Is There Dark Matter in Orbit around the Earth?

For slightly over a year, I have largely put aside my longtime interests in the foundations of quantum mechanics and in particle physics, and have been working on dark matter. This interest came about in two different ways. The first was a paper on models for modifications of the Schrödinger equation on which I was working with my frequent collaborator Angelo Bassi during the 2007–08 academic year. The models called for a noise source to act weakly on ordinary matter, and one of the mechanisms I decided to try was dark matter collisions with ordinary matter. I starting learning about dark matter through a lunch with IAS Visitor Masataka Fukugita, whom I had met at a dinner at Peter and Sue Goldreich’s. Following this, I made what I considered a “toy model” for the paper I was writing with Angelo—not a realistic mechanism for our purposes, but it got me learning and thinking about dark matter.

The second way was a news item that my wife Sarah showed me in the Economist about the so-called “flyby” anomalies. When spacecraft are put in “flyby” orbits, passing close to the earth to produce large changes in direction, the outgoing velocity is found to deviate from expectations by about a part in a million. (A review for a general physics audience is given in M. M. Nieto and J. D. Anderson, “Earth Flyby Anomalies,” Physics Today, October 2009.) Sometimes the spacecraft slows down slightly (as would be expected from normal drag), but in some cases it speeds up, a really weird effect if true. I made a mental note to look for the article when published in a journal, and when a detailed report appeared in Physical Review Letters[1], I asked Scott Tremaine what he thought. He said that the group at Jet Propulsion Lab that wrote it has a reputation for careful work, so one couldn’t just dismiss it. So I started to think about possible explanations.

Having been in physics for nearly fifty years, I have seen many purported new effects be discounted as improvements have been made in the experiments or the analysis. The most probable explanation of the flyby anomalies is that they are the result of an inadvertent omission of some conventional physics from the analysis. People are still actively pursuing this route, but so far nothing convincing has emerged, and many things have been ruled out. Hence there is a chance that the effect is an indicator of new physics. I personally believe that if new physics is involved, it is very unlikely to implicate Maxwell’s equations for electromagnetism, because these, and their relativistic extension to quantum electrodynamics, have been tested to fantastic precision. I am also skeptical that the flyby anomalies can be attributed to changes in Einstein’s theory of general relativity, which has also been well-tested in the framework of metric theories that obey the equivalence principle (in a freely falling elevator, you feel no gravity), and within this framework one can show that possible effects are at least a factor of one hundred too small to be relevant. So deviations from gravity would have to take the form of a theory that does not obey Einstein’s equivalence principle, and this too has been tested to great accuracy.

This leaves another possibility: effects of dark matter. We now know that ordinary matter is only a minor component of the universe; for every gram of ordinary matter, there are five grams of a mysterious “dark matter,” which participates in Newtonian gravitational forces, but is electrically neutral and so does not readily emit or absorb light (hence the “dark”). So far there are no firm experimental indications of its properties, beyond what is inferred from astrophysics and cosmology. The question I have been investigating is whether dark matter gravitationally bound to the earth could be responsible for the flyby anomalies. Could collisions of the spacecraft with dark matter near the earth cause the observed velocity changes?

I have written three papers addressing this question. In the first[2], I showed that if there were a dark matter component that underwent an exothermic (energy releasing) reaction when colliding with a spacecraft proton or neutron, by converting to a lower mass particle, then the spacecraft nucleon would get a “kick” and the observed velocity increases could result. In this paper, I also studied various physical constraints, to see whether there is an allowed range of dark matter particle masses and interaction cross sections that could also explain the magnitude of the observed flyby anomalies. The answer is that there is a small window, but it requires dark matter masses much lighter than conventionally assumed in the standard “cold dark matter” model, and larger interaction cross sections with ordinary matter than conventionally assumed.

As a result of circulating the preprint on this, I was invited to give a talk at a space science conference in the summer of 2008, and had some very useful conversations with people there. This led to my second paper[3], which was a determination of an upper limit of how much dark matter could be in orbit around the earth, by using current tracking data for satellites, the moon, and asteroids. By comparing lunar laser ranging of the moon, which gives the sum of the gravitational masses of the earth, the moon, and everything in between, with ranging of the LAGEOS geodetic satellite, which gives the earth gravitational mass, and ranging of a spacecraft tracking the Eros asteroid, which gives an accurate lunar mass, one gets an upper limit for the amount of dark matter that can lie between the earth and the moon. It turns out to be four billionths of the earth’s mass—not much mass, but enough, it turns out, to be compatible with a dark matter explanation for the flyby anomalies.

The third paper[4], which I just finished this summer, involves making a detailed model for dark matter in orbit around the earth and using it to fit the flyby anomaly data. It is easy to see that Saturn-like rings of dark matter in the earth’s equatorial plane don’t work—most of the flybys would pass inside the rings, so there would be no scattering. So I tried the next simplest model, which was to consider a bunch of dark matter in a circular orbit, with its orbital plane tilted with respect to the earth’s equator. Because the earth has an equatorial bulge, its gravitational field differs from the spherically symmetric field of a point particle, and this symmetry deviation causes a tilted orbit to precess (i.e., slowly rotate) in time around the earth’s axis, with the angle between the orbit plane and the earth’s equatorial plane remaining fixed. Over a long period of time, this traces out a shell. My model then consists of two dark matter shells, one composed of elastic scatterers (to give flyby velocity decreases) and one composed of inelastic exothermic scatterers (to give the velocity increases). Parameters of the model are the radius, width, density times interaction cross section, and tilt angles of the two shells—eight parameters in all. The six known flybys are very well fitted—better than I had expected­­—with shell radii in the 30,000 to 35,000 kilometer range. One might suspect “overfitting,” but if one attempts to use just an inelastic shell and fit only the two flybys with the smallest estimated errors, one cannot get a good fit. And if one omits any one flyby from the two-shell fit, the model still gives a reasonable prediction for the omitted one. So the model has a certain “rigidity,” and is not just a case of fitting a wiggly curve through data points.

Incisive tests for the future will include putting in possible constraints from satellites in high-lying orbits, making predictions for new flybys as data becomes available, looking for spacecraft temperature increases caused by the postulated dark matter scattering, and of course seeing the results of earth-bound experiments trying to detect dark matter and determine its properties. In the meantime, others will continue to look for conventional physics explanations of the mysterious flyby anomalies. I think it will take several years, at least, for things to be clarified, and in the meantime I have other projects to pursue, but this has been a fascinating exercise that has taught me new topics in physics, and brought me in touch with a space science community with which I had no previous contact.