Astronomy & Celestial Mechanics
the sky, computed
Gravitation

A constant bounded by the oceans

In 1937 Dirac proposed that the gravitational constant falls as the universe ages. Eleven years later Teller pointed out what that would have done to the Earth: a star's luminosity depends on G to about the seventh power, the Earth's orbit shrinks as G grows, and the sunlight reaching the Earth goes as the ninth power. Run backwards, Dirac's G boils the oceans less than two billion years ago. The rocks were laid down in water, so G cannot have changed that fast — the first bound on a changing constant came from geology.

Assumes Gravitational constant and The mass–luminosity relation.

15 min read 6 figures Measuring the unreachableScales that defeat intuition

The gravitational constant is the least precisely known of the fundamental constants, measured in the laboratory to about twenty parts per million, and astronomy cannot help with its value: nothing in the sky is weighed in kilograms, and every orbit measures the product GM rather than G. What astronomy can do is ask whether G changes. The essay on a Sun that is getting lighter followed the modern version of that test, in which the planets’ orbits measure the rate of change of GM☉ to a few parts in a hundred thousand billion per year, and the Sun’s mass loss has to be subtracted before anything is left to attribute to G.

The question is older than any of those measurements, and its first answer came from somewhere else entirely. In 1937 Paul Dirac noticed that several enormous dimensionless numbers in physics — the ratio of the electric to the gravitational force between a proton and an electron, the age of the universe in atomic units — were all about 104010^{40}, and proposed that the coincidence was a law: that they were equal because they were linked, and that since one of them grows with the age of the universe, the others must too. The simplest way for the force ratio to grow was for G to fall, in proportion to 1/t.

In 1948 Edward Teller wrote a short paper pointing out what that would have done to the Earth.

Why a constant would change at all

The numbers Dirac noticed are worth writing down, because their size is the whole of the argument. The electric force between a proton and an electron is 2.3×10392.3\times10^{39} times their gravitational attraction. The age of the universe, measured in the time light takes to cross a classical electron radius, is about 5×10405\times10^{40}. The number of protons in the observable universe is roughly the square of either. Three numbers of no obvious relation, two of them equal to within a factor of twenty and the third their square, in a range where an accidental coincidence seems implausible.

Dirac’s large-number hypothesis said they were equal because they were related, and that the relation held at every epoch. Since the age grows, the force ratio must grow with it, and the only one of its ingredients with no other reason to be fixed was G. So G would fall in proportion to the universe’s age: G˙/G=−1/t\dot G/G = -1/t, which today is about 7×10−117\times10^{-11} per year — the same order as the expansion rate itself, which is why the essay on the Sun’s mass loss took the expansion rate as the natural scale for any variation worth seeking.

The idea was taken seriously for decades. It motivated the scalar-tensor theory of gravity that Carl Brans and Robert Dicke published in 1961, in which G is set by a field that evolves as the universe expands, and it made a changing G one of the first predictions of a fundamental theory that astronomy could test. The tests that followed — radar ranging to the planets, laser ranging to the Moon, the timing of pulsars — were designed in part to find it. They have not, and the oldest of the tests is the one Teller made with no data beyond the existence of the sea.

A ninth power hidden in the sunlight

Teller’s argument has two pieces, and both multiply.

Why a one per cent change in G is a nine per cent change in sunlight. The power of G in each factor that sets the sunlight reaching the Earth. A star's luminosity goes as G⁷ in the homology Teller used and as G⁴ if electron scattering sets its opacity; the Earth's orbit, keeping its angular momentum while G changes slowly, shrinks as 1/G, which brightens the Sun by G². The flux therefore goes as G⁹ or G⁶: a G larger by one per cent means 9.4% or 6.2% more sunlight.
Fig. 1 The power of G in each factor setting the sunlight at the Earth: luminosity as G7G^7 in Teller’s homology or G4G^4 for electron-scattering opacity, and the orbit shrinking as 1/G, which adds G2G^2. The flux goes as G9G^9 or G6G^6; one per cent more G is 9.4% or 6.2% more sunlight.

The first is the Sun’s luminosity. A star is a ball of gas whose weight is held up by the pressure of its hot interior, and whose luminosity is the rate at which heat leaks out through its opacity. Both depend on G. A stronger gravity squeezes the star harder, needs a hotter interior to hold it up, and drives a larger temperature gradient through the same opacity, so the star shines more brightly at the same mass. The dependence of luminosity on mass is a steep power for exactly this reason, and the dependence on G is steeper still: in the homology Teller used, the luminosity goes as G7M5G^7 M^5; for a star whose opacity is set by electron scattering, as G4M3G^4 M^3. Either way, a few per cent in G is tens of per cent in sunlight.

The second piece is the orbit. If G changes slowly compared with the Earth’s year, the orbit’s angular momentum, mGMam\sqrt{GMa}, is conserved — it is an adiabatic invariant — and so a stronger G means a smaller orbit, a∝1/Ga \propto 1/G. The Earth sat closer to the Sun when G was larger, and the flux it received rose as the inverse square of the distance, adding two more powers. The sunlight at the Earth goes as G9G^9.

The orbit closes in

The second piece alone would be noticeable.

An Earth closer to the Sun while gravity was stronger. The Earth's orbital radius in astronomical units against time before the present, if G changes slowly enough for the orbit's angular momentum to be conserved, so that a ∝ 1/G. Under Dirac's G ∝ 1/t the Earth was at 0.72 AU 3.8 billion years ago; with Ġ/G = −10⁻¹¹ per year at 0.963 AU; at about today's bounds, 10⁻¹³ a year, the orbit has moved by 0.04% in the age of the oldest rocks.
Fig. 2 The Earth’s orbital radius against time before the present if a ∝ 1/G. Under Dirac’s G ∝ 1/t it was 0.72 AU 3.8 billion years ago; with G falling at 10⁻¹¹ a year, 0.963 AU; at about today’s bounds, the orbit has moved by 0.04%.

Under Dirac’s hypothesis, with the universe’s present age of 13.8 billion years, G was larger by a third 3.8 billion years ago, and the Earth was at 0.72 astronomical units — where Venus is now. Teller worked with the age the universe was then thought to have, about two billion years, which made Dirac’s G change much faster and his conclusion much stronger; the modern age softens it, and it still holds.

Teller’s own number was more dramatic and slightly wrong. With a universe two billion years old, Dirac’s G would have been a third larger five hundred million years ago, and Teller concluded that the oceans would have been at the boiling point during the Cambrian, when the rocks are full of marine fossils. When the age of the universe was revised upward by a factor of several in the 1950s, the date at which Dirac’s G boils the oceans moved back by the same factor, and George Gamow, among others, briefly tried to rescue the hypothesis by having the charge of the electron change instead. The revisions moved the date of the catastrophe without removing it. That is a property worth noticing in a bound: an argument that survives a factor of seven in its main input is measuring something robust.

The adiabatic invariance is not an approximation worth worrying about here. G would change by a third over billions of years, and the orbit has completed billions of revolutions in that time; the invariant is conserved to a precision far below anything that matters. The same argument applies to every orbit, so all the planets would have been closer in by the same factor, and the shape of the solar system would have been preserved while its size contracted — which is why no measurement of the planets’ relative positions could reveal a change in G, only measurements of their absolute motion against a clock.

The oceans

Put the two pieces together with the Sun’s own slow brightening, and run the Earth’s climate backwards.

Oceans that a changing G would have boiled. The Earth's surface temperature in a crude climate — equilibrium temperature for a fixed albedo, scaled as the received flux to the quarter power, plus a fixed greenhouse warming of 33 K — against time before the present, if the Sun's luminosity goes as G⁷ and the Earth's orbit as 1/G, so the flux goes as G⁹, on top of the Sun's own brightening. With Dirac's G ∝ 1/t and an age of 13.8 Gyr, the oceans boil 1.9 Gyr ago; with G falling by 10⁻¹¹ of itself a year, the surface is 292 K at 3.8 Gyr. Rocks of that age were laid down in water. Constant G gives the faint young Sun's cooling, 270 K in this climate — a separate problem, which the real atmosphere solved.
Fig. 3 The Earth’s surface temperature in a crude climate against time before the present, with the sunlight going as G9G^9 on top of the Sun’s own brightening. Dirac’s G ∝ 1/t boils the oceans 1.9 billion years ago; G falling at 10⁻¹¹ a year gives 292 K at 3.8 billion; constant G gives the faint young Sun’s 270 K. Shaded: liquid water.

The climate drawn is deliberately crude: the Earth’s equilibrium temperature for its present reflectivity, scaled as the fourth root of the sunlight, plus a fixed thirty-three kelvin of greenhouse warming. It is not a model of the early Earth. It is a way of showing where the sunlight would have put the surface relative to the boiling and freezing points of water, and under Dirac’s hypothesis the answer is unambiguous. The sunlight two billion years ago would have been more than three times today’s, the surface above the boiling point, and by three and a half billion years ago five hundred kelvin.

The geological record says otherwise. Sedimentary rocks — layers laid down in standing water — are found in Greenland dated to 3.8 billion years ago, pillow lavas that erupted under water nearly as old, and zircon crystals from more than four billion years ago whose oxygen isotopes record contact with liquid water. Life is recorded in rocks 3.5 billion years old. The Earth has had oceans for essentially its whole history, and Dirac’s G would have boiled them. Teller concluded that the hypothesis was untenable, and the conclusion, with the modern age of the universe, still stands by a wide margin.

The argument works in the other direction too, with less force. A G that has been growing would have made the young Sun dimmer and the Earth more distant, adding to the faint young Sun’s deficit; at a rate of 10−1110^{-11} a year the sunlight four billion years ago would have been about half of today’s, and the crude climate’s surface falls towards 250 kelvin. The Earth did pass through episodes of near-global glaciation, the last of them about 650 million years ago, and the climate’s thermostat brought it out of each; but those episodes are short against the long record of open water, so a growing G is bounded by the same rocks, at a similar rate, as a falling one.

A bound with a climate inside it

Turning the argument into a number needs a decision about how far the sunlight could have strayed from its standard history without the oceans boiling or freezing, and that is a statement about the climate rather than about gravity.

A constant bounded by the oceans, and by better things since. Upper bounds on |Ġ/G| per year, logarithmic, with the rate Dirac's hypothesis requires today at the top: 1/t, 7.2·10⁻¹¹. Requiring that the sunlight reaching the Earth has stayed within ±30% of its standard history for 3.8 billion years — long enough for liquid water, at the ninth power — bounds the rate below 7.7·10⁻¹², 9 times below Dirac's. The bars below are the bounds of the other methods, rounded to their order: the light elements' abundances, the Sun's oscillation frequencies, the orbital decay of binary pulsars, the Moon's distance by laser, and the planets' orbits, which now reach a few parts in 10¹⁴.
Fig. 4 Upper bounds on |Ġ/G| per year. Liquid water for 3.8 billion years with the sunlight within ±30% of its standard history bounds the rate below 7.7 × 10⁻¹², nine times below Dirac’s 7.2 × 10⁻¹¹. The other bars are the light elements, the Sun’s oscillations, binary pulsars, lunar laser ranging and the planetary ephemerides.

Allowing the sunlight to have been thirty per cent higher or lower than the standard history, at the ninth power and over 3.8 billion years, bounds the rate of change of G below about 8×10−128\times10^{-12} per year — nine times smaller than Dirac’s rate. A tighter climate tolerance gives a tighter bound, but in proportion to the logarithm of the tolerance, so the bound is robust: allowing a factor of two in sunlight weakens it by only a factor of two and a half. The softness is real and it is acknowledged; the Earth’s climate has been regulated by the slow cycling of carbon dioxide between the atmosphere and the rocks, which responds to warming by drawing the gas down, and how much change that thermostat could absorb is not known to better than tens of per cent.

The bar chart puts the geological bound in its place. Every method that has come since is better. The abundances of the light elements made in the first few minutes, averaged over the age of the universe, give a similar bound; the oscillation frequencies of the Sun and the decay of binary pulsars’ orbits a few times better; and the laser ranging that tests whether the Moon would lean towards the Sun and the planetary ephemerides one to three hundred times better, at parts in 101310^{13} and 101410^{14}. The oceans were the first bound and are now the weakest. They are also the only one that integrates over almost the whole age of the Earth with a measurement — the presence of water — that involves no instrument at all.

A fuel gauge in the core

The Sun’s luminosity is also its fuel consumption, and a brighter past Sun would have burned more of its hydrogen.

The fuel a stronger past G would have burned. The hydrogen the Sun has burned in its 4.57 billion years, relative to a Sun with constant G, if its luminosity went as G⁷ and G has been falling at the rate on the horizontal axis. Under Dirac's G ∝ 1/t the Sun would have burned 5.1 times as much — more than the hydrogen its core holds, so it would have left the main sequence. A Sun that has burned 30% more than the standard model allows is reached at |Ġ/G| ≈ 1.6·10⁻¹¹ per year; the oscillation frequencies, which measure the helium already made in the core, test the burn to a few per cent.
Fig. 5 The hydrogen the Sun has burned in 4.57 billion years relative to a Sun with constant G, if its luminosity went as G7G^7. Dirac’s G gives 5.1 times as much — more than its core holds. Thirty per cent more than standard is reached at |Ġ/G| ≈ 1.6 × 10⁻¹¹ a year.

Under Dirac’s G the Sun would have burned five times as much hydrogen as it has, which is more than its core held: it would have left the main sequence and become a red giant before the Earth was half its present age. That is a second, independent refutation, and it has a modern descendant. The helium made in the Sun’s core changes its interior structure, and the frequencies of the Sun’s oscillations measure that structure well enough to say how much hydrogen has been burned, to a few per cent. A stronger G in the past would have left more helium in the core than a constant one, and the oscillations’ agreement with standard solar models bounds the difference.

That bound depends on the standard model’s other ingredients — the opacities, the abundances of heavy elements, the treatment of diffusion — which are themselves uncertain at the level of a few per cent, and a change in G can be partly absorbed by adjusting them. It is about five times better than the oceans, and it sits on the same physics: the steep dependence of a star’s luminosity on the strength of the gravity holding it together.

The paradox a falling G would have solved

There is a twist, and it is the surprising connection the argument makes. The Sun has been brightening throughout its life, as the helium accumulating in its core raises the core’s mean molecular weight and forces it to contract and heat. Four billion years ago it was about three quarters as bright as today, and with the present atmosphere the Earth would have frozen — the faint young Sun paradox, pointed out by Carl Sagan and George Mullen in 1972, which the oceans also contradict, from the other side.

A rate that would have dissolved the faint young Sun. The sunlight reaching the Earth relative to today's against time before the present. With G constant it follows the Sun's brightening and was 74% of today's four billion years ago — the faint young Sun, too dim, with the present atmosphere, for the liquid water the oldest rocks record. A G that was larger in the past by just enough, falling now at 8.3 × 10⁻¹² of itself a year, would have held the flux at today's value through the whole span. That is 83 times the bounds the Moon's and the planets' orbits now set, about 10⁻¹³; at that rate the faint young Sun is untouched.
Fig. 6 The sunlight at the Earth relative to today’s against time before the present. With constant G it was 74% of today’s four billion years ago. A G falling at 8.3 × 10⁻¹² a year would have held it at today’s value throughout — 83 times about today’s bounds. At those bounds the faint young Sun is untouched.

A falling G makes the past Sun brighter and the Earth closer, the opposite of the faint young Sun’s effect, and there is a rate at which the two cancel almost exactly: G falling by about 8×10−128\times10^{-12} of itself a year would have held the sunlight at the Earth at today’s value for four billion years. Dirac’s G overshoots and boils the oceans; a G changing ten times more slowly would have solved the paradox and been invisible in the geological record. It is a curious place for the two problems to meet. The one constraint on a varying G that the oceans provide is matched, at nearly the same rate, by the one problem in the Earth’s early climate that a varying G could have removed.

The modern bounds close the door. The rate needed is eighty times larger than the ones the Moon’s and the planets’ orbits allow, and at the bounds the faint young Sun is exactly as faint as in standard models. The paradox has to be solved by the atmosphere — by more carbon dioxide or methane, a lower reflectivity, or an early Earth whose air and water were set by the young Sun’s activity — and the habitable zone has to be understood, as it now is, as a band that moves while the orbit stays put.

What the argument assumes

The luminosity’s power of G is a homology. The exponents seven and four belong to idealised stars with a single opacity law and no change of structure. A real star whose G changed would also change its convective zones, its nuclear burning and its radius, and its luminosity would follow a path the homology only approximates. Computed stellar models with a varying G give exponents in the same range.

G changes slowly. The orbit’s response, a∝1/Ga \propto 1/G, assumes the change is gradual over many orbits. A sudden change would leave the orbit eccentric instead. Every proposed variation is slow by many orders of magnitude.

The climate is a sketch. The surface temperatures drawn come from a fixed reflectivity and a fixed greenhouse warming. The real climate changed both, through clouds, ice and the carbon cycle, and the bound’s tolerance of thirty per cent is a judgement about how much it could absorb, not a measurement.

The Sun’s mass is constant. The Sun has lost a fraction of a per cent of its mass to its wind and its light over its life, and a Sun a few per cent heavier when young, losing mass faster through a stronger early wind, has been proposed as another solution to the faint young Sun paradox, since luminosity rises steeply with mass. A heavier young Sun and a stronger young G act through the same powers, and a bound on one from the climate is a bound on a combination of both.

Only G changes. A theory in which G varies usually varies other things too — the masses of particles, the strength of other forces — and the Sun’s luminosity depends on those as well. Dirac’s original proposal was a statement about G alone, and so is Teller’s refutation.

Still open: what made the large numbers equal

Teller’s argument removed the simplest version of Dirac’s hypothesis, and the measurements since have removed every version in which G changes at anything like the rate of the universe’s expansion. What they have not removed is the coincidence that prompted it. The large numbers are about equal, and the explanation that replaced Dirac’s, put forward by Robert Dicke in 1961, is that the age of the universe at which anyone can measure it is set by how long stars take to make the carbon observers are built from — a statement about when the measurement is possible rather than a law tying the numbers together.

The bounds on a changing G now reach a part in 101310^{13} or 101410^{14} a year, and theories with a scalar field coupled to gravity predict rates that are smaller still, tied to the field’s slow evolution. The geology’s bound is three hundred times weaker than the best one. It remains the only one written by water, and the only one that would have been possible without a telescope.

Where this goes next

A band and a strip, and the constant where they cross. The plane of the two quantities a planetary ephemeris cannot separate: the fractional rate of change of the gravitational constant, vertically, and of the Sun's mass, horizontally, both in units of 10⁻¹⁴ per year. The orbits respond only to their sum, the rate of change of GM☉, so a measurement of that sum — drawn here as −9.0 ± 3.0 × 10⁻¹⁴ per year, the precision recent ephemerides reach, not any one published value — is a diagonal band. On its own it allows any Ġ/G at all, provided the Sun's mass is changing to compensate. The vertical strip is the Sun's mass loss computed from its luminosity and its wind, −9.9 to −8.8 × 10⁻¹⁴ per year. Where the two overlap, Ġ/G lies between −3.2 and 3.9 × 10⁻¹⁴ per year. The bound on a change in a fundamental constant is therefore set as much by how well the solar wind is measured as by how well the planets are. A constant measured through a Sun that is getting lighter Whether the gravitational constant changes with time is tested by watching the planets, and the planets report only the product of G with the Sun's mass. The Sun is losing mass — four million tonnes a second as light, a million and a half more as wind — at nearly a ten-trillionth of itself a year, and every bound on a changing G is a statement about the solar wind as much as about gravity. G: 14 determinations in two families. Published determinations of G, each with its quoted one-sigma interval, sorted into two families — torsion balance, in one form or another, against beam balance, pendulum, atom interferometry. The shaded band behind each family is that family's inverse-variance weighted mean: 6.67435 ± 0.00004 across 11 of them, against 6.67343 ± 0.00009 across 3. The difference is 0.00092 ± 0.00010 10⁻¹¹ m³ kg⁻¹ s⁻², which is 9.3 standard deviations, computed here from the quoted errors alone. The arithmetic is the same one the Hubble figure uses and here it should be distrusted, because the scatter inside each family already exceeds what the intervals allow: eleven torsion-balance determinations spread over 500 parts per million with quoted intervals of 12 to 130 cannot all be right, whatever the difference between the families comes to. That is why the recommended value's uncertainty is expanded far beyond any single experiment's rather than being the weighted combination drawn here — the disagreement is between laboratories using the same method, not between methods. Nothing in the sky is weighed in kilograms The Sun's gravitational parameter is known to eleven significant figures. The Sun's mass is known to five. The two statements are about the same object and the difference between them is a constant measured in basements, which is the worst-determined fundamental constant in physics. Luminosity against mass, against a slope of 3.5. Main-sequence luminosity against mass, both in solar units, on logarithmic axes, over the range 0.079 to 63 solar masses. The measured curve comes from the eclipsing binaries and is the same in every drawing of it; what changes here is what it is compared against. The dashed line is a pure power law of exponent 3.5, and the curve crosses it rather than following it — the local slope runs from about 2.3 at the bottom of the range, where the interiors are convective, through nearly 4 near a solar mass where bound-free opacity dominates, to about 3 among the massive stars where electron scattering does. Quoting one exponent across the whole sequence is a convenience and the places it fails are the places the interior physics changes. Because the slope is between three and four across most of the range, a small spread in mass becomes an enormous spread in output: the 63-solar-mass end is 3.0e+9 times brighter than the 0.079-solar-mass end. Mass decides everything, by a power of three and a half Two stars of the same mass are almost the same star. Double the mass and the output multiplies by eleven — which is why a modest range of masses produces a colossal range of stars.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Adiabatic invariantFaint young sunGravitational constantHelioseismologyLunar laser rangingThe mass–luminosity relationSolar evolutionVarying constants