A Moon that would lean towards the Sun

The equivalence principle in its everyday form says that everything falls the same way in a gravitational field: a feather and a hammer on the Moon, a lead ball and a wooden one from a tower. Experiments with torsion balances have tested it for different materials to a part in ten trillion, and a satellite in orbit has carried it to a part in a quadrillion. The principle that light and heavy, iron and aluminium, fall alike is among the best-tested statements in physics.

There is a stronger version that those experiments cannot reach. A massive body is held together by its own gravity, and that binding energy is part of its mass: the Earth weighs less than the sum of its parts by the energy it would take to pull them apart, about four parts in ten billion of the total. General relativity says that this gravitational binding energy falls like every other form of mass-energy. Many alternative theories of gravity say it does not. The difference is invisible in a laboratory, where the test masses’ self-gravity is negligible. It needs bodies big enough to bind themselves appreciably, and the nearest such pair is the Earth and the Moon, falling together around the Sun.

Kenneth Nordtvedt worked out in 1968 what a violation would look like. The Earth carries much more self-energy per unit mass than the Moon. If self-energy fell towards the Sun even slightly more or less readily than ordinary mass, the Earth and the Moon would accelerate towards the Sun at slightly different rates, and the Moon’s orbit around the Earth would be pushed bodily towards the Sun or away from it.

A difference in falling becomes a lean #

The Earth and the Moon orbit the Sun together, and in the frame that moves with their common centre of mass the Sun’s pull is almost entirely cancelled — it accelerates both bodies equally, and only its tidal part acts on the Moon’s orbit around the Earth. A violation of the strong equivalence principle adds one thing that does not cancel: a small extra acceleration of the Moon relative to the Earth, directed along the line to the Sun, of size

$$\Delta a = \eta\left(\frac{U_M}{M_Mc^2}-\frac{U_E}{M_Ec^2}\right)g_\odot,$$

where $U/Mc^2$ is each body’s gravitational self-energy as a fraction of its mass, $g_\odot$ is the Sun’s gravitational acceleration at the Earth — six millimetres per second squared — and η measures how far self-energy departs from falling normally. For the Earth the fraction is $-4.6\times10^{-10}$ and for the Moon $-1.9\times10^{-11}$, so the difference is about $4.4\times10^{-10}$, and the extra acceleration is about $2.6\times10^{-12}\,\eta$ metres per second squared.

That acceleration always points at the Sun, while the Moon goes round the Earth once a month. In the Moon’s own frame the push rotates once per synodic month — the cycle from new Moon to new Moon — and a force at that frequency drives the orbit at that frequency: the Moon is pushed outward when it is between the Earth and the Sun and inward a fortnight later, or the reverse, and its orbit is displaced bodily towards or away from the Sun. The displacement is a change in the Earth–Moon distance with the synodic period and a definite phase, tied to the Sun’s direction.

The size of the response follows from the orbit’s own dynamics. A perturbing force at a frequency close to the orbit’s natural frequency produces a large response, and the synodic frequency is close to the Moon’s orbital frequency — they differ only by the Earth’s motion round the Sun. For a Moon on a circular orbit with nothing else acting, the forced response is about eight metres for each unit of η.

The Sun makes the lean bigger #

The Moon’s orbit is not a circle with nothing else acting. The Sun’s tide distorts it strongly — the variation and the evection are oscillations of thousands of kilometres in the Moon’s distance, driven by the Sun — and an orbit already distorted by the Sun responds to an extra force differently from an undistorted one.

The full calculation with the Sun’s tide included gives a coefficient larger than the bare estimate by more than half. The integration here, starting from a circular orbit, gives 11.4 metres per unit η; the careful analytical treatment with the real lunar orbit gives about thirteen. Nordtvedt’s first estimate left the enhancement out, and it was found when the calculation was done properly — a reminder that in the Moon’s orbit, nothing is small enough to ignore until it has been computed.

The enhancement matters for the experiment’s sensitivity. A larger response to a given η is a smaller η detectable at a given ranging precision, and the factor of 1.6 is a factor of 1.6 in the bound.

Why the Moon, and not a satellite #

The experiment needs two bodies falling together towards a third, with one of them carrying appreciable self-energy, and with nothing else pushing them about at the millimetre level. The Earth and the Moon are almost uniquely suited. An artificial satellite has no self-energy worth the name, and it is pushed by the solar wind, by sunlight, by the thin upper atmosphere and by its own thermal radiation by amounts far larger than the effect. The Moon is seventy million million million tonnes, and every non-gravitational force on it is negligible: sunlight’s pressure moves it by less than the width of a reflector.

The geometry is also favourable. The Earth–Moon–Sun system is a three-body problem with no general solution, but it is a strongly hierarchical one: the Moon is four hundred times closer to the Earth than the Earth is to the Sun, so the motion can be expanded in that ratio, and the expansion converges. Lunar theory has been developed in exactly that form since the eighteenth century, and the parallactic inequality is the first term in the expansion that depends on the ratio itself rather than on its square. A violation of the equivalence principle produces a term of the same form, and the century and a half of lunar theory that went before is what makes it possible to say that the Moon’s orbit has no such term.

What the alternatives predict #

General relativity gives η = 0 exactly, because in it every form of energy, including gravitational binding, couples to gravity identically. Theories that add a scalar field to gravity generally do not. In the parametrised framework used to compare such theories with experiment, the Nordtvedt parameter is a combination of the two numbers that describe how much space curvature unit mass produces and how non-linear gravity is: $\eta = 4\beta - \gamma - 3$, which vanishes when both take their relativistic value of one.

That combination is what makes the lunar test useful alongside the others. The parameter γ is measured by the deflection and delay of light passing the Sun — the delay that no orbit can imitate, measured by a spacecraft near Saturn to a few parts in a hundred thousand. The parameter β enters the precession of planetary perihelia, Mercury’s forty-three arcseconds chief among them, but there it is mixed with the Sun’s own oblateness. The lunar bound on η, combined with the delay’s bound on γ, gives β independently of the Sun’s shape, and the result agrees with general relativity to a few parts in ten thousand.

A scalar–tensor theory of the simplest kind predicts η equal to one over a coupling constant plus two; the lunar bound alone requires that constant to be at least a few thousand, and the delay measurement requires it to be at least forty thousand. In that family of theories the lunar test is now the weaker one. In others, where the scalar field is screened in the solar system or couples differently to self-energy, it is the stronger, and that is why it continues to be made.

A sign that would say which way #

The bound is quoted as an interval around zero, but a detection would carry a sign, and the sign would mean something. If self-energy fell more readily than other mass — η positive in the usual convention — the Earth, with more self-energy per unit mass, would fall faster than the Moon, and the Moon’s orbit would be displaced away from the Sun: farther at new Moon, nearer at full. If self-energy fell less readily, the displacement would reverse. The phase of the synodic term relative to the Sun’s direction is the sign of η, and it is fixed by geometry, which is one more thing a spurious signal from a modelling error would have to get right by accident.

A frequency the Sun has already filled #

The difficulty of the measurement is not the size of the signal but its company. The Earth–Moon distance varies at the synodic frequency anyway, because of the Sun’s ordinary gravity.

The Sun’s tidal field is usually treated to first order, as a stretch along the Sun–Earth line that is symmetric between the near and far sides of the Moon’s orbit. That symmetric stretch acts at twice the synodic frequency — the variation. But the Sun is not infinitely far away: it pulls a little harder on the Moon when the Moon is on the sunward side of its orbit than when it is on the far side. That asymmetry acts once per synodic month, at exactly the frequency of the Nordtvedt signal, and its effect on the Moon’s distance is called the parallactic inequality.

The parallactic inequality is a hundred kilometres, and the signal being sought is a millimetre. The experiment therefore consists almost entirely of computing the parallactic term — and everything else at that frequency — to a part in a hundred million, and looking at what is left. The parallactic term depends on the ratio of the Earth–Moon distance to the Earth–Sun distance, so it is computed from the same ephemeris that the ranging is fitting; an error in the astronomical unit expressed in lunar units would appear as a spurious Nordtvedt signal. The phase helps: the Nordtvedt term is tied to the Sun’s direction in a specific way, and so is the parallactic term, but they enter the distance with different dependence on the orbit’s eccentricity and on the Earth’s distance from the Sun through the year, and a fit over many years separates them.

Fifty years of light sent to the Moon #

The measurement is made by firing laser pulses at retroreflectors left on the Moon by three Apollo crews and two Soviet rovers, and timing the few photons that return. In 1969 the round-trip time was measured to a few nanoseconds, a range of about twenty centimetres. The precision improved by a factor of ten in the following decades and by another factor of ten in the 2000s, when a large telescope in New Mexico began collecting enough returned photons per shot to time each normal point to a few millimetres.

Each improvement in precision is an improvement in the bound on η, but not in simple proportion. The signal is a single sinusoid at a known frequency, and with years of data its amplitude can be determined far more precisely than any single range: the error on the amplitude falls as the square root of the number of independent normal points. What does not fall is the systematic error in modelling the rest of the orbit — the parallactic inequality, the tidal dissipation that slows the Earth and drives the Moon outward, the Moon’s own rotation and the rocking of its face, the motion of the reflectors as the lunar surface flexes under tides, the station’s position on a deforming Earth, and the delay of the laser pulse in the atmosphere. Every one of those is modelled in the same fit as the orbit, and the bound on η is ultimately set by how well they are modelled rather than by how many photons came back.

The analyses of the full record find η consistent with zero at a few parts in ten thousand. General relativity predicts exactly zero. Many of its rivals — theories in which gravity is carried partly by an extra scalar field — predict a non-zero η, and the lunar bound has removed large regions of their parameters.

Separating composition from self-energy #

There is a subtlety in reading the bound, and it is why laboratory experiments matter to it. The Earth and the Moon differ not only in self-energy but in composition: the Earth has an iron core and the Moon very little iron. A violation of the ordinary, composition-dependent equivalence principle would also make them fall differently towards the Sun and would produce exactly the same monthly polarisation. The lunar measurement alone cannot tell the two apart.

Torsion-balance experiments supply the separation. A laboratory pendulum carrying test masses made to mimic the Earth’s core and the Moon’s mantle in composition, but with negligible self-gravity, measures how differently those compositions fall towards the Sun. That difference is bounded at a part in ten trillion, far below what the lunar measurement can see. So any polarisation found in the Moon’s orbit would have to be due to self-energy, and the absence of one bounds the self-energy violation directly. The two experiments, one on a bench and one spanning four hundred thousand kilometres, are halves of a single test.

Bodies with more self-energy to test #

The lunar test’s weakness is its lever arm. The Earth’s self-energy is four parts in ten billion of its mass, so a violation η is multiplied by that tiny fraction before it produces any effect. Bodies with more self-energy test the principle with a longer lever.

A neutron star is the extreme. About a tenth of its mass is gravitational binding, so a violation of the strong equivalence principle would make it fall towards a third body at a rate differing by a tenth of η from an ordinary star. In 2014 a millisecond pulsar was found in a stellar triple: the pulsar and a white dwarf in a tight orbit, and a second white dwarf orbiting the pair at a wider distance. The inner pair falls together towards the outer white dwarf exactly as the Earth and the Moon fall together towards the Sun, and a difference in their rates of fall would polarise the inner orbit towards the outer companion. The pulsar’s timing, measured to microseconds over years, shows no polarisation, and bounds the fractional difference in fall to a few parts in a million — a bound on η tighter than the lunar one by a large factor, purchased by the neutron star’s enormous self-energy.

The two tests are not redundant. The lunar test probes the principle in the weak gravitational field of the Earth, where any violation would be a small correction to Newtonian gravity; the pulsar test probes it in the strong field inside a neutron star, where alternative theories can behave quite differently — some predict violations that grow sharply with compactness, others ones that are suppressed. A theory has to pass both.

The ephemeris as an instrument #

The Nordtvedt test is the clearest example of what an ephemeris that is a fit can be made to do. The quantity being measured, η, is one more parameter in a model of the Moon’s orbit that already contains the masses of the Earth and Moon, the orbit’s elements, the Sun’s position, the tidal dissipation, the lunar interior and the station coordinates. It is not observed directly; it is the coefficient of a particular sinusoid, and it is determined by fitting all the other parameters at the same time and seeing whether that coefficient is needed.

The same structure underlies the bound on a distant planet from Saturn’s orbit, and it carries the same hazard. A parameter fitted alongside many others is only as clean as its correlations with them, and a model error at the same frequency will be absorbed into it and reported as a signal. A real detection of a Nordtvedt term would have to survive changes to every other part of the lunar model, and the analyses that report bounds test exactly that — varying the treatment of the parallactic inequality, the tides, the lunar interior — before they quote a number.

Still open: how much smaller the bound can get #

The lunar bound is now limited by modelling rather than by photons. Ranging at the millimetre level has outrun the models of the Earth’s deforming crust, the atmosphere’s delay and the Moon’s interior, and the next factor of ten in η needs those models improved more than it needs a better laser. New reflectors designed to return more photons with less spread are planned for the lunar surface, and they will help only if the rest of the system is understood at the same level. Whether the lunar test has more to give than a few more years of averaging — or whether the strong equivalence principle is now tested better in the sky’s neutron stars than in the Earth’s own satellite — depends on how far the model of everything else the Moon does can be pushed.