Measuring Newton's Constant with a Space-Borne Gravity Train

I recently came across a paper which was published in Classical and Quantum Gravity, a respected journal, after it had initially appeared on arXiv. It proposes a space mission consisting of a metal sphere with a cylindrical hole, floating through space as a smaller reflective object oscillates back and forth along the hole, pulled by the gravitational field of the sphere. The position of the the smaller object can be monitored by another space probe, and the period of oscillation can be used to measure Newton's gravitational constant, big G. I like this idea, it draws upon some of my recent work and I place it firmly in the "just crazy enough to work" paper category, which are my favourite papers to read.

Diagram of the proposed experiment, from the arXiv version of the paper.

The proposal is motivated by some recent analysis (by some of the current proposal's authors) of independent measurements of the gravitational constant, which showed that even though they are measuring the constant with smaller and smaller uncertainty, the different measurements are not in precise agreement of each other, sometimes deviating by 40 times the standard error on the measurements. The analysis makes the bolder claim that the difference between the measurements has the same periodicity as Earth's length-of-day variations, which are due to large-scale seismic effects. They conclude that there are systematic effects caused by the fact that all these experiments (which typically involve monitoring a rotating pendulum near large masses) take place on the Earth, and desire a way to measure this constant away from the Earth. The National Science Foundation has put out a call for proposals for more accurately measuring big G; I recommend reading its three paragraphs if you're wondering why anyone would bother caring about this.

The various non-agreeing measurements of G over the years, from Anderson et al. Pay more attention to the red than the black. The black is the "bold proposal" I mentioned.

To make this measurement, the authors, lead by independent researcher Michael Feldman, suggest sending a miniature gravity train into space. A gravity train, something I have written about in great detail, consists of a spherical mass (often taken to be the Earth, but not here) with a tunnel through it (often through its center), with a smaller object falling through the hole. It builds up speed due to gravitational attraction to the sphere, passes the halfway point, and then starts decelerating, coming to a rest on the other side. Inside the Earth, it would take 38 minutes to fall through this tunnel. Feldman and friends propose a small metal sphere, roughly 10 cm in diameter and 1.3 kg in mass, that would take about two hours for the small object to fall through.

How could G be measured from such a device? For a uniform density sphere, it can be shown that the period of a gravity train is $T=2\pi\sqrt{\frac{R^3}{MG}}$. If the mass and radius are known, and the time is measured, G can be extracted. In the proposed setup, the position of the small object will be monitored by a laser aimed at the tunnel from another nearby space probe, and from these periodic measurements of position, the time can be measured, and sent back to Earth by an antenna on the probe. The paper consists of more detailed derivations of the G-T measurement, unique to the proposed design (which consists of two layers of different materials).

Zoomed out schematic of the probe and sphere, from the arXiv version of the paper.

The authors are concerned with the precision of such a device, and which systematic errors contribute to the overall uncertainty in G. These include the metallurgy of the sphere and hole (uncertainties in R, M, and the uniformity thereof), the initial placement of the small reflecting object in the hole (which must be extremely gentle), the ideal place to position sphere with respect to the host probe so that the probe is close enough to block the sun but not so close that its own gravity affects the experiment, the radiation pressure from the probe laser on the device, deformations of the sphere due to the tidal influence of the sun, possible charging of the tunnel's interior due to cosmic rays, and more. They even calculate the change in the period if general relativity is taken into account, which is something I was curious about for my gravity tunnel research, but didn't have the tools to solve. The hypothetical uncertainty analysis was probably the most fun part of the paper to read.

They estimate that, given advances in metallurgy and aerospace deftness, they can get the precision of the G measurement down to an optimistic 63 parts per billion. The current record for Earth-based G measurements is 13,000 parts per billion. This would be a huge improvement if it actually worked, and would eliminate some of the systematic issues with measuring the strength of gravity in Earth's gravitational field.

The question, of course, is whether this thing will actually exist, and whether the budgetary will exists to make it so. The authors suggest that the experiment would not be the main mission payload of a space launch, but rather would piggyback on a larger, more important probe headed out of the solar system.

I was interested in this paper because I like crazy yet scientifically rigorous ideas, and it draws upon a system that is close to my scientific heart [disclaimer: the paper cites mine]. It was a pleasure to read about all the potential effects that could skew the time measurement, and how they planned to deal with them. I hope the available metallurgy, metrology, and money becomes sufficient to launch this thing into space.