Why Doesn't the Moon Fall to Earth?
― It Really Does Fall, by 1.4 Millimetres Every Second
Let go of an apple and it falls. So why doesn't the Moon, floating in the sky, fall too? The answer is a little surprising. The Moon is falling. It's just that the ground it is falling toward never arrives.
You are walking home at night. A child points at the sky and asks, "Hey, why doesn't the Moon fall down?"
It is tempting to say, "Because there's no gravity in space." But that is wrong. The Moon goes around Earth because Earth's gravity is pulling on it.
It is being pulled, yet it doesn't come down. Let's untangle this apparent contradiction.
There are only two reasons
The Moon moves sideways at about 1 kilometre every second. In that time it falls about 1.4 millimetres toward Earth. Because Earth is round, the ground curves away by just as much as the Moon falls.
The Moon is about 60 times Earth's radius away. Gravity weakens with distance, and at the Moon it is about 1/3600 of its strength on the ground. This weak pull and the sideways speed are in exact balance.
So the Moon isn't "not falling". It is "falling and missing Earth, again and again". Let's take it step by step.
Is the Moon really falling?
Throw a ball sideways and it traces a curve and drops to the ground. The harder you throw, the farther it lands. So what if you threw it with absurd force?
Earth is round, so the ground curves downward as you go farther. If the ball is fast enough, the ground curves away faster than the ball falls. The ball keeps falling, yet never reaches the ground. It goes all the way around Earth and comes back to where it started.
The Moon is in exactly this state. Look at Figure 1. The slider changes the Moon's sideways speed.
At zero speed, the Moon drops straight onto Earth. A little faster, and it falls at an angle and still hits. At 1.02 kilometres per second, the curve of its fall matches the curve of Earth exactly, and the path becomes a neat circle. The Moon today travels at precisely this speed.
Newton compared the apple and the Moon
The first person to check this idea with numbers is said to be the 17th-century English scientist Isaac Newton. He wondered, "Could the force that makes an apple fall and the force that pulls on the Moon be the same thing?"
Near the ground, an object falls about 4.9 metres in its first second. The Moon is about 60 times Earth's radius away. If gravity weakens in proportion to the square of the distance, then at the Moon it should be 60×60, or 1/3600 as strong. See Figure 2.
Next, we work out how far the Moon falls in one second from its own motion. The Moon circles Earth in about 27.3 days. From the size of that circle and the number of days, we find that it bends inward from a straight line by about 1.4 millimetres every second. And 1/3600 of 4.9 metres is also about 1.4 millimetres. The two numbers agree.
The force that drops the apple and the force that keeps the Moon circling were one and the same gravity. This discovery is said to have led to the law of universal gravitation.
If its sideways speed dropped to zero, the Moon would fall straight toward Earth. By calculation, it would hit in about five days. What protects the Moon is not its height but its sideways speed.
Far from falling in, the Moon is moving away from Earth by about 3.8 centimetres a year. This was found by firing laser light at mirrors left on the Moon during the Apollo missions and timing the return. It is thought that, through the ocean tides, Earth's spin is slowly passing some of its momentum to the Moon.
Summary
Earth's gravity pulls the Moon, and it falls about 1.4 millimetres every second. But it also moves sideways at about 1 kilometre per second, so it keeps falling along Earth's curve and never reaches the ground. Too slow, and it would fall in. Too fast, and it would fly away. The Moon today moves at just the right speed in between.
The Moon isn't failing to fall.
It is falling, and missing Earth, again and again.
The reason bodies float inside a space station is the same idea of "continuous falling" (Is it true that people float on the space station because there's no gravity?). For how far away the Moon really is, see How far away is the Moon, really?, and for how the Moon's gravity moves the oceans, see Why are there two tides a day?.
- Tie an eraser firmly to the end of a string about 50 centimetres long. Do this in a wide space with no people or breakable things nearby.
- Hold the other end and swing it slowly over your head. The string pulls the eraser inward, playing the part of gravity for the Moon.
- If you slow down, the eraser can't keep its circle and sags inward. The faster you swing it, the harder your hand has to pull on the string.
You can feel with your hand that a steady circle needs sideways speed and an inward pull in balance. Letting go while it is spinning fast is dangerous, so don't.
For those who want more ― terms, formulas and links to your textbooksEach part is labelled by level, from middle-school science to university courses
- Middle schoolCovered in middle-school science
- High schoolCovered in high-school physics (circular motion, universal gravitation)
- High school+Advanced high-school material, or textbook sidebars
- UniversityNot taught in high school; university courses (mechanics, celestial mechanics)
- ResearchNot yet settled textbook fact; topics researchers are still investigating
Middle schoolTerms: this phenomenon has names
- Revolution (orbit): one body going around another. The Moon circles Earth once in about 27.3 days.
- Gravity (universal gravitation): the force by which things with mass attract each other. The force that drops an apple and the force that pulls the Moon are the same.
- Centripetal force: the force pointing toward the centre that is needed to keep moving in a circle. For the Moon, Earth's gravity plays this role.
Middle schoolHigh schoolChecking with a formula: how many millimetres does the Moon fall per second?
We work out the inward acceleration the Moon needs to move in a circle, and the value of ground gravity weakened by distance, and compare them. This is the calculation known as Newton's "Moon test".
| In symbols | v² ÷ r ＝ g × ( R ÷ r )² |
| In words | (Moon's speed squared) ÷ (distance to the Moon) = gravitational acceleration on the ground × (Earth's radius ÷ distance to the Moon) squared |
| Where it comes from | The left side is the inward acceleration needed to move in a circle (centripetal acceleration). The right side is the law of universal gravitation, where gravity weakens with the square of the distance. When the two are equal, the Moon keeps circling. |
| v | The Moon's orbital speed (in m/s) |
| r | Distance from Earth's centre to the Moon (in m) |
| R | Earth's radius (in m) |
| g | Gravitational acceleration on the ground (in m/s²) |
| Average distance from Earth's centre to the Moon | about 384000 km |
| Earth's radius | about 6370 km |
| The Moon's orbital period (one turn relative to the stars) | about 27.3 days |
| Gravitational acceleration on the ground | about 9.8 m/s² |
| Diameter of the orbit | 384000 × 2 ＝ 768000 km |
| Length of one orbit (π = 3.14) | 768000 × 3.14 ≒ 2411520 km |
| Seconds in one orbit | 27.3 × 86400 ≒ 2358720 s |
| The Moon's speed (per second) | 2411520 ÷ 2358720 ≒ 1.02 km |
| Speed squared (taking 1020 m/s) | 1020 × 1020 ＝ 1040400 |
| Centripetal acceleration (left side) | 1040400 ÷ 384000000 ≒ 0.00271 m/s² |
| Distance in Earth radii | 384000 ÷ 6370 ≒ 60.3 |
| That squared (taking 60) | 60 × 60 ＝ 3600 |
| Gravity at the Moon (right side) | 9.8 ÷ 3600 ≒ 0.00272 m/s² |
| Distance fallen in 1 second (half the acceleration) | 0.00271 ÷ 2 ≒ 0.00136 m |
The left side, 0.00271, and the right side, 0.00272, almost match. The Moon falls toward Earth by about 1.4 millimetres each second. In that same second it moves about 1 kilometre sideways, so the fall is about 1/700,000 of the distance travelled. Add up these tiny bends and you get exactly a circle.
High schoolHigh school+What happens to the path when the speed changes?
High schoolThe speed for a circular orbit is fixed by the condition that centripetal force equals gravitational force. Worked out just above Earth's surface, it gives the first cosmic velocity, about 7.9 kilometres per second. At the Moon's distance, far from Earth, about 1.02 kilometres per second is enough.
High school+If the speed exceeds √2 times (about 1.41 times) the circular-orbit speed, the kinetic energy beats the energy needed to climb out of gravity's pull, and the body never returns. This is the escape velocity. It is why, in Figure 1, the Moon flies away from about 1.44 kilometres per second. At lower speeds the path is an ellipse (Kepler's first law).
UniversityThe two-body problem, and how the Moon drifts away
The motion of two bodies attracting each other, like Earth and the Moon, is called the two-body problem. From the law of universal gravitation and the equations of motion, it follows that the orbit is a circle, an ellipse, a parabola or a hyperbola. The real Moon's orbit wobbles in complicated ways because of perturbations from the Sun's gravity. In addition, the tidal bulge is carried forward by Earth's spin, and its gravity pulls the Moon forward, so angular momentum from Earth's spin passes into the Moon's orbit. The recession of about 3.8 centimetres a year measured by lunar laser ranging is thought to be the result of this tidal acceleration.
📖 Derivations and further reading: Universal gravitation (Wikipedia, in Japanese) ／ Lunar laser ranging experiment (Wikipedia, in Japanese)
ResearchWhat is still not clear
- How did the Moon get just the right speed? The leading idea is that a body about the size of Mars hit the young Earth, and the Moon formed from the debris. But details are still debated, such as why Moon rocks and Earth rocks are so similar in composition.
- Has the Moon always drifted away at the same rate? Run today's recession speed backward and the result doesn't match the Moon's age. Tides act differently depending on the shape and depth of the oceans, so the drift was probably slower long ago, but that history is still under study.
- What gravity actually is. The law of universal gravitation explains the Moon's motion superbly, but the question of why gravity exists has no answer yet that links it to the quantum world, even in general relativity.
In other words, this article too is "an explanation within what we know today". How the Moon "falls" was solved over 300 years ago, but the Moon's origin and the nature of gravity remain open questions.
Links to your textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Middle school | Science: "Earth and the Universe" (the Moon's orbit) | The Moon circles Earth in about 27.3 days |
| High school | Physics: "Circular motion", "Universal gravitation" | Comparing centripetal acceleration with gravity weakening by the inverse square |
| High school+ | Physics (advanced): first cosmic velocity, escape velocity | The paths when the speed is changed in Figure 1 |
| University | Mechanics, celestial mechanics (two-body problem, tidal acceleration) | Classifying orbits, and how the Moon drifts away |
| Research | Planetary science, theories of gravity | The Moon's origin, the history of its recession, the nature of gravity |
| ― | Links to daily life | Satellites and space stations also orbit by the same mechanism of continuous falling |
- NASA NSSDCA | Moon Fact Sheet (Moon's distance, orbital period and orbital speed)
- Lunar laser ranging experiment (measurement of the Moon receding about 3.8 cm a year)
- National Institute of Standards and Technology, Fundamental Physical Constants (recommended value and uncertainty of the gravitational constant)
- National Astronomical Observatory of Japan (国立天文台), ed., Rika Nenpyo (理科年表, Chronological Scientific Tables), Maruzen Publishing (Earth and Moon data)
- Isaac Newton, Philosophiæ Naturalis Principia Mathematica (自然哲学の数学的諸原理), Book III (the Moon test)
※This article is a general-audience science explainer. The figures given are rough guides to help you understand the mechanism. The Moon's distance and speed vary by a few percent depending on where it is in its orbit.