Equation of time and analemma

If you ever happened to observe the dial of a Sundial, perhaps you may have noticed a line in the shape of "8" which takes the name of Analemma.
This line has the role of "translating" the time naturally dictated by the sun into the artificial one we are used to reading on watches. The meaning and the shape of the analemma come from the motion of the earth and the concept of the equation of time.

How is it possible that parameters of the terrestrial orbit can take shape in this particular track? And what is the equation of time?
On this page, I will try to answer these questions.

Introduction

It is hard to show offhand how to read the time on a Sundial: sometimes, the time indicated on the dial is tens of minutes from that of our watches, so it seems that the sundial is poorly built. Yet it is not a mistake, but simply a difference in the definition of the time: it is necessary nothing but a few practice to understand how to "correct" this time lag, and with an analemma drawn on the dial it becomes even easier.
However, once one has understood how to read the time, it is natural to ask "why?". Why do we need so much effort to read the time on a sundial? Why do we need such a strange shape (the analemma) to translate the sun position?

When I finished building my sundial, I had to answer these questions made by friends, relatives, and curious. So, since they are not easy topics, I decided to write this page that wants to respond to the question "Why?", from which the answer to the "how?" is immediate.

The discussion on this topic is scientific and includes some specific terms. Nevertheless, everyone should be able to read and understand this article if you search for some words and phenomena I only name that you don't know. If you are not used to astronomical coordinates or gravitation, read the document very calmly: the arguments we are treating are not obvious.

Enjoy the reading!

The equation of time

The equation of time is a value changing during the year which expresses the difference between true solar time ^[1] (the one commonly indicated by the lines of the sundials) and mean solar time, the one we are all used to and for which every day lasts 24h.

The equation of time can be represented graphically depending on the day of the year or the sun's declination. In this second case, the curve thus designed takes the name of the analemma. That's why the analemma is a curve with the shape of the "8".

equation of time as a function of the date

Equation of time according to the declination

Neglecting long-term effects, the gap between solar time and mean time (the equation of time) is not null due to two phenomena:

The eccentricity of Earth's orbit, that is, the fact that the orbit of Earth is an ellipse and not a perfect circle.
The inclination of the ecliptic plan compared to that of the celestial equator, which is the commonly named Earth's axial tilt (23° 27')

Now we'll focus separately on the two phenomena to understand how they are involved in determining the value of the equation of time.

The eccentricity

The terrestrial orbit is an ellipse, whose eccentricity is ε = 0.0167. ε = 0 corresponds to a circular orbit. By Kepler's first law, the Sun is one focus of this ellipse. Therefore, there is a point in earth's orbit in which it is at the shortest distance from the sun, this point is called perihelion. After half a year, the earth is at the aphelion, which is its most distant point from the sun.

For Kepler's second law, the angular velocity of the earth along the orbit is not constant (the conserved one is the areal velocity) therefore the apparent velocity of the Sun seen from the earth is ^[2] not constant.

Since the solar day is defined as the time interval between two successive transit by the sun of the meridian for a given location, that is, the time from when the sun is higher in the sky to when it is again. The elliptical orbit makes this time vary.

Suppose we are on Earth, and we observe a sun's transit. In 23h 56m 4S the earth make a complete rotation to fixed stars, but since in the meantime it's also moved on its orbit, it must still carry out a small rotation on itself before we can observe the next sun's transit. When the earth is located near the perihelion, this effect makes the day last more than 24h, near the aphelion less: in the first case in one day the earth moved more along the orbit and takes more time to find the sun in the same position in the sky, in the second case we need a smaller rotation of Earth to compensate its movement around the Sun

Equation of time eccentricity orbit drawing

If you consider only the eccentricity of the terrestrial orbit, then the time equation is in the first approximation a sinusoid whose period is 1 year as in the figure^[3].

Orbital inclination

We'll now suppose a perfectly circular terrestrial orbit, and we'll study the second phenomenon that affects the equation of time.

The orbits of the planets are planar, and the projection of the Earth's orbit plane on the celestial sphere takes the name of the ecliptic. From the point of view of an observer on Earth, the sun moves along the ecliptic in one year

Suppose, for simplicity, the year is made up of 365 days ^[4] , then, since we are assuming the orbit to be circular, every 24h the sun moves α =

360°/365

≈ 59' along the ecliptic. The ecliptic plane is however inclined to ψ = 23 ° 27' compared to the celestial equator, which is the projection in the sky of the Earth's equator.

The shift of the sun projected to the celestial equator will therefore not always be equal to α, but will always be between α cosψ, the value that it takes on on the equinoxes, and

α/cosψ

, which happens on the solstices ^[5]. A couple of images can clarify the idea:

The solar movement and its projection on the celestial equator on the solstices

The solar movement and its projection on the celestial equator on the equinoxes

The two images show the shift of the sun (S) and its projection (s _proj) on the equator with a 1-day delay (1^d

). In these pictures, to better show the points, the solar path along the ecliptic is 15° (instead of 59 '), and the inclination of the ecliptic is greater than reality. As you can see, the shift of the sun is the same in the two images, but the movement of his projection on the equator is different.

Since the important quantity that determines the midday is the instant in which the sun transit the meridian ^[6] And this instant is determined by the right ascension ^[7] of the Sun, the important quantity to determine the length of the day isn't the speed along the ecliptic (which is a constant in this model), but the one of the sun projection on the celestial equator. Therefore, on the solstices, in the same amount of time, the projection moves more compared to the average, while on the equinoxes it moves less.

It follows that the duration of the day on the solstices is greater than on the equinoxes (the difference is about 20 seconds). The contribution made by this phenomenon to the equation of time is a sinusoid with a mid-year period, such as the one in figure ^[8].

The equation of time is nothing else ^[9] than the sum of the two graphics previously displayed for the two phenomena:

The analemma

We should have understood what the equation of time is, but you might be wondering: why draw an analemma on a sundial?

It's common to draw analemmas to read the mean time directly on the dial.

In fact, in a sundial, the straight hourly lines mark the solar time, directly associated with the solar position, while if you want to read the mean time ^[10] you have to read the time on the analemma. This is why the time indicated on the dial seems wrong: the difference is precisely the equation of time of that day.

The equation of time on a Sundial takes the form of an analemma because the length of the gnomon's shadow indicates the solar declination, which is the same two days of the year temporally equidistant from the solstices.

The result is that it's possible to read the mean time on a sundial by drawing an analemma.

If you have advice, you found mistakes or anything else, write me!

^[1] It's also called apparent solar time or simply solar time.

^[2] It is implied that the apparent velocity of the sun is an angular velocity as the apparent ones of all of the sky objects seen from Earth

^[3] Note that the two instants of the year in which the Earth is in perihelion and aphelion correspond with the zeros of the function shown. This is because the time equation is a cumulative function day by day (e.g. If the equation of time today is zero and every day would last 24h 2m, then the equation of time in three days will be +6min). The value of the daily difference is therefore readable in the graph as the variation of the equation of time in one day. Approximately this difference can be read as the derivative of the function which is a sinusoid with maximum and minimum when the earth is at perihelion or aphelion.

^[4] In reality, it's 365.24 days. See "Tropical year".

^[5] Looking at the image it is quite clear that the minimum is just α cosψ but it may seem less obvious that the maximum is α/cosψ. To see it, you can think that at the solstices the sun moves as if it moved along a circumference parallel to the equator but passing through the point where the sun is located. The ratio of an arc of this circle to its projection on the equator is the ratio of these circumferences. You can see this is cosψ

^[6] This is true if you consider the local midday and not one of the watches (civil time). For further information, you are looking for the difference between average time and civil time. For the other hours, the idea is the same, but you have to look at other meridians which differ from the local one by multiples of 15°.

^[7] The right ascension of a star and the sidereal time, which is a measure of time calculated only on the rotational position of the earth to the fixed stars and therefore independent of the position of the earth around the sun, is enough information to determine when that star passes across the meridian or at any angle from it. For more details, you are looking for "right ascension" and "hour angle"

^[8] If you believed that the amplitude of the function should be 20 seconds or that the maximums and the zeros should be switched, then look at the note ^[3]. The equation of time is a cumulative function of the differences between the times we are talking about in the article.

^[9] As a first approximation.

^[10] Or, even more interesting, directly the civil time if the Sundial was built considering the longitude difference between its location and the terrestrial meridian chosen as the main one for the civil time of its time zone (for Italy and all the countries with time zone +1 the 15th meridian east)