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1.9 — Gravitation, and the Three Places Newton Fails
Tycho Brahe spent twenty years, from 1576 to 1596, measuring the positions of the planets by eye to an accuracy of about one arcminute — a thirtieth of the width of the full Moon — using instruments the size of buildings and no telescope, because the telescope had not been invented. When he died in 1601 his assistant Johannes Kepler inherited the data.
Kepler spent the next eight years trying to fit Mars to a circle. He got within eight arcminutes, which was better than anyone had ever managed, and he threw it away, because he knew Tycho's data was better than eight arcminutes and he refused to accept a fit he knew was wrong. That refusal is one of the great moments in science, and it led him to the ellipse.
Kepler had three laws and no explanation for any of them. Eighty years later Newton showed all three follow from a single inverse-square force, and that same force is what makes an apple fall. This chapter does that derivation.
Kepler's three laws
First law (1609). Every planet moves in an ellipse with the Sun at one focus.
An ellipse is the set of points whose distances to two fixed foci add to a constant. Its longest diameter is 2a, where a is the semi-major axis. How squashed it is is measured by the eccentricity e, between 0 (a circle) and just under 1 (a long thin cigar). The Earth's orbit has e = 0.0167, which is so nearly circular that a drawing of it to scale looks like a circle — the seasons are not caused by it. Mercury's is e = 0.206, and Halley's Comet is e = 0.967.
Second law (1609). The line from the Sun to a planet sweeps out equal areas in equal times.
So a planet moves fastest at its closest point (perihelion) and slowest at its furthest (aphelion). The Earth is closest to the Sun in early January and moving about 3.4% faster than in July, which is why northern-hemisphere winter is a few days shorter than northern summer.
Third law (1619). The square of a planet's orbital period is proportional to the cube of its semi-major axis:
T^2 \propto a^3
Kepler found this by trial and error, after nine years, and recorded the date he saw it. It is the law that ties the whole solar system into one system rather than a set of unrelated orbits.
Deriving Newton's law from Kepler's third
Newton's question was: what force law would produce these? Take the simple case of a circular orbit, which is an ellipse with e = 0 and a = r, and where the third law reads T^2 = kr^3 for some constant k.
Step 1 — the planet is in circular motion, so it needs a centripetal force, from Chapter 1.5:
F = \frac{mv^2}{r}
Step 2 — express the speed using the period. The planet covers one circumference 2\pi r in one period T:
v = \frac{2\pi r}{T}
Step 3 — substitute:
F = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2 = \frac{m}{r}\cdot\frac{4\pi^2r^2}{T^2} = \frac{4\pi^2 m r}{T^2}
Step 4 — use Kepler's third law to replace T^2 with kr^3:
F = \frac{4\pi^2 m r}{kr^3} = \frac{4\pi^2}{k}\cdot\frac{m}{r^2}
\boxed{F \propto \frac{m}{r^2}}
The inverse-square law falls straight out of Kepler's third law. Nothing was assumed except circular motion and Kepler's observation. That is the derivation Newton did, and it is the reason he could claim the force is inverse-square rather than merely propose it.
Step 5 — bring in the third law. The force on the planet from the Sun is proportional to the planet's mass m. By Newton's third law the planet pulls the Sun equally hard, and by the same argument that force must be proportional to the Sun's mass M. A single force cannot be proportional to m but not to M, so it is proportional to both:
\boxed{F = G\frac{Mm}{r^2}}
with G a universal constant, the same for every pair of masses anywhere. This is the law of universal gravitation, published in 1687, and the word universal is the radical part. Before Newton, the heavens and the Earth ran on different rules. Newton's claim was that the force pulling the apple down and the force holding the Moon in orbit are the same force, differing only in r.
Newton's Moon test
He checked it, and the check is beautiful in its simplicity.
The Moon is about 60 Earth radii away. If gravity really falls off as 1/r^2, the Moon's acceleration should be 1/60^2 = 1/3600 of the acceleration at the Earth's surface:
a_{\text{Moon, predicted}} = \frac{9.81}{3600} = 2.72\times10^{-3}\ \text{m/s}^2
Now compute the Moon's actual centripetal acceleration from its orbit. Its distance is 3.84\times10^8 m and its period is 27.3 days, which is 27.3 \times 24 \times 3600 = 2.36\times10^6 s:
\omega = \frac{2\pi}{T} = \frac{6.283}{2.36\times10^6} = 2.66\times10^{-6}\ \text{rad/s}
a = \omega^2 r = (2.66\times10^{-6})^2 \times 3.84\times10^8 = 7.08\times10^{-12}\times3.84\times10^8 = 2.72\times10^{-3}\ \text{m/s}^2
The two agree to three significant figures. One number computed from a falling apple and the size of the Earth, one computed from the Moon's orbit, and they match. That is the moment the heavens and the Earth became one subject.
The shell theorem: why a planet counts as a point
There is a hole in everything above. The inverse-square law is written for point masses, and the Earth is not a point — it is 6.4\times10^6 m across, and the apple is three metres from the surface and 6.4\times10^6 m from the centre. Which distance goes in the formula?
Newton proved the answer, and it took him years, and it is one of the reasons the Principia was late.
A uniform spherical shell of matter attracts an outside body exactly as if all its mass were concentrated at its centre. And it exerts no net force at all on a body inside it.
A solid sphere is a stack of shells, so the same holds for a planet: from outside, use the distance to the centre.
The outside result comes from an integration Newton did geometrically. The inside result can be seen without any calculation, and the argument is worth having. Stand anywhere inside a hollow shell and draw a narrow double cone outwards in both directions, meeting at you. Each cone cuts a patch out of the shell. The nearer patch is closer, so its pull is stronger by 1/r^2 — but it is also smaller, and its area goes as r^2, because that is how cones widen. The two effects cancel exactly. Every pair of opposite patches cancels, so the total is zero, everywhere inside.
That gives you the field inside a planet. At depth, only the mass in the sphere below you pulls; every shell above you contributes nothing. If the density were uniform, the mass below at radius r would be M(r) = \rho\cdot\tfrac43\pi r^3, so
g(r) = \frac{GM(r)}{r^2} = \frac{G\rho\tfrac43\pi r^3}{r^2} = \tfrac43\pi G\rho\,r
Gravity inside a uniform planet is proportional to r, falling linearly to exactly zero at the centre. (The real Earth is denser in the core, so g actually rises slightly to about 10.7\ \text{m/s}^2 at the core–mantle boundary before falling to zero.)
A tunnel through the centre of the Earth would therefore be a simple harmonic oscillator, because a restoring force proportional to displacement is exactly the condition of Chapter 2.1. Drop something in and it would oscillate with a period of about 84 minutes — the same period as a satellite skimming the surface, which is not a coincidence.
Measuring G: the experiment that weighed the Earth
Newton's law has G in it, and nobody knew its value for over a century. You cannot get it from planetary motion, because every astronomical measurement gives you the product GM and never the two separately. To get G you must measure the gravitational pull between two masses you can weigh yourself — and gravity between laboratory-sized objects is absurdly feeble.
Henry Cavendish did it in 1798, at the age of 67, using an apparatus designed by John Michell.
Two 0.73 kg lead spheres hung from a thin wire on a horizontal rod. Two 158 kg lead spheres were brought close, and their gravitational pull twisted the wire. The twist was measured by a light beam reflected off a mirror on the wire, read through a telescope from outside the room, because the heat of Cavendish's own body would have set up air currents big enough to swamp the effect. The force he was measuring was about 1.7\times10^{-7} N — roughly the weight of a grain of sand.
G = 6.674\times10^{-11}\ \text{N·m}^2/\text{kg}^2
That number is why gravity feels strong and is in fact by far the weakest of the four forces. It takes an entire planet, 6\times10^{24} kg, to pull on you with the modest force you call your weight — and a fridge magnet the size of a coin beats all of it when you lift a paperclip.
Cavendish did not present his result as a measurement of G; he called his paper Weighing the Earth, because once you have G you can invert g = GM/R^2:
M_{\oplus} = \frac{gR^2}{G} = \frac{9.81 \times (6.371\times10^6)^2}{6.674\times10^{-11}} = \frac{9.81\times4.059\times10^{13}}{6.674\times10^{-11}} = 5.97\times10^{24}\ \text{kg}
Dividing by the volume gives an average density of 5510 kg/m³, about twice that of surface rock — which is how we first knew the Earth has a dense metallic core.
G remains, by a wide margin, the least accurately known of all the fundamental constants, at about 22 parts per million. Every other constant is known thousands of times better. Gravity is simply too weak to measure precisely, and different laboratories still disagree with each other by more than their stated error bars.
Gravitational potential energy, done properly
Chapter 1.6 used U = mgh. That assumed constant g, which is only true near the surface. The general form comes from integrating the force.
Work done by gravity moving a mass from r_1 to r_2, with the force always attractive (pointing towards decreasing r):
W = \int_{r_1}^{r_2}\left(-\frac{GMm}{r^2}\right)dr = -GMm\left[-\frac{1}{r}\right]_{r_1}^{r_2} = GMm\left(\frac{1}{r_2}-\frac{1}{r_1}\right)
Since \Delta U = -W:
U(r_2) - U(r_1) = GMm\left(\frac{1}{r_1}-\frac{1}{r_2}\right)
Choose the zero of potential energy at infinity, U(\infty) = 0, which is the natural choice because at infinite separation the two masses do not interact. Setting r_1 = \infty:
\boxed{U(r) = -\frac{GMm}{r}}
The minus sign is not a bookkeeping accident. It says every bound gravitating pair has negative energy relative to being infinitely far apart, which means you must add energy to separate them. Everything gravitationally bound — a moon, a solar system, a galaxy — sits in a negative-energy well, and the depth of the well is exactly what you would have to pay to get out.
Recovering mgh
Near the surface, at height h above radius R:
U = -\frac{GMm}{R+h} = -\frac{GMm}{R}\cdot\frac{1}{1+h/R}
For h \ll R, use the binomial approximation \dfrac{1}{1+x}\approx 1-x (Volume II, Chapter 5.8):
U \approx -\frac{GMm}{R}\left(1-\frac{h}{R}\right) = -\frac{GMm}{R} + \frac{GMm}{R^2}h
The first term is a constant, and constants in potential energy never matter. The second contains GM/R^2, which is exactly g:
U \approx \text{constant} + mgh
So mgh is the low-altitude approximation of -GMm/r, and now you know precisely when it stops being good: when h is no longer small compared with 6371 km. At the altitude of the International Space Station, 400 km, the error is about 6%.
Escape velocity
How fast must you throw something so that it never comes back?
"Never comes back" means it reaches infinity, and just barely — arriving with zero speed left. So set the total energy at the surface equal to the total energy at infinity, which is zero:
\tfrac12mv_e^2 - \frac{GMm}{R} = 0
\boxed{v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}}
For Earth:
v_e = \sqrt{2\times9.81\times6.371\times10^6} = \sqrt{1.250\times10^8} = 11\,200\ \text{m/s} = 11.2\ \text{km/s}
About 40 000 km/h, or Mach 33.
Three things about this that are commonly misunderstood.
The mass of the escaping object cancelled. A pebble and a spaceship need the same escape speed.
Direction does not matter (ignoring the ground and the atmosphere). Fire it sideways, upwards, at 45° — the energy equation contains only speed. This is because gravity is a conservative force and only the amount of energy matters, not how it is pointed.
A rocket does not need to reach escape velocity to leave. Escape velocity is the speed for an object given one instantaneous kick and then left alone. A rocket with a continuously burning engine can climb away at walking pace given enough fuel. Escape velocity is a statement about energy budgets, not a speed limit at a gate.
| Body | Escape velocity |
|---|---|
| Moon | 2.4 km/s |
| Mars | 5.0 km/s |
| Earth | 11.2 km/s |
| Jupiter | 59.5 km/s |
| Sun | 618 km/s |
| Sun's surface, if compressed to 3 km | 300 000 km/s |
That last row is the point where escape velocity reaches the speed of light, so nothing at all can leave. John Michell worked out in 1783 that such an object would be invisible, and called it a dark star. Chapter 12.3 shows that the Schwarzschild radius from general relativity gives exactly the same number, by completely different reasoning — one of the strangest coincidences in physics.
Orbits, and why astronauts float
A satellite in a circular orbit of radius r has gravity supplying its centripetal force:
\frac{GMm}{r^2} = \frac{mv^2}{r} \quad\Longrightarrow\quad \boxed{v = \sqrt{\frac{GM}{r}}}
The orbital speed depends only on r, not on the satellite's mass. Lower orbits are faster. The ISS at 400 km altitude does 7.66 km/s and circles the Earth every 92 minutes; a geostationary satellite at 35 786 km does 3.07 km/s and takes exactly 24 hours, which is what makes it hang over one spot.
Now the question everybody asks: why do astronauts float?
Not because gravity is absent. At the ISS's altitude, r = 6371+400 = 6771 km, so
g = 9.81\times\left(\frac{6371}{6771}\right)^2 = 9.81\times0.885 = 8.68\ \text{m/s}^2
That is 88% of surface gravity. If gravity were switched off up there the station would fly off in a straight line and never come back.
They float because they are in free fall, and so is the station, and they are falling together. Recall the lift example in Chapter 1.4: when the cable snaps and the lift falls freely, the scale reads zero, because nothing is pushing on you. An orbiting station is a lift whose cable has snapped permanently — it is falling towards the Earth continuously, and missing, because it also has enough sideways speed to keep the ground curving away beneath it at exactly the rate it falls.
Newton drew this himself. Imagine a cannon on a very high mountain. Fire it gently and the ball lands nearby. Fire it harder and it lands further. Fire it hard enough and the ground curves away as fast as the ball drops, and it never lands — it is in orbit. An orbit is a projectile trajectory that misses the planet.
Tides
The Moon pulls on the Earth. If that were the whole story there would be one high tide a day, on the side facing the Moon. There are two, twelve hours apart, and the second one is on the side facing away.
The cause is not gravity but the difference in gravity across the Earth's diameter. The near side is 6371 km closer to the Moon than the centre is, so it is pulled harder. The far side is 6371 km further, so it is pulled less. Working in the Earth's own frame, where the centre is the reference, the near side is pulled towards the Moon and the far side is left behind — and being left behind, relative to the centre, is the same as being pushed away. The Earth is stretched along the Moon line, in both directions equally.
The size of the effect follows from differentiating the force:
F = \frac{GMm}{r^2} \quad\Longrightarrow\quad \frac{dF}{dr} = -\frac{2GMm}{r^3}
so the tidal force across a body of size d goes as
F_{\text{tidal}} \approx \frac{2GMmd}{r^3}
Inverse cube, not inverse square. That single fact settles the Sun-versus-Moon question. The Sun's mass is 27 million times the Moon's, but it is 390 times further away, and 390^3 = 5.9\times10^7. So the Sun's tidal effect is 2.7\times10^7 / 5.9\times10^7 = 0.46 of the Moon's — under half. The Moon wins on tides despite the Sun winning enormously on raw gravitational force, purely because tides fall off faster with distance.
When Sun and Moon line up (new and full Moon) their tidal bulges add, and you get spring tides, the highest of the month. At the quarter Moons they are at right angles and partly cancel, giving neap tides. The word "spring" here has nothing to do with the season; it means to leap up.
The same inverse-cube law explains why a moon that wanders too close to a planet is torn apart — the tidal stretching exceeds the moon's own gravity holding it together. The distance where that happens is the Roche limit, and Saturn's rings sit inside it, which is very probably what they are: a moon that got too close.
The three places Newton fails
Newton's law has now survived two chapters unchallenged, and it deserves to have — it lands spacecraft on Mars. But it is not exactly right, and the discrepancies are measured, not theoretical.
1. Mercury's perihelion
An isolated two-body orbit is a closed ellipse that retraces itself forever. Mercury's does not: the ellipse itself slowly rotates, so the point of closest approach creeps around the Sun. This is called perihelion precession, and it is 5600 arcseconds per century.
Almost all of that is explained by Newton perfectly well: 5025 arcseconds is the drift of our own coordinate system as the Earth's axis precesses, and 532 arcseconds is the tug of the other planets, chiefly Jupiter and Venus. Adding them up leaves:
5600 - 5025 - 532 = 43\ \text{arcseconds per century}
Forty-three arcseconds. That is about a hundredth of a degree per century, and it is completely, stubbornly unexplained by Newtonian gravity. Urbain Le Verrier found it in 1859 and it stayed open for 56 years. Astronomers proposed an unseen planet closer to the Sun, named it Vulcan, and searched for it for decades. It does not exist.
In November 1915 Einstein computed the perihelion shift from his new field equations and got 43 arcseconds per century out, with no free parameters to adjust. He later wrote that it gave him heart palpitations. Chapter 6.9 does that calculation.
2. Light bends, and by twice the amount Newton allows
You can get light to bend in Newtonian gravity by pretending a photon is a particle with mass — the mass cancels, so you get an answer without knowing it. Doing so predicts starlight grazing the Sun should be deflected by 0.87 arcseconds.
General relativity predicts 1.75 arcseconds — exactly twice — because spacetime curvature bends light through the curvature of space as well as of time, and Newton only has the second effect.
Arthur Eddington's expedition to Príncipe photographed stars near the Sun during the total eclipse of 29 May 1919. The measured deflection matched Einstein. It made the front page of The Times on 7 November 1919 and made Einstein world-famous overnight.
3. Clocks run at different rates, and GPS proves it every day
Newtonian gravity has nothing whatsoever to say about time. In general relativity, a clock deeper in a gravitational well runs slower.
GPS satellites orbit at 20 200 km. Two effects apply, and they pull in opposite directions:
- Special relativity: they move at 3.87 km/s, so their clocks run slow by about 7.2 microseconds per day.
- General relativity: they are higher in the Earth's gravitational well, where time runs faster, so their clocks run fast by about 45.9 microseconds per day.
The net is 45.9 - 7.2 = 38.7 microseconds per day fast.
That sounds negligible until you turn it into distance. Light travels 300 metres in a microsecond, so 38.7 microseconds is:
38.7\times10^{-6}\ \text{s} \times 3\times10^8\ \text{m/s} = 11\,600\ \text{m}
Uncorrected GPS would drift by 11.6 kilometres per day. The satellite clocks are deliberately built to tick at a slightly wrong rate on the ground so that they tick correctly in orbit. Every phone with a map in it is running on a correction that only exists because Newton is wrong, and Chapter 6.9 derives that number.
What this does and does not mean
Newton is not overturned. He is contained. General relativity reduces exactly to Newtonian gravity when gravity is weak and speeds are small, which is essentially everywhere you will ever be, and Chapter 6.8 shows that reduction line by line. The relationship is not right-versus-wrong; it is a special case inside a bigger theory, in the same way that Chapter 1.2's four kinematic equations are a special case of calculus.
Newton's law is still what NASA uses to plan a Mars transfer, because the relativistic correction is smaller than the errors in the thrust model. It is enough for everything except Mercury, starlight, and clocks — and those three exceptions were enough to remake physics.
Where this shows up in your life
Every satellite phone call, weather forecast and mapped route depends on orbits computed from this chapter's equations. Every tide table is the inverse-cube calculation run for your specific coastline, with local geography making the difference between the 0.5 m tidal range of the Mediterranean and the 16 m of the Bay of Fundy, where the basin's natural sloshing period happens to match the tidal period and resonance amplifies it (Chapter 2.2).
Oil and mineral prospecting uses a gravimeter — an instrument that measures g to nine decimal places. A salt dome is less dense than the surrounding rock, so g above it is very slightly lower, and mapping those parts-per-million variations reveals the geology kilometres down without drilling.
And the reason a plumb line does not point exactly at the Earth's centre is that a nearby mountain pulls it sideways. Nevil Maskelyne measured this deflection on the Scottish mountain Schiehallion in 1774, estimated the mountain's mass from its shape, and got a value for the Earth's density that was within 20% of the modern figure — 24 years before Cavendish, using nothing but a telescope, a plumb line and a very large hill.
What the next chapter fixes
We can now handle a body that moves. What we have not touched is the far more common case of a body that does not: a bridge, a ladder, a crane, a bone. Chapter 1.10 does statics — the conditions for a rigid body to stay put, which turn out to be two equations rather than one — and then what happens when the forces get large enough that the body stops being rigid and starts to stretch, bend and break.