Galpy MWPotential2014: Milky Way Model

• MWPotential2014 is a composite, axisymmetric galactic potential model representing the Milky Way's gravitational field through a power-law spherical bulge, Miyamoto–Nagai disk, and NFW dark-matter halo.
• The model employs analytical formulations with a power-law bulge featuring an exponential cutoff, a Miyamoto–Nagai disk potential, and an NFW halo to replicate observed Galactic dynamics.
• Galpy’s implementation enables rapid numerical orbit integration, circular velocity curve analysis, and flexible adjustment with dimensionless units (R0 = 8 kpc, V0 = 220 km/s).

MWPotential2014 is a composite, axisymmetric galactic potential implemented in the galpy library to represent the Milky Way’s time-independent gravitational field. This three-component model consists of a power-law spherical bulge with exponential cutoff, a Miyamoto–Nagai disk, and a Navarro–Frenk–White (NFW) dark-matter halo. All components are physically motivated and parameterized using observed Galactic properties, with dimensional scaling fixed by the solar circle: $R_0 = 8$ kpc, $V_0 = 220$ km s$^{-1}$ (Bovy, 2014).

1. Analytical Formulation of Components

MWPotential2014 is constructed as the sum of three axisymmetric potentials, each corresponding to a distinct Milky Way component:

• Bulge: Realized as a power-law spherical density profile with exponential cutoff (PowerSphericalPotentialwCutoff). The density is

$$\rho_b(r) = \rho_0 \left( \frac{r}{r_c} \right)^{-\alpha} \exp[-r/r_c]$$

where $\alpha$ is the inner power-law slope, $r_c$ the cutoff radius, and $\rho_0$ the normalization. The analytic potential is

$$\Phi_b(r) = -4\pi G \rho_0 r_c^2 \frac{1}{2-\alpha} \left[ 1 - \left( \frac{r}{r + r_c} \right)^{2-\alpha} \right].$$

• Disk: Modeled as a Miyamoto–Nagai potential:

$$\Phi_d(R,z) = - \frac{G M_d}{\sqrt{R^2 + \left[a + \sqrt{z^2 + b^2}\right]^2}}$$

with $M_d$ the disk mass, $V_0 = 220$0 the radial scale length, and $V_0 = 220$1 the vertical scale parameter.

• Halo: Represented by a standard NFW profile (NFWPotential):

$V_0 = 220$2

where $V_0 = 220$3 is the scale radius and $V_0 = 220$4 the characteristic density.

Each term is implemented in galpy in dimensionless internal units, as described below.

2. Numerical Parameters and Units

The galpy package establishes its dimensionless unit system by setting:

• $V_0 = 220$5 kpc (default length unit)
• $V_0 = 220$6 km s$V_0 = 220$7 (default velocity unit)
• $V_0 = 220$8 Gyr (time unit)

Masses and amplitudes (“amp”) are specified so that

$V_0 = 220$9

Parameter values for MWPotential2014:

Component	Parameter	Value in code units	Physical interpretation
Bulge (Powerlaw+Cutoff)	$^{-1}$0	1.8	Inner slope
	$^{-1}$1	0.075	$^{-1}$2 kpc
	amp	0.053823…	$^{-1}$3
Disk (Miyamoto–Nagai)	$^{-1}$4	0.5	$^{-1}$5 kpc
	$^{-1}$6	0.0375	$^{-1}$7 kpc
	amp	0.6097…	$^{-1}$8
Halo (NFW)	$^{-1}$9	2.0	$$\rho_b(r) = \rho_0 \left( \frac{r}{r_c} \right)^{-\alpha} \exp[-r/r_c]$$0 kpc
	amp	0.3314…	$$\rho_b(r) = \rho_0 \left( \frac{r}{r_c} \right)^{-\alpha} \exp[-r/r_c]$$1

These values are hard-coded in galpy's MWPotential2014 object (Bovy, 2014).

3. Construction and Circular Velocity Curve

The total gravitational potential is formed by summing the three components:

$$\rho_b(r) = \rho_0 \left( \frac{r}{r_c} \right)^{-\alpha} \exp[-r/r_c]$$2

For axisymmetric systems, the circular velocity in the disk mid-plane is

$$\rho_b(r) = \rho_0 \left( \frac{r}{r_c} \right)^{-\alpha} \exp[-r/r_c]$$3

with $$\rho_b(r) = \rho_0 \left( \frac{r}{r_c} \right)^{-\alpha} \exp[-r/r_c]$$4. In galpy, this is computed via vcirc(MWPotential2014, R) where $$\rho_b(r) = \rho_0 \left( \frac{r}{r_c} \right)^{-\alpha} \exp[-r/r_c]$$5 is in units of $$\rho_b(r) = \rho_0 \left( \frac{r}{r_c} \right)^{-\alpha} \exp[-r/r_c]$$6.

For the preset parameter choices, the model yields $$\rho_b(r) = \rho_0 \left( \frac{r}{r_c} \right)^{-\alpha} \exp[-r/r_c]$$7 km s$$\rho_b(r) = \rho_0 \left( \frac{r}{r_c} \right)^{-\alpha} \exp[-r/r_c]$$8 at the solar circle (by construction) and an NFW-like decline at large radii.

4. Observational Constraints and Reference Motivations

MWPotential2014 parameter values rely on key Galactic observations and literature:

• Solar location: Adopted $$\rho_b(r) = \rho_0 \left( \frac{r}{r_c} \right)^{-\alpha} \exp[-r/r_c]$$9 kpc, $\alpha$0 km s$\alpha$1.
• Bulge: Inner slope $\alpha$2 fits low-latitude star count data (e.g., Binney et al. 2011).
• Disk: $\alpha$3 kpc, $\alpha$4 kpc, consistent with wide-field photometric surveys (Jurić et al. 2008).
• Halo: Scale radius $\alpha$5 kpc; normalization so $\alpha$6 at $\alpha$7 matches observed rotation, with mass profile matching NFW at large radii.
• Masses: $\alpha$8, $\alpha$9, $r_c$0.

Principal references include Miyamoto & Nagai (1975), Hernquist & Ostriker (1992), Navarro, Frenk & White (1996), Jurić et al. (2008), Binney & Tremaine (2008), and Bovy (2015) (Bovy, 2014).

5. Programmatic Usage and Example

The MWPotential2014 model is a built-in object in galpy. Python users can instantiate, diagnose, and manipulate it in code. An example workflow:

$r_c$1

Parameter attributes are accessible directly, e.g.,

$r_c$2 Normalization (amp) and unit scales (ro, vo) can be adjusted on instantiation for model variants. The full codebase, documentation, and reference test suite are publicly available (Bovy, 2014).

6. Related Methodological Considerations

MWPotential2014’s design supports rapid numerical orbit integration using various Runge–Kutta and symplectic schemes available within galpy. The package framework permits arbitrary combinations of built-in and user-defined potential components, facilitating investigation of time-independence, axisymmetry, and parameter sensitivity. Calculations of action–angle coordinates and orbital frequencies are also supported for analysis within general axisymmetric contexts (Bovy, 2014).

The MWPotential2014 model constitutes a reference Milky Way potential compatible with contemporary large-scale observations, supporting dynamical modeling, rotation curve analysis, and kinematic studies across the Galactic disk, bulge, and halo.