# Watt (Kibble) balance and the kilogram - how does the dependence on $g$ get eliminated?

The standard ${kg}$ is now in the process of being redefined by the watt balance (rather than the lump of metal in Paris)

A watt balance is very simple, you measure the force needed to support a mass against gravity (by accurately measuring the current/voltage in an electromagnet).

But in order to translate this to mass you obviously need to know local $g$. This varies by a few 0.1% due to latitude and local geology so how do you measure mass to a few parts per billion without a method to calibrate out the local $g$?

The obvious point being that if you had a sufficiently accurate test mass to take around all the labs in the world to calibrate their balances - isn't that the standard mass?

• Here is a 1967 paper to show how you might measure g accurately. No doubt the techniques have improved. nvlpubs.nist.gov/nistpubs/jres/72C/jresv72Cn1p1_A1b.pdf – Farcher Jul 4 '17 at 16:15
• Another interesting and more recent article. pdfs.semanticscholar.org/1fa9/… – Farcher Jul 4 '17 at 16:21
• @LucJ.Bourhis - no it doesn't, it says with 'g' accurately measured. I might as well say with 'm' accurately measured I can build a standard kg – Martin Beckett Jul 4 '17 at 16:48
• You realise that any mass falls with an acceleration g, don't you? And that measuring an acceleration only requires measuring distances and times… – user154997 Jul 4 '17 at 18:30

You are quite correct that the determination of mass requires measuring the local value of the $g$. However this is routine these days with instruments like the FG5 gravimeter giving accuracy of $2$ $\mu$Gals which is getting on for $1$ part in $10^9$. The gravimeters measure freefall speed so they measure the acceleration directly and do not depend on a standard mass.

I can't find anything on the NIST web site saying which gravimeter they plan to use, but presumably it will be at least as good as if not better than commercial gravimeters like the FG5. There is a reference to the NIST gravimeters here but it is frustratingly vague.

• I wonder how much 'g' varies? I remember one of the interferometry groups claiming they could detect tides in the magma chamber under Hawaii. At 1 ppb even ground water levels could have an effect. – Martin Beckett Jul 4 '17 at 16:55
• @MartinBeckett: I think the timescales for the gravimeter measurement are minutes not hours or days. So changes caused by tides wouldn't be a problem. – John Rennie Jul 4 '17 at 17:08
• of course - assuming you measure 'g' as part of the experiment rather than an annual calibration. – Martin Beckett Jul 4 '17 at 17:16
• @Martin Surface gravity varies at the $\text{few} \times 10^{-4}$ level even over fairly short distances (a fact widely used to help with mapping subsurface features—including aquifers) and by more than tenths of a percent globally. – dmckee --- ex-moderator kitten Jul 4 '17 at 17:39
• @MartinBeckett $g$ has relative daily variations of the order of $10^{-{-7}}$, mainly due to moon tides. There are also yearly variations due to sun tides. Moreover, there are variations due to micro-seisms, water table variations and solid earth tides (the Earth is an elastic body). In Boulder, CO, NOAA has a facility with a number of gravimeters, both absolute, like the FG5 type, and relative: when the ocean tides grow, the differential gravimeters can detect the phenomenon. It's a nice facility in the middle of nowhere. – Massimo Ortolano Jul 4 '17 at 19:31

how do you measure mass to a few parts per billion without a method to calibrate out the local $g$?

You don't. The local $g$ is an acceleration and it can therefore be measured directly using only access to length and time standards, which ─ given how the meter is defined ─ boils down to an optical experiment with an accurate timing reference.

Thus, as a simplistic approach, you can essentially make do with a ruler and a stopwatch, but if you want reasonable accuracy then you probably want a proper instrument like the FG5 gravimeter that John linked to. Luckily, gravimetry is a very sturdy branch of metrology, because changes in $g$ can be traced to deposits of random stuff like some kinds of rock and decayed plant matter, which for some reason tend to bring huge clouds of money along with them, so there is a lot of (relatively) cheap metrology you can do rather accurately.

On the other hand, if you want to do primary metrology, then the game changes somewhat, because your measurement of $g$ needs to have as many significant figures as your final measurement: you're trying to upstage a ${\sim}50\:\mu\mathrm{g}/\mathrm{kg}$ variation in the existing IPK system, so you want all of those eight significant figures ─ parts per billion ─ in your measurement of $g$. Quoting from

Watt balance experiments for the determination of the Planck constant and the redefinition of the kilogram. M. Stock. Metrologia 50, R1–R16 (2013).

The value of the gravitational acceleration $g$ needs to be known at the centre of mass of the test mass, which is inaccessible once the experiment is set up. One technique to achieve this is to establish a map of the variation of $g$ in the laboratory with a relative gravimeter before the watt balance is installed. In addition, the absolute value needs to be known at least at one point. The absolute value of the gravitational acceleration at the centre of mass of the test mass can then be obtained by interpolation. The gravitational acceleration also varies in time by as much as $2.5$ parts in $10^7$ due to tidal forces from external bodies. This needs to be taken into account either by permanent g measurements or by modelling of the tidal effects. A correction needs to be applied for the gravitational effect of the watt balance itself.
That last point is important, because here the $1/r^2$ dependence of newtonian gravity plays against you: the effects of the balance itself are small but nonnegligible, and they are measurable with some variation from the outside of the balance, but their strength and variation increase by squares when you get to the inside of the device.
• I'm not sure that the difference in $g$ between the bottom and the top makes a difference, but the point is that whatever effect the masses close to those two points have is going to be amplified dramatically as compared to just the overall gradient coming from underlying deposits or even other parts of the building, so you need to estimate the overall gradient from external masses but also think carefully about how the masses close up to the experiment will affect it. – Emilio Pisanty Jul 4 '17 at 19:30
• Emilio, I recently attended a workshop where the Kibble balance was also discussed. The uncertainty on g is one of the main sources of uncertainty, at present is usually known in a few parts in $10^9$. I asked if the fact that g is not measured along the balance axis is an issue, and they've told me that g inhomogeneities in the labs remain pretty constant in time, and so you can map the lab's gravity just at the beginning, and then you can measure it in a single place when you operate the balance. – Massimo Ortolano Sep 30 '17 at 14:15