> Can the Sun / Earth have a dark matter core?
Perhaps a more interesting question is: What is the approximate distribution of Dark Matter within the Sun (I haven't researched the Earth portion of your question yet).
> If dark matter interacts with ordinary matter at all, it should most likely occur where ordinary matter is densest.
That does not appear to be a certainty in the case of the Sun, the center is so dense that cosmic rays are restricted by the core; altering or obscuring the measurement of Dark Matter.
> ... Sun, or even the Earth can have dark matter cores too - they're just less likely to have such cores. If the Sun / Earth does have such a core dark matter would be much easier to study since they're so nearby.
The Sun is studied using instruments pointing into space, using neutrino telescopes. Since these instruments point up rather than down (at the Earth), and additional space-borne instruments are available, it's possible that studying the Sun would be easier than studying the Earth (with it's weaker gravity) - but I'm not far enough into researching your answer to say for certain.
> Is there any observational evidence that the Sun / Earth is 100% ordinary matter? If no, what is the observational limit of the Sun / Earth's dark matter fraction?
No, see below.
> I've seen popular-level articles (example) about such theories, but they're all rather speculative.
It's not entirely speculative, it seems rather creditable to me.
> If dark matter interacts with ordinary matter at all, it should most likely occur where ordinary matter is densest. Hence we have papers about neutron stars possibly containing dark matter cores (example).
For each analysis there are two kinds of background, the “reducible” backgrounds where particles fake the particles we are looking for (for example, a high energy electron can look just like a high energy photon) and the “irreducible” backgrounds where particles are the same kind as the ones we are looking for.
In "[Neutrino telescope searches for dark matter in the Sun][1]" (14 Oct 2017), by Dr. Pat Scott, he writes on page 1:
> "**Current status**
> High-energy neutrinos from the Sun provide one of the cleanest potential discovery channels for weakly-interacting dark matter (DM). Weakly-interacting DM particles passing through the Sun are expected to scatter on solar nuclei. Some of these collisions reduce the kinetic energy of the DM particle enough for it to become gravitationally bound to the Sun, causing it to return on a bound orbit and undergo subsequent scattering, eventually thermalising and settling down to the solar core. If DM is able to annihilate, either with itself of with anti-DM captured in a similar manner, high-energy SM particles will be produced in the solar core. Even if neutrinos are not amongst those particles produced in the annihilation hard process, they will still be generated with quite high energies in the decay and subsequent interaction of other SM particles with nuclei in the Sun. Unlike the other SM particles, these GeV-scale neutrinos are then able to travel unhindered from the centre of the Sun to the surface, and across space to Earth, where they may be detected with terrestrial experiments.
> The directionality of the signal is the primary means by which it can be distinguished from the atmospheric neutrino background, caused by cosmic ray interactions with the Earth’s atmosphere. The only known background to the signal is therefore the analogous production of high-energy neutrinos in the atmosphere of the Sun, due to interactions of cosmic rays with solar nuclei.
> The capture of dark matter by the Sun typically becomes the rate-limiting step in the production of any signal, rather than the annihilation. Searches for high energy neutrinos from the Sun are therefore most useful for constraining the interaction cross section of dark matter with nuclei. Spin-dependent interactions are particularly relevant, as the Sun consists mostly of hydrogen, which possesses nuclear spin.".
> ...
> Page 3:
> "**Improved background calculations**
> Previous predictions of the background rate of high-energy neutrinos from the Sun, due to interactions of cosmic rays with nuclei in the solar atmosphere, were computed more than a decade ago.$^{6,7,8}$ However, two more recent recalculations have appeared.$^{9,10}$ Compared to the older predictions, the new calculations make use of modern knowledge on neutrino oscillations, production and interaction cross-sections. One of these$^{10}$ also makes use of up-to-date models of the solar composition and structure, and carries out extensive Monte Carlo simulations of neutrino production, interaction and oscillation. Both studies (and another at the same time, based on the old flux estimates$^{11}$) show that the solar atmospheric background lies barely an order of magnitude below current sensitivity limits for some models (Fig. 2). This suggests that future neutrino telescopes might be able to directly measure this irreducible ‘neutrino floor’, and that the improved calculations of the background rates should be included in future phenomenological studies of DM scattering and annihilation in the Sun.".
> 9. C. A. Argüelles, G. de Wasseige, A. Fedynitch, and B. J. P. Jones, Solar atmospheric neutrinos and the sensitivity floor for solar dark matter annihilation searches, JCAP 7 (2017) 024, [arXiv:1703.07798].
> 10. J. Edsjö, J. Elevant, R. Enberg, and C. Niblaeus, Neutrinos from cosmic ray interactions in the Sun, JCAP 6 (2017) 033, [arXiv:1704.02892].
In "[Neutrinos from cosmic ray interactions in the Sun][2]" (10 Jul 2017), by Edsjö, Elevant, Enberg, and Niblaeus, on page 23, they write and include this image:
> "Finally, for illustration we show in figure 12 how the Sun wold look like if we could see the neutrino-induced muons from solar cosmic ray interactions. Even if the neutrino-nucleon scattering angle smears the neutrino fluxes somewhat, we still see the ring like signature of this signal with a slight dip in the centre.".
[![Figure 12. The Sun as it could be seen in neutrino-induced muons at a neutrino telescope_Tile_Org+6-color][3]][3]
The lower image is a posterized slice of the original image, to better show the shells.
> "Figure 12. The Sun as it could be seen in neutrino-induced muons at a neutrino telescope. Note the dip in the centre.".
As you can see, the density in the center is slightly less, that is explained on page 15:
> "If we compare the he neutrino fluxes at production (solid lines) to the ones after passage through the Sun (dashed lines), we see that we get a dip at low impact parameters. This is the effect of the attenuation that happens due to interactions when the neutrinos pass through the Sun, as we saw already in figure 4. As the density of the Sun is significantly higher in the centre, the effect is very strongly pronounced for low impact parameters. We can also see that the effect of attenuation is higher for higher energies as expected.
> We can also see how the production fluxes depend on the impact parameter. We can see that for higher energies these are quite peaked at large impact parameters, which is expected as the density where the cascade happens is lower for these Sun grazing CRs, and hence the fluxes are higher. We also get a small contribution from muons decaying outside of the Sun at high impact parameters and high energies.3 The total flux from the Sun is obtained by integrating over the impact parameters including the fact that the solid angle is larger for large impact parameters. Hence, the high impact parameter part of these figures will be most important for the total flux from the Sun.".
While that paper also discusses the Earth I have not prepared that portion of the answer to your question. As usual more information is available in the quoted papers.
The paper "Neutrinos from cosmic ray interactions in the Sun" is also recommended on page 70 of "[The IceCube Neutrino Observatory Contributions to ICRC 2017 Part III: Cosmic Rays][4]" (over 300 authors), in the article "Solar atmospheric neutrino search with IceCube", by Seongjin In and Carsten Rott:
> "Solar atmospheric neutrinos provide a natural back-
ground to solar dark matter searches and limit their sensitivity as recently pointed out $^{[5, 6, 7]}$.".
> 7. J.Edsjö, J.Elevant, R.Enberg and C.Niblaeus, [astro-ph/1704.02892v1].
In "[The distribution of inelastic dark matter in the Sun][5]" (19 Feb 2018), by Blennow, Clementz, and Herrero-Garcia, on page 17 they write:
> "We now use Eq. (36) to translate the distribution in E-L space into a radial distribution. The results for elastic scattering are shown in Fig. 5 for the times $t = 10^{−10} t\odot$ (left), $t = 10^{−8} t\odot$ (middle) and $t = 10^{-6} t\odot$ (right). The distribution is compared to the isothermal one of Eq. (33), with the angular degrees of freedom integrated over. One can see that the distribution has essentially reached equilibrium already at $t = 10^{−8} t\odot$, changing only slightly at $t = 10^{-6} t\odot$. The Boltzmann distribution gives a fairly accurate description of the distribution, although the numerically computed one is slightly shifted towards larger radii, and its peak is not as pronounced.
> Moving on to the case of inelastic DM, we will again focus our discussion on the illustrative case of m$_χ$ = 100 GeV and δ = 100 keV.".
[![Figure 5.][6]][6]
On page 20 they write:
> "Finally, we obtain the radial distributions for inelastic scattering. The results are shown in Fig. 7. At very early times, the distribution extends up to large radii. At $t = 10^{−9} t\odot$, a large concentration starts to form, shown below $r/R\odot \simeq 0.3$, that very slowly moves towards lower radii, forming a distribution centred at $r/R\odot \simeq 0.1$ at $t = 10−5 t\odot$. However, even at $t = t\odot$, the distribution has not reached a stationary state. Another important observation is that the Boltzmann distribution is now a very poor description of the final distribution. This is entirely due to particles being trapped with no possibility of scattering further, in particular those in the region at low $E$ which are the ones that contribute to $f_{num}(r)$ at smaller radii. Since the number of $χ^∗$ particles is completely negligible, the total DM distribution is identical to the $χ$ distribution.
[![Figure 7.][7]][7]
[1]: https://arxiv.org/abs/1710.05190
[2]: https://arxiv.org/abs/1704.02892
[3]: https://i.sstatic.net/eESbR.png
[4]: https://arxiv.org/abs/1710.01194
[5]: https://arxiv.org/abs/1802.06880
[6]: https://i.sstatic.net/g0JM1.jpg
[7]: https://i.sstatic.net/ouHQv.jpg