In this `jupyter`

recipe, we will explore the HAWC+ data cube and describe some of the basic analysis techniques involving imaging polarimetry data.

This cookbook recipe follows the SOFIA press release of 30 Doradus observations: SOFIA Reveals Never-Before-Seen Magnetic Field Details.

The Level 4 reduced data from this program has been released immediately to the public and is available on the SOFIA Data Cycle System (DCS). This notebook will guide the reader through downloading the 30 Doradus data with a walkthrough of basic analysis techniques with `python`

.

The Strategic Director's Discretionary Time (S-DDT) for SOFIA is aimed at providing the astronomical community with data sets of high scientific interest over a broadrange of potential research topics without any proprietary period. These observingsessions allow the general user community access to high-level data products that aremeant not only for general understanding of SOFIA data and its packaging but also for inclusion in published scientific work. The S-DDT target have been selected on a non-interference basis with existing programs and in terms of SOFIA flight planning.

The 76_0001 program, "Community Science: HAWC+ Polarimetry of 30 Dor," was designed and scheduled to provide the community with SOFIA polarimetry data of an important and relatively bright source. The observing strategy also provided significantly increased scheduling efficiency for the OC6I (HAWC+) flights in July 2018. The west-bound observing legs for 30 Doradus allowed a larger fraction of the highest ranked Cycle 6 targets, predominantly in the inner Galaxy, to be scheduled and flown.

To enhance the scientific exploitation of these data products, we present here an overview of the observations, visualizations of the data, and preliminary analysis of their quality.

The purpose of this document is to provide a template for a jupyter notebook session, or even just some code snippets, that will aid in analysis of SOFIA/HAWC+ data.

In this notebook, we focus on imaging polarimetry data; however, some of the techniques mentioned here would also apply to HAWC+ total intensity imaging as well. For more information on HAWC+ observing modes, visit our observer's handbook at https://www.sofia.usra.edu/science/instruments/hawc.

- If you do not yet have a DCS account, register for one at https://dcs.arc.nasa.gov/userSupport/registration.jsp
- Log into DCS: https://dcs.arc.nasa.gov
- Go to Search Science Archive
- Fill in:
- Instrument:
`HAWC_PLUS`

from drop-down menu - Processing State:
`LEVEL_4`

from drop-down menu - Target: 30Dor
- Click the
`SIMBAD Position`

button - Change
`Spatial Search Radius`

to`600`

arcsec - Click the
`Submit`

button

- Instrument:
- After the results load, select the checkboxes next to each of the six rows of the table. Each row represents a single observation in this data set.
- Click
`Get Selected Data in Current Page`

- Click
`Request Data Bundle`

- After a few minutes, an email with a download link will be sent to the email address associated with your DCS account.
- For more information, consult the HAWC+ Data Handbook accessible at https://www.sofia.usra.edu/science/proposing-and-observing/data-products/data-resources

After downloading the SOFIA DCS bundle to your working directory you will want to unzip it, which will produce a directory structure like this:

```
.
└── sofia_data
├── level4
│ └── p5813
│ └── F0484_HA_POL_7600018_HAWCHWPC_PMP_022-114.fits
└── missions
├── 2018-07-05_HA_F481
│ └── p5827
│ └── F0481_HA_POL_7600012_HAWDHWPD_PMP_050-083.fits
├── 2018-07-07_HA_F483
│ └── p5646
│ └── F0483_HA_POL_7600014_HAWCHWPC_PMP_022-065.fits
├── 2018-07-11_HA_F484
│ └── p5648
│ └── F0484_HA_POL_7600017_HAWCHWPC_PMP_065-114.fits
└── 2018-07-12_HA_F485
└── p5658
├── g1
│ └── F0485_HA_POL_76000110_HAWAHWPA_PMP_043-052.fits
└── g2
└── F0485_HA_POL_7600019_HAWEHWPE_PMP_055-075.fits
```

Note the following features of this data bundle.

- Each
`fits`

file in the 'missions' directory corresponds to data from a single AOR (or a different filter element) obtained on a single flight - Each subdirectory under missions corresponds to a single flight
`fits`

files under 'level4' correspond to data combined from several flights- If multiple filters were observed on the same flight, they will be further divided into subdirectories (g1/g2 on the last line)

Note that two observations were made with the same filter (HAWC C, $89\,\mathrm{\mu m}$). These files, `F0483_HA_POL_7600014_HAWCHWPC_PMP_022-065.fits`

and `F0484_HA_POL_7600017_HAWCHWPC_PMP_065-114.fits`

, were combined into one:

`level4->p5813->F0484_HA_POL_7600018_HAWCHWPC_PMP_022-114.fits`

.

You can choose to keep the `fits`

files nested, or copy them into one directory.

For the purpose of this basic analysis, though, let us dump all the files into one `sofia_data`

directory:

```
.
└── sofia_data
├── F0481_HA_POL_7600012_HAWDHWPD_PMP_050-083.fits
├── F0483_HA_POL_7600014_HAWCHWPC_PMP_022-065.fits
├── F0484_HA_POL_7600017_HAWCHWPC_PMP_065-114.fits
├── F0484_HA_POL_7600018_HAWCHWPC_PMP_022-114.fits
├── F0485_HA_POL_76000110_HAWAHWPA_PMP_043-052.fits
└── F0485_HA_POL_7600019_HAWEHWPE_PMP_055-075.fits
```

For this analysis, we require the standard numpy/scipy/matplotlib stack as well the astropy and aplpy modules.

With just a few lines of code, we can explore the HAWC+ `fits`

data cubes and plot the images.

In [1]:

```
from astropy.io import fits
efile = 'sofia_data/F0485_HA_POL_7600019_HAWEHWPE_PMP_055-075.fits'
dfile = 'sofia_data/F0481_HA_POL_7600012_HAWDHWPD_PMP_050-083.fits'
cfile = 'sofia_data/F0484_HA_POL_7600018_HAWCHWPC_PMP_022-114.fits'
afile = 'sofia_data/F0485_HA_POL_76000110_HAWAHWPA_PMP_043-052.fits'
hawc = fits.open(afile)
hawc.info()
```

We can see above the data structure of the multi-extension `fits`

files. Each file contains 19 extensions which encapsulates all of the measurable Stokes parameters, derived polarization information, and associated errors in a single package.

Stokes $I$---the zeroth extension in the `fits`

file---represents the total intensity of the image.

Let us go ahead and plot this extension:

In [2]:

```
import matplotlib.pyplot as plt
%matplotlib notebook
# ^jupyter magic for inline plots
# leave this out if working in a non-interactive environment
from aplpy import FITSFigure
# set colormap for all plots
cmap = 'rainbow'
stokes_i = hawc['STOKES I'] # or hawc[0]. Note the extension is from the hawc.info() table above
axs = FITSFigure(stokes_i) # load HDU into aplpy figure
axs.show_colorscale(cmap=cmap) # display the data with WCS projection and chosen colormap
# FORMATTING
axs.tick_labels.set_font(size='small')
# Add colorbar
axs.add_colorbar()
axs.colorbar.set_axis_label_text('Flux (Jy/pix)')
```

Similarly, we can plot the Stokes Q and Stokes U images:

In [3]:

```
stokes_q = hawc['STOKES Q']
stokes_u = hawc['STOKES U']
axq = FITSFigure(stokes_q, subplot=(1,2,1)) # generate FITSFigure as subplot to have two axes together
axq.show_colorscale(cmap=cmap) # show Q
axu = FITSFigure(stokes_u, subplot=(1,2,2),
figure=plt.gcf())
axu.show_colorscale(cmap=cmap) # show U
# FORMATTING
axq.set_title('Stokes Q')
axu.set_title('Stokes U')
axu.axis_labels.set_yposition('right')
axu.tick_labels.set_yposition('right')
axq.tick_labels.set_font(size='small')
axq.axis_labels.set_font(size='small')
axu.tick_labels.set_font(size='small')
axu.axis_labels.set_font(size='small')
```

We can additionally plot the associated error maps for each extension.

In [4]:

```
stokes_q = hawc['STOKES Q']
error_q = hawc['ERROR Q']
axq = FITSFigure(stokes_q, subplot=(1,2,1)) # generate FITSFigure as subplot to have two axes together
axq.show_colorscale(cmap=cmap) # show Q
axe = FITSFigure(error_q, subplot=(1,2,2), figure=plt.gcf())
axe.show_colorscale(cmap=cmap) # show error
# FORMATTING
axq.set_title('Stokes Q')
axe.set_title('Error Q')
axq.axis_labels.hide() # hide axis/tick labels
axe.axis_labels.hide()
axq.tick_labels.hide()
axe.tick_labels.hide()
```

Level 4 HAWC+ additionally provides extensions with the linear polarization percentage ($p$), angle ($\theta$), and their associated errors ($\sigma$).

Percent polarization ($p$) and error ($\sigma_p$) are calculated as:

\begin{align} p & = 100\sqrt{\left(\frac{Q}{I}\right)^2+\left(\frac{U}{I}\right)^2} \\ \sigma_p & = \frac{100}{I}\sqrt{\frac{1}{(Q^2+U^2)}\left[(Q\,\sigma_Q)^2+(U\,\sigma_U)^2+2QU\,\sigma_{QU}\right]+\left[\left(\frac{Q}{I}\right)^2+\left(\frac{U}{I}\right)^2\right]\sigma_I^2-2\frac{Q}{I}\sigma_{QI}-2\frac{U}{I}\sigma_{UI}} \end{align}Note that $p$ here represents the **percent** polarization as opposed to the more typical convention for $p$ as the **fractional** polarization.

Maps of these data are found in extensions 7 (PERCENT POL) and 9 (ERROR PERCENT POL).

Polarized intensity, $I_p$, can then be calculated as $I_p = \frac{I\times p}{100}$, which is included in extension 13 (POL FLUX).

Also included is the debiased polarization percentage ($p^\prime$) calculated as:

$p^\prime=\sqrt{p^2-\sigma_p^2}$, found in extension 8 (DEBIASED PERCENT POL).

We similarly define the debiased polarized intensity as $I_{p^\prime} = \frac{I\times p^\prime}{100}$, which is included in extension 15 (DEBIASED POL FLUX).

These values are also included in table form in extension 17 (POL DATA).

In [5]:

```
stokes_ip = hawc['DEBIASED POL FLUX']
axi = FITSFigure(stokes_i, subplot=(1,2,1))
axi.show_colorscale(cmap=cmap) # show I
axp = FITSFigure(stokes_ip, subplot=(1,2,2), figure=plt.gcf())
axp.show_colorscale(cmap=cmap) # show Ip
# FORMATTING
axi.set_title(r'$I$')
axp.set_title(r'$I_{p^\prime}$')
axp.axis_labels.set_yposition('right')
axp.tick_labels.set_yposition('right')
axi.tick_labels.set_font(size='small')
axi.axis_labels.set_font(size='small')
axp.tick_labels.set_font(size='small')
axp.axis_labels.set_font(size='small')
```

From the $Q$ and $U$ maps, the polarization angle $\theta$ is calculated in the standard way:

$\theta = \frac{90}{\pi}\,\mathrm{tan}^{-1}\left(\frac{U}{Q}\right)$

with associated error:

$\sigma_\theta = \frac{90}{\pi\left(Q^2+U^2\right)}\sqrt{\left(Q\sigma_Q\right)^2+\left(U\sigma_U\right)^2-2QU\sigma_{QU}}$

The angle map is stored in extension 10 (POL ANGLE) in degrees, with its error in extension 12 (ERROR POL ANGLE).

As part of the HAWC+ reduction pipeline, $\theta$ is corrected for the vertical position angle of the instrument on the sky, the angle of the HWP plate, as well as an offset angle that is calibrated to each filter configuration. $\theta=0^\circ$ corresponds to the North-South direction, $\theta=90^\circ$ corresponds to the East-West direction, and positive values follow counterclockwise rotation.

We also provide the PA of polarization rotated by $90^\circ$, $\theta_{90}$, in extension 11 (ROTATED POL ANGLE). This PA of polarization needs to be used with caution. If the measured polarization is dominated by magnetically-aligned dust grains, then the PA of polarization, $\theta$, can be rotated by $90^\circ$ to visualize the magnetic field morphology. For more details, see Hildebrand et al. 2000; Andersson et al. 2015.

We can now use the $p^\prime$ and $\theta_{90}$ maps to plot the polarization vectors. First, however, let us make a quality cut. Rather than defining a $\sigma$ cut on the polarization vectors themselves, it is more useful to define a signal-to-noise cut on total intensity, $I$, the measured quantity.

Starting with the propagated error on the polarization fraction:

\begin{equation*} \sigma_p = \frac{100}{I}\sqrt{\frac{1}{(Q^2+U^2)}\left[(Q\,\sigma_Q)^2+(U\,\sigma_U)^2+2QU\,\sigma_{QU}\right]+\left[\left(\frac{Q}{I}\right)^2+\left(\frac{U}{I}\right)^2\right]\sigma_I^2-2\frac{Q}{I}\sigma_{QI}-2\frac{U}{I}\sigma_{UI}} \end{equation*}Let's assume the errors in $Q$, $U$, and $I$ are comparable so that there are no covariant (cross) terms in the error expansion.

Therefore, \begin{equation*} \sigma_Q = \sigma_U = \sigma_{Q,U} \\ \sigma_{QI} = \sigma_{QU} = \sigma_{UI} = 0 \end{equation*}

$\require{cancel}$ \begin{align} \sigma_p & = \frac{100}{I}\sqrt{\frac{1}{(\cancel{Q^2+U^2})}\left[\sigma_{Q,U}^2\left(\cancel{Q^2+U^2}\right)\right]+ \sigma_I^2\left(\frac{Q^2+U^2}{I^2}\right)} \\ \sigma_p & = \frac{100}{I}\sqrt{\sigma_{Q,U}^2+\sigma_I^2\left(\frac{Q^2+U^2}{I^2}\right)}= \frac{1}{I}\sqrt{\sigma_{Q,U}^2+\sigma_I^2\,p^2} \end{align}

If we assume that $p$ is relatively small (e.g. the source is not highly polarized), and that the errors in $I$ are small, then the second term ($\sigma_I^2\,p^2$) is negligible.

\begin{equation*} \sigma_p = \frac{\sigma_{Q,U}}{I} \end{equation*}By design, the HAWC+ optics split the incident radiation into two orthogonal linear polarization states that are measured with two independent detector arrays. The total intensity, Stokes $I$, is recovered by linearly adding both polarization states. If the data is taken at four equally-spaced HWP angles, and assuming 100% efficiency of the instrument, then the uncertainty in $I$ is related to the uncertanties in $Q$ and $U$: \begin{equation*} \sigma_Q\sim\sigma_U\sim\sqrt{2}\,\sigma_I \end{equation*}

This simplifies our error on $p$ to: \begin{align} \sigma_p &\sim \sqrt{2}\frac{\sigma_I}{I} \\ \sigma_p &\sim \frac{\sqrt{2}}{\left(S/N\right)_I} \end{align}

If we desire an error in $p$ of $\sim0.5\%$, what is the required signal-to-noise in $I$?

\begin{align} \left(\mathrm{S/N}\right)_I & \sim \sqrt{2}\left(\frac{1}{\sigma_p}\right) \sim \sqrt{2}\frac{1}{0.5\%} \\ & \sim \frac{\sqrt{2}}{0.005} \sim 283 \end{align}So, therefore if we desire an accuracy of $\sigma_p\sim0.5\%$, we require a S/N in total intensity $I$ of $\sim283$.

This S/N cut in $I$ is very conservative. In the Level 4 HAWC+ data, the last extension, FINAL POL DATA, contains a table of values similar to POL DATA, with somewhat less restrictive quality cuts applied. This extension includes vectors satisfying the following three criteria:

- (S/N$_I$) > 200
- (S/N$_p$) > 3
- $p<50\%$

Since we include maps of all measurable polarization information with the full data set, you are free to decide on any quality cuts that satisfy your scientific needs.

In this next panel, we include a single quality cut on S/N > 100, by performing the following steps:

- use the Stokes $I$ image as the background for the vector plot
- perform a quality cut on Stokes $I/\sigma_I > 100$ to make a mask
- mask out low S/N vectors
- plot remaining polarization vectors
- add contours to better visualize changes in flux across the map

In [6]:

```
from astropy.io import fits
import numpy as np
from aplpy import FITSFigure
def make_polmap(filename, title=None, figure=None, subplot=(1,1,1)):
hawc = fits.open(filename)
p = hawc['DEBIASED PERCENT POL'] # %
theta = hawc['ROTATED POL ANGLE'] # deg
stokes_i = hawc['STOKES I'] # I
error_i = hawc['ERROR I'] # error I
# 1. plot Stokes I
# convert from Jy/pix to Jy/sq. arcsec
pxscale = stokes_i.header['CDELT2']*3600 # map scale in arcsec/pix. CDELT2 always in deg
stokes_i.data /= pxscale**2
error_i.data /= pxscale**2
fig = FITSFigure(stokes_i, figure=figure, subplot=subplot)
# 2. perform S/N cuts on I/sigma_I, and p/sigma_p
i_err_lim = 100
mask = np.where(stokes_i.data/error_i.data < i_err_lim)
# 3. mask out low S/N vectors by setting masked indices to NaN
p.data[mask] = np.nan
# 4. plot vectors
scalevec = 0.4 # 1pix = scalevec * 1% pol # scale vectors to make it easier to see
fig.show_vectors(p, theta, scale=scalevec, step=2) # step size = display every 'step' vectors
# step size of 2 is effectively Nyquist sampling
# --close to the beam size
# 5. plot contours
ncontours = 30
fig.show_contour(stokes_i, cmap=cmap, levels=ncontours,
filled=True, smooth=1, kernel='box')
fig.show_contour(stokes_i, colors='gray', levels=ncontours,
smooth=1, kernel='box', linewidths=0.3)
# Show image
fig.show_colorscale(cmap=cmap)
# If title, set it
if title:
fig.set_title(title)
# Add colorbar
fig.add_colorbar()
fig.colorbar.set_axis_label_text('Flux (Jy/arcsec$^2$)')
# Add beam indicator
fig.add_beam(facecolor='red', edgecolor='black',
linewidth=2, pad=1, corner='bottom left')
fig.add_label(0.02, 0.02, 'Beam FWHM',
horizontalalignment='left', weight='bold',
relative=True, size='small')
# Add vector scale
# polarization vectors are displayed such that 'scalevec' * 1% pol is 1 pix long
# must translate pixel size to angular degrees since the 'add_scalebar' function assumes a physical scale
vectscale = scalevec * pxscale/3600
fig.add_scalebar(5 * vectscale, "p = 5%",corner='top right',frame=True)
# FORMATTING
fig.tick_labels.set_font(size='small')
fig.axis_labels.set_font(size='small')
return stokes_i, p, mask, fig
```

In [7]:

```
stokes_i, p, mask, fig = make_polmap(afile, title='A')
```

We can also plot the polarization fraction $p$ to better visualize the structure of 30 Doradus. We plot the same contours from total intensity $I$ in the background.

In [8]:

```
fig = FITSFigure(p)
# Show image
fig.show_colorscale(cmap=cmap)
# Plot contours
ncontours = 30
fig.show_contour(stokes_i, colors='gray', levels=ncontours,
smooth=1, kernel='box', linewidths=0.3)
# Add colorbar
fig.add_colorbar()
fig.colorbar.set_axis_label_text('$p^\prime$ (%)')
```

Finally, using the function defined above, we plot all four HAWC+ observations of 30 Doradus.

In [9]:

```
files = [afile,cfile,dfile,efile]
titles = ['A','C','D','E']
for file, title in zip(files,titles):
make_polmap(file,title)
```