# A multispectral analysis of the northeastern shell of IC 443

Alexandre Alarie,1,2,3 Laurent Drissen,2,3
1 Instituto de Astronomía, Universidad National Autónoma de México, Apdo.
Postal 70264, 04510 Mexico D.F., Mexico
2 Département de physique, de génie physique et d’optique, Université Laval,
Québec, QC, G1V 0A6, Canada
3 Centre de Recherche en Astrophysique du Québec

###### Abstract

We have carried out optical observations of the north-eastern part of the
supernova remnant IC 443 using the CFHT imaging spectrograph SITELLE. The
observations consist of three multispectral cubes covering an 11′ $\times$11′
area allowing the investigation of both the spatial and spectral variation of
9 emission lines : [O ii] $\lambda\lambda$3726+3729, [O iii]
$\lambda\lambda$4959,5007, H$\beta$, H$\alpha$, [N ii]
$\lambda\lambda$6548,6583 and [S ii] $\lambda\lambda$6716,6731. Extinction
measurement from the H$\alpha$/ H$\beta$ shows significant variation across
the observed region with E(B-V) = 0.8-1.1. Electron density measurements using
[S ii] lines indicate densities ranging from 100 up to 2500 cm-3. Models
computed with the shock modelling code mappings are presented and compared
with the observations. A combination of complete shock model and truncated
ones are required in order to explain the observed spectrum. The shock
velocities found in IC 443 are between 20 and 150 km s-1 with 75 km s-1 being
the most prominent velocity. The pre-shock number density varies from 20 to 60
cm-3. A single set of abundances close to solar values combined with varying
shock parameters (shock velocity, pre-shock density and shock age) are
sufficient to explain to great variation of lines intensities observed in IC
443. Despite the relatively modest spectral resolution of the data (R$\sim
1500$ at H$\alpha$), we clearly separate the red and blue velocity components
of the expanding nebula, which show significant morphological differences.

###### keywords:

techniques: imaging spectroscopy - proper motions - ISM: individual objects:
IC443 - ISM: supernova remnants

††pubyear: 2019††pagerange: A multispectral analysis of the northeastern shell
of IC 443–30

##  1 Introduction

IC 443 (G189.1+3.0) is an evolved galactic supernova remnant (SNR) located in
the direction of the galactic anti-center at a distance between 1.5 and 2 kpc
(Petre et al., 1988; Welsh & Sallmen, 2003; Ambrocio-Cruz et al., 2017) and an
age estimated at around 3000-30,000 years (Petre et al., 1988; Olbert et al.,
2001). Because of its large scale ($\approx$0.5∘) and clear view without any
obstruction in the direction of the remnant, IC 443 is known to be an
excellent laboratory to study the complex interaction between a SNR and the
interstellar and circumstellar medium.

IC 443 has been classified as a mixed-morphology remnant: shell-like and
bright structure in radio continuum and in the optical (Lee et al., 2008) and
centrally filled in X-ray. Braun & Strom (1986) suggested that the remnant has
evolved inside a preexisting wind-blown bubble, which likely was formed by the
remnant’s massive progenitor star. This proposed explanation was supported by
Troja et al. (2006); Troja et al. (2008) based on the analysis of X-ray
observations obtained with XMM-Newton. IC 443 is apparently interacting with
both atomic and molecular gas (Snell et al., 2005). In the northeastern part,
the SNR shock is encountering mostly an atomic medium, and only the atomic
lines expected in the post-shock recombining gas are detected (Fesen &
Kirshner, 1980). The southern boundary, on the other hand, shows various
molecular lines with broad wings suggesting that the shock is propagating into
mostly molecular medium (Burton et al., 1988; Dickman et al., 1992; van
Dishoeck et al., 1993).

In recent years, there has been considerable interest in studying the complex
interaction between the IC 443 SNR with its surrounding molecular clouds at
radio and infrared wavelengths (Hezareh et al., 2013; Fang et al., 2013; Su et
al., 2014) as well as the determination of the abundances of heavy elements in
some part of the remnant from X-ray observations (Troja et al., 2008; Bocchino
et al., 2009; Swartz et al., 2015).

Detections of high and very high energy gamma rays have also been reported in
the vicinity of IC 443, which overlap with the object’s structure seen in the
radio, IR, optical and X-ray bands. First with the MAGIC telescope (Albert et
al., 2007), and then early in the Fermi-LAT mission (Abdo et al., 2010;
Ackermann et al., 2013). Is is assumed that these emissions are associated
with locally accelerated cosmic rays.

Despite all these observations made in recent years, optical spectroscopy,
which samples very different physical phenomena and chemical el- ements, are
limited to long-slit spectroscopy (Parker, 1964; Esipov et al., 1972; Fesen &
Kirshner, 1980).

This present work was undertaken to improve the spectroscopic data available
for the north-eastern part of IC 443 where optical lines are present. Multi-
spectral cubes obtained with the imaging Fourier transform spectrometer
SITELLE are presented and analyzed alongside with shock models to infer shock
properties. The paper is organized as follows: in section 2 we present the
observations and data reduction; in section 3 we discuss the shock modelling
technique and make direct comparisons with observations. A kinematical
analysis of the data is presented in section 4. We conclude this paper with
remarks in section 5.

##  2 Observations and data reduction

###  2.1 Observations

The data presented here consist of three multispectral data cubes obtained
with the optical imaging Fourier Transform Spectrometer (iFTS), SITELLE,
attached to the 3.6-meter Canada-France-Hawaii Telescope (Drissen et al.,
2019). SITELLE delivers multispectral cubes covering a 11′ $\times$11′ field-
of-view, sampled with $0.32^{\prime\prime}$ pixels, covering specific
bandpasses of the visible band using filters . A single region of IC 443,
centered at RA 6h17m36.25s and DEC +22∘49′15′′, was observed during the nights
of 2016 March 4, 6 & 7 using three filters targeting bright emission lines :
H$\alpha$, H$\beta$, [O ii] $\lambda$3726+3729, [O iii] $\lambda$4959,5007, [N
ii] $\lambda$6548,6583 and [S ii] $\lambda$6716,6731. SITELLE’s spectral
resolution R is adjustable to the need of the observer; we selected relatively
modest values of R (see Table 1) allowing us to clearly separate the emission
lines from their neighbors as our primary goal was not to study the
kinematics. However, as we show in section 4, the high signal-to-noise reached
in the SN3 filter (around H$\alpha$) allowed us to perform a preliminary
kinematical analysis of the target.

Table 1: Characteristics of the multispectral cubes Characteristics | SN1 | SN2 | SN3
---|---|---|---
Spectral range (Å) | 3650-3850 | 4800-5200 | 6510-6850
Spectral resolution (R) | 500 | 800 | 1500
Exposure time per image (s) | 30 | 30 | 20
Number of images | 88 | 180 | 266
Total observing time | 44 min | 1h30m | 1h28m40s

###  2.2 Data reduction and line fitting

The data were reduced using SITELLE’s dedicated data reduction software orbs
(Martin et al., 2015). Standard reduction procedures were applied (flat, bias,
dark) and the flux calibration was performed using a datacube of the
spectrophotometric standard GD71 for each filter to define the transmission
function at each wavelength. The spectral calibration was done using a helium-
neon laser datacube acquired during daytime. A detailed description regarding
the reduction process is provided by Martin & Drissen (2017).

An individual spectrum is shown in Figure 1. It has to be noted that the
apparent wiggles on each side of the emission lines are not due to noise but
rather to the cardinal sine function (sinc) typical of all Fourier transform
spectrometers, as we chose not to apodize the signal. This peculiar instrument
line shape is taken into account by the dedicated line-fitting software orcs
(Martin et al., 2016) which was specifically created to deal with SITELLE data
and therefore uses sinc-shaped line profiles to accurately fit each emission
line. In order to improve the signal to noise ratio, the line flux extraction
was performed using a 3$\times$3 pixel binning, except for the kinematical
analysis, where no binning was done. This manipulation reduced the spatial
resolution to approximately 1′′/pixel.

Figure 1: Spectra extracted from the data cubes using a circular aperture
(2.22′′) centered at RA 6h17m57s and DEC 22∘44′42′′. Figure 2: Composite image
of IC 443, using SITELLE’s SN2 and SN3 data cubes. H$\alpha$ is coded in red,
[S ii] $\lambda$$\lambda$6716,31 in green and [O iii] $\lambda$5007 in blue.
The field of view is $11^{\prime}\times 11^{\prime}$, with East to the left
and North at the top.

###  2.3 Flux and line ratio maps

Using the flux and error maps generated by orcs, we have created ratio maps
and only kept pixels for which the ratios error is smaller than 10%. Flux maps
in all emission lines are shown the Appendix A : [O ii] 18, [O iii] 19,
H$\alpha$ 20, [N ii] 21 and [S ii] 22. An unbinned color composite image of
the field, using the H$\alpha$, [S ii] and [O iii] flux maps, is shown in
Figure 2.

The followings ratios also generated and presented in Appendix A : [O ii]/
H$\alpha$ (24), [O iii]/ H$\beta$ (25), [O iii]/ [O ii] (26), [N ii]/
H$\alpha$ (27), [S ii]/ H$\alpha$ (28), [N ii]/ [S ii] (29) and [S ii]
$\lambda$6731/$\lambda$6716 (30). Due to insufficient exposure time during the
acquisition of the SN1 cube, [O ii] emissions could not be accurately measured
at many locations leading to incomplete [O ii]/ H$\alpha$ and [O iii]/ [O ii]
maps.

###  2.4 Extinction correction

IC 443 is well known to contain patchy obscuration over the whole remnant and
its surroundings. Since the observations cover an appreciable fraction of the
northeastern shell, it is essential to measure the interstellar reddening at
various locations and correct the observed line intensities accordingly. For
interstellar radiative shocks, the Balmer lines of hydrogen are thought to be
primarily produced by recombination with only a small amount of direct
collisional excitation from ground level to bound excited states. Shock models
calculated by Raymond (1979) have shown that the H$\alpha$/ H$\beta$ ratio
fall between 2.9 and 3.3 with larger values occurring for the slower shocks.
Since we do not have the information about the shock velocity prior to a
reduction, we used the standard value of H$\alpha$/ H$\beta$ of 3.0 to correct
our observations. With this value, we estimated the colour excess E(B-V) for
every pixel where the error on the H$\alpha$/ H$\beta$ ratio is smaller than
10%. The extinction was evaluated using pyneb (Luridiana et al., 2012) and the
RedCorr class which makes use of the Fitzpatrick (1999) correction curve.
Since the extinction is most likely diffuse in front of the remnant, we used a
Gaussian blur with a standard deviation of 6$\sigma$ in order to smooth the
extinction map and fill regions where the extinction could not be directly
measured. The map used to correct the line intensities pixel by pixel is shown
in Figure 23111The map can be found in the supplementary material in FITS
format.. The values found are in agreement with earlier measurements, such as
those reported by Fesen & Kirshner (1980): E(B-V) = 0.8 - 1.1.

###  2.5 Faint nebulosities from the HII region S249

Some faint nebulosities visible at the northern edge of our field are due to
the H II region S249. These emissions undoubtedly contaminate some of IC 443
spectra overlapping these regions. We estimate that the contamination from the
brightest H$\alpha$ and [O iii] emission from S249 could contribute up to 10%
and 8% respectively of the measured emission from IC 443. Despite this
contamination, we do not attempt the subtract it from the emissions coming
from IC 443. Our decision is based on the fact that both the intensity and
shape of the emissions coming from S249 behind IC 443 are unpredictable. It
has to be noted that the contamination does not seem to affect the whole
observed region, but is rather localized in the northern part of IC 443.

##  3 Line ratio analysis

###  3.1 Comparison with previous observations

Very few spectroscopic observations in the optical regime can be found in the
literature. The earliest measurements are presented by Parker (1964) where
multiple spectra were taken on the surface of IC 443. Table 2 of their paper
shows the line intensities as well as a few line ratios which could be
compared with our observations. One position noted as filament 7′ is located
in the north-eastern part of the nebula. The exact location of that filament
is unknown since its coordinates were not provided. By approximating its
location from their finding charts, we compared visually their [N ii]/ [S ii],
[S ii]/ H$\alpha$, [N ii]/ H$\alpha$ and [O iii]/ H$\beta$ ratios with our
ratio maps and found a very similar set of values for multiple locations
around the same area.

So far, the most extensive spectroscopic analysis in the optical was carried
out by Fesen & Kirshner (1980). They targetted six positions in IC 443, among
which two (their numbers 1 and 4) are located inside our field of view. The
positions were sufficiently well indicated to allow a good comparison. As
mentioned earlier, their observed H$\alpha$/ H$\beta$ ratio is very similar to
ours, which lead to the same measurement of the extinction. Also, the line
ratios at those places such as [O iii]/ H$\alpha$, [O ii]/ H$\alpha$, [N ii]/
H$\alpha$, [S ii]/ H$\alpha$ and [S ii] $\lambda$6716/$\lambda$6731 are in
very good agreement with our measurements. Furthermore, Fesen & Kirshner
(1980) compared their observations with shock models to investigate both the
shock conditions and elemental abundances.

The data presented in this paper can also be compared with shock models in
order to investigate shock conditions at the surface of IC 443. Using
SITELLE’s data with both the spectral and spatial informations, we can attempt
to associate spatial line intensity variations directly with shock conditions.
The next section is dedicated to the analysis of shock models combined with
our observations. Further comparisons are made between the analysis of Fesen &
Kirshner (1980) and previous modelling of shock-wave spectra.

###  3.2 Comparison with new shock models

Several shock models have been published (Shull & McKee, 1979; Raymond, 1979;
Dopita & Sutherland, 1996; Allen et al., 2008) and proved useful to study the
properties of a great variety of astrophysical objects, especially evolved
supernova remnants such as the Cygnus Loop and IC 443. Despite their
undeniable usefulness, each grid only covers a very limited set of shock
parameters and limit the analysis that can be done for a specific object. To
circumvent this problem, we built our own shock models in an attempt to
reconcile physical condition found in IC 443 and emission line variations
derived from our observations.

We used the shock modelling code mappings v (Sutherland & Dopita, 2017). This
is an updated version of previous releases: mappings iv (Dopita et al., 2013),
mappings iii (Groves et al., 2004a, b; Allen et al., 2008), mappings ii
(Sutherland & Dopita, 1993) and mappings i (Binette et al., 1982; Dopita et
al., 1982; Binette et al., 1985) which were themselves based on an earlier
version of the code created by Michael A. Dopita (Dopita, 1976, 1977; Dopita
et al., 1977; Dopita, 1978, 1979; Dopita et al., 1980). Details concerning the
intricate physics and computational recipes inside the code can be found among
the cited papers.

Modelling shock wave spectra require the knowledge of several shock parameters
used as input during computation. In the next sections, each parameter is
explored independently in order to simplify the overall analysis and
discussion. Comparisons with published shock models are made as well as
comparisons between the conditions derived from previous analysis on IC 443
and those derived from our observations.

####  3.2.1 Preionization

The ionization state of the precursor gas is a strong function of shock
velocity and its value greatly affects the detailed structure of the shock,
hence the prediction of the emissivities as mentioned by Raymond (1979). Since
this is not a parameter that can be determined directly from our observations,
we assume that the shocks present in IC 443 are fast enough to ionise the gas
in front of them and no external source other than shocks exists. The
computation of the pre-ionization was calculated self-consistently by mappings
v as described in (Sutherland & Dopita, 2017). The pre-ionization was
evaluated only once using solar abundance for shock between 50 and 200 km s-1
with pre-shock density of 20 cm-3 and a transverse magnetic field of 1 $\mu$G.
The result is shown in Fig.3. Since the pre-ionization changes very little
under similar shock parameters, we have re-used the same pre-ionization values
function of the shock velocity for all models computed in this paper.
Preventing the evaluation of the pre-ionization for each model allows faster
computation, therefore resulting in virtually the same results concerning
emissivity predictions. The models presented in this paper do not include the
emission emanating from the precursor gas since the shock velocities are not
fast enough (< 150 km s-1) to produce any appreciable emission, as mentioned
in Dopita & Sutherland (1996).

Figure 3: Ionization fraction of H+, He+ and He++ versus the shock velocity used as pre-ionization for all shock models computed and shown in this paper. Table 2: Solar abundances with respect to hydrogen used during model calculation based on the compilation of Asplund et al. 2009, Scott et al. 2014 and Grevesse et al. 2014. The abundance of nitrogen and sulfur was modified according to our observations and explained in section 3.2.5. Atom # | Abundances | log(X/H)
---|---|---
1 | Hydrogen | 0
2 | Helium | -1.07
3 | Lithium | -10.95
4 | Beryllium | -10.62
5 | Boron | -9.30
6 | Carbon | -3.57
7 | Nitrogen | -3.95
8 | Oxygen | -3.31
9 | Fluorine | -7.44
10 | Neon | -4.07
11 | Sodium | -5.79
12 | Magnesium | -4.41
13 | Aluminum | -5.57
14 | Silicon | -4.49
15 | Phosphorus | -6.59
16 | Sulfur | -4.50
17 | Chlorine | -6.50
18 | Argon | -5.60
19 | Potassium | -6.96
20 | Calcium | -5.68
21 | Scandium | -8.84
22 | Titanium | -7.10
23 | Vanadium | -8.11
24 | Chromium | -6.38
25 | Manganese | -6.58
26 | Iron | -4.53
27 | Cobalt | -7.07
28 | Nickel | -5.80
29 | Copper | -7.82
30 | Zinc | -7.44

####  3.2.2 Line intensity evolution

Fast shocks are effective at ionizing the gas; the faster the shock, the
larger the volume of gas is ionized and the longer it will take to recombine.
The recombination time depends on several parameters such as the shock
velocity, the pre-shock density and the abundances. Whatever the set of
parameters, the highest ionization species recombine first, followed by the
lowest. This means that the emission lines intensities vary with time. The
rate at which the recombination occurs depends on many complex processes and
is determined by shock modelling codes such as mappings. As an example, Fig. 4
shows the emission line intensity variation for [O iii] $\lambda$5007, [O ii]
$\lambda\lambda$3727+3729, [N ii] $\lambda$6584, [S ii]
$\lambda\lambda$6716+6731 and H$\alpha$ for a 100 km s-1 shock propagating
into a region with an electron density of 20 cm-3. It can be seen that [O iii]
recombines before H$\alpha$ and at a certain time both emission are
superimposed. This time delay thus leads to a variation of the [O iii]/
H$\alpha$ ratio with time. A high [O iii]/ H$\beta$ is thus a good indicator
of a recent shock and can be used as a compass showing the direction of the
shock front. Fig. 25 clearly shows this phenomenon, where the [O iii]/
H$\beta$ ratio indicates the regions where shocks are the most recent and at
the same time indicates the direction of the blast wave. The same happens with
the [O ii]/ H$\alpha$ ratio (Fig. 24). This effect is far less obvious in our
observations with the [N ii]/ H$\alpha$ and [S ii]/ H$\alpha$ ratios since the
physical separation between [N ii], [S ii] and H$\alpha$ are not as large as
those for [O iii] and [O ii].

It must be mentioned that the emission line structure generated behind a shock
can barely be observed from our observations considering their spatial
resolution. At the distance of IC 443, estimated to be 1.9 kpc (Ambrocio-Cruz
et al., 2017), our seeing-limited spatial resolution of 1′′ corresponds to a
spatial distance of 2.8$\times$1014 cm. With the example given in Fig. 4 and
for similar shock parameters, nearly all of the observed emission lines are
located within a single pixel in our intensity and ratios maps. Therefore, we
suppose that the emission coming from one pixel can be interpreted with the
help of one shock model in which the total emissivities can be summed.

The parameter used in this paper to compute truncated models of incomplete
shocks is the cut-off temperature (Tcut). It corresponds to the temperature of
a region located a certain distance behind the shock at which the computation
of a model has been stopped. A model can be considered complete when Tcut
reaches 1000 K below which no further significant emission in the considered
species is produced. Very few truncated models can be found in the literature
and were computed to match very precise object such as filaments in the Cygnus
Loop (Raymond et al., 1980; Contini & Shaviv, 1982; Raymond et al., 1988) and
a low-excitation Herbig-Haro object (Binette et al., 1985).

The large [O iii]/ H$\beta$ ratios found in some regions of IC 443 suggest the
presence of incomplete shocks and the absence of grids with truncated models
re-enforced the need to compute one specially adapted to IC 443, taking in
consideration the other shock parameters covered in the next sections.

Figure 4: Emissivity profile of [O iii] $\lambda$5007, [O ii]
$\lambda\lambda$3726+3729, [N ii] $\lambda$6583, [S ii]
$\lambda\lambda$6716+6731 and H$\alpha$ versus the distance (bottom) and the
distance (top) for a 100 km s-1 shock propagating into a density of n0 = 20
cm-3 and a magnetic field of B0 = 1 $\mu$G. The abundances used are given
Table 2.

####  3.2.3 Shock velocity

When only observations in the optical regime are available, the shock velocity
can derived from the [O iii] $\lambda$5007 to [O ii] $\lambda$3727 ratio, as
shown by Dopita (1977) (Fig. 7). His figure can be slightly modified to take
into account incomplete shock models. Fig. 4 shows that the [O iii] and [O ii]
emissivities vary differently with time, therefore influencing the [O iii]/ [O
ii] ratio at a given moment. Fig. 5 shows the variation of the [O iii]/ [O ii]
ratio as a function of the shock velocity and fixed cut-off temperature during
model evaluation. Two important points can be made from this figure:

Figure 5: Variation of the [O iii] $\lambda$5007/ [O ii]
$\lambda\lambda$3726+3729 ratio as a function of the shock velocity and cut-
off temperatures for n0 = 20 cm-3 and B0 = 1 $\mu$G. The abundances used are
given Table 2.

  * –

The [O iii]/ [O ii] ratio becomes useful at a shock velocity around 75 km
s-1,coinciding with the emergence of [O iii] $\lambda\lambda$4959,5007
emission. It has to be noted that the velocity at which [O iii] appears depend
on the ionization state of the precursor gas during model evaluation. For
example, a shock propagating into a neutral medium requires at least a 100 km
s-1 shock to produce [O iii]. In either case, fast shock can be identified
directly by observing filaments emitting [O iii], as seen in Fig. 19. Regions
without [O iii], while emitting lower ionization lines such as H$\alpha$, [N
ii] and [S ii], indicate the presence of slower shocks (< 75 km s-1).

  * –

Deriving a shock velocity with the [O iii]/ [O ii] ratio requires a precise
knowledge of the shock age. Even then, when the shock can be considered as
complete (Tcut=103 K), the usefulness of the [O iii]/ [O ii] ratio is limited
to a shock velocity range between 75 and 200 km s-1, faster shocks producing
[O iv] at the expense of [O iii]. Despite these limitations, the [O iii]/ [O
ii] ratio can still be useful to approximate element abundances and will be
discussed in section 3.2.5.

As mentioned above, the absence of [O iii], along with the presence of lower
ionization lines is a good indication of regions being excited by slower
shocks. Another indicator that this is the case is the presence of a
particularly high [S ii]/ H$\alpha$ ratio, as seen in Fig. 28. Shock models
presented in Fig. 6 indicate that shocks with a velocity lower than 30 km s-1
could be responsible for the observed emissions and does not depend on the
pre-shock density.

Figure 6: Variation of the [S ii] $\lambda\lambda$6716+6731/ H$\alpha$ ratio
as a function of the shock velocity and pre-shock densities for B0 = 1 $\mu$G.
The abundances used are given Table 2.

The [N ii]/ H$\alpha$ ratio is also linked to the shock velocity as shown in
Fig 7 alongside with other ratios. We can see from Fig. 27 that the [N ii]/
H$\alpha$ ratios are lower in locations where the [S ii]/ H$\alpha$ are higher
and [O iii] absent, therefore, where the shock velocities are the slowest.

Figure 7: Variation of the [O iii]/ H$\alpha$, [O ii]/ H$\alpha$, [N ii]/
H$\alpha$ and [S ii]/ H$\alpha$ ratios for shock velocities between 50 and 150
km s-1, n0 = 20 cm-3 and B0 = 1 $\mu$G. The abundances used are given Table 2.

####  3.2.4 Pre-shock density

Electron densities can be found using the [S ii] $\lambda$6716/$\lambda$6731
(Weedman, 1968) which are available from our data. Measurements obtained by
others, using [O ii] and [S ii] lines, show clear variations across the
nebula. Measurements of brights filaments (Osterbrock, 1958) and three
positions by Parker (1964) show density fluctuations within the range of
100-600 cm-3 with an average of 350 cm-3. Fesen & Kirshner (1980) found
slightly different values, with electron densities ranging from 100 to 400
cm-3, with an average of 200 cm-3. Using infrared observations, Rho et al.
(2001) found an electronic density of 500 cm-3.

Figure 30 shows the variation of the [S ii] ratio from our observations. The
ratio varies from 0.70 to 1.30, with an average of 1.22. Using pyneb
(Luridiana et al., 2012, 2015) with the atomic data of sulphur taken from the
chianti database version 8 (Dere et al., 1997; Del Zanna et al., 2015), we
calculated the corresponding electronic densities, which translate a minimum
and maximum being respectively 106 and 2477 cm-3. These values are in
agreements between previous measurements, but ours show regions with
significantly higher electronic densities. Most of them are located in the
northeastern part of IC 443, but also in some localised filaments as well.
This indicates that the blast wave is encountering a higher pre-shock density
in those regions.

Values of the pre-shock density can be estimated with the help of Fig. 8.
Assuming a constant magnetic field of 1 $\mu$G throughout IC 443, the pre-
shock density can be estimated when the shock velocity is known. Using the [S
ii] $\lambda$6716/$\lambda$6731 ratio, and [O iii]/ [O ii] ratio to determine
the shock velocity, we found that most pre-shock densities vary between 20 and
60 cm-3. These values are higher than those measured by Fesen & Kirshner
(1980), but this is not surprising since we measured higher electronic
densities. We could not measure lower pre-shock densities since the velocity
could not be measured at all locations. Measuring the pre-shock density for
slow shocks (< 40 km s-1) could prove challenging since the [S ii]
$\lambda$6731/$\lambda$6716 ratio then varies very little, as shown in Fig. 8.
This effect can be seen by comparing the locations of high [S ii]/ H$\alpha$
ratio (slow-shocks-Fig. 28) and the [S ii] ratio map (Fig. 30), where no clear
relation can be seen between [S ii]/ H$\alpha$ line and [S ii]
$\lambda$6731/$\lambda$6716 line ratios morphologies.

Figure 8: Variation of the [S ii] $\lambda$6731/$\lambda$6716 ratio as a
function of the preshock density and the shock velocity for solar abundances
and B0 = 1 $\mu$G. This figure was made using 16384 individual shock models
evaluated with mappings.  Figure 9: Diagnostic diagram of [O ii]/ H$\alpha$,
[O iii]/ H$\alpha$, [N ii]/ H$\alpha$ and [S ii]/ H$\alpha$ versus [O iii]/ [O
ii] ratio. The point cloud represent the observations and the colored lines
shock models evaluated with different metallicities for shock velocity between
70 km s-1 and 200 km s-1, n0 = 20 cm-3 and B0 = 1 $\mu$G.

####  3.2.5 Abundance range estimates of O, N and S

With the emissions of [O iii], [O ii], [N ii], [S ii] and H$\alpha$, it is
possible to estimate the probable abundances of oxygen, nitrogen and sulphur.
In order to achieve this, it is required to estimate the shock velocity, which
can be derived from the [O iii]/ [O ii] ratio as mentioned in section 3.2.3.
Inspired by the work of Hester et al. (1983), the [O iii]/ [O ii] ratio can be
used as an abscissa while the [O ii]/ H$\alpha$, [O iii]/ H$\alpha$, [N ii]/
H$\alpha$ and [S ii]/ H$\alpha$ can be used as the ordinate. This
configuration allows putting a physical meaning on each axis, the abscissa,
the shock velocity and the ordinates a mean to adjust the abundances. This
convenient setup works only under some circumstances which are going to be
explained in detail in this section.

Fig. 9 shows diagrams using the axes configuration described above, showing
both the observations made with SITELLE (grey dots) taken from the ratio maps
and complete shock models calculated with different abundances (coloured
lines). We used a fixed pre-shock density of 20 cm-3 and a magnetic field of 1
$\mu$G for all models. It has to be noted that a different pre-shock density
would have little effect on the models shown in Fig. 9 as long as the selected
value remains inside the given pre-shock density range measured in IC 443 from
the [S ii] ratio in section 3.2.4. The shock velocity chosen run from 70 km
s-1 to 200 km s-1 varying from left to right on the diagram. By selecting the
[O iii]/ [O ii] ratio, only fast shocks were selected at the expense of slower
shocks. Therefore, the data points do not include all the observed spectra
with SITELLE.

An interesting feature coming from these diagrams is how well the slope on the
[O iii]/ H$\alpha$ diagram is reproduced, clearly showing the relation between
the intensity of [O iii] and the shock velocity. On the contrary, the slopes
on the other diagrams ( [O ii]/ H$\alpha$, [N ii]/ H$\alpha$ and [S ii]/
H$\alpha$) are very low, meaning that these ratios are affected very little
inside a narrow shock velocity range ($\approx$ 80 to 100 km s-1). This
feature can be exploited in order to grossly estimate plausible abundance
ranges of O, N and S, since the [O ii]/ H$\alpha$, [N ii]/ H$\alpha$ and [S
ii]/ H$\alpha$ vary linearly with abundances change. This method is by no
means to be used to determine precisely the abundances in IC 443. The
variation of [O ii]/ H$\alpha$, [N ii]/ H$\alpha$ and [S ii]/ H$\alpha$ are
most likely the result of an intricate combination of several other physical
parameters and will be discussed in section 3.3. What can be said with
reasonable certainty with these diagrams is that some abundances could be
discarded. For example, an abundance of nitrogen with log(N/H)=-4.17 seems to
be a little high; as for sulphur, log(S/H)=-4.88 seems a little too low for
the shock velocity range considered. As for oxygen, it seems that a wide range
of abundances could explain the observed emissions.

The derived abundances range derived from Fig. 9 are somewhat close the solar
values determined by Asplund et al. (2009) with the exception of nitrogen and
sulphur being too low with values evaluated at log(N/H) = -4.17 and log(S/H) =
-4.88. As for the solar abundance of oxygen, its derived value of log(O/H) =
-3.31 could explain the observed spectrum. The abundances adopted in this
paper from helium to zinc (excepting N and S) were taken from the compiled
solar abundances of Asplund et al. (2009), Scott et al. (2015a), Scott et al.
(2015b) and Grevesse et al. (2015). The abundance of nitrogen and sulphur were
chosen from Fig. 9 fitting the model on the point clouds and the list of
abundances used to compute all presented models is given in Table 2.

###  3.3 Bi-shock models with multiple parameters

Previous sections have demonstrated that many shock parameters are responsible
for the observed spectra in IC 443. So far in this paper, as well as in
previous studies, the observations were mainly compared with single shock
models that include only one set of parameters. While this rather simplistic
technique allows to derive important parameters, such as the shock velocity,
pre-shock density and even approximate the abundances, it does not always
explain all the observed spectra. Fig. 9 exposes several flaws when it comes
to compare shock models with observations too simplistically. The most obvious
one is the inability to reproduce all observations above log( [O iii]/ [O
ii])$\approx$-0.22. The presence of elevated ratios can be explained by the
presence of incomplete shocks, as mentioned in section 3.2.3 (Fig. 5), which
demonstrates the sensitivity of the [O iii]/ [O ii] ratio to shock
completeness. However, it does not suffice to run a grid of truncated models
in order to improve the match between the models and observations. Fig. 9
shows that several shock models with different abundances allow the
overlapping of models over observations. While these diagrams allow
approximating the plausible abundances in IC 443, it does not explain the
reason for the point dispersion and therefore the origin of the great variety
of line ratios observed in IC 443. The variation of abundances alone would not
make sense. Ideally, each spectrum should be represented by one model with its
very own set of parameters.

The filamentary structure in IC 443 is pretty intricate with many filaments
criss-crossing, as viewed from the observer’s point of view. Upon close
inspection, many distinct interlaced filaments show different intensities for
a given emission line. As we show in section 4, the receding and approaching
components of the expanding bubble show very different morphological
structures, suggesting that shock conditions on a given line of sight could be
different and that the summation of both components on the same line of sight
can lead to a spectrum that cannot be interpreted with the help of a single
shock model. Therefore, many locations obviously required more than one shock
model.

In an attempt to better reconcile models and observations, we have computed a
grid of 57,032 shock models with the parameters and intervals listed in Table
3. The shock velocity range was chosen in order to reflect the measurements
presented in section 3.2.3 and the pre-shock densities with those derived in
section 3.2.4. We assume that they are shocks with different ages, as
mentioned in section 3.2.2. We also assume that the abundances do not vary
significantly in the observed region and selected one set of abundances as
determined in section 3.2.5.

Table 3: Shock parameters used to compute a grid of 57032 models combined with abundances given in table 2. Shock parameters | Value range | Interval
---|---|---
Shock velocity | (20 - 150) km s-1 | 1 km s-1
Pre-shock density | (14 - 60) cm-3 | 2 cm-3
Cut-off temperature | (1000 - 20000) K | 500 K
Magnetic field | 1 $\mu$G for all model |

To simulate the effect of multiple filaments on the same line of sight, we
have to sum a given number of distinct shock models according to the number of
components. Since we do not always have a clear idea about the exact shock
parameters for each component, we can generate a new grid of models based on
another grid using permutations. The total number of permutations of such
shock models is given by the following equation:

Figure 10: Bi-shock models in black on top of observations in gray. The
combination of shock parameters for the shock velocity, cut-off temperature
and pre-shock density are shown in a histogram in Fig.11. The abundances used
are shown in Table 2

$C\left(n,r\right)=\binom{n}{r}=\frac{\left(r+n-1\right)!}{r!\left(n-1\right)!}$
(1)

where $n$ is the total number of shock models available in a grid and $r$ the
number of models to be summed, or in other words the number of components on
the same line of sight. Fabry-Perot observations conducted by Ambrocio-Cruz et
al. (2017) have indeed shown that up to four velocity components can be found
along the line of sight at some positions in the nebula.

From equation 1, we see that the total number of permutations can become very
large when $r$ is significantly larger than 2, depending on the total number
of shock models available in a grid. Since our observations do not allow us to
determine the exact number of components at every point in IC 443 and due to
computational limitations, we choose a value of $r=2$. This leads to a total
number of of 1,626,353,028 models permutations which were systematically
compared with every spectrum shown in Fig. 10 (gray points) made from 35,479
spectra taken from our ratio maps. In total, 5.3$\times$1013 comparisons where
made and models who predicted all 10 line ratios presented in Fig. 10 with a
margin of error smaller than 5% were kept. The result is given on the same
figure (black points) which shows 16,215 models overlapping the observations.
About 45 % of the selected spectra where fitted by our models. Most of the
spectra can be represented by more then one model with different parameters
leading to degeneracies.

Fig. 11 shows the frequency range of parameters that best fit the
observations. These diagrams were made including the degeneracies. From that
figure, we see that the spectra could be explained by the presences of shocks
velocities range between 20 and 140 km s-1 with a clear preference toward
velocity around 75 km s-1. Interestingly, there is a dip around that velocity
where there should be a peak. This can be explained by the fact that the [O
iii]/ [O ii] ratio quite steeply above 75 km s-1 as shown in Fig. 5. Any small
error on the [O iii]/ [O ii] ratio can easily "shift" the velocity on the
other side of the dip as shown in Fig. 11. The same figure also shows a
variety of cut-off temperatures, and thus shocks with different ages, favoring
complete shocks. Models tend to favour pre-shock densities around 20 cm-3 but
lower densities could not be excluded.

The most important result from Fig. 10 is that the combination of two shock
models allows us to reproduce some observed spectra with log( [O iii]/ [O
ii])$\geq$0.60, which were could not be explained with a simple model. This
suggests that the individual spectra are indeed the combination of at least
two components along the line of sight. Despite this success, some observation
could not be reproduced. For example, there is a clear gap in the upper left
for the [S ii]/ H$\alpha$ diagram and the lower left for the [N ii]/ [S ii]
diagram in Fig. 10 for which none of our summed models were able to fit the
observations. However, single shock models can easily reproduce the observed
ratios. It is probably the case with many other spectra for which one or more
than three components can potentially provided a better fit to the spectra.
However, more components will undeniably bring more degeneracies at which
point the shock parameters found might lose their meaning and cannot provide
additional informations.

Figure 11: Histogram showing the distribution of the combination of two models
for the shock velocity (top), cut-off temperature (middle) and pre-shock
density (bottom) shown in Figure 10.

.

It seems that one set of abundances is sufficient in order to reproduce the
line intensities dispersion. In order words, that is no need to vary the
abundances for specific spectrum in order to better fit the observations. We
tested this with different abundances of O, N and S around the ranges that can
be inferred from Fig. 3.2.5 and saw no significant improvements. However, we
noticed that an abundance of oxygen closer to solar value was preferable in
order to better fit the observations.

We notice however high [N ii]/ H$\alpha$ ratios (Fig. 27 south of the field of
view) which might suggest a region enriched with nitrogen. Since the [O iii]/
[O ii] ratio cannot be obtained for these locations, we were unable to
estimate a shock velocity and therefore accurately determine the nitrogen
abundance. However, the absence of [O iii] for most of the location where [N
ii]/ H$\alpha$ is elevated suggests shocks slower than 75 km s-1. In that
case, models calculated with a nitrogen abundance close to solar are unable to
reproduce the observed [N ii]/ H$\alpha$ and higher abundance must be
considered. Figure 12 shows the variation of the [N ii]/ H$\alpha$ ratios in
function of different nitrogen abundances. It can been seen that for given [N
ii]/ H$\alpha$ ratio an accurate determination of the abundances is very
sensitive to the shock velocity. When the velocity is unknown, multiple
plausible abundances could be responsible for a unique ratio.

From figure 27 we measured a ratio of [N ii]/ H$\alpha$ as high as 1.2. From
figure 12 we can see that a shock velocity around 75 km s-1 with an abundance
of nitrogen of 3 times the solar could explain the observation. However, if
the shock velocity is slower, a higher abundances of nitrogen must be
considered.

The location of this enrichment in nitrogen is interesting since it is located
toward the center of the nebula and might be ejecta from the progenitor which
was blown outward by its stellar winds prior to its explosion.

Figure 12: Variation of the [N ii]/ H$\alpha$ ratio for different nitrogen
abundances vs the shock velocity. 1$\times$solar corresponds to a nitrogen
abundance of log(N/H) = -4.17. The other abundances used to compute the curves
are given in table 2

.

##  4 Kinematics

A few measurements of the radial velocities across IC 443 have been obtained
using long-slit spectroscopy (Lozinskaya & Esipov, 1969) and Fabry-Perot
interferometer (Lozinskaya, 1969; Pismis & Rosado, 1974; Lozinskaia, 1979;
Ambrocio-Cruz et al., 2017). The radial velocities measured by the various
observations reported velocities ranging from approximately $-180$ to $+135$
km s-1 and vary quite significantly across the nebula.

While our primary intent was not to study the kinematics (hence the relatively
low spectral resolution of our cubes: 130 km/s/channel at H$\alpha$), the high
signal-to-noise of the SN3 cube, as well as the relatively similar intensity
of four emission lines within the limits of this filter have allowed us to
separate the ionized gas into two distinct and morphologically very different
components. Two approaches, leading to very similar results, were followed.

The first one is very simple: the spectral sampling and the expansion velocity
of the nebula are such that emission from the nebula is visible on three
consecutive frames of the data cube for each line: H$\alpha$-red,
H$\alpha$-green and H$\alpha$-blue; [N ii] $\lambda 6584$-red, and so on. The
line centroid (and hence maximum emission) does not always coincide with the
middle frame and the balance between the red and the blue components is not
identical from one line to the next. However, combining all the red frames
together and doing the same for all the blue frames results in two emission-
line maps sampling velocities separated by 260 km s-1. These images, which
essentially sample the blue and red wings of all emission lines, are shown in
Fig. 13, and a color composite is presented in Fig. 14 . While the upper left
part of the nebula in this field looks pretty similar in the red and blue
channels, the red and blue components of the lower right part, closer to the
center of the supernova remnant and thus where the Doppler shift is expected
to be larger, are dramatically different. While this method is relatively
straightforward and provides a good glimpse at the three-dimensional structure
of the nebula, it does not offer a quantitative distribution of the velocities
at play.

We have therefore used ORCS to quantify this. Among the fitting procedures
offered by ORCS, two are particularly adapted to the IC 443 data: as a first
approach, the lines can be fitted using a sincgauss function (Martin et al.,
2016), which is a convolution of the instrument line shape – a sinc – with a
gaussian function characterizing the Doppler broadening of the line; or they
can be fitted with two sinc components attributable to two distinct
(approaching and receding) components. As expected, the sincgauss fit revealed
a clear difference between the upper section of the field, where the sincgauss
fit was not significantly different from a single sinc fit (given the low
spectral resolution of our cube), and the lower right part, where a
significant broadening – between 75 and 200 km s-1– was clearly detected. We
have thus targeted a $7.3\times 4.5$ arcmin region in the lower right corner
of the field (see Fig. 13) to perform a two-component sinc fit. While ORCS
allows to attribute different velocities to the individual lines, we chose to
force all the lines present in the SN3 filter to share the same velocity. An
example of these two fits on the spectrum of a single pixel is presented in
Fig. 15, and the maps resulting from individual fits to the entire region are
shown in Fig. 16. The result is strikingly similar to that using the simple
approach described above. It is important to note that the limitations of this
fitting technique become apparent in the upper left section of the blue
component, where very subtle, but nevertheless present, stripes are visible.
In this region, the width of the sincgauss function approaches 50 km s-1, one
fourth of the instrumental line function’s width. An histogram of the
velocities resulting from the fits is presented in Fig. 17: the two components
are clearly separated, by an average of 150 km s-1. While the receding
component of the nebula presents a simple distribution, the approaching one is
clearly bi-modal. There is also a very obvious difference in the morphologies
of the two components (Fig. 16): the approaching side (very likely the front
of the nebula, closer to us) presents small-scale honeycomb-like structures,
while the other side (presumably the back, more distant) is smoother and more
regular. This dichotomy very likely reflects the structure of the interstellar
medium in which the blast wave is expanding (such as the pre-shock density).
We also note that the combined flux of all the SN3 emission lines is twice as
large in the red component as in the blue one.

While ORCS also provides the flux in the individual lines for both components,
we do not wish to over-interpret our rather low spectral-resolution data at
this time, but note that a cube with a higher spectral resolution could
provide a means to determine the physical properties of both components
individually. In the previous sections, we have demonstrated that a single
shock model could not reproduce the observed line ratios. The different
morphologies of the red and blue components indeed suggest a dichotomy in
their shock properties.

Figure 13: Red and blue components of all emission lines present in the SN3
filter, obtained by selecting the frames in the data cube corresponding to the
wings of the lines (see text). The region which was analyzed kinematically
with ORCS is indicated by a white rectangle. Figure 14: Color-coded
combination of the two images shown in Fig. 13. Figure 15: Example of the
fitting functions used by ORCS, for the [S ii] 6717,31 doublet in a single
pixel. On the top left panel, a sincgauss function was tried, returning a
velocity dispersion, $\sigma=120$ km/s. On the top right, the result of two
sinc functions. The lower panel shows the two components. Figure 16: Red and
blue components in the SW region ($7.3^{\prime}\times 4.5^{\prime}$) of the
SITELLE field of view, as extracted with ORCS (see text). The similarity
between these images and Fig. 13 is striking. The limitations of this method
are however visible in the upper left corner of the blue component image,
where low-amplitude fringes can be seen. Figure 17: Histogram of the
velocities obtained with ORCS in the region shown in Fig. 16.

##  5 Concluding remarks

We used the optical imaging Fourier transform spectrometer SITELLE to obtain
multi-spectral cubes of the north-eastern part of IC 443. From these cubes, we
generated maps allowing the investigation of the spatial variations in
spectral line intensity of seven emission lines: H$\alpha$, H$\beta$, [O ii]
$\lambda$3726+3729, [O iii] $\lambda$5007, [N ii] $\lambda$6583 and [S ii]
$\lambda$6716,6731. We compared the observations with shock models computed
with mappings and estimates important physical properties found in IC 443.
Specific results are as follows:

  1. 1.

The reddening has been measured from the H$\alpha$/ H$\beta$ ratio for the
observed region and found to vary significantly at its surface with E(B-V) of
0.8-1.1.

  2. 2.

The electron density observed in the filaments ranges from less than 100 to
2500 cm-3 with higher densities found at the extremity of the north-eastern
part of IC 443.

  3. 3.

The line intensities vary quite significantly across the field of view and can
be traced to the presence of different shock velocities at different stage of
recombination. High [O iii]/ H$\beta$ and [O ii]/ H$\beta$ ratios are higher
towards the direction where the shock front is propagating, due to the
presence of incomplete shocks. High [S ii]/ H$\alpha$ with no emissions in [O
iii] toward the center of the observed region indicates the location of shocks
with velocity slower then 40 km s-1.

  4. 4.

Shock models with a unique set of parameters are unable to reproduce the great
majority of the observed emissions. The summation of two shock models (bi-
models) with different shock parameters allow a better prediction of the
variability of the observed emission line ratios. Bi-models comparisons allow
to constraint the probable shock condition inside IC 443 for the observed
region. The most probable shock velocities varying from 20 to 150 km s-1 with
75 km s-1 being the most common velocity found, the pre-shock density between
20-60 cm-3 favouring lower densities and shocks at different stages of
recombination as suggested by wide range of the [O iii]/ H$\beta$ ratios.

  5. 5.

A single set of abundances close to solar values combined with varying shock
velocity, pre-shock density and shocks with different ages are sufficient to
explained the measured ratios.

  6. 6.

Despite the modest spectral resolution of the SN3 cube, we were able to
separate the approaching and receding components of the expanding nebula.
These two components harbor very different morphological structures, probably
resulting from different properties of the interstellar medium in which the
blast wave propagates. Higher spectral resolution data cubes across the
visible range with SITELLE would allow us to better probe the physical
condition on both sides of the nebula.

## Acknowledgements

Based on observations obtained with SITELLE, a joint project of Université
Laval, ABB, Université de Montréal, and the Canada-France-Hawaii Telescope
(CFHT) which is operated by the National Research Council of Canada, the
Institut National des Sciences de l’Univers of the Centre National de la
Recherche Scientifique of France, and the University of Hawaii. The authors
wish to recognize and acknowledge the very significant cultural role that the
summit of Mauna Kea has always had within the indigenous Hawaiian community.
We are most grateful to have the opportunity to conduct observations from this
mountain. AA thanks CONACyT-CB2015-254132 project for supporting his postdoc
position at IA-UNAM. Models were performed using computers from projects
DGAPA/PAPIIT-107215 and CONACyT-CB2015-254132. LD is grateful to the Natural
Sciences and Engineering Research Council of Canada, the Fonds de Recherche du
Québec, and the Canada Foundation for Innovation for funding. LD is very
grateful to the Department of Physics and Astronomy, University of Hawaii
Hilo, as well as the CFHT, for having provided him with an ideal and very
enjoyable working environment during his sabbatical stay on the Big Island.

All computations from data reduction to analysis were conducted using open
source software : python programming language and the GNU Data Language.
Special thanks to the people who developed and maintain vital libraries used
in this work: numpy & scipy (van der Walt et al., 2011), cython (Behnel et
al., 2011) and matplotlib (Hunter, 2007). This work also made extensive use of
the gnu parallel shell tool (Tange, 2011).

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##  Appendix A Figures

Figure 18: The observed region in [O ii] $\lambda$3727+$\lambda$3729\. FOV is
11${}^{\prime}\times 11^{\prime}$ with North at the top and East to the left.
Figure 19: Same as Fig. 18, for [O iii] $\lambda$5007. Figure 20: Same as Fig.
18, for H$\alpha$. Note the presence of two emission-line stars near the
central part of the image. Figure 21: Same as Fig. 18, for [N ii]
$\lambda$6583. Figure 22: Same as Fig. 18, for [S ii]
$\lambda$6716+$\lambda$6731. Figure 23: Extinction map of north-eastern part
of IC 443 using the H$\alpha$/ H$\beta$ line ratio. The map was smoothed with
a Gaussian distribution with $\sigma$= 5.0 arcsec.  Figure 24: Map of the [O
ii] $\lambda$3727+$\lambda$3729/ H$\alpha$ line ratio. Background image :
H$\alpha$ emission. Figure 25: Same as Fig. 24, for [O iii] $\lambda$5007/
H$\beta$. Figure 26: Same as Fig. 24, for [O iii] $\lambda$5007/ [O ii]
$\lambda$3727+$\lambda$3729. Figure 27: Same as Fig. 24, for [N ii]
$\lambda$6583/ H$\alpha$. Figure 28: Same as Fig. 24, for [S ii]
$\lambda$6716+$\lambda$6731/ H$\alpha$. Figure 29: Same as Fig. 24, for [N ii]
$\lambda$6583/ [S ii] $\lambda$6716+$\lambda$6731. Figure 30: Same as Fig. 24,
for [S ii] $\lambda$6731/ [S ii] $\lambda$6716.