Point Spread Function

Overview on available profiles

PampelMuse uses analytical PSF profiles. Most of the available profiles are based on either a Gaussian or a Moffat profile. An extensive discussion of the profiles and all associated formulae can be found in here. Here, only the main formulae and most important properties are presented.

The analytical profile of the Gaussian profile as a function of the radius \(r\) is given by

\[G(r) = \Sigma_\text{0}\exp\left\{\frac{-r^2}{2\sigma^2}\right\}\]

whereas the Moffat profile is described via

\[M(r) = \Sigma_\text{0}\left[1+\left(\frac{r}{r_{\text d}}\right)^2\right]^{-\beta}\]

For either profile, \(\Sigma_\mathrm{0}\) is the central intensity. The width of the Gaussian is controlled by the standard deviation \(\sigma\), while the effective radius \(r_{\text d}\) defines the width of the Moffat. However, instead of \(\sigma\) or \(r_{\text d}\), PampelMuse uses the full width at half maximum (FWHM) as a measure for the width of either profile. The conversion formulae from \(\sigma\) and \(r_{\text d}\) to \({\text FWHM}\) can be found here).

Note

The \({\text FWHM}\) is always measured in units of spaxels.

Besides the \({\text FWHM}\), the Moffat profile has a second shape parameter, \(\beta\). It measures the kurtosis of the profile, i.e. the strengths of the wings of the PSF. Higher values of \(\beta\) correspond to less pronounced wings and vice versa.

Important

From my experience, \(\beta\sim2.5\) is a typical value for a seeing-limited PSF. For this value, a Moffat profile has much more pronounced wings than a Gaussian profile. Therefore I strongly recommend using the Moffat profile because the Gaussian profile will almost certainly underestimate the wings of the PSF.

Besides the plain Gaussian and Moffat profiles, PampelMuse also supports combinations of the two:

A Moffat-Gauss profile, consisting of an outer Moffat profile (with parameters \(beta\) and \({\text FWHM}_1\)) and a central Gaussian profile (with parameter \({\text FWHM}_1\)). An additional parameter \(f_{12}\) controls the fractional flux in the Moffat component.

A double-Gauss profile, consisting of an outer and a central Gaussian component (with parameters \({\text FWHM}_1\) and \({\text FWHM}_2\), respectively). Again, an additional parameter \(f_{12}\) is used, controlling the fractional flux in the outer component.

A double-Moffat profile, consisting of an outer and a central Moffat component (with parameters \({\text FWHM}_1\), \(\beta_1\) and \({\text FWHM}_2\), \(\beta_2\)), respectively. As for the double-Gauss profile, the parameter \(f_{12}\) controls the fractional flux of the outer component.

PampelMuse supports elliptical PSF profiles. The radius \(r\) incorporates the ellipticity \(e\) and the position angle \(\theta\) of a PSF profile. Thus, besides the actual centroid coordinates, the Moffat profile has 4 free parameters, the Gaussian profile has 3. For combined profiles, only a global ellipticity and position angle can be used.

Adaptive optics data

New in version 1.0: To facilitate the handling of adaptive optics (AO) data, in particular MUSE narrow field mode (NFM) observations, the MAOPPY PSF model developed by [Fetick2019] is made available in PampelMuse. Besides the parameters \(e\) and \(\theta\) that it shares with the other PSF models, the MAOPPY PSF has five parameters that can be optimized during the analysis. They are the Fried parameter \(r_0\), the kurtosis \(\beta\) of the Moffat contribution, the effective radius \(\alpha\) of the Moffat contribution, the AO-corrected phase power spectral density (PSD) background \(bck\), and the residual variance \(amp\). Note that some parameters have been renamed compared to the names provided in Table 2 of [Fetick2019] in order to avoid confusion with other fitting parameters.

Important

In case you use this PSF model, please cite [Fetick2019].

Specifying the PSF to use

To specify the properties of the PSF during the analysis, one needs to modify the psf section of the PampelMuse configuration.

Setting the PSF type and parameters

First, one needs to specify which type of profile should be used. This can be using the following configuration parameter.

psf|profilestring: The analytical PSF profile used during the analysis. Currently supported are gauss, moffat, double_gauss, double_moffat, moffat_gauss, and maoppy.

Next, initial guesses for the PSF parameters must be defined. As already mentioned, the available profiles are described by different sets parameters. This is summarized in the following table.

Overview of PSF parameters
Parameter index \(n\)	gauss	moffat	double_gauss	double_moffat	moffat_gauss	maoppy
1	fwhm	beta	f12	f12	f12	r0
2	e	fwhm	fwhm1	beta1	beta	bck
3	theta	e	fwhm2	fwhm1	fwhm1	amp
4		theta	e	beta2	fwhm2	alpha
5			theta	fwhm2	e	beta
6				e	theta	e
7				theta		theta

For each parameter of the chosen PSF profile, an initial guess can be defined in the psf section of the configuration, e.g.:

psf|fwhmfloat: The full width at half maximum of the selected PSF profile. Must be \(>0\).

Finally, one needs to define which of the PSF parameters should be optimized during the analysis and which remain fixed at their initial guesses. To this aim, PampelMuse uses a single configuration parameter:

psf|fitinteger: Defines which PSF parameters are optimized during the analysis. The value should set to \(\Sigma_{n} 2^n\), with the sum including the index \(n\) of every PSF parameter to be optimized (see table).

Assuming one wants to model the PSF as a Moffat profile, with a fixed value of \(\beta=2.5\), and free parameters \({\text FWHM}\), \(e\), and \(\theta\), the psf section of the configuration could look as follows.

{
    "psf": {
        "profile": "moffat",
        "beta": 2.0,
        "fwhm": 2.5,
        "e": 0.2,
        "theta": 0.5,
        "fit": 28
    }
}

The values provided for \({\text FWHM}\), \(e\), and \(\theta\) serve as initial guesses for the fit.

Setting the PSF radii

Each PSF profile used during the analysis will be defined out to a maximum radius, beyond which the surface intensity of the profile is believed to be negligible. This radius can be changed using the following parameter.

psf|radiusfloat: The radius out to which a PSF profile is calculated, given in units of spaxels.

As a rule of thumb, the radius parameter should be large enough so that no bright halos remain around the bright stars when creating the residuals of the analysis with the SUBTRES routine.

New in version 1.0: PampelMuse also allows one to scale the PSF radius according to the flux of a star, such that a larger radius is used for brighter sources. This is useful for example for the analysis of MUSE narrow field mode data, where the PSF halos around the brightest stars can extend out to \(\sim50-100\) spaxels. As using such a large value for all sources will significantly slow down the analysis, one can set the following configuration parameter.

psf|scalingfloat: Scaling factor for the PSF radius, must be \(>0\) and \(<1\). It is defined such that the PSF radius will shrink by a factor \(2\) for every drop in flux by the provided value. A flux reference, the brightest star in the field is used, for which the PSF profile will be defined out to psf|radius.

In-depth analysis of fitted PSF

PampelMuse offers the option to store detailed information about the the PSF fits it performed when running the CUBEFIT routine. To this aim, the following parameter should be enabled.

psf|inspectboolean: If enabled, a FITS-file will be created that contains the details of the individual PSF fits. For each source used to fit the PSF, four extensions will be created, containing the input data of the fits (named DAT_<ID>), radial profiles of the input data (named PR_<ID>), the fit residuals (named RES_<ID>), and the fitted parameters (named PM_<ID>). The IDs used to construct the extension names are the same as in the reference catalogue. All data will be stored as a function of wavelength. The file is saved under <prefix>.psf.fits, where the same prefix is used that was provided in the user configuration file to identify the integral field data.

Note

This setting only has an effect only if PampelMuse is run with the global|smallifu key set to False or if the analysis has been started using the SINGLESRC routine routine.

Further information about the content of the FITS file that is created can be found here.