Phenomenological constraints on accretion of non-annihilating dark matter on the PSR B1257+12 pulsar from orbital dynamics of its planets

Abstract

We analytically compute the effects that a pulsar's mass variation, whatever its physical origin may be, has on the standard Keplerian changes ΔτKep in the times of arrival of its pulses due to potential test particle companions, and on their orbital dynamics over long time scales. We apply our results to the planetary system of the PSR B1257+12 pulsar, located in the Galaxy at ∼ 600 pc from us, to phenomenologically constrain a putative accretion of non-annihilating dark matter on the hosting neutron star. By comparing our prediction for ΔτṀ/M to the root-mean-square accuracy of the timing residuals δ(Δτ) = 3.0μs we find for the mass variation rate Ṁ/M ⩽ 1.3 × 10−6 yr−1. Actually, considerations related to the pulsar's lifetime, of the order of Δt ∼ 0.8 Gyr, and to the currently accepted picture of the formation of its planets point toward a tighter constrain on the mass accretion rate, i.e. Ṁ/M ⩽ 10−9 yr−1. Otherwise, the planets would have formed at about 300–700 au from PSR B1257+12, i.e. too far with respect to the expected extension of 1–2 au of the part of the protoplanetary disk containing the solid constituents from which they likely originated. In fact, an even smaller upper limit, Ṁ/M ⩽ 10−11 yr−1, would likely be more realistic to avoid certain technical inconsistencies with the quality of the fit of the timing data, performed by keeping the standard value M = 1.4M⊙ fixed for the neutron star's mass. Anyway, the entire pulsar data set should be re-processed by explicitly modeling the mass variation rate and solving for it. Model-dependent theoretical predictions for the pulsar's mass accretion, in the framework of the mirror matter scenario, yield a mass increment rate of about 10−16 yr−1 for a value of the density of mirror matter ρdm as large as 10−17 g cm−3 = 5.6 × 106 GeV cm−3. Such a rate corresponds to a fractional mass variation of ΔM/M ∼ 10−7 over the pulsar's lifetime. It would imply a formation of a black hole from the accreted dark matter inner core for values of the dark matter particle's mass mdm larger than 3 × 103 Gev, which are, thus, excluded since PSR B1257+12 is actually not such a kind of compact object. Instead, by assuming ρdm ∼ 10−24 g cm−3 = 0.56 GeV cm−3, the mass accretion rate would be Ṁ/M ∼ 10−23 yr−1, with a fractional mass variation of the order of ΔM/M ∼ 10−14. It rules out mdm ⩾ 8 × 106 Gev. Extreme values ρdm = 1.8 × 10−13 g cm−3 = 1011 GeV cm−3 for non-annihilating dark matter in central spike may yield the constraint Ṁ/M ⩽ 10−11 yr−1; over Δt = 0.8 Gyr, it rules out mdm ⩾ 12 Gev.