Disk potential using SCF basis-function-expansion¶
- class galpy.potential.DiskSCFPotential(amp=1.0, normalize=False, dens=<function DiskSCFPotential.<lambda>>, Sigma={'amp': 1.0, 'h': 0.3333333333333333, 'type': 'exp'}, hz={'h': 0.037037037037037035, 'type': 'exp'}, Sigma_amp=None, dSigmadR=None, d2SigmadR2=None, Hz=None, dHzdz=None, N=10, L=10, a=1.0, radial_order=None, costheta_order=None, phi_order=None, ro=None, vo=None)[source]¶
Class that implements a basis-function-expansion technique for solving the Poisson equation for disk (+halo) systems. We solve the Poisson equation for a given density \(\rho(R,\phi,z)\) by introducing K helper function pairs \([\Sigma_i(R),h_i(z)]\), with \(h_i(z) = \mathrm{d}^2 H(z) / \mathrm{d} z^2\) and search for solutions of the form
\[\Phi(R,\phi,z = \Phi_{\mathrm{ME}}(R,\phi,z) + 4\pi G\sum_i \Sigma_i(r)\,H_i(z)\,,\]where \(r\) is the spherical radius \(r^2 = R^2+z^2\). We can solve for \(\Phi_{\mathrm{ME}}(R,\phi,z)\) by solving
\[\frac{\Delta \Phi_{\mathrm{ME}}(R,\phi,z)}{4\pi G} = \rho(R,\phi,z) - \sum_i\left\{ \Sigma_i(r)\,h_i(z) + \frac{\mathrm{d}^2 \Sigma_i(r)}{\mathrm{d} r^2}\,H_i(z)+\frac{2}{r}\,\frac{\mathrm{d} \Sigma_i(r)}{\mathrm{d} r}\left[H_i(z)+z\,\frac{\mathrm{d}H_i(z)}{\mathrm{d} z}\right]\right\}\,.\]We solve this equation by using the SCFPotential class and methods (scf_compute_coeffs_axi or scf_compute_coeffs depending on whether \(\rho(R,\phi,z)\) is axisymmetric or not). This technique works very well if the disk portion of the potential can be exactly written as \(\rho_{\mathrm{disk}} = \sum_i \Sigma_i(R)\,h_i(z)\), because the effective density on the right-hand side of this new Poisson equation is then not ‘disky’ and can be well represented using spherical harmonics. But the technique is general and can be used to compute the potential of any disk+halo potential; the closer the disk is to \(\rho_{\mathrm{disk}} \approx \sum_i \Sigma_i(R)\,h_i(z)\), the better the technique works.
This technique was introduced by Kuijken & Dubinski (1995) and was popularized by Dehnen & Binney (1998). The current implementation is a slight generalization of the technique in those papers and uses the SCF approach of Hernquist & Ostriker (1992) to solve the Poisson equation for \(\Phi_{\mathrm{ME}}(R,\phi,z)\) rather than solving it on a grid using spherical harmonics and interpolating the solution (as done in Dehnen & Binney 1998).
- __init__(amp=1.0, normalize=False, dens=<function DiskSCFPotential.<lambda>>, Sigma={'amp': 1.0, 'h': 0.3333333333333333, 'type': 'exp'}, hz={'h': 0.037037037037037035, 'type': 'exp'}, Sigma_amp=None, dSigmadR=None, d2SigmadR2=None, Hz=None, dHzdz=None, N=10, L=10, a=1.0, radial_order=None, costheta_order=None, phi_order=None, ro=None, vo=None)[source]¶
Initialize a DiskSCFPotential.
- Parameters:
amp (float, optional) – Amplitude to be applied to the potential (default: 1); cannot have units currently.
normalize (bool or float, optional) – If True, normalize such that vc(1.,0.)=1., or, if given as a number, such that the force is this fraction of the force necessary to make vc(1.,0.)=1.
dens (callable) – Function of R,z[,phi optional] that gives the density [in natural units, cannot return a Quantity currently].
N (int, optional) – Number of radial basis functions to use in the SCF expansion.
L (int, optional) – Number of angular basis functions to use in the SCF expansion.
a (float or Quantity, optional) – Scale radius for the SCF expansion.
radial_order (int, optional) – Order of the radial basis functions to use in the SCF expansion.
costheta_order (int, optional) – Order of the angular basis functions to use in the SCF expansion.
phi_order (int, optional) – Order of the azimuthal basis functions to use in the SCF expansion.
Sigma (dict or callable) – Either a dictionary of surface density (example: {‘type’:’exp’,’h’:1./3.,’amp’:1.,’Rhole’:0.} for amp x exp(-Rhole/R-R/h) ) or a function of R that gives the surface density.
hz (dict or callable) – Either a dictionary of vertical profile, either ‘exp’ or ‘sech2’ (example {‘type’:’exp’,’h’:1./27.} for exp(-|z|/h)/[2h], sech2 is sech^2(z/[2h])/[4h]) or a function of z that gives the vertical profile.
Sigma_amp (float, optional) – Amplitude to apply to all Sigma functions.
dSigmadR (callable, optional) – Function that gives d Sigma / d R.
d2SigmadR2 (callable, optional) – Function that gives d^2 Sigma / d R^2.
Hz (callable, optional) – Function of z such that d^2 Hz(z) / d z^2 = hz.
dHzdz (callable, optional) – Function of z that gives d Hz(z) / d z.
ro (float or Quantity, optional) – Distance scale for translation into internal units (default from configuration file).
vo (float or Quantity, optional) – Velocity scale for translation into internal units (default from configuration file).
Notes
Either specify (Sigma,hz) or (Sigma_amp,Sigma,dSigmadR,d2SigmadR2,hz,Hz,dHzdz)
Written - Bovy (UofT) - 2016-12-26