A constant measured through a Sun that is getting lighter

The Sun’s mass is known to five significant figures and its gravitational parameter to eleven, because an orbit reports the product of Newton’s constant with the mass it orbits and nothing about either one alone. The product has its own name, GM☉, and the whole of celestial mechanics is written in it. That is a convenience while both factors are constant. It becomes the central difficulty of one of the cleanest tests in fundamental physics: whether the gravitational constant is constant at all.

The question is not idle. Nothing in general relativity requires G to vary, but many of its extensions do. Theories in which gravity is carried partly by a scalar field make the effective strength of gravity a property of that field, which evolves as the universe expands; and the idea that the constants of nature might drift with cosmic time has a long history, beginning with Dirac’s observation in 1937 that the ratio of the electric to the gravitational force between two protons is about the same large number as the age of the universe in atomic units — a coincidence that would be explained, he suggested, if G fell in proportion to the age. That predicts a fractional rate of about one part in $10^{10}$ per year. If G varies at anything like that rate, the planets should show it.

A rate that would be tied to the expansion #

The size of rate worth looking for is set by the one timescale a varying constant would naturally follow. In the theories that allow it, G is controlled by a field that evolves as the universe expands, and the expansion’s own fractional rate today is the Hubble constant — about seven parts in $10^{11}$ per year, the same number that sets the universe’s age. A theory in which G changes as fast as the universe does would give a rate of that size; one in which the controlling field is only weakly coupled to the expansion would give that rate multiplied by some small coupling. A measurement at a part in $10^{14}$ per year therefore constrains the coupling to below about one part in a thousand, which is a statement about how strongly any such field could influence gravity today.

That framing explains why the test is worth the effort even though nobody expects G to vary. A null result at this precision rules out whole families of cosmological models in which the acceleration of the expansion is driven by an evolving field that also touches gravity, because such a field would be expected to drag G along with it. The planets are, in that sense, a probe of dark energy’s couplings rather than of Newton’s constant alone.

What an ephemeris can see #

A planetary ephemeris is a numerical integration of the solar system, fitted to every position and distance measurement of the planets and the spacecraft around them. The equations it integrates contain the Sun’s gravitational parameter, and if that parameter is allowed to change slowly — a fractional rate ε per year, $\mathrm{d}\ln(GM_\odot)/\mathrm{d}t = \varepsilon$ — the fit returns a value of ε along with everything else.

What it cannot return is a split. The rate of change of a product is the sum of the rates of change of its factors:

$$\frac{\mathrm{d}\ln(GM_\odot)}{\mathrm{d}t} = \frac{\dot G}{G} + \frac{\dot M_\odot}{M_\odot}.$$

A measured ε is a statement about that sum. On its own it allows any rate of change of G whatever, provided the Sun’s mass is changing to compensate. The diagonal band in the opening figure is that statement. To turn it into a bound on G, the Sun’s rate of mass loss has to be known independently.

A Sun that is getting lighter #

It is known — and it is large enough to matter.

The larger term is the light itself. The Sun radiates $3.8\times10^{26}$ watts, and energy has mass: dividing by the square of the speed of light gives 4.26 million tonnes a second converted from mass into photons that leave the solar system for good. That is 6.76 parts in $10^{14}$ of the Sun per year, and it is known as well as the Sun’s luminosity is, which is to a fraction of a per cent.

The smaller term is the solar wind, the stream of ionised gas flowing outward from the corona at several hundred kilometres a second. Its mass flux is measured by spacecraft sampling it in situ, and it amounts to between about 1.3 and 2 million tonnes a second, varying with the eleven-year activity cycle and with the kind of wind — the fast wind from coronal holes, the slow wind from the streamer belt. Most of those measurements were made in the plane of the planets, and the mass flux over the Sun’s poles was measured directly by only one spacecraft, which found it broadly similar. The wind adds between 2.1 and 3.2 parts in $10^{14}$ per year, and its uncertainty — about a part in $10^{14}$ — is the largest single uncertainty in the whole budget.

Everything else is smaller. Coronal mass ejections carry off a few per cent of what the steady wind does. Dust and comets falling into the Sun add mass rather than removing it, at a rate many orders of magnitude below the wind. So the Sun’s gravitational parameter should be falling by about nine parts in $10^{14}$ per year even if G is perfectly constant, and any test of G through the planets has to find that fall first.

Orbits that widen as the Sun gets lighter #

The planets respond to a slowly weakening Sun by moving outward, and the size of the response is set by a conserved quantity.

A mass loss that is slow compared with an orbital period, and spherically symmetric, exerts no torque on a planet. The planet’s orbital angular momentum, which for a circular orbit is $m\sqrt{GM_\odot a}$, is therefore unchanged, and if GM☉ falls the semi-major axis a must rise in inverse proportion. The same argument, with the same result, applies if the change is in G. This is an adiabatic invariant at work — the same principle that makes a slowly contracting cloud spin faster — and it gives the rate of widening directly.

The Earth moves away from the Sun by about a centimetre and a half a year. That is not measurable as a distance — nobody measures the Earth’s distance from the Sun to a centimetre — but it is measurable as a change in the timing of the orbit, because a wider orbit is a slower one. With the angular momentum fixed, the mean motion falls twice as fast as the semi-major axis rises, and the planet falls steadily behind where a constant-GM☉ ephemeris would place it.

There is a small historical irony in this. The astronomical unit was for two centuries defined through the Gaussian constant, a fixed value of $\sqrt{GM_\odot}$ in units of astronomical units and days, so that the length of the astronomical unit in metres was whatever made GM☉ come out at that fixed value. A slowly falling GM☉ meant a slowly changing astronomical unit — a unit of length that shrank by a centimetre or so a year as the Sun lost mass. When the astronomical unit was redefined in 2012 as an exact number of metres, one of the reasons given was precisely that a unit tied to the Sun’s mass was a unit tied to something that is not constant.

An error that grows as the square of the time #

The planet’s lag behind a constant-GM☉ ephemeris is the observable, and its growth with time decides how the measurement is made.

A fractional rate ε changes a planet’s mean motion by $2\varepsilon t$ after a time t, and the planet’s longitude lags by the integral of that, $n\varepsilon t^2$ — growing as the square of the time. Doubling the span of the data quadruples the signal. A secular drift of this kind is therefore measured from the longest baselines, and the precision of the individual measurements matters less than how long they have been accumulating.

That is why the modern bounds come from spacecraft. A spacecraft orbiting Mars or Mercury is ranged from the Earth by timing radio signals, and a distance measured from a radio round trip is good to about a metre — far better than any optical position of a planet. A decade of ranging to a Mars orbiter or lander fixes the Earth–Mars distance so tightly that a drift of a few metres is detectable. The spacecraft that orbited Mercury for four years constrained the rate of change of GM☉ further still, because Mercury’s short period gives it the largest mean motion and so the largest drift for a given ε.

The published analyses of those data have found a rate of change of GM☉ consistent with the Sun’s expected mass loss, at a precision of a few parts in $10^{14}$ per year. Subtracting the mass loss leaves a rate of change of G consistent with zero, bounded at a few parts in $10^{14}$ per year in either direction. Dirac’s prediction was a part in $10^{10}$. It is excluded by a factor of several thousand.

Put in terms of the solar system’s own history, a rate of three parts in $10^{14}$ per year, held for the 4.6 thousand million years since the planets formed, would change G by about one part in seven thousand. The orbits say that whatever G has done over the last few decades, it has done less than that — and since any plausible variation would be smooth on cosmological timescales, a bound on the present rate is also, with that assumption, a bound on the change since the Earth formed. The assumption is the weak point, and it is why the bounds from the helium abundance and the Sun’s interior, which measure the change directly over billions of years rather than extrapolating a rate, are worth having despite being weaker.

The same test around the Earth #

The Moon offers an independent version of the measurement, with a different confound.

Lasers fired from the Earth at reflectors left on the Moon by the Apollo astronauts and the Soviet Lunokhod rovers have measured the Earth–Moon distance since 1969, now to a few millimetres. The Earth and the Moon lose mass at a negligible rate, so a change in the lunar orbit’s timing is a change in G directly. But the Moon has a much larger reason to drift: the tides it raises on the Earth are carried slightly ahead of it by the Earth’s rotation, and they pull it forward, transferring angular momentum from the Earth’s spin to the Moon’s orbit. The Moon recedes by 3.8 centimetres a year — more than a thousand times what a change in G at $10^{-13}$ per year would produce — and the same exchange has lengthened the Earth’s day over geological time.

The two can be separated because they are different kinds of change. The tide adds angular momentum to the orbit; a change in G leaves the orbit’s angular momentum alone and changes only the relation between its size and its period. A fit that models the tidal torque explicitly, with its own parameters, can distinguish the two, and the lunar-ranging analyses report a rate of change of G consistent with zero at a precision of about $10^{-13}$ per year — comparable to the planetary bound, with a completely different set of systematics. That two such different methods agree is worth more than the precision of either.

Bounds from the stars and the first minutes #

Orbits in the solar system measure the change of G now, over decades. Other methods measure it over much longer spans and compare different epochs.

A binary pulsar’s orbital period changes if G does, and the pulsar is a clock stable enough to measure it — the same clock that has measured an orbit shrinking by gravitational radiation, so the radiation’s contribution has to be computed and subtracted first; the bounds are weaker than the solar system’s, around $10^{-12}$ per year, but they come from a system with gravitational fields a million times stronger, which matters in theories where the variation depends on the field’s strength. The Sun itself is a constraint: its structure depends on G, a stronger G in the past would have burned its hydrogen faster, and the frequencies of its oscillations say how much hydrogen has been burned. And the abundances of the light elements made in the first few minutes depend on how fast the universe was expanding then, which depends on G: the helium abundance limits the difference between G then and G now to about ten per cent over thirteen billion years, an average of about $10^{-11}$ per year.

None of these methods has the same confound as the planetary one. That is exactly why they are useful: a variation of G that was masked in the solar system by an error in the solar wind would not be masked in the helium abundance.

When the ephemeris outruns the wind #

The arithmetic of the opening figure has a consequence that is easy to state and slightly startling.

As the ephemeris improves, the bound on a changing G stops improving with it. The band narrows and the strip stays the same width, and once the band is narrower than the strip the width of the allowed interval is set entirely by the uncertainty in the Sun’s mass loss — which is about a part in $10^{14}$ per year, almost all of it from the wind. Beyond that point a better ephemeris is not a better test of gravity. It is a measurement of the solar wind’s total mass flux, integrated over all directions and averaged over a decade, which no spacecraft in the wind itself can make.

That reversal is the most interesting thing about the measurement. A test designed to probe a fundamental constant turns, at sufficient precision, into a probe of a mundane astrophysical flow, and the reason is nothing more than that orbits report a product.

What the picture leaves out #

The figures treat the change of GM☉ as the only unmodelled effect in the ephemeris, and it is not. The masses of the largest asteroids perturb the inner planets and are fitted along with everything else; the Sun’s oblateness and the relativistic precession of perihelia change orbital timing in ways that are partly degenerate with a secular drift; and the ephemeris is fitted to data of very uneven quality spread over a century. The published rates come with uncertainties that include these correlations, and the uncertainties are larger than the purely statistical precision of the ranging.

The mass-loss budget is also a present-day number. The Sun’s wind was stronger when the Sun was young and rotating faster, and its luminosity was lower; a bound on the change of G over decades needs only today’s mass loss, but any argument that integrates the change over the Sun’s lifetime needs a history of both. And the budget assumes the loss is spherically symmetric. A wind that carried away momentum preferentially in one direction would push the Sun as well as lighten it, which is a different effect with a different signature, and it is small enough to be ignored at present precision but not by an arbitrary margin.

Still open: whether the wind can be weighed well enough #

The bound on a changing G from the planets is now limited at about the level where the solar wind’s uncertainty begins to dominate, and the next improvement depends on the wind rather than on the orbits. The spacecraft now sampling the corona from close in, and the ones that have flown over the Sun’s poles, measure the wind’s mass flux in directions never sampled before, and combining them into a total over the whole sphere and a whole cycle would shrink the vertical strip in the opening figure. The next Mercury orbiter will shrink the diagonal band. Whether the intersection closes on zero — and whether any residual, if one appears, belongs to gravity or to the Sun — is the question that the two measurements together are positioned to answer.