Negligible impact of dark matter on the Solar System

Mike has asked me about the following preprint:

Constraints on Dark Matter in the Solar System by N.P. Pitjev and E.V. Pitjeva (Leningrad, Russia)

This article was celebrated by an impressive title and a "more than just uncritical" article at the Physics arXiv Blog:

The Incredible Dark Matter Mystery: Why Astronomers Say it is Missing in Action

Wow. The only comment over there that isn't preposterous is the comment by S. Seibert. Thankfully, Sean Carroll presents the same stance as your humble correspondent: in his opinion, the expected impact of the dark matter on the Solar System is comparable to the dark matter's influence on NBA three-pointers.

Why?




The Russian authors review some observations of the Solar System in general and planets in particular which lead them to the conclusion that no gravitational impact of the dark matter (which in principle affects the trajectories of everything, including the planets) has been seen. This may be translated to an upper bound on the density of dark matter. The strongest one that we can find in the Table 1 of the paper says\[

\rho_{\rm dm}\lt 3\times 10^{-19}{\rm g/cm}^3

\] For the dark matter around the Earth, the estimate is later improved (i.e. lowered) by another factor of two. Is this upper bound capable of identifying a contradiction between the dark matter theory and the observations?




First of all, the critical density of the Universe for which the Universe is spatially flat – and our Universe seems to be, with an amazing accuracy – is equal to \(9.9 \times 10^{-30}{\rm g/cm}^3\). The dark matter density is about 25% of this figure.

If this were the density in the Solar System, it would clearly be hopeless to try to find an influence of the dark matter on the Solar System: the upper bound derived by the Russian paper only says that the Solar-System-observed dark matter is smaller than 10-100 billion times the correct value. Well, it almost certainly is. One is indeed smaller than 10-100 billion.

However, unlike dark energy, dark matter does clump. It is mostly located in halos around the galaxies – which are somewhat larger than the distribution of the bright stars in the same galaxy. Because of this concentration, the average density of dark matter inside a galaxy such as the Milky Way is larger than one quarter of the aforementioned figure \(9.9\times 10^{-30}{\rm g/cm}^3\).

What is the density of dark matter in the galaxy? The density may depend on the location and these "profiles" aren't quite well-known and the proposed models are somewhat complicated. But to get an order-of-magnitude estimate, it's enough to ask Wolfram Alpha to calculate the following ratio:

mass of milky way / volume of milky way in cubic cm

The result is \(1.3\times 10^{-23}{\rm g/cm}^3\). I tried to use this unit of density (which is the Russian folks' favorite unit) and no other unit throughout this article although it's not my preferred unit in any sense.

So you may see that the upper bound derived from the planetary dynamics is still about four orders of magnitude too weak (i.e. the number is large) relatively to the actual density. The Russian paper says that the planetary-dynamics-observed density of the dark matter is smaller than 10,000 times the right value. Well, 1 is almost certainly smaller than 10,000, indeed.

It shouldn't be hard for you to intuitively understand why dark matter shouldn't have a detectable (or large) gravitational impact on the events inside the Solar System. The main reason is that the total mass of the dark matter inside the Solar System is much much smaller than the total mass of the visible matter in the same volume (which is dominated by the Sun). Why? Because the visible mass is severely clumped while the dark matter isn't that clumped – it is mostly diluted into the vast interstellar regions (volumes in between the stars) whose overwhelming majority is much further from any star than the Saturn-Sun distance.

(There could be some increase in the density of dark matter even in the vicinity of the stars – just like there is an increase around the galaxies – because the concentration of the visible matter that was needed during the birth of the Sun and the Solar System probably depended on a peak in the dark matter distribution. But it's fair to say that this extra increase isn't changing the qualitative story i.e. that the dark matter density inside the galaxy may be assumed to be more or less uncorrelated with the positions of the stars, at least if we only want order-of-magnitude estimates.)

So while the total mass of dark matter in the Milky Way halo is 5 times larger than the total mass of the visible matter, the total mass of dark matter inside the Solar System is incomparably smaller than the total mass of the visible matter: most of dark matter is outside all "solar systems".

If you allow me to advocate the latter point in one more way, note that the closest next star to the Sun is Proxima Centauri, several (4.24) light years from us, while Saturn is just about 1 light hour away from the Sun. One year is about 9,000 hours but to compare the volumes, you have to calculate the third power of this number. You get 729 billion. So the mostly empty interstellar volume that may be "attributed" to the Sun is something like 1 trillion times larger than the ball of the Saturn-Sun-distance radius. This fact means that only 1 trillionth of the solar (or average stellar: but the Sun isn't too far from an average star) mass may be expected in this relatively small ball and this relatively small mass (or density) is compatible with the upper bounds derived from the planetary dynamics even though the possibility to find an impact (or discrepancy) sometime in the future can't be ruled out entirely because we're "just" 4 (or so) orders of magnitude away from the goal.

One may be able to derive or guess the relevant numbers more accurately or less accurately, more quickly or less quickly, but I am disappointed that just a single commenter under the Physics arXiv Blog was able to pinpoint the qualitative idea implying that the hype is completely unjustified – namely that there can't be any observable contradiction because the expected density of the dark matter in the Solar System is way too low. Almost everyone else added his or her own interpretation of the preposterous statement by the blogger that we're facing an "incredible dark matter mystery".

Let me mention one widespread laymen's mistake that makes them believe similar conspiracy theories. The laymen tend to think that if a scientific theory XY predicts an object or phenomenon UV, then UV should be visible by pretty much every experiment CD. Most of the uneducated laymen's criticism of string theory reduces to the opinion of this sort. However, this opinion is completely misguided. It is completely normal for UV to be invisible by CD. In many or most cases, CD is just too unrefined or weak to see UV and theories about UV (whether UV is a string or dark matter) often unambiguously predict that UV is invisible by a CD or (almost) all CDs. There's absolutely nothing wrong about a theory just because it predicts that a relevant object or phenomenon will be unobservable. There may exist other reasons why the theory is a good idea or convincing and the only scientific way to eliminate a hypothesis is falsification – the discovery of a measurable contradiction between the theory's predictions and observations. Not observing something that should be unobservable according to the precise predictions of a theory (even if it is very important in the theory!) surely doesn't count as falsification! Too bad that most laymen are incapable of understanding this trivial point.

See also Ethan Siegel.



Just one picture advertising a new CMS paper based on the 2011 data. As you can see, there seems to be a 3-sigma excess in dijet (two jets) events indicating a resonance with a mass near \(300\GeV\), not to mention the 2-sigma excesses near \(1,100\GeV\) and elsewhere. But don't be excessively certain that the former comes from new physics. 3 sigma is not much. Moreover, some extra operations had to be applied to remove some background near \(300\GeV\). Nevertheless, the very fact that no 2012 collisions were incorporated to this 2013 study may look... strange.

BTW the best TRF-rated paper today is Karch-Jensen showing that the Maldacena-Susskind Einstein-Rosen bridges appear for an entangled (color-singlet) quark-antiquark pair in AdS as well because one gets a world sheet with two boundaries along two branches of a hyperbola, so they're causally disconnected from each other. I think that I know how to show the analogous thing for M2-branes in M-theory, or any brane with a wormhole shape, for that matter.