Fuzzy-dark-matter dynamical friction¶

As an example of using the FDMDynamicalFrictionForce detailed below, consider a dwarf galaxy:

import numpy
from astropy import units as u
from galpy.potential import NFWPotential
M = 1e9  # Msol
rs = 1.57  # kpc
c = 18.4
A_NFW = numpy.log(1 + c) - c / (1 + c)
NFWHalo = NFWPotential(amp=M / A_NFW * u.Msun, a=rs * u.kpc)

and a globular cluster within it:

Mobj = 1e6  # Msol
rhm = 1e-2  # kpc

Regular Chandrasekhar dynamical friction force can be computed using:

from galpy.potential import ChandrasekharDynamicalFrictionForce
cdf = ChandrasekharDynamicalFrictionForce(GMs=Mobj * u.Msun, rhm=rhm * u.kpc, dens=NFWHalo)

The FDM dynamical friction force for a mass of \(3\times 10^{-22}\,\mathrm{eV}\) can be computed using:

from galpy.potential import FDMDynamicalFrictionForce
m = 3e-22  # mass of the fuzzy particle in eV
fdf = FDMDynamicalFrictionForce(GMs=Mobj * u.Msun, rhm=rhm * u.kpc, dens=NFWHalo, m=m * u.eV)

We then set up an orbit and integrate it with both the CDM Chandrasekhar friction force and the FDM friction force as well as with no friction at all:

from galpy.orbit import Orbit
# initial conditions
R = 0.5
z = 0.0
phi = 0.0
Menclosed = NFWHalo.mass(R * u.kpc)
vR = 0.0
vT = NFWHalo.vcirc(R * u.kpc, use_physical=True, quantity= False)
vz = 0.0
o = Orbit([R * u.kpc,vR * u.km / u.s,vT * u.km / u.s,z * u.kpc,vz * u.km / u.s,phi * u.rad])
o_cdf = o()  # copy of orbit for dynamical friction
o_fdf = o()  # copy of orbit for fuzzy dynamical friction
# Integrate the orbit
t = numpy.linspace(0, 1, 300) * u.Gyr
o.integrate(t, NFWHalo)
o_cdf.integrate(t, NFWHalo + cdf)
o_fdf.integrate(t, NFWHalo + fdf)

The results can be plotted using:

import matplotlib.pyplot as plt
plt.figure(figsize=(10, 4))
plt.subplot(121)
plt.plot(o.x(t), o.y(t), label="No DF")
plt.plot(o_cdf.x(t), o_cdf.y(t), label="CDM DF")
plt.plot(o_fdf.x(t), o_fdf.y(t), label="Fuzzy DF")
plt.xlabel("x [kpc]", fontsize=14, fontweight="bold")
plt.ylabel("y [kpc]", fontsize=14, fontweight="bold")
plt.gca().set_facecolor("whitesmoke")
plt.grid(True, which="both", linestyle="--", linewidth=0.7, alpha=0.7)
plt.legend(fontsize=12)

plt.subplot(122)
plt.plot(t, o.R(t), label="No DF")
plt.plot(t, o_cdf.R(t), label="CDM DF")
plt.plot(t, o_fdf.R(t), label="Fuzzy DF")
plt.xlabel("t [Gyr]", fontsize=14, fontweight="bold")
plt.ylabel("R [kpc]", fontsize=14, fontweight="bold")
plt.legend(fontsize=12)
plt.tight_layout(pad=2.5)
plt.subplots_adjust(left=0.1, right=0.9, top=0.9, bottom=0.1, wspace=0.3)
plt.gca().set_facecolor("whitesmoke")
plt.grid(True, which="both", linestyle="--", linewidth=0.7, alpha=0.7)

plt.suptitle("Dynamical Friction in NFW halo", fontsize=16, fontweight="bold")
plt.show()

which gives

../_images/FDMDynamicalFrictionForce_example.png

class galpy.potential.FDMDynamicalFrictionForce(amp=1.0, GMs=0.1, gamma=1.0, rhm=0.0, m=1e-99, dens=None, sigmar=None, const_lnLambda=False, const_FDMfactor=False, minr=0.0001, maxr=25.0, nr=501, ro=None, vo=None)[source]¶

Implements the fuzzy dark matter (FDM) dynamical friction force.

The force is given by:

\[\vec{F}_\mathrm{FDM} = -\frac{4\pi\mathcal{G}^2 M_\mathrm{obj}^2 \rho}{v^3} C_\mathrm{FDM}(kr) \vec{v}\]

where the coefficient \(C_\mathrm{FDM}(kr)\) depends on \(kr = \frac{m v r}{\hbar}\). There are three regimes for the coefficient, depending on the value of \(kr\):

Zero-velocity regime: for \(kr < M_\sigma / 2\), where \(M_\sigma = v / \sigma(r)\) is the classical Mach number, the coefficient is given by

\[C_\mathrm{FDM}(kr) = \mathrm{Cin}(2kr) + \frac{\sin(2kr)}{2kr} - 1\]

with

\[\mathrm{Cin}(z) = \int_0^z \frac{1 - \cos(t)}{t} \, \mathrm{d}t\]
Dispersion regime: for \(kr > 2 M_\sigma\), the coefficient is given by

\[C_\mathrm{FDM}(kr) = \ln\left(\frac{2kr}{M_\sigma}\right) \left[ \mathrm{erf}(X) - \frac{2X}{\sqrt{\pi}} \exp(-X^2) \right]\]

where \(X = \frac{v}{\sqrt{2} \sigma(r)}\).
Intermediate regime: for \(M_\sigma / 2 < kr < 2 M_\sigma\), the coefficient is given by linear interpolation between the zero-velocity and dispersion regimes.

If the FDM coefficient \(C_\mathrm{FDM}(kr)\) is larger than the classical coefficient \(C_\mathrm{CDM}\), we use the classical coefficient as a cutoff, because it means that we are in the classical regime. The classical coefficient is given by:

\[C_\mathrm{CDM} = {\ln[ 1+\Lambda^2]\over 2} \left[ \mathrm{erf}(X) - \frac{2X}{\sqrt{\pi}} \exp(-X^2) \right]\]

Fuzzy-dark-matter dynamical friction¶

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