Moon – Bartosz Ciechanowski
In the vastness of empty space surrounding Earth, the Moon is our closest celestial neighbor. Its face, periodically filled with light and devoured by darkness, has an ever-changing, but dependable presence in our skies.
In this article, we’ll learn about the Moon and its path around our planet, but to experience that journey first-hand, we have to enter the cosmos itself.
Let’s take a look at the Moon as seen from space in all its sunlit glory. You can drag it around to change your point of view, and you can also use the slider to control the **date and time**:
In this convenient view, we can freely pan the camera around to see the Moon and its marvelous craters and mountains from various angles. Unfortunately, we don’t have that freedom of motion in our daily experience – the Moon wanders on its own path across the daily and nightly skies.
We can simulate these travels below, where you can see the current position of the Moon in the sky. You can drag that panorama around to adjust your viewing direction – this lets you see the breadth of the sky both above and below the horizon. By dragging the sliders you can witness how the position of the Moon changes in the sky across **days** and **hours** of your local time. As the Moon’s placement in the sky shifts, the little arrow will guide you to its position.
You can also drag the little **figurine** on the globe in the bottom-right corner to see how the sky looks at that location on Earth. If your browser allows it, clicking tapping the button will automatically put the **figurine** at your current location. This may all feel quite overwhelming at the moment, but we’ll eventually see how all these pieces fit together:
Over the course of **one day**, the Moon travels on an arc in the sky _almost_ completing a loop around the Earth. As the **days** pass, the Moon’s illumination also visibly changes.
You’ll probably admit that it’s a little hard to focus on the tiny Moon as it shifts its position in the sky. To make things easier to see, I’ll zoom in the camera and lock its position on the Moon:
Notice that across a **single day** the Moon seems to rotate, and over **many days** it quite visibly wobbles. These wobbly variations let us occasionally see some hidden parts on the “edges” of the Moon, but our neighbor ultimately shows us only one of its sides. In our space-floating demo we could easily see the Moon from all sides, but on Earth we can never see most of the _far side_ of the Moon.
Over the course of **days**, the lighting on the Moon also changes dramatically. The line between the lit and unlit parts of the Moon, known as the _terminator_), sweeps across the Moon, revealing the details of its surface. Although the Moon has a spherical shape, the fully lit Moon looks more like a flat disk.
In this article I’ll explain all the effects we’ve just seen, and we’ll also learn about gravity, ocean tides, and eclipses. Let’s begin by exploring how celestial bodies move through space and how their mere presence influences the motion of their neighbors.
Motion in Space
Let me introduce a little cosmic playground in which we’ll do our experiments. Inside it, I put a little **planet** that floats freely in space. You can drag the **planet** around to change its position. The arrow symbolizes the initial velocity of this **body** – you can tweak this velocity by dragging the dashed outline at the end of the arrow. To get things going, you can press the button in the bottom-left corner:
Notice that I’m drawing a ghost trail behind the moving **planet**, making it easier to track its motion. As you can see, once you let the **planet** go, it travels through space in a straight line, only to eventually get out of visible bounds.
Let’s complicate things a little by adding **another body** to this sandbox. You can tweak the positions and velocities of **both****bodies** to see how their mutual presence impacts one another. I’m also marking the thin lines of trajectories that the bodies will take even _before_ you let things go, making it easier to plan their motion:
The motion we see now isn’t as straightforward as before. In some scenarios, the **two****bodies** travel past each other after tweaking their initial trajectories. In other configurations, **both****objects** roam through space together, permanently locked in a swinging dance.
You may have also managed to make the **two****bodies**run into each other. We’ll eventually see a more realistic visualization of that scenario, but in this simplified simulation when two objects collide, they just stick together and continue their coupled journey.
What’s responsible for all these effects is the force of _gravity_ acting on the objects. Let’s explore that interaction up close. As before, you can drag the **two****bodies** around, and you can also change their masses using the sliders below:
The **arrows** represent the force of gravity acting on the **two****bodies** – the longer the arrow, the larger the **force**. For completeness, I’m displaying the values and units of masses and distances, but the numbers aren’t particularly important here. What matters is that when we increase either the mass of the first body **m 1** or the mass of the second body **m 2**, the **force of gravity** grows too.
Moreover, the magnitude of **gravity** also depends on the distance **r** between the objects. As bodies move farther apart, the **gravity** weakens. Notice how the **forces** acting on each body have the same magnitude, but they point towards the other body, which indicates an _attractive_ force.
If you paid close attention to the lengths of the arrows, you might have noticed that the **force** decreases quite rapidly with **distance**. We can visualize this with a plot, in which the **white** line shows the magnitude of **gravity** as a function of **distance**. More precisely, it shows that **gravity** is inversely proportional to the _square_ of that **distance**:
Let’s take a very brief mathematical interlude to describe what we’ve seen in more detail. All these dependencies are captured in the following equation for the **force** of gravity **F**, between **two****objects** with masses **m 1** and **m 2** separated by distance **r**:
F = G × m 1 × m 2/r 2
The _gravitational constant_**G** seen in front of the right-hand side of the equation is incredibly small, making gravity a very weak force. We have no issues lifting everyday objects despite the might of the mass of the entire Earth pulling them down.
While the strength of gravity between any two bodies is equal, the resulting change in motion is not. You may recall from elementary physics classes that force **F** is equal to mass **m** times acceleration **a**. We can encapsulate this idea in a pair of simple formulas that tie these values for the **first** and **second** body:
F = m 1 × a 1
F = m 2 × a 2
By plugging in the equation for the force of gravity **F** and reducing the masses, we end up with a set of two equations for _accelerations_ of the bodies:
a 1 = G × m 2/r 2
a 2 = G × m 1/r 2
Notice that the acceleration of the first body **a 1** depends on the mass of the _second_ body **m 2**. Similarly, the acceleration of the second body **a 2** depends on the mass of the first body **m 1**. Let’s see this in practice in the demonstration below, where I’m temporarily making the **big body**_twenty_ times more massive than the **small body**:
Notice that the body with **smaller mass** drastically changes its course, while the motion of the **larger body** is only marginally affected. This tracks with our day-to-day experience, where every item left hanging in the air very visibly accelerates towards the staggeringly massive Earth, but our planet doesn’t jump out of its way to meet the falling object.
Now that we understand that it’s the force of gravity that makes the bodies move towards each other, let’s do a better job of tracking the motions of these objects over time. Right now our camera is fixed in space, so the **two****bodies** often fly out of visible bounds. Thankfully, we can easily fix this by moving the camera _with_ the bodies.
In the demonstration below, I’m presenting _the same_ scenario from two different vantage points. On the left, I’m showing the scene from the familiar point of view that’s fixed in space – you can plan the trajectories of the **two****bodies** on that side.
On the right, you can see this simulation from the point of view of the camera that’s tied to the motion of the **these****objects**. I’m marking the position of that camera with a white dot on the thin line joining the bodies. By dragging the slider you can **move the camera** between them:
With the camera following the bodies we can now track their motion forever. More importantly, we can also see the _relative_ motion of the two objects. When you make the bodies move together, you can witness how from the perspective of the **teal body**, it’s the **yellow body** that orbits around the **teal body**, but from the perspective of the **yellow body**, it’s the other way around.
Better yet, if we position the camera halfway, or even anywhere else between the **two****bodies**, _both_ objects seem to orbit the camera. The perception of relative motion depends on the point of view, but there is one point that’s particularly useful for observation. In this next demonstration, I’ve added a little white trail to the camera itself. Watch how the path of the camera in space changes as you **reposition it** with the slider:
In general, the camera traverses some squiggly path in space. However, there is one special position between the **two****bodies** for which the camera travels in a perfectly straight line. This point is known as the _barycenter_), and it’s located at the _center of mass_ of **these****objects**.
Let’s explore the concept of the barycenter a little closer. In the demonstration below, you can once again drag the bodies around to change the distance between them, and you can also use the sliders to tweak their masses. The center of mass of these **two****bodies** is marked with a black and white symbol:
The equation in the bottom part explains the placement of the center of mass of these two objects – it is located at a point where its distance from the first body **r 1** multiplied by that body’s mass **m 1**, equals that point’s distance from the second body **r 2** multiplied by its mass **m 2**.
This simple rule becomes slightly more complicated when more than two bodies are involved. In those scenarios, the position of the center of mass is the _weighted average_ of the positions of all the bodies, where the masses of these bodies serve, very appropriately, as weights.
We’ll only be interested in the center of mass of two bodies, so the demonstration we’ve just seen fits our needs well. Notice that as the bodies move farther away, the barycenter also migrates to stay in the constant proportion of the distance separating the objects. Moreover, if one of the bodies is much more massive than the other, the center of mass could lie inside that larger body.
In our space simulator, the mass of the **teal body** is three times the mass of the **yellow body**, so the barycenter of this system lies three-quarters of the way between the **yellow** and **teal** objects:
The motion of the barycenter shows us that the tangled dance of two celestial bodies hides a much simpler linear motion through space _and_ some additional motion of the **two****bodies** around that barycenter .
Let’s try to see that other motion more clearly by making one more modification to the right side of the demonstrations we’ve seen. Notice that the trails left by the bodies linger in space, but ideally, we’d also want to see the paths taken by the bodies relative to the moving camera.
To make this work we can attach a little drawing plane to the camera itself – I’m outlining that plane below with a thin rectangle. Then, as the bodies move around, they can trace their trails on that plane as well:
With this new method we can see the paths the bodies took _relative_ to the moving camera . When seen from this perspective, we can finally reveal that, in most practical scenarios, the two orbiting bodies trace ellipses relative to each other.
Depending on the initial conditions, some of those ellipses are larger, and some are smaller. Some are almost circular, and some are quite elongated. Changing the **position of the camera** with the slider changes the relative sizes of these two ellipses, but they maintain their overall proportions. The ellipse of motion of one body seen from the perspective of the other is _the same_ for bothbodies, it just shifts in space.
As you may have seen on this blog before, an ellipse can be more formally characterized by its **eccentricity** and the size of its **semi-major axis**, which you can control using the sliders below:
**Eccentricity** specifies how elongated an ellipse is. It can be defined as the ratio of the length of the **dark pink segment** to the length of the **semi-major axis**. That **segment** spans the distance between the center of the ellipse and one of the two **focus points**, which are also jointly known as _foci_. When we watch orbital motion from the perspective of the **orbited body**, that **body** is always in one of the **focus points** of the orbital ellipse of the **orbiting body**.
I’ve also marked two special points on the orbital ellipse. At **apoapsis**, the **orbiting body** is at its farthest distance from the **orbited body**, and at **periapsis** the **orbiting body** is closest to that **body**. These two points are collectively known as _apsides_, and the line joining them is known as the _line of apsides_. The simple rule for remembering which apsis is which is that **a poapsis** is the one that’s farther a way from the **orbited body**.
We’ve just described the orbital ellipse and its apsides as seen from the point of view of the larger body, but in our cosmic playground we’ve seen how moving the camera around with a slider can change the perception of motion:
With a two-body system like this one we actually have some flexibility in describing which body orbits which. We typically say that it’s the **less massive object** that orbits the **more massive one**, but the observer on the **smaller body** would just see the motion of the **larger neighbor** around it.
For us, it will be often useful to describe things from the point of view of the barycenter – we’ve seen earlier how that special point lets us decompose the motion of two solitary bodies into the movement on a straight line and the orbiting motion around that barycenter.
That particular viewpoint also lets us explain another irregular motion we can see in these elliptical orbits. Notice that as the two bodies are close to each other, they swing across their trajectories much faster.
You can see it best when looking at the dashed segments I’ve drawn on the elliptical orbits – traversal of each brighter or darker section takes the same amount of time. These lines are visibly longer when the bodies are close, which reflects their faster motion as they travel longer distance over the same period.
This non-uniform motion can also be seen in the angular velocity of the orbital motion, which describes how many degrees per second an orbiting body sweeps through. In this next demonstration the **blue line** rotates with constant angular velocity, so in every second it goes across the same number of degrees. As you can see, the **orange line** joining two bodies rotates with varying speed:
Notice how the **orange line** is sometimes ahead of and sometimes behind the **blue line**, which shows that the orbital motion doesn’t have a constant angular velocity.
This unusual behavior is more easily explained with the following contraption, where I put the **two****bodies** on a **giant bar** that spins around on an axis placed right at the center of mass of the two bodies. Using the slider you can change the **distance** between these objects:
As the bodies get closer, the rotation speeds up. Conversely, as the bodies move farther apart, the rotation slows down. You can easily recreate a version of this experiment by holding heavy items in your hands and spinning on a desk chair with your arms spread out. As you pull them towards your torso, your rotation will speed up.
These are examples of _conservation of angular momentum_ in which the speed of revolution and the mass distribution of a system are inherently tied together. Broadly speaking, when we double the distance from the axis of rotation, the angular velocity becomes _four_ times smaller.
The space playgrounds we’ve looked at earlier work just like the demonstration with the bar, but instead of a slider, it’s the force of gravity that determines the distance between the bodies. Gravity pulls the objects closer together, increasing the speeds at which they swing by each other. As the bodies move past their closest distance, that increased speed shoots them out away from each other and the cycle continues.
The details on how this action creates elliptical paths are beautifully covered in the video on Feynman’s Lost Lecture, but for our needs it will be enough to just witness once more how all the initial values of masses, positions, and velocities of the **two****bodies** decide everything about their motion:
With a firmer grasp on orbital motion in space, we can finally see how everything we’ve learned affects movement of our planet and its closest celestial neighbor.
Moon and Earth
Let’s first look at the Moon and Earth side by side to compare their masses and sizes in imperial units metric units:
Moon Earth mass 0.01619 0.07346 1.317 5.972× 10 25 lb 24 kg mean radius 1079.6 1737.4 3958.8 6371.0 mi km volume 0.5270 2.1968 25.9876 108.321× 10 10 mi km 3 mean density 208.8 3344 344.2 5513 lb/ft 3 kg/m 3
The Earth’s mean radius is only around 3.67 times larger than that of the Moon. Since the volume of a sphere grows with the third power of its radius, and the Earth is on average much denser, our planet’s mass ends up being around 81.3 times larger than the Moon’s.
Let’s try to replicate this table in our space simulator, where I added **two****bodies** with sizes and masses matching those of the **Earth** and the **Moon**. Let’s see how these values affect the motion of the two objects:
With our **simulated Earth** being so massive, we can quite easily make this **Moon** orbit the **Earth** with various ellipses. Unfortunately, while this simulation correctly mimics the relative sizes of the real Earth and Moon, it doesn’t reflect the cosmic scale of the distance between these two bodies.
Let’s see how far away the Moon really is. In the demonstration below, you can use the slider to **zoom away** from the Earth until the Moon’s position becomes visible:
If you drag the slider all the way to the right, you’ll notice that I’m actually marking _three_ distances between the centers of the Earth and the Moon. The orbit of the Moon doesn’t form a perfect circle, so the separating distance varies as the Moon gets closest to the Earth at periapsis, and farthest away at apoapsis. The values shown here in miles kilometers are the predicted **maximum**, **mean**, and **minimum** of that distance in the 21 st century.
Let’s see the orbit of the Moon in more detail. The following demonstration shows the motion of our neighbor from the perspective of the Earth itself. You can drag around the following demonstration to change the viewing angle. The slider lets you control the **speed of time**:
With all the sizes and distances replicated realistically, it may be hard to see these tiny bodies. To make things more legible, you can press the button in the bottom right corner to toggle between the real and _ten_ times larger artificial sizing of these bodies.
With this three dimensional view we can now see that the Moon’s motion lies in the _orbital plane_ that I’m marking with a **faint gray disc**. To help us orient ourselves in space, I’ve also added a **line** that marks a fixed reference direction pointing at some very distant stars.
On average, it takes the Moon 27.322 days 27 days, 7 hours, and 44 minutes to complete the whole orbit, as measured by crossings of the **reference line**. That period is known as the _sidereal month_, where _sidereal_ means “with respect to stars”. This is only one of the four different types of _lunar months_ that we’ll explore in this article.
As the Moon orbits the Earth, it traces the familiar elliptical shape. We can quite clearly see how the elliptical eccentricity shifts the Moon’s path relative to the perfect circle of the visualization of the **orbital plane** that I’ve drawn above.
Let’s take a closer look at some of the parameters of the Moon’s orbit. In this next demonstration I’m using the current position and velocity of the Moon to calculate an ellipse that best describes the Moon’s orbit at _that_ moment of time. I’m drawing this ellipse with a dashed line, while the solid trail shows the actual path the Moon took:
Since we’re making the ellipse fit the current orbital motion, this idealized ellipse matches the actual trail very well in the vicinity of the orbiting Moon. However, farther away from the Moon this best-fitting ellipse _diverges_ from the path the Moon actually took. This shows us that while it’s pretty close, the Moon’s trajectory doesn’t form a perfect ellipse.
As we see in the labels, both eccentricity and the length of the semi-major axis of this “currently best-fitting” ellipse vary over time. Measured over a long period, the eccentricity of the Moon’s orbit has the average value of 0.0549, while the semi-major axis has the average length of 239,071 mi 384,748 km.
Moreover, the fitted orbital ellipse not only changes its shape, but also its orientation. The **line of apsides** of the ellipse which joins the **apoapsis** and the **periapsis** wobbles over time in a quite chaotic manner.
These effects happen because the Earth and the Moon aren’t the sole bodies in space – they’re both part of the Solar System. True to its name, the Solar System is dominated by the Sun itself, and it’s primarily the effects of the Sun’s gravity that cause all these _perturbations_) of the Moon’s orbit.
We’ll soon explore the influence of the Sun in more detail, but for now let’s focus on the changes of the positions of **apoapsis** and **periapsis**. In the demonstration below, I’ve made time flow even faster than before. Additionally, every time the Moon is at its closest to the Earth, that is when it’s at the **periapsis**, I’m leaving a little **marker** on the orbital plane:
Notice how the **line of apsides** wobbles back and forth, but across many months it overall makes steady progress rotating, when seen from above, in the counter-clockwise direction. Averaged over long time, this **line of apsides** makes a full rotation in 8.85 years 8 years and 310 days, which defines the period of the Moon’s _apsidal precession_.
The **markers** that I drop when the Moon crosses the **periapsis** measure the _anomalistic month_. Notice that the lengths of anomalistic months vary a lot as they happen on different parts of the orbit. Sometimes it takes the Moon less than 25 days to get closest to the Earth again, but sometimes it takes it over 28 days to reach **periapsis** again. Over long time the anomalistic month has a _mean_ length of 27.554 days 27 days, 13 hours, and 3 minutes.
This period is a bit longer than the 27.322 days 27 days, 7 hours, and 44 minutes of the sidereal month, which is tracked by the crossings of the **reference line**. When averaged over time, the **line of apsides** rotates steadily in the same direction as the Moon’s orbital motion, so it takes the Moon a bit more time to catch up to **periapsis**.
All the demonstrations we’ve seen also show one more effect that we didn’t account for in our simple playground simulations – both the Earth and the Moon spin around their axes. You can see this more clearly in the demonstration below where I glued a **blue arrow** to the surface of the Earth, and a **gray arrow** to the surface of the Moon:
When viewed from the side, we can see that the axes of rotations of these two bodies aren’t neatly perpendicular to the **orbital plane**, and they also spin at very different rates. Our planet takes roughly 23.93 hours 23 hours and 56 minutes or _almost_ one day to complete a full revolution and point towards the **reference direction** again. The Moon rotates much slower, taking 27.322 days 27 days, 7 hours, and 44 minutes to revolve just once and align with **that direction** again.
From above we can see that the **gray arrow** fixed to the Moon’s surface generally points towards the Earth, as indicated by the thin line joining the two bodies. If you pay close attention, you’ll notice that this **arrow** is sometimes pointing a bit ahead of that direction and sometimes a bit behind that direction.
This is a consequence of the Moon’s non-circular orbit – we’ve seen earlier how the angular velocity of an orbiting body changes as it sweeps through its orbital ellipse. The Moon rotates around its axis with more or less constant speed, but the Moon’s angular position relative to the Earth doesn’t advance at a constant rate. As a result, the two rotating motions don’t always perfectly cancel each other out.
In a close-up view of the bodies you might have also noticed that the rotation axis of the Moon is tilted relative to its **orbital plane**. Similarly, the axis of rotation of our planet is also tilted relative to that **plane**. Let’s briefly switch our point of view to align ourselves straight-up with the Earth’s rotation axis:
From this perspective we can see that the Moon’s **orbital plane** is _inclined_ to our planet. Notice how the Moon’s position relative to the Earth changes during its orbital motion – it is sometimes “above” and sometimes “below” our planet, revealing the truly three dimensional aspects of the Moon’s motion.
All the orbital observations we’ve made will help to explain some of the effects we’ve seen at the beginning of this article, where we looked at the Moon through the eyes of an observer on the ground. Before we investigate these effects, we need to build a bit more intuition on how objects in space look to someone viewing them from the surface of Earth.
Eyes on the Heavens
Let’s first place ourselves on Earth and look at the sky in which I artificially put three **colorful****celestial****bodies**. You can drag the demonstration around to change which part of the sky you’re looking at. If you lose track of these bodies, the little arrows will guide you back to their area of the sky:
Although the markers of the compass directions are of some help, it may be quite hard to grasp how this view from the Earth’s surface corresponds to the more external view from space we’ve gotten used to.
Let me clarify things in the next demonstration, where the left side shows the same view we’ve just seen, and the right side shows the same scene, but _as seen from space_. I’ve also outlined the sky view on the left with **the****four****colored****lines** – as you pan around the landscape on the left, you can see that square outline reflected on the right. I’ve also added a **figurine** that represents a _vastly_ enlarged observer standing on the ground. The **figurine’s** body and its _right_ hand always point in the current direction of observation:
With that external view, we can see how the observer on the ground can’t see the sky in _every_ possible direction. Half of it is obscured by the Earth itself, with the horizon clipping the whole breadth of the surrounding sky to only the visible hemisphere.
Moreover, notice how the actual size of an object doesn’t match its size seen in the Earthly observer’s sky. For example, both **yellow** and **teal** bodies are of the same _physical_ size, but the **latter** looks smaller in the sky. Similarly, the **pink** body is physically larger than the **yellow** one, but they share similar size from the observer’s point of view.
We can understand these sizing effects with the help of cones that shoot out from the position of the observer towards the bodies in space. Note that these cones start on the ground here, because the actual observer is much smaller than the gigantic illustrative **figurine**.
The size of the intersection of those cones with the hemisphere of the sky, or the size of the _projected area_, determines the _visible_ size. Intuitively, the farther away the object, the smaller it appears. If the projection occupies a larger fraction of the total hemisphere, the object will look larger as well.
We can conveniently describe the size of objects in the sky by measuring the _angle_ spanned by the visible cone. In the demonstration below, I’m showing a flat side view of this **cone**. You can drag the **yellow body** around to change its distance from the **observer**. You can also use the slider to change the size of that **body**:
The closer the object is to the **observer**, or the larger the body, the greater the **angle** of the visible cone. That **angle** is known as the _angular diameter_ or _angular size_ of the observed object.
Having experienced how objects in the night sky may look at a fixed moment in time, let’s see how the Earth’s rotation affects observations done from the ground. In the demonstration below, you can **scrub through time** with the slider to witness the effects of the spin of our planet:
This scene may seem a bit contrived, because **the****three****objects** are just magically floating in space at fixed positions. Fortunately, it’s a decent representation of how all the stars in the night sky appear to Earthly observers – they’re distant enough that over the course of a day they essentially don’t move relative to the Earth’s center. As our planet spins, **these****three****objects** seem to rise over the horizon, travel across the visible sky, and then set below the horizon again.
You’ll probably agree that it’s a little annoying to have to manually keep panning through the night sky to look at these objects, so on the left side of this next demonstration I’m automatically adjusting the viewing angle to track the **teal** body. On the right side, I’m locking the camera on the **figurine** itself. Don’t be misled by what you see here – the Earth is still rotating around its axis, the camera just rotates with it:
As seen through the observer’s eyes on the left side, the **other****objects** now seem to rotate around the **teal** one, but this is purely a consequence of the **observer** turning on the ground to keep facing the **teal** body.
You may have experienced something similar when watching an airplane flying over your head. As the plane is approaching, its front is closer to you and its tail is in the back, but after the plane has passed over, you see the plane’s tail as being closer to you, and its front is more distant. In your eyes the plane has rotated, but in fact the plane has kept its course the entire time, and it was _you_ who turned to keep an eye on it.
When these celestial bodies disappear beneath the horizon, it becomes impossible to track them, but thankfully in these computer simulations I can make the Earth transparent, giving us an unobstructed view of the full sphere of the surrounding space:
With this approach we can now see the entire trajectory of **the****three****objects** as an observer on Earth sees them. Because these objects don’t move relative to the center of our planet, they travel on closed paths, returning to where they came from after the Earth completes one revolution around its axis over the course of 23.93 hours 23 hours and 56 minutes.
Let’s bring back the Moon into the picture. In the simulation below, we can see the Moon in the starry sky as seen from the surface of the Earth. Note that I removed all the visual effects related to sunlight, including the daytime blue sky and any illumination changes on the surface of the Moon itself.
We’ll bring in those effects later on, but for now we’ll just look at the _artificially_ lit Moon over the course of the next 24 hours – you can **scrub through this time** with the slider. You can now also drag the little **figurine** around the globe to change the observer’s location, or click tap the button in the corner to jump to your location:
Notice how small the Moon actually is in the sky – it only spans around 0.5° of the viewing angle. Just like our colorful objects did, the Moon also travels across the sky as the Earth rotates. However, because the Moon _moves_ relative to the center of our planet, it doesn’t quite close up its path. This is easily observable from space:
Notice that over the course of 24 hours the Moon moves ahead on its orbit, so a bit more time has to pass for the Earth to rotate to have our neighbor be over roughly the same spot on the Earth again. Moreover, the inclined **orbital plane** shifts the Moon to be a little lower or higher relative to the Earth, so its arc in the sky shifts too.
Let’s go back to observing the Moon from Earth. To see our neighbor more clearly I’ll increase the zoom level of the camera, and I’ll lock it on the Moon:
When viewed this way, the Moon seems to rotate over the course of one day, but this effect is purely a consequence of the **observer** turning around to face the Moon – we’ve already seen this behavior with colorful objects seemingly rotating in space.
The observer stands up vertically on the ground, so as the Earth rotates, the **observer’s** “up” and “towards the Moon” directions change in space. How much these directions change depends on the **latitude** of the **observer’s** location. When seen from the equator, the Moon “rotates” quite rapidly as it passes over the **observer’s** head. On the North and South Poles, the “up” direction is fixed in space, which removes that daily rotation. Notice that even on the poles the Moon still visibly turns a little over the course of a day.
To investigate these subtler aspects of the Moon’s motion in the sky we have to give ourselves a bit more time for observations. In the demonstration below, you can track the Moon over the next 30 days. You can still drag the **figurine** to some other location, but the observer’s Moon-facing rotation can make things pretty nauseating outside of the poles, so you can always get back to that stationary location:
Notice that over the course of a month the Moon wobbles visibly, and it also changes its size. The oscillations we see here are caused by the Moon’s orbital motion.
Let’s see this more clearly from space by drawing the cone of visibility of the Moon for the observer on the surface of the Earth. Unlike in previous examples, where I’ve aligned the rotation of the camera to the **reference line**, this time I’ve synchronized our perspective with the orbital motion of the Moon, giving us an unchanging perspective on that body:
As we’ve discussed earlier, the Moon’s orbit around the Earth isn’t perfectly circular. The Moon changes its distance to our planet, which affects how large it looks in the sky. Below you can see a side-by-side comparison of the Moon’s visible size when it’s at **apoapsis** and **periapsis**:
The Moon’s orbital motion is also responsible for the periodic wobbles, which we can see clearly by once more gluing an arrow to its surface:
We can see from above that the Moon appears to wobble from side to side, because the angular speed with which it sweeps the orbit varies over time, while its angular speed with which it rotates around its axis is almost constant. Similarly, in a side view we can see that the axis of rotation of the Moon is tilted relative to its orbital plane, so we sometimes see more of the Moon’s top, and sometimes more of its bottom.
All the effects we’ve seen here are known as _librations_. Over the course of many days, librations make it possible to see around 59% of the Moon’s surface. However, because of the Moon’s synchronized spin and orbital motion, a large part of the Moon’s surface is never visible from Earth. It’s finally time to investigate how the Moon got locked into that motion by taking a more detailed look at gravity and the structure of celestial bodies.
Gravity at Scale
So far we’ve only been experimenting with gravitational interactions between two objects, but it’s time we vastly increased the number of participating entities. In the demonstration below, I randomly distributed over **1200 bodies** – they all gravitationally attract each other:
We’re watching this scene from afar, so the individual **bodies** we see here are very big – each on the order of dozens of miles kilometers across. Initially, these **objects** move very slowly, but the mutual gravitational forces consistently accelerate them towards each other, increasing their speed and kinetic energy.
This simple simulation doesn’t reflect this, but once these **bodies** collide, this energy gets released by heating up the matter constituting the **objects**. When hot enough, the matter loses its solid form and starts to behave more like a fluid that can relatively easily change its shape. The pushing pressure from the surrounding neighbors and the heat from the decay of radioactive isotopes also help to maintain that liquid form.
When in this state, this mass of matter can’t really maintain any rigid shape, and after wobbling for a while, it reaches an equilibrium forming a sphere. When no other forces are involved, this liquid spherical shape balances itself perfectly – any mountain that stands out gets gravitationally pulled towards the center, and any valley gets squeezed out by the surrounding matter trying to fill the empty space.
In the simulation we’ve just seen, all the **bodies** started frozen in space. Let’s see what happens when we give these **objects** some initial random velocity:
After a while we end up with the similar spherical shape, but this time this blob rotates. What we’re witnessing here is another example of the conservation of angular momentum in action.
From our previous examples you may associate angular momentum with some kind of spinning or orbital motion, but even the simplest movement on a straight line contains a rotational component when seen from an appropriate point. Below you’ll find a replica of the very first space simulation we’ve played with in this article, but this time I’m also drawing an additional dashed line spanned between the **yellow planet** and the central **blue point**:
That dashed line turns as the **body** moves in a straight line, revealing the rotational motion relative to the **blue point**. Even in this scenario the angular momentum of the system is maintained.
Through all the collisions in our complex system the velocity and the angular momentum of each of the hundreds of bodies constantly change, but, relative to some fixed point, the _sum_ of the angular momenta of all the bodies remains constant. Whatever original value of angular momentum this system had, persists forever.
In the initially chaotic motion of all the bodies there is some average amount of rotational motion. Once all these bodies get closer to each other, the angular velocity grows high enough to be visible. This is the exact equivalent of two planets orbiting each other more quickly as the distance between them decreases, but it happens here at much larger scale.
There is one more aspect of these self-aggregating blobs that we should explore. In this next demonstration, one fourth of the objects is **colored blue**. These **bodies** are much denser than the **others**, therefore, each is also more massive:
Notice that in the final liquefied planet these **denser objects** have a tendency to aggregate at the center of the body. We’re basically observing buoyancy in action, where this **denser material** sinks to the “bottom” of the planetary blob and the **lighter one** floats to the surface.
These accumulation, or _accretion_) processes that I’ve crudely simulated with a small number of bodies, happened on an absolutely massive scale during the formation of Earth and other planets. The whole fascinating history of the early Solar System is beyond the scope of our discussions, but the simple simulations we’ve seen highlight the origins of the Earth’s rotation, and illustrate why it’s _differentiated_ with a heavy iron core in the middle.
A few different theories have been suggested to explain the origin of the Moon itself. These days the leading one is the _giant impact hypothesis_, in which a large body hit the early Earth around 4.5 billion years ago.
Scientific opinions differ not only on the size, speed, and composition of the impacting body, but also on the subsequent process of formation of the Moon from the resulting debris.
Some earlier papers assume the Moon simply formed from the matter scattered into space after the collision. Other authors suggest that the energy released during impact created a huge, partially vaporized cloud of matter from which small moonlets condensed and accreted to create the Moon. Some other recent research shows with beautiful computer simulations that the proto-Moon may have formed immediately after the impact.
Any theory of the Moon’s origin has to end up with a similar state as the Earth and the Moon are in right now. For example, if we estimate that during the collision only a small amount of matter got ejected out of reach of the Moon’s and Earth’s gravity, the total mass of the two bodies before and after the impact should be more or less the same.
Moreover, the Moon is on average much less dense than Earth, because the Moon’s iron core is comparatively much smaller than that of Earth’s. If we assume that the colliding body and proto-Earth formed in the same area of the Solar System and therefore had similar composition, then this implies that a large part of the impacting object’s iron core must have transferred to our planet.
The Moon and Earth also share very similar ratios of isotopes of some elements, suggesting that the ejected material that formed the Moon was a mix of the proto-Earth and the other proto-planet.
Let’s try to recreate some simple collision scenarios using our rudimentary simulations. In the demonstration below, you can drag the impacting body around and change its initial velocity, similarly to how we did this in the introductory orbital simulations:
We don’t know what the initial conditions of this collision actually were, but when everything finally settled, we most likely ended up with the Moon orbiting Earth and both bodies spinning around their axes.
The simulation below gives a rough overview of this situation. Note that it doesn’t try to accurately reflect the distances, speeds, or surface details involved in those early stages, but it will be enough to help us explore the other details of gravitational effects between the **Earth** and the **Moon**:
Let’s try to first understand what forces the Earth imposed on this early Moon when we incorporate the more fine-grained scale of gravitational interactions we’ve been playing with. In the demonstration below, I put **three****small****bodies** far away from the **Earth**. Initially, **these****three****objects** are evenly spaced, but notice what happens to the distances between them over time:
The dashed circles show the original positions of the **pink** and **teal** objects relative to the central **yellow** body. Quite clearly, the three bodies seem to drift apart.
Recall that the force of gravity is proportional to the inverse of the square of distance between the objects. When an object is close to its **massive neighbor**, it’s also close to each tiny parcel of matter that makes up that **neighbor**. We can visualize this with a plot and arrows that show the Earth’s **gravitational forces** acting on these **equally****spaced****objects** placed at a varying offset – you can control it with the slider below:
The **pink body** is closest to **Earth**, so it experiences the strongest **force** and the strongest acceleration. Conversely, the **teal body** is the most distant, so it feels the weakest **force** and it doesn’t increase its velocity as fast as the **closer****bodies**. It’s this _variation_ in **forces** that increased the distance separating the objects in the previous simulation.
Even though **all****three****bodies** were moving towards the planet, from the perspective of the **central body** both its neighbors moved _away_ from it. It’s easiest to understand this effect by calculating the _difference_ between the **forces** acting on that **central body** and its neighbors. I’m drawing these force differences with **yellow arrows** that have been scaled up to be more legible:
These **yellow arrows** show actual forces on the **pink** and **teal** body _relative_ to the **yellow body**. As seen by the **yellow body**, its **two****neighbors** are pulled away by these so-called _tidal forces_, which arise from differences in gravity experienced by the bodies.
Our early Moon isn’t immune to these effects either. Because of its orbital motion it doesn’t crash into our planet, but its parts closest to the Earth feel a stronger pull than the Moon’s central sections. Those central parts are in turn pulled more forcefully than the Moon’s parts most distant from the Earth. We can visualize these forces by putting the **gravity arrows** on small sections of the Moon:
If we then calculate the difference between the **force** acting on each of those small sections and the **force** acting on the center of mass of the Moon, we can visualize the **tidal forces** acting on the Moon itself. For clarity, I’m drawing the arrows much larger than the gravity differences actually are:
As you can see, the **tidal forces** are trying to flatten and stretch the Moon both towards and away from the Earth. Thankfully, the self-gravity of the Moon is strong enough and the Moon is far away enough that these differences in the Earth’s gravity don’t pull the Moon’s body apart. However, the Moon does stretch a little, forming an elongated shape.
In the demonstration below, you can visualize this **stretching** with the second slider, but be aware that the distortion you’re playing with here is _vastly_ exaggerated:
As this early Moon keeps spinning around its axis, its different parts get closer and farther away from the proto-Earth. The elongation travels across the Moon’s surface, continuously morphing its shape. As you can imagine, it takes a lot of energy to deform a celestial body, and some of this energy inevitably gets lost due to friction.
These losses introduce a delay to the entire deformation process, and the maximum elongation is reached a little _after_ that area of the Moon has been closest to the Earth. As a result, the elongated bulge doesn’t point directly _at_ the Earth, but it’s carried ahead by the spin of the Moon – the elongated shape is a bit _off-axis_. In the demonstration below, you can play with an overemphasized degree of this **delay**:
Let’s pause here for a minute to understand what effect the **tidal forces** may have on this stretched and skewed body. In the demonstration below, I’m drawing a **long bar** that gets pulled by **two ropes** attached to its ends. Using the slider you can scrub through time to see how this contraption would behave when pulled by these **forces**:
Notice that initially the **bar** is rotated, so it’s a little off-axis with the directions of the **pulling forces**. This gives the **ropes** some leverage, and the **pulling forces** rotate the entire **bar** clockwise until it’s aligned _with_ those **forces**.
This is very similar to what happens to our spinning early Moon deformed by **tidal forces**. Since the elongation is slightly off-axis, the **tidal forces** rotate the Moon clockwise, which gently decreases the Moon’s existing counterclockwise spin!
The actual elongation and the off-axis skew of the early Moon were much smaller than what I’ve depicted here, and its non-circular orbit complicated things even more, but the net result of **tidal forces** was to slow down the Moon’s spin until it was synchronized with the Moon’s average orbital motion.
In this whole process the angular momentum of the Moon had to be conserved, so as the Moon’s spinning motion slowed down, its distance from the Earth increased. This kept the overall balance of how quickly all the matter rotated and how far from the center of rotation it all was.
In the last few paragraphs we’ve only focused on the effects of Earth’s gravity on the Moon, but everything we’ve discussed also manifests in the influence of the Moon’s gravity on Earth. The Moon also creates a slightly elongating bulge on Earth that travels across our planet as Earth rotates.
Earth also used to rotate faster, but tidal forces slowed down its rotation, transferring some of the angular momentum from the rotational motion to the joint orbital motion. This has also increased the distance between the two bodies.
Even today, Earth is very gently slowing its rotation, with the average day getting longer by about 2 milliseconds per century. As a result, the Moon is also moving away from our planet at the rate of around 1.5 inches 3.8 centimeters per year. Part of the present-day energy dissipation is caused by the deformation of Earth itself, but most of the energy gets lost in the oceans in the form of _tides_.
The forces driving the ocean tides have the very same nature as the ones we’ve just discussed. To understand how they work, let’s look at a **fictional planet** completely covered with a deep layer of water and orbited by a **smaller neighbor**. The **white arrows** symbolize the **neighbor’s** gravity forces acting on the water at that location. The slider lets you control the **speed of time**:
Water closer to the **orange body** experiences a stronger pull than the water farther away. The solid part of the water-covered **planet** also gets gravitationally pulled by the **neighbor** with some force. Like before, we can calculate the difference between the **force** acting on each parcel of water and the **force** acting on the center of the solid body, which will show us the **tidal forces** acting on the water:
Subject to these **tidal forces** the surface of the water will deform until it reaches a new balance with the gravitational forces of the **planet** itself. It’s actually pretty hard for **tidal forces** to just raise the water against the force of gravity of the **planet**. The tidal deformation is primarily caused by “sliding” the water away from the regions where the **tidal forces** act tangentially to the surface.
Here we’ll make an idealized assumption that, on this **planet**, water can very quickly travel and deform under the influence of gravity. It’s not super realistic, but this simplification will help illustrate some fundamentals of tidal motions. The demonstration below shows an exaggerated view of that tidally deformed ocean. I’ve also added a little **figurine** that you can drag around to more easily see the water level at that location:
The plot below shows the water level over time at the **observer’s** location. As the **planet** spins, different areas of the ocean are directly in line with the **orbiting body**, so the water level oscillates over time. Notice that the tidal forces create _two_ bulges, so during a single