Title: Can a Dark Inferno Melt Earth’s Core?
Authors: Christopher Cappiello and Tansu Daylan
First author institution: Department of Physics and McDonnell Center for the Space Sciences, Washington University, USA
Status: Available on ArXiv
Dark matter – it’s a mysterious substance that we expect probably exists, but we can’t really see its effects (except through gravity) since it doesn’t seem to interact via electromagnetism. The problem of finding dark matter has been discussed more in various Astrobites – see for example here, here and here. We think it is the reason that galaxies spin faster than they should be able to (given their visible mass) without flying apart, and its presence would also significantly impact the growth of massive structures, stars and galaxies in our Universe. However, we have yet to find concrete evidence of a dark matter candidate through either laboratory experiments that try to detect dark matter interactions or astrophysical probes which try to detect the products of dark matter interactions.
In today’s bite, we look at an approach to set limits on the possible characteristics of dark matter candidates (in terms of the possible mass or interaction strength) by understanding how its presence affects dense structures. Hypothetically, if dark matter is able to decay or annihilate into Standard Model particles, via some kind of interaction or collision with another particle, it is much more likely to do so in a region of dense matter. As such astrophysicists look for signatures in the galactic centre, in dwarf galaxies, or even in the centres of stars and planets. The authors of today’s paper look at the impact that dark matter might have if it accumulated in the Earth’s core If the mass of a dark matter particle is comparable to that of a target nucleus in a collision, it can become gravitationally bound, and then start decaying or annihilating. This process is expected to influence the temperature of the Earth, particularly if the dark matter decays to photons or charged particles, which could stay trapped and heat the planet.
The authors note in previous research on dark matter accumulation in a planet, the flow of heat and the effect on the planetary surface has been studied. In general, it is assumed the system is in thermal equilibrium – the heat carried out through the planet is equal to the energy captured from interactions via dark matter particles (the energy injection rate). However, the impact of the heat flow on the interior of the planet has been neglected. In this work, they especially try to understand how the temperature is impacted at different radii within the core of the Earth.
Heat flow through the Earth from a Dark Inferno
Firstly, the authors study how the heat from the dark matter interactions would flow through the core of the Earth over time, by calculating a radial temperature profile of the Earth and subsequently the flow of heat through the radius at \(r = 1000\) km. This is because initially, the dark matter density distribution (with density decreasing with radius) confines most of the interactions to more central regions of the Earth’s core, so it takes some time for the heat to flow out from the centre, given the interactions could only have started when the Earth first formed. Depending on the dark matter properties, they find in some cases it takes about a billion years for the maximum amount of heat to flow through this radius. In other planets with varying interiors, the heat conduction may be much more efficient. Figure 1 below shows the power output, sourced by dark matter, at this radius as a function of time, for a dark matter particle of mass = 1 TeV and various cross-sections (which basically tells you the chance of dark matter interacting with a particle and directly influences the energy injection rate).
Seismic analyses allow us to probe somewhat into the core of the Earth, and we know down to a depth of ~400 km from the centre, that the core is not totally melted; there are regions of heterogeneity that suggest otherwise. So at most, we can rule out any dark matter candidates that create sufficient heat to melt anything beyond the innermost 400 km radius of the Earth’s core within the length of time that Earth has existed (~4.5 billion years). To calculate how much different dark matter candidates (in terms of mass and cross-section) might melt the Earth, the authors need to make a few assumptions; they assume a rough value for the thermal conductivity of the material (iron alloys) in the centre under very high pressure and temperatures, and they also roughly assume the temperature at which these materials would melt.
They compute the temperature at \(r = 400\) km as a function of time, which can be seen below in Figure 2, and use this to determine if the core at this radius would be fully melted. For a particle of m = 1 TeV and a cross-section that results in an energy injection rate of 5 TW (terrawatts), they find within about a billion years the core would be melted at \(r = 400\) km; given this melts this part of the core so quickly, this candidate can be ruled out.
Finally, Figure 3 below shows a range of dark matter masses and cross-sections that can be ruled out based on the constraint that the core should not have melted beyond 400 km, at least within a time less than the age of the Earth. For very small masses, the constraints weaken because the heat output from the dark matter does not stay confined to the inner part of the planet (for lighter dark matter candidates the characteristic scale of the dark matter density distribution in the core is larger), and the radii at the centre don’t heat up as much. For very large masses, the constraints also weaken because the interaction cross-section needs to be much larger in order for dark matter candidates to become gravitationally bound.
Summary
This work allows for more sensitive constraints on dark matter candidates to be derived by considering the heat flow through Earth’s core, and a similar procedure can be applied to different planets to derive new limits. Potentially, it may be even possible to start deriving interesting new limits from exoplanet populations, which may help us understand the nature of dark matter.
Astrobite edited by: Alexandra Masegian
Image credit: Adapted from figure by Kelvinsong, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons
