Summary of Methods
By Elijah Bernstein-Cooper, July 28, 2015, 0 comments.

Table of Contents



Selecting Regions

We select regions by hand. Figure 1 shows the selected regions. We divide Taurus into a North and a South region as Pineda et al. (2010) did. We also divide Perseus into two regions for testing the dependence of our analysis on region selection, shown later.


Figure 1

Top: map of the three clouds, integrated from -5 to 15 km/s. Bottom: Planck map of the same region. The white lines outline the regions selected for each cloud.


Masking

Method

To determine the DGR, velocity range over which to integrate (hereafter width), and the background, we must select only diffuse pixels without any varying background IR emission or .

We follow the methods of Planck et al. (2011) paper (see Section 4.3) to mask pixels around each cloud.

Consider a region where all lines of sight contained only with associated dust. If we derived a DGR and intercept to model the with the , then the residuals between the model and the data would consist only of the measurement error, i.e. white, Gaussian, noise.

However in reality many lines of sight contain excess from associated dust or from an IR background. We can exclude these pixels however if we take the faintest pixels in a region, corresponding to diffuse lines of sight with only , derive a model , then add the brighter pixels, subtract the model from the data and examine the residuals. Residuals with excess will show up as high positive residuals. We therefore fit a Gaussian to the rising, negative residuals, and mask pixels whose residual value is above a few times the width of the Gaussian. An example of this is in the Planck paper.

We include more of the brighter pixels and perform this process again. Initially diffuse pixels will be added, which the Gaussian fitting will not exclude. Later on however, pixels with excess will be added, resulting in positive residuals, and will be masked. We perform this iterative process until the DGR used to make the model converges to a 1% of the previous iteration.

Below is an outline of our masking process.

  1. Mask all but faintest 10% of pixels.

  2. Fit a model using a linear least squares fit, deriving a DGR and an intercept.

  3. Unmask the next faintest 5% of pixels.

  4. Mask the residuals by fitting residuals mag with a Gaussian. Mask all pixels more than 3 times the standard deviation of the fitted Gaussian.

  5. Repeat steps 2 through 5 until the fitted DGR converges to within 1%. Supply the mask from step 4 to step 2.

Application

In all subsequent analysis we bin the data onto 30 scale pixels. This is in order to mitigate the correlated relationship between pixels from the cosmic infrared background on scales of 5.

We should examine the mask maps for each iteration of the masking (steps 2 - 5). Below is a figure showing the progression of masks for all three clouds. We can see that for each cloud the residual masking begins to dominate after a few iterations. Restated, at some point when including brighter pixels, we start including only lines of sight with excess emission which will be masked by the residual masking.


Perseus

Taurus

California

Figure 2

Masked maps of each cloud for each iteration. Left: The fractional-masked map from step 3. Right: The residual-masked map from step 4. The residual mask begins excluding most added pixels from the fractional mask after just a few iterations.


We should also examine the residual distributions at each iteration. Below are the Kernel density estimates of each cloud. Just before the masking converges, the residuals near 0 mag, i.e., the white noise, have similar distributions. This means that the masking has included the diffuse pixels available and including only pixels with excess in further iterations.


Figure 3

Residual kernel density plots for Perseus, Taurus, and California (clockwise from top left) at each iteration. The residual PDF is in black, and the fitted Gaussian in purple. The residual cut-off is the dashed black line. The figure shows that the majority of data points are masked by residual masking.


Selecting the Velocity Range

Imara et al. (2011a) searched for cloud emission across an width of km/s around the CO line center for clouds in the Milky Way. In their study of M33 clouds (Imara et al. 2011b), they consider a larger range, km/s, over which to search for a GMC’s extent.

We therefore allow the MLE calculation widths less than 70 km/s.

Selecting the center is tricky however. In previous iterations we were using the average velocity weighted by the square of the brightness temperature. This favors an center near the peak emission, but can be skewed by a large tail, e.g. in California.

Previous results for Taurus favored an extremely small width of order < 10 km/s. However, we are now using the peak of the average spectrum as the center. This results in vastly different results for Taurus, where now, the widths are on order 50 km/s, see below for more details.

Results

Likelihoods

Perseus

Taurus

California

Figure 4

The likelihood spaces for the DGR, velocity width, and intercept for each cloud. The contour represents the 95% confidence level. The plots on the side show the marginalized distribution for each parameter, where the dashed line is the best estimate, and the shaded region is the 68% confidence intervel. California likelihoods with Planck data. A high DGR, wide width, and negative intercept are favored.

We can see that the width agrees well with Lee+12, seeing as there is little emission in high negative or positive velocities around Perseus, thus doubling the width does not change the much. However we can see for Taurus that the width is much larger than discussed earlier, about 50 km/s, a change due to moving the center 5 km/s.


Regions in Taurus

Pineda et al. (2010) divided Taurus into a North and a South component. They identify the diffuse background in the North-Eastern side of Taurus in the appendix B. They select a velocity range of 0 to 20 km/s which they associate with in Taurus. The rest of the emission is used to estimate the foreground/background in Taurus of about mag. Our finding from the previous post is consistent with this range and background.


Figure 5

Clockwise from top-left: Taurus, Taurus-South, and Taurus-North maps. For each plot, top: Original resolution map overlaid with mask contour, bottom: binned image, with pixels used to calculate the width, DGR and intercept in color, and masked pixels in gray. These plots show that when both the north and south regions are considered together, the masking favors the lower- region in the South-West as the diffuse pixels.

We can see that California has a negative intercept, however this is not necessarily a worrying thing. See previous discussion on the negative intercept.


Results

widths

We can examine the and CO spectra of each cloud. Below show the total, unmasked median and CO spectra, and the masked median spectra. The masked and the unmasked spectra hardly differ.


Figure 6

and CO spectra of each cloud with the MLE for the width. The errors in the widths for each cloud hardly affect the derived considering the spectra are quite shallow at these widths.


vs. Distributions

Below are distributions of vs. , with the MLE and polynomial fits overlaid. The top plots of each figure are the binned, masked data, to which the DGR and intercept were fit. The bottom plots are the unbinned unmasked data.


Perseus, Planck

Figure 7

Perseus vs. using Planck data.



Taurus

Figure 8

Taurus vs. .



California

Figure 9

California vs. .


We should also consider what the distribution is as a function of to make sure the fit between and worked correctly.


Perseus

Taurus

California

Figure 10

Perseus, Taurus, and California vs. distributions derived from Planck data.