Separable triaxial potential-density pairs in modified Newtonian dynamics

Luca Ciotti*, Hongsheng Zhao, P. Tim de Zeeuw

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


We study mass models that correspond to modified Newtonian dynamics (MOND) (triaxial) potentials for which the Hamilton-Jacobi equation separates in ellipsoidal coordinates. The problem is first discussed in the simpler case of deep-MOND systems, and then generalized to the full MOND regime. We prove that the Kuzmin property for Newtonian gravity still holds, i.e. that the density distribution of separable potentials is fully determined once the density profile along the minor axis is assigned. At variance with the Newtonian case, the fact that a positive density along the minor axis leads to a positive density everywhere remains unproven. We also prove that (i) all regular separable models in MOND have a vanishing density at the origin, so that they would correspond to centrally dark-matter-dominated systems in Newtonian gravity; (ii) triaxial separable potentials regular at large radii and associated with finite total mass leads to density distributions that at large radii are not spherical and decline as ln(r)/r 5; (iii) when the triaxial potentials admit a genuine Frobenius expansion with exponent 0 < ε < 1, the density distributions become spherical at large radii, with the profile ln(r)/r 3 + 2ε. After presenting a suite of positive density distributions associated with MOND separable potentials, we also consider the important family of (non-separable) triaxial potentials V 1 introduced by de Zeeuw & Pfenniger, and we show that, as already known for Newtonian gravity, they obey the Kuzmin property also in MOND. The ordinary differential equation relating their potential and density along the z-axis is an Abel equation of the second kind that, in the oblate case, can be explicitly reduced to canonical form.

Original languageEnglish
Pages (from-to)2058-2071
Number of pages14
JournalMonthly Notices of the Royal Astronomical Society
Issue number3
Publication statusPublished - 1 May 2012


  • Dark matter
  • Galaxies: kinematics and dynamics
  • Galaxies: structure
  • Methods: analytical


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