# Equation of Time - Analemma

## Earth's Tilt Effect

We will begin this section by making two assumptions:

1. The Earth's orbit around the Sun is circular.
2. The observer is standing on the equator.

From the previous section, we know that if we observe the position of the Sun at the same time every day starting from 2nd January, the Sun appears to move slowly to the east, in comparison to what your watch reads and takes one year to return to its starting position. Let's take the stars behind the Sun as markers, and imagine the Sun as being a little less bright so that we can look at it and still see the stars behind it. The Sun would then appear to drift slowly to the east against the background of the stars. At the end of one year, the Sun would return to its original position.

Let's look at this motion over a period of one day. The animation shows the Sun between 21st March and 22nd March. The middle panel shows what we would see if the Earth was not tipped on its axis. If this were so, the motion of the Sun against the stars would be in a horizontal motion only. Every day at noon, the Sun would appear to be at the highest point in the sky or what is known as culmination. After 24 hours, the Sun would again culminate in the sky at noon, however having drifted slightly to the east in relation to the background stars. This Sun represents the mean sun that would travel on the celestial equator.

The bottom panel shows what we will see in reality because the Earth is tipped on its axis. The path of the Sun will follow a slightly different path throughout the year. Not only will the Sun drift slightly to the east (or west) but also to the north (or south) depending on the season. This Sun is the true sun that will travel on the ecliptic. Notice that the true sun is not on the 'noon' line on 22nd March.

Let's look at the motion of the Sun around the Earth for one year.

Remember that we are assuming the Earth's orbit around the Sun is circular. The velocity of the mean sun and the true sun are constant, each one taking one year to make a complete trip around the celestial sphere. We also notice that at the vernal and autumnal equinoxes, the true sun and the mean sun meet.

Looking at the top view, observe the motion of the true sun and the mean sun on the celestial sphere. Again, at the vernal equinox, the true sun and the mean sun are in the same position. Now watch what happens as the two Suns move toward the summer solstice on 21st June. They start together but then the true sun lags slightly behind the mean sun until sometime in May, and then starts to catch up to the mean sun, catching it at the summer solstice. If we look at the position of the true sun at noon in July, we should "expect" it to be in the position of the mean sun (remember that the mean sun at noon will be at culmination). As the animation shows, however, it is slightly to the right (as viewed from Earth) of where we would expect it to be. In other words, the true sun would have culminated a few minutes before the mean sun.

We will now explain why the true sun lags behind or moves ahead of the mean sun. Looking at the top view of the celestial sphere, we can see that the path of the ecliptic is nothing more than the path of the celestial equator that has been "tipped" toward us. Therefore, some of the motion of the true sun will be toward or away from us around the time of the vernal equinox or the autumnal equinox respectively. At these times, we will not have the perception of the true sun moving as fast as the mean sun. Imagine someone throwing a ball straight at you to catch. You may not perceive any motion of the ball because it is coming straight at you. However, if you were an observer standing on the sidelines, you would see the ball moving forward toward you.In other words, on top of the general eastward drift among the background stars, the true sun is moving along the ecliptic northward or southward with respect to the celestial equator. Thus during some periods the true sun appears to move eastward faster than during others. Looking at the graph below, we notice that only at the solstices, all of the true sun's motion is parallel to the celestial equator. At other times of the year, the Sun is also either moving north or south.

Figure 16