The Near-Infrared Polarization of the Pre-Planetary Nebula Frosty Leo

The intensity emission (fig. 1, left panel) reaches its peak at the central star location and then it decreases almost evenly in all directions, except at tips of the bipolar lobes, where a small increase is detected.
The NIR polarization in Frosty Leo was previously studied and reported by Dougados et al. (1990) in J band, although their research was only focused whithin a radius of  5″. They found $P\%$ as high as  $40\%$ in the lobes close to the central star and a low polarized region across the equator. Our polarization map is in good agreement for the zones in common and extend this study to 3 times more radius.
The first noticeable feature in our maps is the extended envelope (EE) that protrudes from the well-known bipolar lobes of this object. Part of the EE is also seen at visible wavelengths, Clemens & Leach (1987) detected it also employing imaging polarimetry. However, our new POLICAN data reveal a larger size. To our knowledge, it is the first detection of this extended feature in the NIR. Since this object is fainter in the J band, the shape of the EE looks elliptical at this wavelength. However, based on our data from the H band we conclude that the actual shape is more likely spherical (see fig. 2). We did not see any variation of the $P\%$ within the EE and we estimated an average of $16\%$. The polarization vectors related to this EE show a clear east-west alignment, and the average PA of these vectors is $79^{\circ}\pm 10^{\circ}$.
Whitin the bipolar outflows, the polarization vectors follow a centrosymmetric pattern. Such vector behavior suggests single scattering as the polarizing mechanism and that the optical depth in the media is low (Gledhill et al., 2001; Gledhill, 2005). Previous observations at visible wavelengths obtained similar results (Scarrott & Scarrott, 1994). Within a radius of 7″ we resolved 3 clumps (2 northwards and 1 southwards from equator), these structures are characterized by their $P\%$ which are between $30\%$ to $40\%$. A narrow and relatively low polarized area is seen between the north clumps. It could be identified with the elongated region found by Murakawa et al. (2008) in H and K bands. The authors attributed the low polarization level to a low dust density, which is in good agreement with our interpretation. Another two clumps are found at the tips of the bipolar jets and because of their appearance, we labeled them NS and SS as in north shell and south shell, respectively. They seem to match the location of the ansae remarked by Sahai et al. (2000) and their $P\%$ ranges between $40\%$ and $62\%$.
Along the equator, there is a narrow region of low polarization, often associated with a disk (Rounan et al., 1988; Hodapp et al., 1988; Dougados et al., 1990; Murakawa et al., 2008). Some of the vectors in this region are aligned and parallel to the equator, these vectors are localized between 2 spots of almost zero polarization: the east local minima (ELM) and the west local minima (WLM) (see fig. 1 right panel). Murakawa et al. (2008) found similar results in H and K bands and also suggested the dichroic mechanism as a possible explanation. A dichroic interpretation would indicate the presence of non-spherical grains with their longer axes preferentially aligned. The electric vector in the transmited beam is thus perpendicular to that preferential direction (Lazarian, 2003; Whittet, 2004; Lazarian, 2007; Crutcher, 2012).
On the eastern edge of the equator, we notice an "arm-like" region of 7″ length that is highlighted by its low polarization ($11\%$) and is directed towards the south. A low polarization trend is also seen towards the northwest, but it looks more diffuse and rapidly reaches our detection limit. Sahai et al. (2000) labeled a structure as northwest lobe in that same direction.

The polarization maps in H band are shown in fig. 2. The total intensity of Frosty Leo in H band follows a trend quite similar to that in J band. Except for the low polarized arm-like area and the equatorial local minima, the polarization in H band looks quite similar to that in the previous section. We emphasize that polarization features such as the EE and the north and south shells are first time detections in the NIR.
Although our resolution is only $1$″, our results are in good agreement with those shown by Murakawa et al. (2008) (adaptive optics psf of $0.16\arcsec\times 0.28\arcsec$ in K band). Whithin a radius of $7$″, the $P\%$ reaches between $40\%$ and $50\%$ in the clumps nearby the central star, and a depolarized fringe along the equator shows an average of  $11\%$. As in J band, some vectors in this fringe are parallel to those in the EE. This depolarized region seems to extend towards the east and west almost as if splitting the whole nebula in half.
Although slightly distorted by the eastern depression, the EE in H band seems mostly spherical and considerably larger than in the J band image. Such difference in size is probably a matter of detection limit since Frosty Leo is brighter in H band by almost one magnitude (see table 1). The diameter of the EE is of 35″ and the vectors linked to this area are clearly aligned on east-west direction. We measured an average PA is $74^{\circ}\pm~{}10^{\circ}$ and the $P\%$ has a mean value of $20\%$, which is slightly larger than in J band (see table 2).
Along the major axis, the vectors follow a centrosymmetric pattern and the NS and SS are clearly seen in polarized light. The $P\%$ ranges from $36\%$ to $70\%$ and their appearance is more spreaded out than in J band. The NS seems to smear down southwards, almost reaching the internal clumps.

K’ band polarization map of Frosty Leo is shown in figure 3. This is (to our knowledge) the first polarization map presented in this band for this object. Given our detection limit, the EE is not seen in this map and only the bipolar nebula was clearly detected. Nevertheless, it still provide some new information on this object. As expected, most of the vectors in the map follow a centrosymmetric pattern. The exception is a small equatorial spot (ES) where several vectors are perpendicular to the nebula’s axis. Although not entirely, the ES seems to match the location of some aligned vectors in the J and H images, this is better seen in fig. 4. The vectors in the ES show an almost constant position angle of $80^{\circ}$ whilst the average degree of polarization is $15\%$.
The polarized structure in this band only shows two clumps near the equator. The "missing clump" seems to be merged with the NS. This could be an effect of poor resolution since the PSF in this band is degraded to ${\sim}$1.5″. The polarization in the central clumps range from $25\%$ and $35\%$.
As in the previous maps, the highest $P\%$ levels are found in the NS and SS, in some points even reaching or exceeding the $80\%$. These high P$\%$ levels are not uncommon since other PPNe such as OH 231.8+4.2 shows similar levels and this is in agreement with a scattering process (Ageorges & Walsh, 2000). The NS looks more extended and is also smearing southwards even more evidently than in H band. The SS is less evident given the limits of our map. Nevertheless, a large amount of material is seen at the edge of the southern side of the map and it could still be associated to the SS.

AGB stars produce a CSE through a slow and tenuous wind (Kwok, 1993), the moderate rate of mass ejection through the AGB-wind is about $10^{-5}M_{\odot}yr^{-1}$ (Loup et al., 1993). At the end of the AGB lifetime, the mass-loss rate and the velocity of the expelled material increase drastically (up to ${\sim}10^{-4}M_{\odot}yr^{-1}$ e.g. Bujarrabal 1999; Bujarrabal et al. 2001). This period of accelerated mass-loss is expected to last a few thousand years (Kaufl et al., 1993) and is known as the superwind phase.
The global shaping of the nebula must occur at this stage of increased mass-loss (Lagadec, 2016). A noteworthy feature seen in many nebulae are the ansae. Ansae are double knot-like structures along the nebula’s axis (Aller, 1941). These knots move in opposite directions and often, their speed is faster than the previously AGB ejected material (e.g. Patriarchi & Perinotto 1991; Redman et al. 2000; Sahai & Patel 2015). The mechanism that leads to ansae formation is still unclear, but, since they are oriented along the major axis, it is probable that they are the remnants of outflow collimation. The most likely sources of collimation are the mass-transfer between a binary system (e.g. Soker & Rappaport 2000; Jones & Boffin 2017), or a magnetic field (also driven by a binary companion) (Nordhaus & Blackman, 2006; Nordhaus et al., 2007). Once collimation has taken place, ansae could be formed by diverse mechanisms (see Balick & Frank 2002 for a discussion of ansae formation).
Although their origin is not fully understood, ansae (or shells) can be used to constrain the superwind time $t_{sw}$ (Gledhill et al., 2001). The outer and inner edges of a shell should indicate the beginning and cessation of mass-loss. Thus, by using the shell’s angular size and assuming expansion velocities from molecular line measurements, we can estimate the superwind time (Meixner et al., 1997; Gledhill et al., 2001).
The bipolar regions within the nebula are optically thin (centrosymmetric pattern), thus, the NS and SS are physical structures illuminated by a central source (Wolstencroft et al., 1986; Dougados et al., 1990; Gledhill et al., 2001). Since these shells are symmetric, detached from the central region and spatially correlated with ansae (see fig. 4), we are assuming that they are the remains of a superwind event.
Our polarization maps show that the NS and SS cover an area of 51 and 42 $arcsec^{2}$, respectively. By inspection of the J band map, where the resolution is better, the average angular size of the shells (along the polar axis) was found to be $8\arcsec$. At a distance of 3 $kpc$ (Mauron et al., 1989; Robinson et al., 1992), their length would be of ${\sim}0.1~{}pc$. To estimate the superwind time $t_{sw}$, we used the CO speed component of 50 $km~{}s^{-1}$ measured by Sahai et al. (2000) and an inclination angle of $15^{\circ}$ (Roddier et al., 1995):

.
where, $l_{sh}$ is the length of the shell and $V_{exp}$ is the projected expansion velocity of the CO along the axis. Given the assumption of a constant speed when it actually may vary with time, this estimation should be considered as a lower limit. The computed error is $200~{}yr$ and it is mostly due to the assumed distance to the object. Nevertheless, this estimation is not unrealistic when compared with the expected duration of this phase ($t_{sw}$   $10^{3}-10^{4}$ $yr$ (Kaufl et al., 1993)).
If a binary is involved in the shaping of the nebula, a common envelope evolution could provide enough rotation to start a dynamo effect. This dynamo would be able to produce an explosive outflow from which ansae could be created (Nordhaus & Blackman, 2006; Nordhaus et al., 2007). Considering this, Nordhaus & Blackman (2006) derived a burst time of ${\sim}100~{}yr$ for the ansae in the planetary nebula NGC 7009. This time seems short when is compared with the $600~{}yr$ that we calculated in the shells of Frosty Leo. However, since Frosty Leo is a younger object and its distance is somewhat uncertain, it might be worth it to investigate if a dynamo could have produced the ansae with further modelling.
A computational model that could fit better with our results was proposed by Steffen et al. (2001). The so-called "stagnation knots" model requires a time-dependent jet that drives into a CSE forming knot-like structures (shells). At some point, the stream of material will stop and the shell slows down as it expands and continues adding mass from the envelope. This model can reproduce morphologies and speeds ($240~{}kms^{-1}$) similar to those we inferred in the NS and SS for an assumed distance of 3 kpc (see model b from Steffen et al. 2001), although, the time required to form shells is a bit larger (${\sim}1100~{}yr$) than our $600~{}yr$ calculation. However, if we regard that our estimation is a lower limit, both ages are still comparable.

The extended envelope in Frosty Leo and its geometry are seen in J and H bands (figs. 1 and 2). The largest angular size of the faint emission was detected in the H band where it shows a diameter of 35″. This EE can be understood as the remains of the previous AGB phase ejecta. Since the halo’s shape seems spherical, the mass-loss in the first stages was probably isotropic. The polarization from J and H bands did not reveal new structures or trends. Nevertheless, the most noteworthy feature is the well-defined alignment from most of its polarization vectors (79^∘ and 74^∘, in J and H bands, respectively).
Gledhill (2005) proposed that the vector alignment is caused by an optically thick media, under this assumption, most of the material in the nebula should concentrate at the equator in a dense disk. The disk is then responsible for the high optical depth and the alignment distribution. However, a high optical depth would have prevented us from detecting any structure in the bipolar regions, thus, this interpretation does not stand for the EE.
Since the EE (35″) is larger than the resolution of our data (1″), the PSF smoothing effect is also an unlikely answer (Piirola et al., 1992; Gledhill, 2005).
Whitney & Wolff (2002) showed that single scattering could also produce non-symmetric polarization, for this, the grains should be aligned by a magnetic field. In this model, a single source illuminating an optically thin cloud of non-spherical grains can account for some asymmetries in the polarization vector pattern. This could fit with our observational results assuming that the EE is illuminated directly by the star but the grains need to be coherently aligned by a strong magnetic field. Observations of OH Zeeman splitting and submillimeter polarimetry have shown magnetic fields of the order of mG in post-AGB stars (Bains et al., 2003; Sabin et al., 2007). However, for this object, more observational evidence is required.
Multiple scattering is another alternative. This could occur when an envelope is not directly illuminated by the star, but by the light that has been already scattered. Previously scattered light reaching the envelope, could be scattered again by the material within and this results in an aligned polarization pattern (Bastien & Menard, 1988; Fischer et al., 1994; Scarrott & Scarrott, 1995; Wolf et al., 2002). A scenario where multiple scattering could account for our observational results is the following: The bipolar lobes are directly illuminated by the central star and scatter the light as a first event. Then, this light later reaches the EE and following scattering events could occur and result in the aligned polarization pattern. This scenario is quite similar to that proposed by Bastien & Menard (1988), only in their case, the subsequent scattering events occur in a circumstellar disk.
We favor an interpretation in which the EE is illuminated by scattered light coming from the lobes. This is the more likely physical situation for the EE since the scattering dominates in the NIR (Scarrott et al., 1991; Bains et al., 2003).

The polarization maps from Murakawa et al. (2008) show a spot of aligned vectors located at the equator in both, H and K bands. The suggested mechanism of polarization in this area was the dichroism. A dichroic interpretation (directional extinction) requires aligned non-spherical grains. Such grains should tend to rotate with their longer axes orientated perpendicularly to the angular momentum vector. If a magnetic field is imposing the alignment, its direction should coincide with the angular momentum of the grains, thus, the transmitted electric vector (the polarization) is parallel to the mean-field direction (Lazarian, 2007; Whitney & Wolff, 2002; Crutcher, 2012). Although our resolution is poorer (1″), our data in J, H and K’ bands also exhibit constant position angles in the same area (the equatorial spot ES) (see figs. 3 and 4). According to Li & Greenberg (1997); Martin et al. (1992), the dichroic ISM polarization at NIR wavelengths should follow the power-law $P(\lambda)\propto\lambda^{-\beta}$, where $\beta=1.8\pm 0.2$. This equation indicates that the polarization must follow a decreasing trend from shorter to larger wavelengths. Therefore, if the NIR polarization of an object follows this trend, it might belong to the dichroic case. If the coherent polarization seen in the ES is dichroic, this would indicate the presence of aligned grains and possibly, an ordering magnetic field with an orientation parallel to the equator. Otherwise, the multiple scattering plane proposed by Bastien & Menard (1988) would be a more suitable solution.

We investigated if the polarization in Frosty Leo follow the $P(\lambda)$ power-law. To do so, we computed the average $P\%$ in the ES for each band and then compared them with the $P(\lambda)$ power-law for several values of $\beta$. We found a decreasing tendency in the average polarization between J and H bands (see table 2 and fig. 5), but the index required to make a fit is $\beta=1.3$ which is smaller than the predicted by Li & Greenberg (1997); Martin et al. (1992). The $P\%$ goes up in the K’ band, also moving away from the curve of theoretical polarization.
Although not entirely conclusive, these results compel us to reconsider the dichroic interpretation in the ES. However, there still are some arguments that should be taken into account before discarding this possibility. For example, the increase in K’ band polarization could be explained by assuming that our map is probing deeper regions than J and H. The K band data from Dougados et al. (1990) also supports this idea. If the light in our K’ image is coming from a different layer, the polarization in this band might not be related to that in J and H or, caused by some other mechanism. Thus, the aligned and decreasing $P\%$ between J and H bands, could still be a hint of a dichroic effect occurring in the outer layers. Furthermore, recent submillimeter polarization studies of this source (Sabin et al., 2019) also suggest the presence of a magnetic field located at the equatorial region. The detection is marginal but the orientation of the inferred field would be parallel to the equator, just as the polarization in the NIR data suggests (Murakawa et al. 2008 and our own data), so we should not hurry in dismissing this idea. Further polarization investigations at far infrared and millimeter wavelengths could help obtaining more insights into this subject.