# Moment maps and statistics¶

## Moment maps¶

Producing moment maps from a SpectralCube instance is straightforward:

>>> moment_0 = cube.moment(order=0)
>>> moment_1 = cube.moment(order=1)
>>> moment_2 = cube.moment(order=2)


By default, moments are computed along the spectral dimension, but it is also possible to pass the axis argument to compute them along a different axis:

>>> moment_0_along_x = cube.moment(order=0, axis=2)


Note

These follow the mathematical definition of moments, so the second moment is computed as the variance. For the actual formulas used for the moments, please see moment. For linewidth maps, see the Linewidth maps section.

You may also want to convert the unit of the datacube into a velocity one before you can obtain a genuine velocity map via a 1st moment map. So first it will be necessary to apply the with_spectral_unit method from this package with the proper attribute settings:

>>> nii_cube = cube.with_spectral_unit(u.km/u.s,
velocity_convention='optical',
rest_value=6584*u.AA)


Note that the rest_value in the above code refers to the wavelength of the targeted line in the 1D spectrum corresponding to the 3rd dimension. Also, since not all velocity values are relevant, next we will use the spectral_slab method to slice out the chunk of the cube that actually contains the line:

>>> nii_cube = cube.with_spectral_unit(u.km/u.s,
velocity_convention='optical',
rest_value=6584*u.AA)
>>> nii_subcube = nii_cube.spectral_slab(-60*u.km/u.s,-20*u.km/u.s)


Finally, we can now generate the 1st moment map containing the expected velocity structure:

>>> moment_1 = nii_subcube.moment(order=1)


The moment maps returned are Projection instances, which act like Quantity objects, and also have convenience methods for writing to a file:

>>> moment_0.write('moment0.fits')
>>> moment_1.write('moment1.fits')


and converting the data and WCS to a FITS HDU:

>>> moment_0.hdu
<astropy.io.fits.hdu.image.PrimaryHDU at 0x10d6ec510>


The conversion to HDU objects makes it very easy to use the moment map with plotting packages such as aplpy:

>>> import aplpy
>>> f = aplpy.FITSFigure(moment_0.hdu)
>>> f.show_colorscale()
>>> f.save('moment_0.png')


There is a shortcut for the above, if you have aplpy installed:

>>> moment_0.quicklook('moment_0.png')


will create the quicklook grayscale image and save it to a png all in one go.

### Moment map equations¶

The moments are defined below, using $$v$$ for the spectral (velocity, frequency, wavelength, or energy) axis and $$I_v$$ as the intensity, or otherwise measured flux, value in a given spectral channel.

The equation for the 0th moment is:

$M_0 = \int I_v dv$

The equation for the 1st moment is:

$M_1 = \frac{\int v I_v dv}{\int I_v dv} = \frac{\int v I_v dv}{M_0}$

Higher-order moments ($$N\geq2$$) are defined as:

$M_N = \frac{\int I_v (v - M_1)^N dv}{M_0}$

Descriptions for the three most common moments used are:

• 0th moment - the integrated intensity over the spectral line. Units are cube unit times spectral axis unit (e.g., K km/s).

• 1st moment - the the intensity-weighted velocity of the spectral line. The unit is the same as the spectral axis unit (e.g., km/s)

• 2nd moment - the velocity dispersion or the width of the spectral line. The unit is the spectral axis unit squared (e.g., $$km^2/s^2$$). To obtain measurements of the linewidth in spectral axis units, see Linewidth maps below

## Linewidth maps¶

Line width maps based on the 2nd moment maps, as defined above, can be made with either of these two commands:

>>> sigma_map = cube.linewidth_sigma()
>>> fwhm_map = cube.linewidth_fwhm()


~spectral_cube.SpectralCube.linewidth_sigma computes a sigma linewidth map along the spectral axis, where sigma is the width of a Gaussian, while ~spectral_cube.SpectralCube.linewidth_fwhm computes a FWHM linewidth map along the same spectral axis.

The linewidth maps are related to the second moment by

$\begin{split}\sigma = \sqrt{M_2} \\ FWHM = \sigma \sqrt{8 ln{2}}\end{split}$

These functions return Projection instances as for the Moment maps.

Other common 2D views of a spectral cube include the maximum/minimum along a dimension and the location of the maximum or the minimum along that dimension.

To produce a 2D map of the maximum along a cube dimension, use max

>>> max_map = cube.max(axis=0)


Along the spectral axis, this will return the peak intensity map.

Similarly we can use min to make a 2D minimum map:

>>> min_map = cube.min(axis=0)


The argmax and argmin operations will return the pixel positions of the max/min along that axis:

>>> argmax_map = cube.argmax(axis=0)
>>> argmin_map = cube.argmin(axis=0)


These maps are useful for identifying where signal is located within the spectral cube, however, it is more useful to return the WCS values of those pixels for comparison with other data sets or for modeling. The argmax_world and argmin_world should be used in this case:

>>> world_argmax_map = cube.argmax_world(axis=0)
>>> world_argmin_map = cube.argmin_world(axis=0)


Along the spectral axis, argmax_world creates the often used “velocity at peak intensity,” which may also be called the “peak velocity.”

Note

cube.argmax_world and cube.argmin_world are currently only defined along the spectral axis, as the example above shows. This is because argmax_world and argmin_world operate along the pixel axes, but they are not independent in WCS coordinates due to the curvature of the sky.