To Be Determined

Why does the moon appear where it does in the sky?

Why can we see it during the daytime sometime?

When my mother asked me these questions while on a walk, with the daytime moon clearly looming some 400,000 kilometers away, I was frustrated by my lack of complete understanding of the path that the moon took around our home planet. I took my first orbital mechanics course at university this semester (Fall 2020) and have been learning about the general properties of the elliptical orbits that Kepler had described in a great leap of an attempt to rectify Copernicus’ heliocentric model with observational data. Copernicus’ model that described the Earth as an orbiting body for the first time in history, interestingly did not produce more accurate predictions of planetary motion than the widely assumed Ptolemaic system (everything orbited around the Earth).

Despite taking the course, I was unable to completely visualize the motion of the moon around the earth primarily because of two key unknowns.

1. I did not know the characteristics of the moon’s elliptical orbit around the earth
2. I could not visualize how this would appear to an observer on the rotating body that is the Earth

On top of that, a quick google search yielded seemingly zero results in a query for the path of the moon as it appears from the Earth. All I could find was several time-lapse images, and I couldn’t be certain what time period exactly they were showing and several appeared to be photoshopped.

So, after my finals finished up, I took my lack of understanding upon myself and created a model in Matlab to show me exactly how the orbit of the moon looked, and most importantly, how it appeared to an observer on Earth.

In order to do this, I needed to create what is called a “ground track” for the satellite (moon), or a plot of the orbiting body’s movement around the larger body. But before I dig into the fruits of my labor and a few cool plots, let’s discuss those two key unknowns in a little more detail.

Disclaimer: for this post, I assume you know a bit about how Newton’s law of gravitational attraction work. If you don’t know a lick about that, go look it up!

Orbital Elements

The first thing that I didn’t know was what are called the “orbital elements” of the moons orbit. The orbital elements are six key parameters, shown in the crazy diagram below, that mathematically describe the unique orbit of a two-body system in Keplerian orbit. I won’t get into N-body systems right now, but we really only get truly Keplerian orbits for two body systems assuming that the masses are point particles and there are no other massive bodies nearby (can you think of a bright massive body relatively close to us). Kepler’s first law is that all planets move in elliptical orbits with the sun at the center.

Anyway, let’s understand the diagram below:

The blue is the path of the orbiting body, and the center of the three axes is the body it is orbiting.

We can see that the orbit is an ellipse, which has a long axis and a short axis. The center of the orbit will always be at one focal point of the ellipse, so there is always a side of the orbit where the satellite will be furthest from the center. This is called apoapsis, and the distance is “a,” or the semi-major axis. The opposite side is called the periapsis.

Next the eccentricity, “e,” is a mathematical property that essentially tells us how much the ellipse deviates from being a circle. Note that any orbit can be circular, it simply depends on the energy the orbiting body has – obvioΘusly there is only one way to make a circle, so this almost never happens in nature, at least in our solar system.

Now we see “i,” or the inclination of the orbit plane from the equatorial plane of the center body. This is basically the angle difference between the two discs (also between the angular momentum vector and the reference z-axis, if you are familiar with momentum).

The last three “Ω,θ,ω,” represent the angles between the x-axis and the ascending node (where the satellite crosses from below the equatorial plane), the angle between the satellite position vector and the vector that points towards the periapsis, and the angle between the line created by the two points where the satellite crosses the equatorial plane (line of nodes) and the vector that points towards the periapsis, respectively. These three are constantly changing for the moon’s orbit, something I’ll get into at the end.

Inertial Reference Frames

Now the other thing we need to understand is a little simpler: inertial frames.

A frame is inertial if its center is still or moving in a line with constant velocity. This means that if the observer is rotating or otherwise accelerating, then their frame is technically not inertial. Why does this matter? If I throw a baseball, you can assume that the ball is going to move so slow it’s as if as if we are standing on a flat earth that isn’t moving. It really doesn’t matter. But if I take super solider serum and throw the ball at a few km/s, suddenly, it matters which direction I throw it because of one key fact: the earth is rotating, so now you and I have to account for the fact that the ball was moving when I threw it and that the earth (and thus you the observer) is moving underneath the ball.

So the difference is, from an inertial frame, it looks like the ball is moving in a curved line, falling back to the earth (like normal), but from a non-inertial frame, the ball could be moving “upwards” and to the side if say, you were rotating 90 degrees from the direction I threw it.

If this is hard to visualize, the groundtrack plots below should help out.

Side note: fun thing about inertial frames! There is no such thing as an inertial frame, everything is always moving with respect to everything else, it all depends on the significance of this to your measurement. Einstein really hammered this home when he broke the entire concept of “simultaneity,” but that’s another post.

Elliptical Orbits & the Moon

One of the key properties of elliptical orbits that Kepler discovered in the 17^th century was a mathematical relationship between the rate of change of area that was swept out by the slice of the ellipse and the angular momentum of the orbit. Because the angular momentum is conserved, the area swept out must be the same per unit time. The upshot of this is that the orbiting body will move faster when it is closer to the center of orbit (each slice has less area) and slower when further away (each slice has more area). This is Kepler’s Second Law

Why is this useful? Kepler also figured out a Third Law: using the area of an ellipse, we can solve for the total time that it takes for an orbiting body to go around it’s orbit once, known as the period. We find that the period is proportional to the 3/2 power of the distance (semi-major axis, a). A lot more math could be done here, but we can also use this and differential calculus to solve for the time it takes the body to orbit from one specific point to another in solving what is called Kepler’s Equation.

Since we now have an understanding of elliptical orbits, let’s look at an example of a ground track for a satellite. This satellite will have orbital elements of

[7000km .01 50˚ 0 0 0] in order [a e i ω Ω θ]

Given that the Earth radius is ~6378 at the equator, this is a very close circular orbit with inclination of 50 degrees.

I plot this for 3 periods by solving Kepler’s equation for orbit at increments along the orbit to be in the time domain and the differential equations for gravity between two bodies, assuming the mass of the Earth is so large that the satellite’s mass is negligible (approximately 0). You can see why this might be a problem for bigger things like the moon that represent a significant fraction of the Earth’s mass. Also, 3 periods means the satellite goes around the Earth 3 times here.

Let’s look at the charts.

Notice that the red lines that represent the path of the satellite shift in the westward direction. This is because the Earth is spinning on its axis west to east or counterclockwise from the north pole, so land moves rapidly underneath the satellite’s orbit in the eastward direction. You and I move under the satellite in this direction!

So now let’s apply this to the orbit of the Moon around the Earth. A quick google search tells us that the moon has orbital elements of

[405,400km .0549 5.15˚ ~ ~ ~]

The last three elements, as I mentioned are always changing along respective timescales, and I’m not going to model how that looks. Interesting effects abound here though, and I’ll discuss those at the end.

Now, solving the differential equations and iterating around a single period of the moon’s orbit gives this plot:

It’s not super helpful for visualizing the path, and if I were to increase the fidelity of my solver for more points, it would just be a solid red band ~5 degrees from the equator. This is because the moon is so far from the Earth that the Earth can rotate under it several times before it completes one revolution! Approximately 27.3 days for one full moon revolution, actually.

So taking that into account, I made this animation of the moon’s orbit over the course of a single day.

Notice how the moon looks like it’s tracking west? It’s not. It’s actually in a prograde orbit, which means that the moon revolves around the earth the same way that it rotates (you can see the moon’s true motion during a solar eclipse because the moon will move west-east in front of the sun, visible because the sun is inertial from this frame). The reason it seems to be moving west from our perspective is because we are spinning under the moon faster than it moves around us because it is so far away!

Why can we see the moon in the sky sometimes during the day? Well, the moon is always reflecting the light of the sun, we can see it if it’s a clear enough day if it’s above us in the sky. There are two common measurements that can give us some insight on where the moon will be over a day. Elevation is the measure in degrees of something above an observer on Earth from the horizon, and Azimuth is the measure in degrees from the North direction. Using a series of transformation matrices, I can find the plot of both for an observer in Austin, Tx.

This is over the course of the day that I plotted in the animation as well. Notice how the Elevation dips below the horizon for part of the day, but less than half? The moon is always in the sky for some part of the day, some part of the night!

What if we were much closer to the north pole of the planet?

This is over a period of several days, but notice that the moon appears much lower in the sky as we get further from the equator.

Let’s look again at the entire orbital period of the moon. Remember inertial and non-inertial frames? We have been looking at non-inertial frames for the ground track, but only in two dimensions. First, lets see what the 3D inertial frame would look like

Notice how the orbit is always in the same band? This isn’t realistic because of the effects of precession that happen to the moon’s orbit that change it’s orientation, but given a timescale of around a month for the single period, this is pretty accurate. This is also roughly to scale, too!

Now if we look at the Earth Centered Earth Fixed frame, we are now in a frame rotating with the Earth and centered at its radial center. This is the path that the moon takes over a period as it appears for the Earth observer.

Notice the toroidal spiral-esque pattern that the moon makes around the planet? That’s because as the moon orbits the Earth spins around several times, so for some turns, the moon will appear further, some, lower. Some, a higher inclination, some, a lower. The key is that this is periodic, which is why we get this pattern in the non-inertial frame.

What’s Wrong Here?

So now that you and I have a much better understanding of how the moon moves around the Earth and why we can see it in the sky during the day, I want to discuss the nuance of the topic just a little bit more.

For starters, how close is my simulation to reality? It’s close! But it’s wrong. I calculate the moon completing an 360 degree orbit in 29.7 days. The moon actually does this in 27.3 days, more than 2 days faster than my calculations. Why is this? It’s because the moon has mass, 1.2% of the Earth’s, which means that the Earth and moon revolve around a common center of mass. This common center is about 72.6% of the radius of Earth into the Earth, and accounting for this and the mass of the moon slightly changes the orbital elements and mechanics of the orbit such that it will be faster.

Another interesting effect that I’m not accounting for or showing is the precession of the moon’s orbit. There are three ways the moon precesses, which is a change in orientation of the rotational axis of a rotating body. They are: Axial, apsidal, and nodal precession.

Axial precession is the precession of the moon’s axis of rotation over about 18.6 years – the north pole effectively traces out a circle. This is caused by the Moon’s non-uniform spherical shape and the significant gravitational effects this has when torqued by the Earth, and it happens to all large orbiting bodies. This isn’t really going to effect our simulation, but it will make the moon appear different in the sky over such a period.

Nodal precession is the precession of the moon’s entire orbital plane to spin around the Earth over a period of about 18.6 years. This is caused by the Earth’s equatorial bulge, which torques the orbit of the moon along its equator causing the entire orbital plane to precess. This means that the point that the moon crosses the Earth’s ecliptic plane rotates around the Earth in this period. This is the point near which solar and lunar eclipses occur and the effect of this in our simulation means that over the course of two decades, we would see the elevations of the moon in the sky approach their extremes.

Now the second orbital precession, apsidal precession of the moon is the most interesting, and the reason my simulation of a ~month appears the most incorrect. Apsidal is the precession of the orbital plane along this plane and it is caused by the tidal effects and the perturbations of other non-spherical bodies that torque the orbit, exchanging angular momentum and changing the eccentricity and causing this precession. If the gravitational attraction was always 1/r^2 (i.e. everything was spherical and non-rotating), this precession wouldn’t occur. This happens to all orbiting bodies of large mass, and for the moon, the precession appears to occur every 8.85 years when viewed from Earth. This means that the point where the moon is furthest from us is constantly changing!

Interestingly, because these bodies are so large and rotating, we must account for relativistic effects here: the correction that General Relativity provided for the apparent anomalous precession of Mercury’s orbit serves as one of the primary confirmations of the theory.

One last discrepancy I’d like to address, and certainly one of the largest: the sun. More accurately, its gravitational effects. The moon and Earth actually orbit a common gravitational barycenter, as I stated, and this common center of mass is what actually orbits around the sun. Or, both the moon and the Earth orbit the sun. Typically, we think about the orbit of the moon from the perspective of the Earth. It’s only natural, but it gives the wrong impression – the moon’s orbital velocity around the earth is 1 km/s while around the sun it is around 30 km/s. The moon doesn’t do this:

Both the sun and the Earth are large masses that effect the acceleration of the moon proportional to their mass and inversely proportional to the distance squared. If we look at this relationship we can see that there is a limit to the distance a satellite can be from the Earth (or any body) before it starts to orbit the sun (or any larger body!). The set of these radii is called the Hill Sphere. The moon is within the Hill sphere because it orbits Earth, but it still is accelerated by the sun. Let’s look at the relationship between the gravity of the sun and Earth to find the threshold radius where the moon will accelerate by the sun more than the Earth and analyze all the planets in our system.

We see something really interesting: the moon is the only satellite that orbits outside of this threshold radius! This means that while it orbits the Earth, it is always falling towards the sun, or the orbit of the moon is always convex when viewed from the sun. The moon is never looping around the earth, creating little “cusps” in its orbit around the sun. Instead, if we zoom in, it looks like this

For a better explanation, check out the wolfram link in the further reading section.

So from this, I can conclude that my decision to exclude the effect of the sun’s acceleration on the moon in my simulation to be incorrect: although it acts on both the Earth and the moon, the magnitude of this force is time and spatial dependent on both bodies and creates a significant effect on the moon. I would need to simulate this three body system from the ECEF inertial frame in order to get a better plot of the moon’s true path over several years.

Or I could go outside and look up.

“The sky puts everything in perspective”

Emily Sarig King

Further Reading

Exact Solution for the Metric and the Motion of Two Bodies in (1+1) Dimensional Gravity

Tidal evolution of the Moon from a high-obliquity, high-angular-momentum Earth

https://en.wikipedia.org/wiki/Milankovitch_cycles

https://en.wikipedia.org/wiki/Milankovitch_cycles

https://www.wired.com/story/what-is-angular-momentum/

https://demonstrations.wolfram.com/EpicyclesRevisitedConvexityOfLunarOrbitAboutTheSun/

Sources

• James Binney (2016) – Astrophysics
• NASA.com
• KhanAcademy.org
•  David M. Kipping (8 August 2011). The Transits of Extrasolar Planets with Moons
• Hawking, Stephen. On the Shoulders of Giants : the Great Works of Physics and Astronomy.
• Schaub, Hanspeter (2003). Analytical Mechanics of Space Systems
• Physics.stackexchange – honeste_vivere