Why Sunrises And Sunsets Change At Uneven Rates

TL;DR: Sunrises and sunsets don't change at the same rate, a phenomenon affected by Earth's axial tilt and its elliptical orbit. This blog presents an intuitive picture to understand why.

The Problem

Just a few days ago we had the autumnal equinox. The autumnal equinox is when the rate of change in daylight is at its maximum. Here in NYC, we are losing about 3 min a day. However, if you look at the schedule, you'll notice something peculiar:

Sunrise and sunset times for NYC. Note that sunrises move later slower than sunsets move earlier.

At this time, we're losing about 1 min a day of morning (later sunrise) and 2 min of evening (earlier sunset). Why are our evenings getting shorter much faster?

Existing explanations mention axial tilt and elliptical orbit, but don't provide an intuitive understanding. This post will.

The Plan

In order to gain an intuition of sunrise and sunset times, we'll need to go through a few topics in order:

• How the sun moves across our sky per day
• How the sun moves across the sky per year
• How this translates to sunrise and sunset changes

How Sun Moves Across Our Sky Per Day

We all know how the Earth revolves around the sun. But not all of us have had experience with how this translates to motion of stars and the sun in our sky. As a day goes by, an observer on Earth will see the sky rotate around a point. This is due to the Earth's rotation around its axis, and rotation around the sun. It will look like this:

The rotation of stars across the night sky, as seen from the Equator. Over an hour, the star Vega has moved along a circle (green arrow), whose center is Earth's axis of rotation.

You can see here, over about an hour, the star Vega has moved along this line. The axis of rotation is the North Pole. Note this was also what would be observed at the Equator.

For a city that is not at the Equator but anywhere else, this axis of rotation simply moves by the latitude. Here is New York City, at a latitude of about 40° for example:

The rotation of stars across the night sky for NYC. The motion is the same as one would observe at the Equator, with the center of rotation shifted. This is because Earth's axis of rotation for this observer has moved.

All objects very far from Earth will follow this path.

What About The Sun?

The sun's position relative to the background stars shifts over the year, while the stars themselves appear to be in fixed positions relative to each other. This is because the sun is close, relative to the stars. However, for short times (about a day), we can assume that it follows the same path as the stars:

The sun's path over the course of a day on Sept 23rd in NYC, showing how it follows a circular arc centered on the Earth's axis of rotation. The Sky has been removed for effect (obviously, it's daytime, so you wouldn't normally see these stars!).

Wherever the sun is in our sky, you can safely assume that it will follow a circle around the Earth's axis of rotation.

What Is A Day?

If you actually studied the motion of the stars, you'd notice a peculiar thing. A full day later, they're not in the same place. This is because they actually rotate every 23h56 min and not 24 hours. Why? Well, that's because Earth undergoes a full rotation every 23h56min! However, our concept of a day is aligned to the sun. Because we orbit it, the motion of the sun is a combination of the earth's rotation around its axis plus its motion along its orbit around the sun. This results in the sun appearing to circle the Earth every 24h. This will be important, because sunrise and sunset times are relative to our notion of a day, not from the actual period of rotation of the Earth.

How much will it move by? Since we're also interested in how the sun will move, we'll select a frame of reference aligned to a day. An easy way to do that is simply take a picture of the sun at the same time of day each day. For a circular orbit, our frame of reference would  roughly be the red arrow:

Every day at solar noon for a circular orbit, we're looking radially inwards.

Our orbit is not circular, but rather elliptical. In order to estimate a day, we need to average the mean position of the sun each day. For low eccentricity (less stretched out) elliptical orbits, this makes sense enough. Note that for Earth, the variation is very small, see here.

Sample elliptic orbit. Higher eccentricities stretches it out. Lower eccentricities compresses it until it looks like a circle. Note that the planet moves faster when closer to the sun but slower further away (Kepler's laws).

For this analysis, we assume the Sun's daily motion is consistent, an assumption that holds true for Earth but not necessarily for planets with more eccentric orbits or more chaotic star systems (multiple suns).

"Looks like we're going to be at peak intensity for a few Dubniums" (Dubnium-270's half life is ~1hour. How else would these creatures measure time?) Image courtesy of Gemini

How The Sun Moves Across Our Sky Per Year

If you took a photo of the sky at noon every day for a year on Earth, you'll notice that the sun moves. As a matter of fact, it will move like this:

Sun's path over a year. Position of sun taken on the 22nd of each month at NYC latitude at 11:56 EST (12:56 EDT during daylight savings).

This picture in particular is called an analemma. This is what you would see if you were to take a photo of the sun at the solar mean (at noon). You can see plenty of them online, and here is a nice example.

This is taken from the latitude of New York City, but note that this would look similar at any other point on earth (well, except for places like the north pole in the winter where the sun is hidden behind Earth itself). Don't worry about exactly how this is caused. We'll get to that later. Let's first take a look at how this motion would tie into changes in sunrise and sunset.

How Does This Tie Into Sunrise and Sunset Time Changes?

I presented this analemma without really describing how we can use it to understand how our sunrise / sunset times might change. This section will explain why we care about it.

If you notice, the sun can move up/down (also called declination) or left/right (also called right ascension). What happens to sunrises/sunsets in each case?

Vertical Motion: Day Shortens/Lengthens As Sun Moves Down/Up

Imagine that the sun's apparent position in the sky moved up or down. What would happen? Well, if it moved down, the day would obviously get shorter as it would follow a smaller arc. Jump down and the day would get longer.

The day shortens when the sun moves down. This is because the angle decreases. 

The diagram above shows an example where the sun moves down in the sky. Because the sun will always follow a circle centered on the Earth's axis (green and blue arcs), by moving down, it will clear a smaller angle above the horizon. Note that if we were at the equator (so the horizon intersects with the point of the axis of rotation), $\theta_1$ would equal $\theta_2$. This is exactly what is expected since the length of a day is always 12 hours at the equator!

Horizontal Jump: Day Lengthens/Shortens And Shifts

If there was horizontal motion, we would both see a change in the length of a day and the day would shift.

The former has again the effect of increasing or decreasing the sunrise and sunset times. However, the latter has the effect of increasing/decreasing the sunrise/sunset times in tandem.

When the sun moves right, it moves to a new arc of rotation where it is not at the maximum. This arc is larger, so the length of day increases. However, it also shifts, since the day will end faster.

Putting It All Together

We've seen now that the motion of the sun at the mean solar noon tells us what happens to sunrises and sunsets. For vertical motions, the length of day lengthens/shortens and for horizontal motions, the length of day lengthens/shortens and also shifts earlier/later.

So what can cause uneven sunrise/sunset times? It comes from both a change in day duration and a shift in day. Let's say the length of day is reduced by $\Delta_{\mathrm{length}}$ and the day is shifted by $\Delta_{\mathrm{shift}}$, the shifts in sunrise/sunset will be:

• $\Delta_{\mathrm{sunrise}} = - \frac{\Delta_{\mathrm{length}}}{2} + \Delta_{\mathrm{shift}}$
• $\Delta_{\mathrm{sunset}} = \frac{\Delta_{\mathrm{length}}}{2} + \Delta_{\mathrm{shift}}$

Let's put some numbers to get a feel for this. Let's say the length of day decreased by 4 min ($\Delta_{\mathrm{length}}= -4 \mathrm{min}$), but also shifted earlier by 2 min ($\Delta_{\mathrm{shift}} = -1 \mathrm{min}$). 

We would then have:

• $\Delta_{\mathrm{sunrise}} = 2 \mathrm{min} -1 \mathrm{min} = +1 \mathrm{min}$
• $\Delta_{\mathrm{sunset}} = - 2 \mathrm{min} - 1 \mathrm{min} = -3 \mathrm{min}$

We see that sunrise shifts later by 1 minute, while sunset shifts earlier by 3 minutes.

So What Would Happen If...

Now we understand how the movement of the sun affects sunrises and sunsets, let's get a feel for what would happen in a few extreme examples. We can do so thanks to M. Tirado's Analemma calculator here (please also read their blog post if you're mathematically inclined, it's worth a read and complementary to this post).

Circular Orbit

A circular orbit is just a point. This makes sense as at every day the sun never moves.

The analemma for a circular orbit with no axis tilt. As you can see, the sun's apparent position does not change over the year.

An observer in this case would never see sunrises and sunsets change!

No Tilted Axis, Just An Ellipse

The analemma for an elliptical orbit with no axis tilt. As you can see, the sun's apparent position only moves left and right.

 

This case is interesting. All we would see is movement left and right. This is because, unlike a circular orbit, the sun will move faster/slower depending on where it is in the orbit (for a good reference, see here). However, this still results in lengthening/shortening of days as well as shift in time. So we would still observe sunrise/sunset asymmetries.

Tilted Axis But Not Elliptical Orbit

If Earth only had an axis tilt and a nice circular orbit, its analemma would look like a nice symmetrical figure 8 shape.

The analemma for a circular orbit with an axis tilt. As you can see, the sun's apparent position changes in a figure 8 shape. An observer here would see sunrises and sunsets change in an asymmetric way as we observed here.

The up/down motion can be easily understood by thinking of what happens in the summer or winter. In the summer, the sun is higher in the sky than in the winter. This can be easily seen by looking at the angle the sun makes with respect to Earth's axis for an observer in the summer or winter.

The angle with respect to earth's axis in the summer ($\theta_s$) versus winter ($\theta_w$).

The sun in the summer versus winter for NYC. The sun moves down from summer to winter.

The left/right motion can be understood by changing the frame of reference to be oriented with the axis tilt:

The orbit of a planet around the sun with an axis tilt. The orbit is assumed to be circular.

If look at this orbit from the top, it begins to look elliptical!

The projected circular orbit as viewed overhead.

The stretched section will appear to move slower, and the non-stretched portion will appear to move faster. This will result in the shift of the solar mean (left/right shift) as explained in the ellipse section. Note that this is different from an elliptical orbit, where the sun would not be in the center and the regions of the planet moving faster are closer to this sun.

Earth's Case

Earth's case is a combination of both. The elliptical orbit warps the analemma for a tilted axis a little: 

The analemma for Earth. As you can see, the sun's apparent position has a figure 8 shape, and it is asymmetric (due to the elliptic orbit).

Notice this looks just like the simulation (except warped).

Conclusion

We were able to gain an intuitive understanding on how sunrise/sunset times were affected by Earth's orbit. We did this by translating the complex motion of the Earth's orbit into an analemma, and using that analemma to understand what might happen to sunrise/sunset times. As we can see, the Earth's tilt is enough to cause this phenomenon. However, the Earth's non-circular orbit does contribute to changing it.

We did not delve into calculating the numbers as this would have made this blog post too long. However, I hope this gives you an intuitive enough understanding to be able to explain why sunrise/sunsets change, and the nuances involved next time someone may ask!