Distribution of Si II $\lambda$6355 Velocities of Type Ia Supernovae and
  Implications for Asymmetric Explosions
  The ejecta velocity is a very important parameter in studying the structure
and properties of Type Ia supernovae (SNe Ia). It is also a candidate key
parameter in improving the utility of SNe Ia for cosmological distance
determinations. Here we study the velocity distribution of a sample of 311 SNe
Ia from the kaepora database. The velocities are derived from the Si II
$\lambda$6355 absorption line in optical spectra measured at (or extrapolated
to) the time of peak brightness. We statistically show that the observed
velocity has a bimodal Gaussian distribution consisting of two groups of SNe
Ia: Group I with a lower but narrower scatter ($\mu_1 = 11000 \text{km
s}^{-1}$, $\sigma_1 = 700 \text{km s}^{-1}$), and Group II with a higher but
broader scatter ($\mu_2 = 12300 \text{km s}^{-1}$, $\sigma_2 = 1800 \text{km
s}^{-1}$). The population ratio of Group I to Group II is 201:110 (65%:35%).
There is substantial degeneracy between the two groups, but for SNe Ia with
velocity $v > 12000 \text{km s}^{-1}$, the distribution is dominated by Group
II. The true origin of the two components is unknown, though there could be
that naturally there exist two intrinsic velocity distributions as observed.
However, we try to use asymmetric geometric models through statistical
simulations to reproduce the observed distribution assuming all SNe Ia share
the same intrinsic distribution. In the two cases we consider, 35\% of SNe Ia
are considered to be asymmetric in Case 1, and all SNe Ia are asymmetric in
Case 2. Simulations for both cases can reproduce the observed velocity
distribution but require a significantly large portion ($>35\%$) of SNe Ia to
be asymmetric. In addition, the Case 1 result is consistent with recent
polarization observations that SNe Ia with higher Si II $\lambda$6355 velocity
tend to be more polarized.
