Exact Expressions for the Pericenter Precession Caused by Some Dark Matter Distributions and Constraints on Them from Orbital Motions in the Solar System, in the Double Pulsar and in the Galactic Center

We analytically calculate the secular precession of the pericenter of a test particle orbiting a central body surrounded by a continuous distribution of Dark Matter (DM) by using some commonly adopted spherically symmetric density profiles for it. We obtain exact expressions without resorting to a-priori simplifying assumptions on the orbital geometry of the test particle. Our formulas allow us to put constraints on the parameters of the DM distributions considered in several local astronomical and astrophysical scenarios, such as the Sun\u27s planetary system, the double pulsar, and the stellar system around the supermassive black hole in Sgr A∗, all characterized by a wide variety of orbital configuratio ns. As far as our Solar System is concerned, latest determinations of the supplementary perihelion precessions ̟˙ with the EPM2011 ephemerides and the common power-law DM density profile ρDM(r) = ρ0r−γ λγ yield 5 × 103 GeV cm−3 (γ = 0) ≤ ρ0 ≤ 8 × 103 GeV cm−3 (γ = 4), corresponding to 8.9 × 10−21 g cm−3 ≤ ρ0 ≤ 1.4 × 10−20 g cm−3, at the Saturn\u27s distance. From the periastron of the pulsar PSR J0737-3039A and the same power-low DM density, one has 1.7 × 1016 GeV cm−3 (γ = 0) ≤ ρ0 ≤ 2 × 1016 (γ = 4) GeV cm−3, corresponding to 3.0 × 10−8 g cm−3 ≤ ρ0 ≤ 3.6 × 10−8 g cm−3. The perinigricon of the S0-2 star in Sgr A∗ and the power-law DM model give 1.2 × 1013 GeV cm−3 (γ = 0) ≤ ρ0 ≤ 1 × 1016 (γ = 4, λ = rmin) GeV cm−3, corresponding to 2.1 × 10−11 g cm−3 ≤ ρ0 ≤ 1.8 × 10−8 g cm−3