Scaling Relations for Dark Matter Annihilation and Decay Profiles in Dwarf Spheroidal Galaxies

Measuring the dark matter distribution in dwarf spheroidal galaxies (dSphs) from stellar kinematics is crucial for indirect dark matter searches, as these distributions set the fluxes for both dark matter annihilation (J-Factor) and decay (D-Factor). Here we produce a compilation of J and D-Factors for dSphs, including new calculations for several newly-discovered Milky Way (MW) satellites, for dSphs outside of the MW virial radius, and for M31 satellites. From this compilation we test for scaling relations between the J and D-factors and physical properties of the dSphs such as the velocity dispersion ($\sigma_{\mathrm{los}}$), the distance ($d$), and the stellar half-light radius ($r_{1/2}$). We find that the following scaling relation minimizes the residuals as compared to different functional dependencies on the observed dSphs properties $J(0.5 {\rm deg}) = 10^{17.72} \left(\sigma_{\mathrm{los}}/5\,{\rm km \, s^{-1}}\right)^4 \left(d / 100\,{\rm kpc}\right)^{-2}\left( r_{1/2}/100 \,{\rm pc} \right)^{-1}$. We find this relation has considerably smaller scatter as compared to the simpler relations that scale only as $1/d^2$. We further explore scalings with luminosity ($L_V$), and find that the data do not strongly prefer a scaling including $L_V$ as compared to a pure $1/d^2$ scaling. The scaling relations we derive can be used to estimate the J-Factor without the full dynamical analysis, and will be useful for estimating limits on particle dark matter properties from new systems that do not have high-quality stellar kinematics.

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