Galactic dark matter and Bertrand space-times

Bertrand space-times (BSTs) are static, spherically symmetric solutions of Einstein’s equations, that admit stable, closed orbits. Starting from the fact that to a good approximation, stars in the disc or halo regions of typical galaxies move in such orbits, we propose that, under certain physical assumptions, the dark matter distribution of some low surface brightness galaxies can seed a particular class of BSTs. In the Newtonian limit, it is shown that for flat rotation curves, our proposal leads to an analytic prediction of the Navarro-Frenk-White dark matter profile of Navarro et al. [Astrophys. J. 462, 563 (1996)]. We further show that the dark matter distribution that seeds the BST is described by a two-fluid anisotropic model, and present its analytic solution. A new solution of the Einstein’s equations, with an internal BST and an external Schwarzschild metric, is also constructed.

Figure 1

Circular velocity ($\mathrm{km}/\sec$) vs radial distance $r$ (kpc) as predicted from Eq. (4). The dashed red, solid blue, and dotted-dashed black curves correspond to $D=3\times {10}^5$, $1.5\times {10}^5$ and ${10}^5$, respectively. For all the curves, $\alpha =4.5\times {10}^6\mathrm{kpc}$.

Figure 2

Circular velocity of Galaxy NGC4395 (blue dots) compared with the prediction of Eq. (4) (red line) with $\alpha /D=1545.45\mathrm{pc}$.

Figure 3

Circular velocity of Galaxy UGC1281 (blue dots) compared with the prediction of Eq. (4) (red line) with $\alpha /D=1785.71\mathrm{pc}$.

Figure 4

Admissible ranges of ${\gamma }_1$ and ${\gamma }_2$, from positivity of ${\rho }_1$ and ${\rho }_2$.

Figure 5

Admissible range for $r$ and ${\gamma }_2$, from reality of $u_{\mu }{v}^{\mu }$, with ${\gamma }_1=0.9$.

Figure 6

Plots of ${\rho }_1$ (solid blue) and ${\rho }_2$ (dashed red) vs $r$ for ${\gamma }_1=0.9$, ${\gamma }_2=-0.4$. See text for more details.

Figure 7

$K$ vs $r$ for ${\gamma }_1=0.9$, ${\gamma }_2=-0.4$ (solid blue) and ${\gamma }_1=0.9$, ${\gamma }_2=-0.6$ (dashed red). See details in text.

Figure 8

Admissible range for $\beta$, for ${\gamma }_1=0.9$, ${\gamma }_2=-0.4$. See text for more details.

Figure 9

Admissible range for $\beta$, for ${\gamma }_1=0.9$, ${\gamma }_2=-0.6$. See text for more details.