Why Does the Moon Follow You When You Walk?
― The Farther Away Something Is, the Less Its Direction Changes
The Moon is not chasing you. It is simply so far away that, however far you walk, the direction you see it in does not change at all. And your brain reads anything whose direction stays the same as something "moving along with you". The same mechanism is rooted in why cars collide at intersections with a clear view.
You are walking home at night, holding a child's hand. The child points at the sky and says, "Look, the Moon is following us!"
It is the same from a train window. Utility poles beside the track whip backward. Distant mountains drift slowly past. But the Moon stays in the same spot in the window.
Stationary poles fly by, while the Moon, 380,000 km away, runs alongside you. When you think about it, that is backwards. Even adults struggle to answer when asked "why?"
Only two reasons
How much the direction of an object changes depends on "your speed ÷ the distance to it". The Moon is 38.4 million times farther away than a tree 10 m off, so its direction barely changes however many kilometres you walk.
Your brain estimates distance from how things shift in view when you move. It cannot judge how far the Moon really is, so it concludes: "it is nearby, yet its direction does not change, so it must be following me."
The Moon is not special. Anything extremely far away follows you in the same way. Let's go through it step by step.
A nearby tree slips behind; the Moon stays in the same direction
Suppose you are walking along a road, looking at a tree beside it. At first it appears ahead and to the side. Then it passes directly beside you, and soon it swings round behind. Within a few dozen steps, the direction of the tree changes a great deal.
Over the same stretch, the direction of the Moon hardly changes. Move the slider in Figure 1 to change the distance walked. The tree 10 m away swings right round, but the line of sight to the Moon (the line going up to the right) stays parallel to the one you started with.
Even after 40 m of walking in Figure 1, the direction of the Moon has changed by only 0.000006°. That is a change far too small for the eye to detect. By calculation, you would have to walk a little over 3,000 km, the length of the Japanese archipelago, before the Moon shifts by even its own width (the calculation is in the fold-out at the end).
How much the direction changes depends on "your speed ÷ distance". At the same speed, ten times the distance means one tenth the change in direction. The Moon is 38.4 million times farther than a tree 10 m away, so its direction changes 38.4 million times less.
Why does the brain feel it is "following"?
If the direction simply did not change, you might expect the Moon to look like it is standing still. It does not, and the reason lies in how the brain judges distance.
When we move, we unconsciously use the fact that near things stream past fast and far things drift slowly. This is what you see from a train window, with poles fast and mountains slow. This clue is called motion parallax.
But the Moon in the sky has no scenery to compare against. Clues to its size are vague too, and the brain is thought to estimate the Moon as much closer than it really is, roughly as far as distant mountains.
"It is as far as the mountains, yet its direction does not change when I walk." The simplest way to resolve this contradiction is "it is moving along with me." The brain is thought to adopt that answer, producing the sensation that the Moon follows you.
Stars are vastly farther than the Moon, so even when the Earth swings to the opposite side of the Sun over half a year, their direction barely changes. In 1838, the German astronomer Bessel measured the tiny back-and-forth shift in the direction of the star 61 Cygni over a year. The angle was less than one ten-thousandth of a degree, and this is said to be the first time the distance to a star was worked out.
A car whose direction does not change is on a collision course
This is not only about gazing at the Moon. The same mechanism also works in life-and-death situations.
At an intersection with a clear view, in the middle of open fields, cars sometimes collide head-on at the corner for no obvious reason. One cause is said to be the "collision course phenomenon". Look at Figure 2.
If you are on a course that gets both cars to the intersection at the same time, the other car keeps appearing in the same direction. Just like the Moon, something whose direction does not change looks as if it is "not moving", and the eye finds it harder to pay attention to.
What is more, if the pillar of your car (the post between the windscreen and the side window) lies exactly in that direction, the other car stays hidden behind it the whole time.
Sailors know this from experience: if the bearing of an approaching ship does not change, they judge there is a risk of collision. Japan's Maritime Collision Prevention Act also sets out this idea.
So what should we do?
- Something that approaches without changing direction is on a collision courseA car or bicycle that stays in the same spot in your window is not "standing still".
- At a clear intersection, move your head back and forth to checkThe other vehicle may be hiding behind the window pillar.
- It is the same on foot or on a bicycleIf a car coming from the side keeps appearing in the same direction, stop and let it go first.
Summary
The Moon seems to follow you because it is so far away that its direction does not change however far you walk. The brain measures distance by changes in direction, so it reads the Moon, whose true distance it cannot estimate, as something "moving along with you". By the same mechanism, a car on a collision course looks as if it is "standing still".
The Moon is not chasing you.
It is just too far away to be left behind.
The Moon looks big near the horizon because of a similar misreading of its distance by the brain. See also Why does the Moon look so big just after it rises? For a feel for the 380,000 km distance, try How far away is the Moon, really?
- Hold up a finger about 30 cm in front of your face and close one eye at a time, alternately. The finger seems to jump left and right.
- Now do the same with your arm stretched out fully. The finger jumps less. The farther away something is, the less its direction changes when you change your viewpoint.
- From a train or bus window, look in turn at a nearby pole, a distant building and a mountain. You will see that their speeds of flow line up in order of distance.
Your eyes are about 6 cm apart, so looking with one eye at a time is the same as "walking 6 cm and comparing the views". When looking outside from a vehicle, stay safely seated inside.
For those who want to know more ― terms, formulas and links to textbooksEach part is labelled by level, from junior-high science to university courses
- Junior highCovered in junior-high science
- High schoolCovered in high-school basic physics and maths
- High school+Advanced high-school material, or textbook sidebars
- UniversityUniversity-level subjects not taught in high school (perception psychology, astronomy)
- ResearchNot yet taught even at university as settled fact; researchers are still investigating
Junior highTerms: this phenomenon has names
- Parallax: the change in the direction in which something is seen when the viewing position changes, and also the angle of that difference.
- Motion parallax: when you move, near things appear to flow fast and far things slowly. One of the brain's clues to distance.
- Annual parallax: the slight yearly cycle in the apparent direction of a nearby star caused by the Earth's orbit round the Sun.
- Collision course phenomenon: because the other party on a collision course keeps appearing in the same direction, it feels as if it is standing still and is easy to overlook.
Junior highHigh schoolChecking with a formula: how much does the Moon's direction change?
We compare how much the direction of a nearby tree and of the Moon change as you walk. For angles we use the easy-to-calculate "radian", then convert to degrees at the end.
| In symbols | ω ＝ v ÷ d (for a walked distance x, the change in direction Δθ ＝ x ÷ d) |
| In words | Rate of change of apparent direction ＝ your speed ÷ distance to the object |
| Where it comes from | It comes from the circle's "arc length ＝ radius × angle (radians)". If you move sideways by x on a circle whose radius is the distance d, the angle changes by x ÷ d (when the object is seen directly to the side; for small angles it holds well in any direction) |
| Symbol | Meaning and unit |
| v | Your speed (m/s) |
| d | Distance to the object (m) |
| ω | Angle by which the apparent direction changes per second (radians/s) |
| x | Distance walked (m) |
| Walking speed | About 1.2 m per second |
| Distance to the nearby tree | 10 m |
| Distance to the Moon (average) | About 384,000 km (384000000 m) |
| 1 radian in degrees | About 57.3° |
| Apparent width of the Moon | About 0.5° (about 0.0087 radians) |
| Angle the tree's direction changes per second (radians) | 1.2 ÷ 10 ＝ 0.12 |
| In degrees (per second) | 0.12 × 57.3 ≒ 6.9 |
| How many times farther the Moon is than the tree | 384000000 ÷ 10 ＝ 38400000 |
| Change in the Moon's direction after walking 1 km (radians) | 1000 ÷ 384000000 ≒ 0.0000026 |
| In degrees | 0.0000026 × 57.3 ≒ 0.00015 |
| Distance to walk to shift by one Moon width (km) | 0.0087 × 384000 ≒ 3340 |
When it passes directly beside you, the nearby tree changes direction by about 7° per second. After 1 km of walking, the Moon has changed by only 0.00015°, and to shift it by one Moon width you would need to walk about 3,340 km, roughly the length of the Japanese archipelago. The Moon's direction changes 38.4 million times less than the tree's, so to the eye it looks "completely unchanged".
High schoolHigh school+Angular velocity and the collision-course condition
High school"Arc length ＝ radius × angle" is exactly the radian measure taught in maths. The rate at which the apparent direction changes is the same idea as "angular velocity" in basic physics: the sideways component of the relative velocity divided by the distance.
High school+When the object is seen at an angle, ω ＝ v × sinθ ÷ d (θ is the angle between your heading and the line of sight). For two cars, if the relative velocity of you as seen from the other car points exactly along the line of sight, the sideways component is zero and ω ＝ 0. This is the collision-course condition, which at sea is put as "constant bearing, decreasing range".
UniversityMotion parallax, optical flow and time to collision
In perception psychology, the whole pattern of flow on the retina of a moving observer is called optical flow. Gibson argued that distance and one's own heading can be read directly from this flow. Motion parallax is part of it.
In 1976, Lee also showed that the time to collision (tau) can be found directly from the rate at which the apparent size of an approaching object grows. An object that only grows in size without changing direction is the one you will hit. In astronomy, the same geometry is used as annual parallax in defining the parsec, the yardstick of stellar distance.
📖 Derivations and further reading: Wikipedia "Parallax" (Japanese edition) ／ Wikipedia "Collision course phenomenon" (Japanese edition) Parallax ／ Collision course
ResearchWhat is still not clear
- How far away does the brain think the Moon is? It is said to differ between people, and between near the horizon and high in the sky. The debate continues, including its link to the illusion that makes the Moon look big.
- At what age do children realise "the Moon does not follow us"? This is a topic the psychologist Piaget recorded as a child's belief, but there are various views on the age at which understanding changes and the path it takes.
- Why is attention so hard to draw to something whose direction does not change? It is thought to be the flip side of the eye being drawn to moving things, but how much of the oversights made while driving it explains is still under study.
In other words, this article too is "an explanation of what is known so far". Especially the part about how the brain works may be rewritten in future.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Junior high | Science "How the Moon and Venus look", maths "sectors" | Distance to the Moon, how direction changes |
| High school | Maths "radian measure", basic physics "relative velocity" | Checking with a formula |
| High school+ | Physics "circular motion and angular velocity" | The collision-course condition |
| University | Perception psychology, astronomy | Motion parallax, optical flow, annual parallax |
| Research | Cognitive science, traffic psychology | Distance estimation, how things get overlooked |
| ― | Link to daily life | How to check at an intersection with a clear view |
- National Astronomical Observatory of Japan (国立天文台), ed., Rika Nenpyo (Chronological Scientific Tables), Maruzen Publishing (average distance to the Moon, apparent diameter of the Moon)
- Gibson, J. J. (1979) The Ecological Approach to Visual Perception. Houghton Mifflin (Japanese translation published by Saiensu-sha)
- Lee, D. N. (1976) A theory of visual control of braking based on information about time-to-collision. Perception 5, 437–459.
- Maritime Collision Prevention Act (海上衝突予防法), Article 7 (risk of collision)
- Wikipedia "Parallax" (Japanese edition)
- Wikipedia "Collision course phenomenon" (Japanese edition)
* This article is a general-audience science explanation. The figures given are rough estimates to help you understand the mechanism. When driving or travelling on foot, follow traffic rules, local signs and the instructions of the police and local authorities.