and contributes to the result of the Equation of Time. When this Equation of Time is graphed with solar declination, it gives us the Analemma which has the shape of figure eight with both the loops equal in size.

When this Equation of Time is combined with the Equation of Time due to the Earth’s elliptical orbit and then graphed with solar declination it gives us an Analemma with a figure eight shape but with both lobes that are unequal in size.

Conclusion.

Because of the axial tilt, the net sidewards motion of the true Sun is unequal to the sideward motion of the mean Sun meaning that the extra time for which the Earth has to rotate in order to bring the Sun into the same position in the sky each day differs for any given day of the year. Due to this difference a sideward shift of the true position of the sun can be seen from its mean (or expected) position in the sky.

The explanation above tells us why the axial tilt of the Earth causes the Sun to shift sidewards from its mean position in the sky. It was also shown by this explanation why the Analemma has the shape of a figure of eight with both the loops equal in size when there is only an axial tilt of the Earth around the Sun and no eccentricity.

Study of the Analemma is important in astronomy as it gives us a better understanding of our Earth and Sun as a system and in general a better understanding of our solar system. The study of the Analemma involves the study of the planet’s motion around the Sun and about their rotation and the tilt of their rotational axis. Every planet in the solar system has its own Analemma which has a different shape depending upon various factors. Since in this article the factors of axial tilt and eccentricity have been discussed in detail, insight can be found into the analemma of different planets.

Also, Analemma can be used as a tool to estimate quantities such as the times of sunrise and sunsets, which depend on the Sun’s position. The time of sunrise and sunsets taken with respect to mean Sun is 6 AM and 6 PM respectively on every day of the year and then with the help of Analemma difference of the position between true Sun and mean Sun is taken and with this positional difference between mean Sun and true Sun time difference between actual sunrise and sunset from 6 AM and 6 PM is calculated and thus providing us with the actual sunrise and sunset times for a particular day in a year.^[11]

Definitions.

(A) Elliptical Orbit – An elliptical orbit is an oval-shaped path in which an object revolves around the other object., an elliptical orbit is one which has non zero orbital eccentricity.

(B) Orbital Eccentricity – Orbital eccentricity is a dimensionless parameter that determines the amount by which the orbit of one body around the other body deviates from a perfect circle. For a perfect circular orbit, the orbital eccentricity is 0 whereas for ellipse eccentricity is non zero.

(C) Solar time – Solar time is the time taken for the Sun to return to the same position in the sky.

(D) True or Apparent Sun – The true sun or apparent Sun is the position of the actual Sun that we see in the sky.

(E) Mean Sun – The mean Sun is an imaginary Sun. The mean Sun moves through the sky with constant speed and is directly overhead the equator throughout the year.

(F) Solar Day – The length of the day as calculated using solar time.

(G) Apparent Solar time – Apparent solar time is based upon the motion of the apparent Sun as seen by observers from the Earth.

(H) Apparent Solar day – The length of the day calculated through the motion of the apparent Sun in the sky. It varies throughout the year depending upon the Earth’s position in the orbit around the Sun. Sometimes it can be more than 24 hours and sometimes it can be less than 24 hours.

(I) Mean Solar time – Mean solar time is calculated from the motion of the mean Sun.

(J) Mean Solar day – Mean solar day is the average length of the solar day or length of the day based on the motion of mean Sun. It is equal to 24 hours.

(K) Equation of Time – The equation of time is a quantity that keeps record of the difference between the mean solar time and apparent solar time. It also shows the sidewards shift which is an eastwards or westwards shift of the true Sun from the mean Sun’s position.

(L) Rotational Axis – The rotational axis is a fixed axis around which an object rotates. For example, the Earth rotating on its own axis.

(M) Orbital Axis – An orbital axis is the straight line around which an object orbits another object. E.g. the Earth orbiting around the Sun.

(N) Solstice – A solstice is an event that occurs when the Sun appears to reach its most northerly or southerly excursion relative to the celestial equator. There are two types of solstice, summer and winter.

(O) Equinoxes – An equinox is an event that occurs when the Sun is directly overhead the Earth’s equator. There are two equinoxes in a year, one vernal equinox and one autumnal equinox.

(P) Declination – Declination is the measure of the angular distance of a body north or south of the celestial equator.

(Q) Celestial equator – A celestial equator is an equator of an imaginary celestial sphere.

(R) Celestial sphere – A celestial sphere is an abstract that has an arbitrarily large radius and is concentric to the Earth. All the objects in the sky are conceived as being projected upon the surface of this sphere. Celestial spheres are a tool in astronomy, allowing astronomers to specify the apparent position of objects in the sky.

Refrences

1 – Scherrer, Deborah et al. \”Viewing and Understanding the Analemma\”. Stanford Solar Center. http://solar-center.stanford.edu/art/analemma.html. [Accessed 1 August 2020].

2,8 – Urschel, Bob. \”the Analemma\”. Analemma.com. http://www.analemma.com/Pages/framesPage.html. [Accesses 1 August 2020].

3 – Ezez. \”Kepler’s Laws of Orbital Motion | How Things Fly\”. Howthingsfly.si.edu. https://howthingsfly.si.edu/flight-dynamics/kepler%E2%80%99s-laws-orbital-motion.

4 – Ezez. \”Equation of time\”. En.wikipedia.org. https://en.wikipedia.org/wiki/Equation_of_time. [Accessed 5 August 2020].

5 – Ezez. \”Axial tilt\”. En.wikipedia.org. https://en.wikipedia.org/wiki/Axial_tilt. [Accessed 6 August 2020].

6 – Ezez. \”Declination\”. En.wikipedia.org. https://en.wikipedia.org/wiki/Declination. [Accessed 6 August 2020].

7 – Siegel, Ethan. 2019. \”This Is How The Sun Moves In The Sky Throughout The Year\”. Forbes. https://www.forbes.com/sites/startswithabang/2019/01/01/this-is-how-the-sun-moves-in-the-sky-throughout-the-year/#6a2079c27303.

9 – WILLIAMS, MATT. 2016. \”What is the Rotation of the Earth? – Universe Today\”. Universe Today. https://www.universetoday.com/47181/earths-rotation.

10 – Astronomy Stack Exchange. 2020. \”Why Would The Earth\’s Solar Analemma Be Still A Figure Eight Even If Earth\’s Orbit Was Circular?\”. [online] Available at: https://astronomy.stackexchange.com/questions/12735/why-would-the-earths-solar-analemma-would-be-still-a-figue-eight-even-if-earth .

11 – Ezez. \”Analemma\”. En.m.wikipedia.org. https://en.m.wikipedia.org/wiki/Analemma. [Accessed 1 August 2020].

Figure References.

1 – Ezez. \”Analemma\”. En.m.wikipedia.org. https://en.m.wikipedia.org/wiki/Analemma. [Accessed 1 August 2020].

2 – Ezez. \”Earth\’s Orbit\”. En. m .Wikipedia.Org. https://en.wikipedia.org/wiki/Earth%27s_orbit.

3,6,11 – Ezez. \”Equation of time\”. En.m.wikipedia.org. https://en.m.wikipedia.org/wiki/Equation_of_time. [Accessed 5 August 2020].

4 – Ezez. \”How does axial tilt affect climate? | Socratic\”. Socratic.org. https://socratic.org/questions/how-does-axial-tilt-affect-climate. [Accessed 6 August 2020].

5 – Ezez. 2020. \”What Is a Solstice? | NOAA SciJinks – All About Weather\”. Scijinks.gov. https://scijinks.gov/solstice.

7,8 – \”Analemma Generator\”. n.d. Alokm.Com. https://alokm.com/astro/analemmagenerator.html.

9,10 – URSCHEL, BOB. \”the Analemma\”. Analemma.com. http://www.analemma.com/Pages/framesPage.html.

Biography

Name: Vaibhav Vajpai

Biography: Vaibhav Vajpai is an 18-year-old boy from India. He is a college student currently pursuing Electronics and communication engineering. He is an Astronomy enthusiast. He loves to read books on astronomy and remains updated about the latest research in the field of astronomy.

He likes to play Cricket and watch comedy movies in his free time.