How reliable is the spin parameter[1], λR, given the assumptions that underpin its functional form and the various observational effects that impact its measurement? We explore:

  1. The observational biases caused by the limitations of real observing (e.g. inclination, projection, seeing conditions, beam smearing, uncertainties in measurement radius, etc.) and
  2. The definition biases between the theoretical inherent value of a galaxy’s spin, λ’, and its observed counterpart, λR.

We created a catalog of 6 galaxy simulations using GalIC[2] and an extended version of Gadget2[3] followed by a series of 2812 synthetic IFU observations using the R package SimSpin[4]. Seeing, beam smearing, surface brightness and spatial resolution limits were incorporated to represent the real observational biases. λR and various flavours of the theoretical spin λ’ inherent to each model were then compared.

The observable spin parameter, λR, is calculated: \[ \lambda_R = \frac{\left< R |V|\right>}{\left< \sqrt{} (V^2 + \sigma^2) \right>}, \] where the angular brackets denote luminosity weighting, \(R\) is the circularised radius, \(V\) is the line-of-sight velocity and \(\sigma\) is the line-of-sight velocity dispersion.

Figure 1: The procedure by which <code>SimSpin</code> creates synethic data cubes from simulated galaxies. For further examples, see <url> https://rpubs.com/kateharborne </url>.

Figure 1: The procedure by which SimSpin creates synethic data cubes from simulated galaxies. For further examples, see https://rpubs.com/kateharborne .

Figure 2: Mock flux, line-of-sight velocity and line-of-sight velocity dispersion maps produced through <code>SimSpin</code> for an Sd galaxy inclined to $70^o$ and blurred with a Moffat PSF FWHM = 1'' to mimic seeing conditions.

Figure 2: Mock flux, line-of-sight velocity and line-of-sight velocity dispersion maps produced through SimSpin for an Sd galaxy inclined to \(70^o\) and blurred with a Moffat PSF FWHM = 1’’ to mimic seeing conditions.

1. Observational bias

Graham et. al’s (2018) empirical correction[5] for reducing the effects of seeing on λR does a good job at recovering the unblurred value for our N-body models. The correction takes the form: \[ \lambda_R^{obs} = \lambda_R^{corr} gM_2 \left(\frac{\sigma_{PSF}}{R^{maj}_{eff}}\right) f_n \left(\frac{\sigma_{PSF}}{R^{maj}_{eff}}\right), \] where, \[ gM_2 \left(\frac{\sigma_{PSF}}{R^{maj}_{eff}}\right) = \left[ 1 + \left( \frac{\sigma_{PSF}/R_{eff}^{maj}}{0.47} \right)^{1.76} \right]^{-0.84}, \] which is a generalised Moffat function that represents the effects of spatial blurring (\(\sigma_{PSF}\)) relative to the semi-major axis of the effective radius of the galaxy (\(R^{maj}_{eff}\)), and,

\[ f_n \left(\frac{\sigma_{PSF}}{R^{maj}_{eff}}\right) = \left[ 1 + (n-2) \left(0.26 \frac{\sigma_{PSF}}{R_{eff}^{maj}}\right) \right]^{-1}, \] which accounts for the variations in the effects of spatial blurring due to galaxy morphology, represented by the sersic index, \(n\).

Figure 3: Considering before (above) and after (below) Graham et. al.'s (2018) correction on our synthetic observations of &#955;$_R$ when taking the difference between the value effected by seeing and the observationally perfect measurement.

Figure 3: Considering before (above) and after (below) Graham et. al.’s (2018) correction on our synthetic observations of λ\(_R\) when taking the difference between the value effected by seeing and the observationally perfect measurement.

We see that there is a reduction in \(\Delta \lambda_R\) from \(\leq 0.2\) to \(\leq 0.07\) at an average seeing of 0.5 when corrected, though this is dependent on the projected inclination (most effective corrections occur at intermediate projections, \(i \sim 50^o\)).

2. Definition bias

We compare various flavours of the theoretical spin, λ’, to the observed λR across a range of seeing conditions. The flavours we consider are:

  1. The Bullock spin parameter measured at the half-mass radius of the galaxy.
  2. The Bullock spin parameter measured at the half-mass of the radius, but adjusted by a factor to account that observationally we use the projected light distribution rather than the 3D mass distribution.
  3. The analytic value derived from the tensor virial theorem by Binney (2005)[6] given intrinsic ellipticity of the galaxy.

1. λ’(Reff)

The Bullock et. al. (2001) spin parameter defines the relative importance of rotation and dispersion support to maintaining the gravitationally bound structure in equilibrium. Traditionally, this is defined at the virial radius, \(\sim 200\)kpc. We define this at the half-mass radius, however, such that it is comparable to the observed λR which we measure at the effective radius.

\[ \lambda'(R_{eff}) = \frac{J}{\sqrt{2} M V_c R_{eff}}. \] Here, \(J\) is the angular momentum, \(M\) is the mass enclosed by the radius \(R_{eff}\) and \(V_c\) is the circular velocity. In the first panel of figure 4, we consider how this value compares to the observed λR: \[ \Delta \lambda^{(1)}_R = \lambda_R^{obs} - \lambda'(R_{eff}) \]

2. λ’R(Reff)

Originally, when Emsellem et. al. (2007) derived the observable spin parameter from the theoretical value, conversion factors were used to account for the fact that with simulations we use the full 3-dimensional mass distribution to describe the dynamics, while observationally we only have the 2D projected light distribution:

\[ \lambda'_R(R_{eff}) = \frac{3}{\sqrt{2}} \times \frac{J}{\sqrt{2} M V_c R_{eff}}, \]

where the values are the same as in the equation for \(\lambda'(R_{eff})\) above. In the second panel of figure 4, we consider how this slightly adjusted value compares to λR across a range of seeing conditions:

\[ \Delta \lambda^{(2)}_R = \lambda_R^{obs} - \lambda'_R(R_{eff}) \]

3. λRanalytic

This inherent value is derived using Binney’s (2005) derivation of the value of λR expected for oblate galaxies of a given intrinsic ellipticity when measured edge on, given by:

\[ \lambda_R^{analytic} = \frac{\kappa V/\sigma}{\sqrt{1 + \kappa^2 (V/\sigma)^2}}, \] where \(\kappa \sim 1.1\) estimated from SAURON observations and two-integral models in Emsellem et. al. (2007) and,

\[ V/\sigma = \sqrt{\frac{\Omega(e) (1 - \beta) - 1}{\alpha \Omega(e)(1-\beta) + 1}}, \] where \(\alpha\) is a parameter that depends on the shape of a galaxy’s intrinsic rotation curve and radial luminosity profile, fixed at \(\alpha \sim 0.15\) to provide accurate representation of real galaxies; \(\beta\) is the anisotropy of the galaxy for which real galaxies follow an approximate linear relation, \(\beta \approx 0.7 \times \epsilon_{intr}\) where \(\epsilon_{intr}\) is the intrinsic ellipticity of the galaxy; \(\Omega(e)\) and \(e\) are then given by the functions:

\[ \Omega(e) = \frac{1}{2} \frac{sin^{-1}(e)/\sqrt{1 - e^2} - e}{e - sin^{-1}(e) \sqrt{1 - e^2}}, \] \[ e = \sqrt{1 - (1 - \epsilon_{intr})^2 }. \]

Using the known intrinsic ellipticities of each model, we can calculate the value we would expect to observe according to these analytic formula. We compare how this analytic value compares to λR in the final panel of Figure 4:

\[ \Delta \lambda^{(3)}_R = \lambda_R^{obs} - \lambda_R^{analytic} \]
Figure 4: Comparing different inherent measures of spin to the observed &#955;$_R$ values.

Figure 4: Comparing different inherent measures of spin to the observed λ\(_R\) values.

Every inherent property of spin is valid to describe the kinematics (spiral galaxies rotate faster than ellipticals in all descriptions) though each flavour is offset from one another by some amount. λR is not a direct proxy for the theoretical Bullock spin parameter intrinsic to a galaxy and so it is important to bear this in mind when comparing simulations to observations.

Citations

  1. Emsellem E., et al., 2007, MNRAS, 379, 401
  2. Yurin D., et al., 2014, MNRAS, 444, 62
  3. Springel V., 2005, MNRAS, 364, 1105
  4. https://github.com/kateharborne/SimSpin
  5. Graham M., et. al., 2018. MNRAS, sty504
  6. Binney J., 2005, MNRAS, 363, 937