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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.


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.

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snail feedback (31) :

reader JollyJoker said...

The link to Sean Carrolls blog gives a 403 Permission denied error. Removing the final slash seems to correct it. This happens on Firefox. I can't reproduce this with Opera, IE or Chrome; the link works fine with all those.

It's pretty sad that the arXiv blog writes drivel like this. Judging by the comments it seems to get them page views mainly from crackpots promoting their own theories who like the idea that there's no dark matter and are incapable of or uninterested in understanding the limits shown in the paper.

reader Luboš Motl said...

I removed the slash. Does it work immediately now?

It would be interesting to know who's the secret writer behind the arXiv blog. It may turn out to be a superficially normal "mainstream" physicist who is living a second life because he or she is afraid to express his or her own real opinions publicly.

reader JollyJoker said...

Yeah, it now works in Firefox and it didn't break in any of the browsers I listed above. Thanks!

I've never realized the blogger is anonymous. I suspect a careful analysis of the knowledge displayed could tell us a lot about who it could be.

reader JollyJoker said...

reader Dilaton said...

Yeah this seems to bee a cool nice blog :-)

Lumo could add him to the list of friends in the right side bar such that I can easier find it for cosmological issues ...
Even before following Lumo's calculations, I was not surprised at all by the fact that DM does not detectably at presence influence the motions in the solar system for some reason, whereas the converse statement would have surprised, delighted, and excited me ...

reader NumCracker said...

Dear Lubos, my compliments to this very lucid post. In particular I would like to understand the following: nonabelian gauge fields transport color as gravitons would do with their "charge" (energy) ... so wouldn't one expect a gravitational potential in the form V(R) = GM/r + a*r, with a tinny "string tension" a ? I understand gluons are spin 1 particles, while gravitons are not ... but I could not really see in simple terms why such "confining" scenario does not apply to gravity also. Thanks

reader JollyJoker said...

At least my own intuition on these matters is pretty bad. I'd have assumed the effect of DM is extremely small and data on the motion of planets is extremely accurate but I couldn't even begin to guess what the relative magnitudes of detectable vs expected effects would be.

The fact that this hasn't been discussed much would indicate the data isn't even close to accurate enough, but that's about it.

reader jitter said...

How much dark matter has been captured by the Sun in the last billion years or so and is now orbiting in the interior of the Sun?

reader Luboš Motl said...

Dear JollyJoker, this blog post actually contained the link to Ethan before you posted it. Search for Ethan here.

I have added his blog to the Blogs led by science feed updates in the right sidebar.

reader Luboš Motl said...

Dear NumCracker, thanks for your compliments, and a good question.

The normal answer is that confinement requires the force to get stronger at longer distances. QCD coupling is tiny (asymptotic freedom) at very short distances but it gets stronger and once it gets stronger at the QCD scale, the fluxtubes confining the charges form and the linear term is added.

On the other hand, electromagnetism is getting weaker at longer distances, so there can't be any confinement: if the force is too weak to confine at normal intermediate scales, it becomes even more hopeless at longer scales. In the case of gravity, it's even more extreme because the dimensionless coupling decreases as a power law with the distance - gravity could only be strong enough to produce confinement at supershort, roughly Planckian distances. The longer distances you consider, the more hopeless it is.

Moreover, it seems that the linear term may violate the equivalence principle although I can't exactly explain why I feel so right now. ;-)

At any rate, your modification of gravity is creative as well as much more brutal than the "MOND" (modified Newton dynamics) theories that have been considered - perhaps your theory is even more justified by your argument than any MOND on the market. I will think about it for some more time.

reader JollyJoker said...

Heh. I saw it some time after I posted and assumed you had added it due to my post. Well, better to repeat a good link than risk missing it completely :)

reader Shannon said...

Excellent post Lubos. And thanks for putting theory vs observation straight again.

reader NumCracker said...

Dear Lubos, it seems me a very interesting explanation! Thanks again. It made me curious in this other aspect: wouldn't in the gravitational case all the positive rest energy of sources (masses) E = mc^2 be completely counterbalanced by the energy stored in the gravitational field, in such a way that the total energy of the system becomes zero? I mean, nobody is able to "deconfine" particles from spacetime, or better saying, to decouple particle sources from the graviton field (there is no Faraday's cage or color anti-screening) ... sorry by my imprecision, but maybe I made myself understandable ;-)

reader Luboš Motl said...

A good try. I was actually tempted to think that your cancellation would be possible, too.

However, you must try to determine what the "mass" exactly means. What you really want to cancel in this way is the mass as measured by the gravitational field.

But observations show that this mass shouldn't be zero - it's the critical density in average, or some fraction of it (if you declare dark energy to be another mistake). So a perfect cancellation will conflict with observations.

Maybe you know how to redefine things very differently so that the spatial flatness will correspond to your color-like cancellation of the charge but I don't know how to do that...

reader Mark said...

Lubos, I know you don't want to write papers anymore, but maybe you could share your thoughts/results on ER bridges in a blogpost. I think many would appreciate further evidence for the MS proposal.

reader Gene Day said...

Why would this Russian couple go to all that trouble when the effect of DM on the planets is clearly too small to be observed?

reader NumCracker said...

Thanks. I see you are a really fast and deep thinker! ;-) So I would ask a last question and after that I am done, right? And, beforehand thanks by your patience and huge didactics! I am not aware of any dS/QCD model who is string-confining ... but in case it exists, would there exist the possibility that this sort of QFT potential can be mapped to the gravitational side while keeping the same functional form? Or conversely, how dark-matter in the bulk (coupled to gravity) would look like at the boundary (CFT) ? Thanks

reader Luboš Motl said...

Perhaps because they haven't appreciated this point yet? Or, when avoiding diplomacy, because they really suck as physicists?

reader Luboš Motl said...

Dear Mark, I don't believe that many would appreciate it, in fact, I have measured how many people appreciated very similar things and the answer is extremely close to zero. Because the work needed to write a paper or a research-level technical blog entry - it's not much different - is about 10x greater than the work needed to write a simpler/popular one, this is a classic case of throwing pearls to the swines.

reader Luboš Motl said...

Dear Numcracker, do you really mean dS/QCD - there is a speculative dS/CFT by andy Strominger - or was it normal AdS/QCD?

This is an exact duality so the forces in the boundary theory correspond to something in the bulk. But the confinement, a long-distance phenomenon, corresponds to a modification of the geometry in the middle of the AdS space. For example, the mass gap means that there's no space "in the very middle" of the AdS space at all.

So (A)dS/QCD isn't just some claim that the same forces seen in QCD become ordinary gravitational forces in the bulk. Because the bulk geometry is so far from the flat one, even the gravitational forces are completely different from what you expect in the flat space, and there's no direct applicability of the duality to the gravitational events in a (near) flat space.

reader Mitchell Porter said...

There are a lot of models now in which dark matter has an interacting
subsector - models according to which most of it is spread out in a
homogeneous galactic halo as in the classic conception, but in which
some fraction of the dark matter does form structures. If such a model
were to predict gravitational association between solar systems and structures made of interacting DM, then this paper provides a relevant bound... It may not be what they had in mind, but it is potentially useful.

reader vincent said...

Hey Lubos can you comment on this interview about the foundation of physics with Tim Maudlin?

reader Luboš Motl said...

Dear Vincent, it is very long. I may write an article later. It is pure rubbish, especially when it comes to his opinions about quantum mechanics.

reader JollyJoker said...

Wouldn't it just be odd if no one had done this and we didn't know of an upper bound?

I think experimentalists cover a lot of ground just because it hasn't been done before, rather than because theory would expect something to be there.

reader Mark said...

I was expecting a similar response :( Given how excited I am about these developments I still thought it's worth trying to persuade you. Probably you know this better, but my feeling is that about a quarter of the hep-th people in the US read your blog, and the news of your post would spread. In the end the post would have the same size of readership as the Jensen-Karch paper. (Maybe if you weigh in on a heated debate (e.g. firewalls) people ignore you, but this post would explain some technical results.)

Cheers, Mark

reader Gene Day said...

Competent experimentalists do not shoot onto the dark unless there is a reasonable chance of discovering something of significance. A little thought here would have shown that a null result was entirely certain.
Publishing this kind of thing just wastes a lot of time.

reader AngularMan said...

Publishing null results is necessary, so that others do know the state of research. Otherwise people would do the same work over and over again.

reader Gene Day said...

Publishing null results is important in the case of well-motivated investigations but this is not one of them. Perhaps it would be worth a single sentence, not more.

reader Mohan said...

this post is very important, and people know very little or nothing about it. it is hard to believe it, I am going to place a link to it on my web

worth seing, mind bending.

love and peace

reader fregocap said...

This has been done long time ago. Those russian guys are just unaware of literature

reader Alison Young said...

Dark matter doesn’t collide in the traditional sense, it has no
way to shed its momentum and angular momentum the way normal matter does.