Where is the Dark Matter in the Earth's Core?

I just finished reading “Dark Matter and the Dinosaurs” by Harvard University physicist Lisa Randall.   Dr. Randall is an excellent popular science writer, as well as being a top-flight theoretical physicist.   Her exposition on dark matter gave me most of my exposure to this arcane topic in modern physics.  

My understanding of the topic is shallow, but I think some common-sense observations provide constraints on the distribution of dark matter, which need to be recognized in models of dark matter and experiments to find it.

My biggest question about dark matter is: Where is the dark matter in the earth's core?  

The Nature of Dark Matter

Dark matter is a form of matter that does not interact with ordinary matter or energy, except through the force of gravity.  A better name for dark matter might be “ghost matter”, as the lack of interaction with ordinary matter means the dark matter can occupy the same space as ordinary matter, or pass right through it, undetected. 

Lisa Randall writes: “Dark matter passes right through our bodies, and resides in the outside world as well…. Every cubic centimeter around you contains about a proton’s mass worth of [dark] matter….if those particles travel at the velocity we expect based on well-understood dynamics, billions of dark matter particles pass through each of us every second.  Yet no one notices that they are there.”

It is unknown whether dark matter can interact with ordinary matter at all, except through gravity.  Nevertheless, some theoretical results suggest there may be very weak interactions, and experiments are in progress seeking to detect dark matter, either directly or indirectly, through some kind of interaction with ordinary matter.  According to Dr. Randall, it is also unknown whether dark-matter interacts with itself. 

It seems to me that dark matter is necessarily self-interacting.  Evidence (the lack of a distinct dark matter core in the earth) indicates that dark matter has a very low maximum density.   Dark matter must exclude other dark matter from occupying the same space.  The density of dark matter is much, much lower than regular matter.  Dr. Randall states that every cubic centimeter around you contains about 1 proton’s worth of dark matter.  Accordingly, the density of dark matter at the surface of the earth is only 1.7 x 10^-24 gm/cc, or 1.7/1,000,000,000,000,000,000,000,000^th of the density of water.

Evidence of Dark Matter

Dark matter is known through its influence on ordinary matter, via the force of gravity.  It is observed only on the scale of galaxies or larger structures.  Observed gravitational effects suggest that dark matter actually comprises 85% of the matter in the universe.  The evidence for dark matter is mostly derived from deep-space astronomy and cosmology, as follows.

1. The orbital velocities of stars in galaxies are much too high, given the quantity of ordinary matter in the galaxy.  Additional mass, in the form of dark matter, is required to explain the cohesion of galaxies.
2. The gravitational lensing of light around galaxies indicates a much greater mass in the galaxy than can be seen in ordinary matter.
3. The background radiation of the universe which formed shortly after the Big Bang shows an irregular distribution, which can only be explained by gravitational accumulation, requiring more mass than is known to exist as ordinary matter.
4. Modeling the development of the universe since the Big Bang shows that the gravitational influence of dark matter is necessary to create galaxies in the primordial universe. 
5. Evidence of dark matter can also be seen in the gravitational lensing of distant objects near colliding galaxies, such as the spectacular Bullet Cluster.  In such events, dark matter becomes separated from ordinary matter, and is revealed by observations of separate patches of magnification by gravity lensing.

Bullet-Cluster Galaxy.  Image credit: NASA

Local Dark Matter and Self-Interaction of Dark Matter

Dark matter reveals itself through the force of gravity on a very large scale – the scale of galaxies or larger structures.  But what about smaller settings?  What can we deduce about dark matter by its small-scale behavior?

Black Holes of Dark Matter

If there was no exclusionary force to dark matter, particles of dark matter (which do interact through gravity) would fall together, presumably to a very high or infinite density.  Without an exclusionary force, dark matter would be particularly prone to forming black holes from small quantities of dark matter, collapsing to very high density.  [Despite the similarity in names and difficulty of observation, dark matter and black holes are quite different things, and should not be confused.]  But we don’t observe the gravitational influence of lots of small black holes, within the galaxy.  If they existed we would notice their presence by abnormalities in the velocities of stars in the Milky Way, by gravitational lensing of distant starlight, and by deflections in clouds of interstellar gas.  We don’t see those things, so there must be an exclusionary force prohibiting the close association of particles of dark matter. 

Dark Matter at the Center of the Earth

Also, since dark matter interacts with normal matter only through gravity, we might expect all of the dark matter in the neighborhood (that is traveling at less than escape velocity) to fall through the crust and mantle of the earth, and accumulate in the earth’s center.  Since dark matter comprises 80% of the matter in the universe, we ought to find a substantial gravity anomaly in the earth’s core, unexplained by the density of normal matter in the core. 

We don’t.

The Earth’s Structure and Core

We have a very good understanding of the structure and composition of the earth’s core.  The structure of the core is revealed by the behavior of earthquake seismic energy as it is transmitted through the earth.  Seismic waves generated by earthquakes travel through the earth, and can be recorded at most places around the earth following a major earthquake.  Compression waves and shear waves travel at different speeds, and behave differently depending on the nature of the transmitting media, whether solid or liquid.   The speed of the waves depends mostly on density, and interfaces between materials of different composition produce both reflections and refraction of the waves.  All of this information allows us to construct the specific solution of layers, mineral composition, and phase (liquid or solid) of the interior of the earth.

Image credit: Charles Sturt University, via Ethan on ScienceBlogs.com

The interior of the earth consists of a number of concentric shells of varying composition and consistency.  Below the atmosphere and oceans, there is the earth’s crust, which occurs as oceanic and continental components.   Below the crust, the upper mantle is divided into the lithosphere and asthenosphere.  The crust and mantle are composed of silicate minerals.  The crust and lithosphere are rigid, and move as plates on the ductile asthenosphere.  The lower mantle is also ductile, and deforms plastically to form convection cells, driving the motions of the shallower plates. 

Image Credit: Wonderopolis.org

The earth’s core is primarily composed of iron and nickel, with a small amount of lighter elements.  A huge clue to the composition of the core exists in form of iron-nickel meteorites, which are derived from some proto-planet in the early solar system.  Iron-nickel meteorites typically contain nickel in concentrations of about 6% to 10%.  Gravity shows that the density of the core is about 3% lighter than pure iron, implying about 10% of lighter constituents, probably silicon, oxygen and sulfur.   Nickel is slightly denser than iron, so higher nickel concentrations would imply correspondingly higher concentrations of light elements to compensate in overall density.

Seismic studies show that the inner core is solid, and the outer core is liquid.  Convection in the liquid outer core accounts for the earth’s magnetic field.  The mineral composition of the inner core can be replicated and studied using high-pressure tools in the laboratory.  The combination of seismic studies, gravity studies, mineral composition studies, meteorite studies, and magnetic studies yields a model that fully explains all observations about the earth’s core.  No dark matter is indicated by the observations; rather, the introduction of dark matter would require unreasonable changes to the most logical interpretation for the composition of the core.  

Model of the Earth's density from the center to the surface.  Image credit Wikipedia.

Model of the Earth's gravity from the center to outer space.  Image credit: Wikipedia

Dark Matter in the Cores of Stars

The sun is a delicately balanced fusion engine.  The heat generated by hydrogen fusion produces an expansion force, which is balanced by the gravity of the star.  When the balance is disrupted by the exhaustion of nuclear fuel, the star becomes unstable, exploding as a nova or supernova, or collapsing into a white dwarf, a neutron star or a black hole.  The processes of nuclear fusion are known and well-quantified as a result of nuclear weapons research and super-collider experiments. 

Any gravitational anomaly in the sun or in the theoretical models of other stars would surely be noticed, and would be glaringly apparent to scientists studying stars.  We have to conclude that there is no dark matter accumulated in the cores of stars.  

Evidence from Spacecraft

Dark matter may exist as a “soup” of uniform density larger than the solar system, so that there is equal gravitational attraction in all directions.  In this case, no anomaly could be detected, because the gravitational influence of dark matter would be the same in all directions.  By analogy, a point at the exact center of the earth would be weightless, subject to equal gravity in all directions.  But even in this case, as objects move in some direction through the soup, differential gravity should be detectable if there are heterogeneities or nearby limits to the dark matter soup. 

Our best experiments to find dark matter in the Solar System are the Pioneer 10, Pioneer 11, Voyager 1 and Voyager 2 spacecraft.  Data from the Pioneer spacecraft was last received in 2002 and 1995, respectively, but the probes lasted long enough to identify a potential gravity anomaly in the solar system, the Pioneer Anomaly.  As I read Lisa Randall’s book about Dark Matter, I initially thought that the Pioneer Anomaly might be the expression of dark matter in the solar system, but subsequent reading revealed that the anomaly was robustly explained in 2012 as a thermal recoil phenomenon relating to the spacecraft itself. 

The Voyager spacecraft are now the most distant man-made objects from earth.  Both are still transmitting, at distances of 12.9 billion miles (19 light-hours) away, and 10.7 billion miles (16 light-hours) away, respectively.  The craft are traveling at roughly a right angle to each other, providing two long baselines to measure any gravity anomalies in those directions, revealed by an unaccounted-for acceleration of the spacecraft.  None have been detected.

Conclusions

Any theory of dark matter must account for the lack of detectable dark matter in the cores of planets and stars.  The lack of a detectable dark matter core in these places is strong evidence that dark matter is self-interacting.  There must be a property of dark matter that prevents dark matter from accumulating at high density.  This exclusionary force must act on at least the scale of a planet, and probably on the scale of the solar system. 

The exclusionary force places a limit on the maximum density of dark matter.  To the best we can now recognize, that limit is the detectable limit of density anomalies in the sun or the earth.  It is a very small density compared to the density of ordinary matter.

The lack of acceleration anomalies in distant spacecraft shows the large-scale homogeneity of dark matter surrounding the solar system.  At this time, I don’t know the limits of velocity determinations of the Voyager spacecraft, but I think that these long-distance measurements would be quite sensitive to a local density anomaly.  It would be a worthwhile exercise to calculate the effect of a dark-matter accumulation (such as Dr. Randall’s posited dark-matter disk within the Milky Way) on the velocities of the spacecraft, and see if the results would be within the tolerance of the spacecraft velocity measurements. 

If not for the robust evidence of dark matter in cosmology, it would be tempting to dismiss the idea of dark matter entirely.  But perhaps the finding that there is an exclusionary force limiting the density of dark matter can be a clue to identifying the true nature of dark matter. 

Maybe dark matter doesn’t fit the particle model of matter at all.  At this time, all we know is that it is very sparsely dispersed gravity.  But the particle theory of matter has proven very useful at explaining most of our reality.  We shouldn’t give up on it too easily.  We should think for a moment about what constraints our observations put on a particle theory of dark matter.

Ordinary matter and dark matter are clearly different in scale.  A proton excludes other protons on the scale of 0.8414 x 10^-15 linear meters.  If we assume a particle of dark matter has the same density as a proton, each dark matter particle must exclude other dark particles on the scale of 0.02 linear meters.  The ordinary proton occupies a volume of about 3 x 10^-46 cubic meters, and a dark matter particle of the same mass would occupy a volume of 1 x 10^-6 cubic meters, a difference of 40 orders of magnitude.

Perhaps the difference in size between ordinary matter and dark matter is the sole reason for the undetectability of dark matter.  It seems to me that electrical and other interactions between normal particles occur because the wave properties of the particles have similar wavelengths.  The waves can interfere, and therefore interact.  With wave properties of vastly different sizes, there is no interference, and therefore no interaction.

I am not optimistic about the present round of experiments looking for dark matter, as described by Dr. Randall.  If you are looking for an elephant with an electron microscope, you are likely to be unsuccessful. 

What kind of experiments could reveal particles which exist at a scale many orders of magnitude larger than ordinary matter?  I don’t know.  The electromagnetic spectrum is well-explored on that scale, and reveals nothing.  Perhaps other forces need to be synthesized, and examined at larger scales.  There may be practical technological benefits if instruments can be developed that directly detect dark matter.  Perhaps, such instruments could provide the ability to manipulate other forces, such as a way to generate, shape and manipulate artificial gravitational fields, in the way that artificial magnetic fields have been generated and used for almost 200 years.

That would be a real advance for mankind.

References

Lisa Randall, Dark Matter and the Dinosaurs; The Astonishing Interconnectedness of the Universe, 2015, 432p.

http://jersey.uoregon.edu/~mstrick/AskGeoMan/geoQuerry57.html

Density of the Outer Core (liquid):  9.9 to 12.2 gm/cc

Density of the Inner Core (solid):  12.6 to 13.0 gm/cc

http://hyperphysics.phy-astr.gsu.edu/hbase/Geophys/earthstruct.html

	Thickness (km)	Density (g/cm3)		Types of rock found
		Top	Bottom
Crust	30	2.2		Silicic rocks
			2.9	Andesite, basalt at base
Upper mantle	720	3.4		Peridotite, eclogite, olivine, spinel, garnet, pyroxene
			4.4	Perovskite, oxides
Lower mantle	2,171	4.4		Magnesium and silicon oxides
			5.6
Outer core	2,259	9.9		Iron + oxygen, sulfur, nickel alloy
			12.2
Inner core	1,221	12.8		Iron + oxygen, sulfur, nickel alloy
			13.1

https://en.wikipedia.org/wiki/Structure_of_the_Earth

http://www.spring8.or.jp/pdf/en/res_fro/11/106-107.pdf

http://slideplayer.com/slide/6951274/

https://www.youtube.com/watch?v=mRtGUCLjQ3w

http://www.sciencedirect.com/science/article/pii/S1068797116000043

Composition of the earth’s core.

https://en.wikipedia.org/wiki/Inner_core

The core is about 3% lighter than pure iron, implying about 10% of lighter constituents, probably silicon, oxygen, and sulfur.

https://www.jpl.nasa.gov/voyager/mission/status/

Voyager 1: 12.9 billion miles (19 light-hours) away. 

Voyager 2: 10.7 billion miles (16 light-hours) away.

https://www.nasa.gov/centers/ames/missions/archive/pioneer.html

The last data received from Pioneer 10 was in 2002.  The last data received from Pioneer 11 was in 1995.

http://www.quantumdiaries.org/2010/07/12/the-size-of-the-proton/

[The new] measurement measured [a proton] to be 0.8418±0.0007 fm.  A femtometer is 10^-15 meters