Tidal disruption of fuzzy dark matter subhalo cores

We study tidal stripping of fuzzy dark matter (FDM) subhalo cores using simulations of the Schrödinger-Poisson equations and analyze the dynamics of tidal disruption, highlighting the differences with standard cold dark matter. Mass loss outside of the tidal radius forces the core to relax into a less compact configuration, lowering the tidal radius. As the characteristic radius of a solitonic core scales inversely with its mass, tidal stripping results in a runaway effect and rapid tidal disruption of the core once its central density drops below 4.5 times the average density of the host within the orbital radius. Additionally, we find that the core is deformed into a tidally locked ellipsoid with increasing eccentricities until it is completely disrupted. Using the core mass loss rate, we compute the minimum mass of cores that can survive several orbits for different FDM particle masses and compare it with observed masses of satellite galaxies in the Milky Way.

Figure 1

Evolution of the core’s central density with unevolved (dots) and evolved (crosses) gravitational potential compared to the prediction in [52].

Figure 2

Regions inside (blank region) and outside (shaded region) the tidal radius with respect to the density ratio $\mu \equiv {\rho }_c/{\overline{\rho }}_{\mathrm{host}}$. The solid line shows the tidal radius computed from Eq. (3). The horizontal lines mark the radii enclosing 95% (top), 50% (middle) and 25% (bottom) of the total soliton mass, respectively. When $\mu <4.5$, the tidal radius is smaller than the core radius. Thus, the solitonic core becomes unstable and is quickly disrupted.

Figure 3

Mass loss rate for different initial density ratios. A spherical tidal field is assumed. The solid black line shows the prediction in [52] using the fitting formula Eq. (7). The vertical dashed lines are the same as in Fig. 2. The solid lines with different colors give the total mass loss rate. The dashed colored lines show the mass loss rate by mass transfer through the tidal radius, while the shaded region corresponds to the mass loss rate from the decreasing tidal radius.

Figure 4

The density profile of the core at the initial time and at $t=4T_{\text{orbit}}$ for a density ratio $\mu =50$. The circles and squares show the average radial density profile obtained from the simulation. The lines display fitted profiles defined in Eq. (8). It can be seen that the cores are well described by soliton profiles even after losing substantial amounts of mass.

Figure 5

Slices through the density field at different times for an initial density ratio $\mu =50$. The circle in the center of each plot indicates the size of the host (for simplicity the host is treated as a small sphere with uniform density). The thick and thin contour lines mark where the density drops to 50% (core radius) and 1% of the maximum density, respectively. The dashed circles show the tidal radii computed from the spherically symmetric approximation.

Figure 6

Core mass loss rate for different initial density ratios. The lines show the prediction from Eqs. (7) and (17) with $\gamma =3/2$ (solid line) and $\gamma =1$ (dashed line). The vertical dashed lines are the same as in Fig. 2.

Figure 7

Slice through the core. The color map indicates the density while the black arrows trace the velocity field relative to the core’s collective motion. Inside the tidal radius (dashed circle) the velocity field is characteristic for an irrotational Riemann-S ellipsoid [55]. Outside the core, vortices can be seen. The thick black circle in the middle represents the core radius while the thin black ellipsoid marks the area where the density drops to one percent of the central density, i.e. almost all the mass lies within the tidal radius. The hollow arrow points towards the host’s center.

Figure 8

Representative run with initial density ratio $\mu =50$. Top: angle between the longest principal axis of the core and $x$-axis (grey circles). The dashed line shows the angle between the line joining the center of the core and the center of host and $x$-axis. It can be seen that the core is tidally locked most of the time. Center: eccentricity ${\epsilon }_{13}$ of the core. The solid line shows the expected values from Eq. (18). Bottom: spin parameter of the core.

Figure 9

Minimum mass of cores that can survive for $N_{\mathrm{sur}}$ orbits assuming different FDM particle masses $m_{22}\equiv m/({10}^{-22}\mathrm{eV})$ versus the distance to the Galactic center $D$. For comparison, we also show the half-light mass $M_{1/2}$ of some satellite galaxies in the Milky Way [63]. The mass of the host is taken to be $10^{12}{M}_{\odot }$. For each particle mass, the solid curve is obtained by assuming $\gamma =1$ and $N_{\mathrm{sur}}=1$ while the dashed curve is obtained by assuming $\gamma =3/2$ and $N_{\mathrm{sur}}=10$.

Figure 10

Subhalo mass function for $m_{22}=10$ with (dashed line) and without (solid line) including the tidal stripping of subhalo cores.

Figure 11

Numerical error of the total energy with respect to time for different time step sizes and different algorithms. The time is in units of the free-fall time scale $t_{\mathrm{ff}}=\sqrt{3\pi /(32G\rho )}$ with $\rho$ equal to the average density over the whole simulated box. Here “O4” refers to the fourth-order algorithm we used in our simulations (Sec. 3). “O2” refers to the second-order kick-drift-kick formulation Eq. (12), which is widely used in previous simulations.

Figure 12

Average numerical error of the total energy with respect to the time step size. Only data with $t>3t_{\mathrm{ff}}$ when the numerical error oscillates around roughly a constant value is included in the analysis.