Physics & Mathematics

Physics & Mathematics#

This page documents the mathematical models implemented in SONGS.

Sérsic Radial Profile (Disk Plane)#

The surface brightness in the disk plane follows the Sérsic profile:

\[S_r(r) = S_e \exp\!\left[-b_n\!\left(\left(\frac{r}{R_e}\right)^{1/n} - 1\right)\right]\]

where

  • \(S_e\) is the flux density at the effective radius \(R_e\),

  • \(n\) is the Sérsic index (controls concentration),

  • \(b_n\) is a series-expansion constant that depends on \(n\):

\[b_n(n) \approx 2n - \tfrac{1}{3} + \frac{4}{405n} + \frac{46}{25515n^2} + \cdots\]

Vertical Exponential Profile#

The vertical structure follows an exponential fall-off from the mid-plane:

\[S_z(z) = \exp\!\left(-\frac{|z|}{h_z}\right)\]

Combining the radial and vertical profiles gives the 3D flux density field:

\[S(x, y, z) = S_e \exp\!\left[-b_n\!\left(\left(\frac{r}{R_e}\right)^{1/n} - 1\right)\right] \exp\!\left(-\frac{|z|}{h_z}\right)\]

with \(r = \sqrt{x^2 + y^2}\) evaluated in the tilted disk plane after the viewing-angle rotation has been applied.

Analytical Rotation Curve#

Tangential velocities are assigned from a compact empirical approximation (milky_way_rot_curve_analytical):

\[v(R) = v_0 \times 1.022 \times \left(\frac{R}{R_0}\right)^{0.0803}\]

where \(v_0\) is a characteristic velocity scale and \(R_0\) is derived from \(R_e\) and \(n\). This form reproduces the gently rising / flat behaviour of typical disk-galaxy rotation curves at IFU-relevant scales.

Beam Convolution#

Each 2D channel map is convolved with an elliptical Gaussian to simulate finite telescope resolution. The FWHM-to-sigma conversion is:

\[\sigma = \frac{\mathrm{FWHM}}{2\sqrt{2\ln 2}} \approx \frac{\mathrm{FWHM}}{2.355}\]

This relation is used when constructing an astropy.convolution.Gaussian2DKernel.

Diffuse Emission Physics#

SONGS adds low-surface-brightness diffuse structure to the disk components during cube assembly. The model contains three additive pieces controlled by DEFAULT_DIFFUSE_PARAMS.

Halo#

The extended halo around the central galaxy uses a spherical Sérsic-like radial profile in the disk plane combined with a vertical exponential:

\[D_\mathrm{halo}(r, z) = A_\mathrm{halo} \exp\!\left[-b_n\!\left(\left(\frac{r}{R_\mathrm{halo}}\right)^{1/n_\mathrm{halo}} - 1\right)\right] \exp\!\left(-\frac{|z|}{h_{z,\mathrm{halo}}}\right)\]

The halo velocity field shares the same kinematic noise realisation as the bridge components so that halo and bridges appear and disappear coherently in the same spectral channels.

Bridges#

Bridges are Gaussian tubes connecting the central galaxy to each satellite. A bridge is parameterised by a unit vector \(\hat{u}\) pointing from the central to the satellite, with arc-length coordinate \(s \in [0, S]\):

\[D_\mathrm{bridge}(s, \rho) = A_\mathrm{bridge} \exp\!\left(-\frac{\rho^2}{2\sigma_\mathrm{bridge}^2}\right) \cdot w(s)\]

where \(\rho\) is the perpendicular distance from the bridge axis and \(w(s)\) is a taper function that smoothly approaches zero at both endpoints. The velocity along the bridge is linearly interpolated between the central systemic velocity \(v_\mathrm{cen}\) and the satellite systemic velocity \(v_\mathrm{sat}\):

\[v_\mathrm{bridge}(s) = v_\mathrm{cen} + \frac{s}{S}\,(v_\mathrm{sat} - v_\mathrm{cen})\]

Tidal Tails#

Tidal tails are physically-motivated Bézier arcs extending from each satellite away from the central galaxy. The control points are constructed so that the arc direction at the root is constrained to the outward hemisphere (opposite to the central-to-satellite vector), preventing unphysical back-folding.

Let \(u \in [0, 1]\) be the normalised arc-length parameter. The tail amplitude decays exponentially from root to tip:

\[A_\mathrm{tail}(u) = A_0 \exp\!\left(-\frac{u}{\lambda_\mathrm{decay}}\right)\]

so emission is densest near the progenitor satellite and thins toward the tip.

The velocity gradient along the tail is derived from the systemic velocity difference between satellite and central:

\[v_\mathrm{tail}(u) = v_\mathrm{sat} + u \cdot \Delta v_\mathrm{sys} \cdot f_\mathrm{scale}\]

where \(\Delta v_\mathrm{sys} = v_\mathrm{sat} - v_\mathrm{cen}\) and \(f_\mathrm{scale}\) is the vel_scale_factor diffuse parameter.

The cross-sectional width tapers linearly from root to tip:

\[\sigma_\mathrm{tail}(u) = \sigma_0 \,(1 - \alpha\, u), \quad 0 \le \alpha < 1\]

so the stream narrows as it stretches away from the galaxy.