We will begin this section by making three assumptions:

1. the Earth's orbit around the sun is circular.
3. while the Earth is actually tilted 23.5° on its axis, for clarity, let’s assume the tilt is 40°.

It will be helpful in the discussion to be familiar with all of the terms shown on the following diagram.

If we observe the position of the sun every day at the same time each day we notice that throughout the year the sun moves slowly to the east against the background of the stars taking one year to return to its starting position. We can’t really "see" the stars behind the sun - it’s just too bright. We can imagine the sun as being a little less bright so we can look at it and still see the stars behind it.

Let’s look at this motion for a period of one day. Since this movie is only 2 frames long, it is best viewed by moving the slider control at the bottom of the window.
show movie
The animation shows the path of the sun between March 21st and March 22nd. The middle panel shows what we would see if the Earth was not tipped 23.5° on its axis. If this was so, the motion of the sun against the stars would be in a horizontal motion only. Everyday at noon the sun will appear to be in the highest point in the sky or what is also 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 upon the season. This sun is the True Sun that will travel on the ecliptic. You have probably noticed that the True Sun is not on the ‘noon’ line on March 22nd. This will be explained.

Let's look at the motion of the sun around the earth for one year.

show movie
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 equinox and the autumnal equinox, the Mean Sun and True Sun are in the same position.

What follows is probably the most important concept of what actually is taking place. Looking at the top view, observe the motion of the True Sun and the Mean Sun on the celestial sphere.
view comparative diagram
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 June 21st. They start together but then the True Sun lags slightly behind the Mean Sun until sometime in May, 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 May, 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.

So... why does the True Sun lag behind or move ahead of the Mean Sun? Looking at the top view of the celestial sphere we can see that the ecliptic is nothing more than 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 autumnal equinox) and at this time will not have the perception of 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 in a forward motion towards the catcher.

Finally, let’s go inside of the celestial sphere and watch the path of the True Sun from the Earth as it makes its way around the celestial sphere.

show movie
Let’s look at a more traditional explanation of the effect of the Earth’s tilt on the equation–of–time.
Show Traditional Movie
This animation is viewed from the equator outside the celestial sphere. Again, we are observing from a point on the equator at noon for an entire year. On the celestial sphere we have added longitudinal lines. Let’s look at a close–up of the positions of the mean sun and the true sun a few days after the vernal equinox.
Remember that both suns are moving at the same speed around the celestial sphere - only their paths are different. Note that the True Sun has farther to travel to the same longitudinal position as the Mean Sun. As viewed from Earth, the True Sun has drifted to the west.

Now, let’s look at a close–up of the positions of the two suns around the summer solstice.
Notice that the True Sun has caught-up with the Mean Sun. Even though at this point in time they are both moving in the same direction horizontally, the True Sun is actually moving faster. Why? Notice that the longitudinal lines are closer together for the True Sun at this position on the celestial sphere.

And finally one last method of looking at the tilt effect on the analemma. We will observe the effect from the sun looking back on Earth.

Show Traditional Movie
The yellow spot is where the sun would appear directly overhead to you, the red spot, standing somewhere on the Earth's equator.
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Elliptical Orbit - Math Earth's Tilt - Math