The scales that work without weight

The floor that cannot be told from gravity began from Einstein’s thought experiment: a sealed laboratory in free fall, in which nothing has weight and no experiment can tell that it is falling. The space station is that laboratory, built. Everything in it floats, and a person standing on a bathroom scale reads zero, because the scale and the person fall together and nothing presses one against the other.

But the person’s mass has not gone anywhere, and on a long mission it matters. Without gravity loading their bones and muscles, astronauts lose both, and body fluids shift upward and are partly excreted. Crews have lost several per cent of their body mass on long flights, and physicians want to track it. The weight is gone; something else has to be measured. What remains is inertia — the resistance to being accelerated — and the equivalence principle says it is the same quantity that weight measures on the ground.

Mass from a period #

A mass on a spring oscillates with a period that depends on the mass and the spring’s stiffness and on nothing else:

$$T = 2\pi\sqrt{\frac{m + m_0}{k}},$$

where $m_0$ is the mass of whatever moves along with the occupant — the chair, the moving parts of the springs. Gravity does not appear. On the ground a spring hanging vertically oscillates with exactly the same period as one lying flat, because gravity only shifts where the spring rests, not how it springs back. In orbit, with gravity cancelled by free fall, the formula is unchanged.

The Skylab space station of 1973 carried exactly this: a seat on springs, with an astronaut strapped tightly in, set swinging and timed electronically. Soviet stations carried a similar device, and the International Space Station has carried both oscillating chairs and a newer instrument that pulls the astronaut with springs of known force over a short distance and measures the acceleration — Newton’s second law, $m = F/a$, applied directly. Every one of them measures inertia. None of them could work by weight.

That they work is not trivial. They rely on inertial mass, the $m$ in $F = ma$, being the same as gravitational mass, the $m$ in the weight $mg$ that a ground scale reads, for the astronaut’s numbers to be comparable with their pre-flight weight. The fall that does not depend on what is falling followed the experiments that have tested that equality to fifteen decimal places. The space-station chair relies on that equality as a matter of routine, and the agreement between its readings and the ground scale’s before and after flight is a small, informal confirmation of it.

Why not a pendulum, and why not a spin #

Two other ways of measuring mass in orbit suggest themselves, and both fail instructively. A pendulum is the obvious oscillator, but its period, $2\pi\sqrt{L/g}$, contains no mass at all. On the ground the bob’s mass cancels because the same mass appears in the weight that drives the swing and in the inertia that resists it — the cancellation that let the length nobody has to measure use a reversible pendulum to find $g$ without knowing the bob. A pendulum measures the ratio of gravitational to inertial mass, times $g$; in orbit, with no effective $g$, it does not swing at all. A spring is the right oscillator precisely because its restoring force has nothing to do with the mass on it, so the mass survives in the period.

The other idea is to restore weight artificially. Spin the astronaut on a centrifuge arm of radius $r$ at angular speed $\omega$, and a scale under them reads $m\omega^2 r$, from which $m$ follows. The forces that are not there described the centrifugal force as a feature of the rotating frame rather than of anything pushing; it is exactly the inertial reaction the scale would read, so a centrifuge-mounted scale measures inertial mass too, by another route. It is just a much clumsier one: a rotating arm big enough to hold a person, spinning in a crowded spacecraft, with the scale’s reading depending on the square of a rotation rate and on exactly where the body’s centre of mass sits along the arm. The spring chair asks for none of that.

Both alternatives, examined, show what the chair is doing. Every way of weighing something involves a force and a response, and the question is which mass each one couples to. Gravity couples to gravitational mass and is absent; a spring couples to nothing and leaves inertia as the only response; a rotation couples to inertia through the acceleration it imposes. In a freely falling laboratory, the measurable kind of mass is inertial, and every working instrument finds a way to apply a known force and time the result.

Calibrated with known masses #

The formula has two unknowns in it besides the astronaut’s mass: the stiffness $k$ and the moving mass $m_0$. Neither needs to be known separately.

Squaring the period makes it linear in mass: $T^2 = (4\pi^2/k)(m + m_0)$. Put in several known masses, measure the periods, draw the line, and both constants come out — the slope gives the stiffness and the intercept gives the moving mass. This is how such instruments are calibrated, and it absorbs every unknown that is the same from one measurement to the next. The third of itself a spring carries found that a spring’s own mass adds a third of itself to the oscillating mass; in the calibration line that third is simply part of $m_0$, measured rather than calculated.

The method is the inertial balance, an old laboratory instrument: a platform on flexible strips, set vibrating with a sample on it, its period compared with those of known masses. It was used for decades in teaching laboratories precisely because it works with no reference to gravity, and the space chair is the same instrument built for a person.

The spacecraft is part of the oscillator #

In orbit there is a correction that has no counterpart on the ground. The springs connect the chair to the spacecraft, and when they push the chair one way, they push the spacecraft the other. The spacecraft is free too, falling in orbit like everything else, so it recoils. The two bodies oscillate against each other about their common centre of mass, and the period is set by the reduced mass $mM/(m + M)$, not by $m$ alone.

On a large station the effect is tiny, a few hundredths of a per cent, comparable to a cup of water in a person’s mass. In a small capsule it is large: a chair pushing against a craft of only a tonne would read a 75-kilogram astronaut as 67 kilograms. On the ground the same correction exists, since a spring fixed to the floor also pushes the Earth, but the Earth’s mass makes it 10⁻²³ and nobody has ever needed to apply it. The point that keeps moving found the centre of mass as the one point that an internal force cannot move; in the spacecraft, the chair and the station oscillate about it, and the station’s oscillation, about ten micrometres on a large station for a chair swinging a few centimetres, is felt by every sensitive experiment on board while the chair is in use.

The correction is easy to apply when the craft’s mass is known, but what matters is not the craft’s total mass so much as the mass that moves rigidly with the chair’s mounting during a swing. A large station is a flexible assembly of modules, and on the time scale of a two-second swing the module the chair is bolted to may respond partly on its own. The cure is the calibration line again, measured in orbit rather than on the ground: known masses swung in the chair where it is mounted give a line that already contains the recoil of whatever the springs actually push against, and an astronaut read from that line needs only a small second-order correction, since the reduced mass is not quite linear in the occupant’s mass when the craft is small. A chair calibrated on Earth and flown would be wrong by exactly the recoil correction, and calibrating it again after installation is part of using it. There is a practical consequence for the spacecraft. Microgravity experiments are disturbed by accelerations of millionths of g, and the recoil of a mass-measuring chair, or of a crew member pushing off a wall, is among the largest disturbances on a station. The measurement of one person’s mass shakes the whole laboratory, slightly, in a way that every other measurement on board can detect.

A body that is not rigid #

The formula assumes that everything in the chair moves together. A human body does not. The skeleton is strapped firmly to the chair, but the organs, the muscles and the fluids are connected to it through soft tissue, which acts as a set of springs with its own resonances, several hertz for the abdomen and chest. When the chair oscillates, those parts follow the skeleton with a lag or a lead depending on the frequency.

The model is crude — a third of the body as one soft mass on one resonance — but the conclusion does not depend on the details. A chair that swings much more slowly than any of the body’s internal resonances sees the whole body move together and reads its whole mass. One that swings near those resonances reads nonsense. One that swings faster reads only the parts rigidly attached. So the chair must swing slowly, with a period of a second or more, well below the few-hertz resonances of the abdomen. That is a constraint the device’s designers had to respect, and it is why the chairs are on soft springs with long periods rather than stiff ones with short, more precisely timed swings.

The same principle governs how vibration affects the body on Earth. Vehicle seats, aircraft cockpits and power tools are designed to keep vibration away from the body’s internal resonances, which lie in the same few-hertz band, because energy there is absorbed by the organs and is what makes vibration uncomfortable and, over long exposure, harmful. The chair simply avoids that band by going far below it.

A clock, not a balance #

Once these corrections are in hand, the precision of the measurement is the precision of the timing.

Since $T^2$ is proportional to the moving mass, a fractional change in mass gives half that fractional change in period. Detecting a hundred grams in ninety kilograms of moving mass means timing two seconds to about a millisecond, easily done electronically and averaged over several swings. The limit in practice is not the clock but the body: a person’s mass changes by hundreds of grams over a day with eating, drinking and the bathroom, so measurements are taken at the same time of day under the same conditions, as they would be on the ground.

The weight a frequency can find followed the same principle at the other extreme of scale: a vibrating cantilever or crystal whose frequency shifts by measurable amounts when a single virus or a few hundred atoms land on it. Mass is measured by frequency from the scale of molecules to the scale of people, and in orbit, where weight is unavailable, it is the only way. A balance compares weights. A frequency measures inertia directly, and a good clock is all the instrument needs.

Two masses that have to be one #

The space-station chair is an everyday experiment on the equivalence principle that nobody thinks of as one. Its readings, in kilograms of inertial mass, are compared with the same person’s weight on the ground, in kilograms of gravitational mass, and the two are expected to agree. They do, to the precision the chair achieves, which is far coarser than the precision of laboratory tests but has the virtue of being a measurement of a living person.

The pull that has to equal the pull back separated a third mass from these two, the active gravitational mass that produces a field. The chair has nothing to say about that one. But it makes the distinction between the first two concrete in a way that ground experiments cannot: on the ground, every measurement of mass is made in a gravitational field, and inertial and gravitational effects are mixed. In orbit, weight is gone, and inertia is the only property left to measure. That the number agrees with what the scale said before launch is the equivalence principle at work.

The antiatom that had to be weighed anyway turned on the difficulty of weighing antimatter, where the equality of inertial and gravitational mass had to be tested rather than assumed. The astronaut is the opposite case: a body whose inertial mass is easy to measure and whose weight is unavailable, and for which the equality is used rather than tested. Between them they show the two directions in which the principle can be put to work.

What the picture cannot show #

The figures use a simple chair: a single mass on linear springs, with the occupant rigidly strapped in, a single soft mass for the body’s tissues, and a spacecraft that responds as a rigid body. Real devices have springs that are not perfectly linear, so the period depends slightly on amplitude, and are operated at a fixed amplitude for that reason. The spacecraft is not rigid either: a large station is a flexible structure with its own vibrations, and its response to the chair’s push depends on where the chair is mounted.

The soft-tissue model is the roughest part. The body has many internal resonances, in different directions, and their frequencies and damping vary between people and with posture, breathing and tension. The figure shows the shape of the effect, not its size for any particular person, and the conclusion that a slow chair is safe rests on the chair being much slower than all of them. Breathing itself moves air and the chest during the measurement; astronauts are asked to hold still and breathe steadily, and the residual effects are among the reasons such measurements scatter by a few hundred grams.

The comparison with ground weight assumes the ground scale is correct and that the astronaut’s mass has not changed between measurements. In practice, ground and orbit measurements are taken days apart, and the agreement between them is limited by the body’s own variation rather than by either instrument.

Still open: how precisely a body can be weighed in flight #

Measuring changes in an astronaut’s body mass precisely matters for long missions, where loss of muscle and bone has to be tracked and countered by exercise and diet. Current devices measure total mass to a few hundred grams, limited by the body’s soft-tissue motions and by how still a person can hold. What is wanted is more: not just total mass but its parts — how much is fluid, how much muscle, how much bone — over months, which current mass measurements cannot distinguish. Methods combining mass measurement with imaging and with the dilution of tracers are being tested, and whether any can be made to work routinely in the small, busy laboratory of a spacecraft, without the large machines used on the ground, is an open engineering problem for missions to Mars.

In free fall weight vanishes and inertia remains, so an astronaut is weighed by the period of a chair on springs, $T = 2\pi\sqrt{(m + m_0)/k}$ — two seconds for 75 kg, read off a calibration line of $T^2$ against known masses — corrected for the spacecraft the springs push against, 0.026 per cent on a 420-tonne station and 10 per cent in a 1-tonne capsule, and swung well below the body’s internal resonances near 4 Hz so that soft tissue moves with the frame. The habit worth carrying is to ask which property of a thing an instrument actually measures, because when one property disappears another, equal by the equivalence principle, can stand in for it.