# Equation of time

In the relaxed talks that I usually have with students or those attending our activities, the issue of the equation of time usually comes up. So why not make a blog post on that topic.

Index

• Introduction
• Factors involved:
• Eccentricity of Earth’s orbit
• Obliquity of the axis of rotation
• Overlapping functions:
• Sum of functions:
• Conclusions:

## Introduction

If you have entered wikipedia, in the article on the equation of time you will have come across this graph:

And you probably haven’t quite understood what it means, that’s why you’re here, right?

Without even going into detail in the graph and analyzing it, what this graph is represented is how many minutes must be subtracted or added to a sundial to equate it to a wristwatch over a year. We can see this graph exactly the other way around:

This representation is perhaps easier to understand, since it shows us how many minutes must be added or subtracted from the synodic period to obtain the sidereal period. That is, the time that must be subtracted from the period that elapses between one solar noon and the next to obtain the period of the Earth’s rotation. Ein? Well, let’s put it another way. Let’s imagine that the upper part of the graph tells us that a wristwatch goes faster than a sundial (they measure time between noon) and the lower part of the graph shows when a wristwatch is behind the sundial.

Note : the graphs shown in this blog post are penciled and not accurate. Its purpose is merely illustrative.

## Factors involved:

• The  obliquity of the ecliptic, that is, how inclined the earth’s axis of rotation is with respect to the ecliptic and, therefore, the apparent movement of the sun with respect to the celestial equator. In other words, throughout the Earth’s orbit, the sun appears to be above and below the celestial equator. To this we owe the seasons, the tropics of Cancer and Capricorn and the Polar Circles.
• The  eccentricity of the Earth’s orbit. Because the Earth has an elliptical orbit and the Sun is located in one of its foci, the speed of translation is not constant along it. This causes that, depending on the time of year, the Earth travels more or less “space” along the orbit. For example, the Earth spends less time in perihelion than during aphelion. All of this is summed up in Kepler’s Laws.

Now, knowing that these two factors exist, let’s go on to describe how they influence the graph.

### Eccentricity of Earth’s orbit

Let us suppose for a moment that the Earth’s axis of rotation is at 90º with respect to the ecliptic. In this case, we would have no stations! That is, the Sun would affect both hemispheres in the same way throughout the year, that is, the sun would always be on the celestial equator. If this were the case, the variation between the mean solar day and the true solar day (synodic and sidereal periods *) would depend exclusively on the eccentricity of the Earth’s orbit. So the previous graph would look like this:

• The sidereal periodis exactly the time it takes for the Earth to rotate on itself ( ~ 23h 56m 4s ). If we take any star other than the Sun as a reference, and measure the time that elapses between the position of said star one day and the same position the next day, we will obtain the Earth’s rotation very precisely and it is an “invariable” period along the throughout the year. I enclose it because we do not take into account other long-term factors that do influence but that go beyond the objective of this explanation and are negligible values ​​on a day-to-day basis.
• The synodic periodis the time that elapses between one noon and the next. That is, the time that elapses since the Sun passes through our local meridian one day and the next day; this time is variable throughout the days. But if we take all the daily values ​​of a year we will obtain an average value of 24 hours .

### Obliquity of the axis of rotation

Now we go with another assumption. Let’s imagine that the Earth’s orbit were circular, that is, it had no eccentricity, or what is the same, the perihelion and aphelion were identical. But the obliquity of the axis of rotation did exist. In this case, the graph would look like this:

### Overlapping functions:

If we superimpose both graphs we can see that the variations over time are different and lagged throughout the year:

### Sum of functions:

By performing the sum of both functions, we obtain the result of the initial graph. As we can see, each of these two factors affects the terrestrial synodic period in a different way, varying its duration throughout the year.

## Conclusions:

I hope that this small contribution allows you to better understand the variation of the length of the day throughout the year and why it is not correct to say that the day lasts 24 hours , nor would it be correct to say that it always lasts 23h 56min. In short, the length of the day varies constantly throughout the year due to the characteristics of our orbit around the Sun.

I have not gone into the precise calculation of the values, nor the mathematical development of the equation itself because it is beyond the scope of this post.

If you have found it useful or if you have any questions about it, you can leave it in the comments and I will be more than happy to try to help you.