.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "user-guide/model-gallery/spatial/plot_gauss.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_user-guide_model-gallery_spatial_plot_gauss.py: .. _gaussian-spatial-model: Gaussian spatial model ====================== This is a spatial model parametrising a Gaussian function. By default, the Gaussian is symmetric: .. math:: \phi(\text{lon}, \text{lat}) = N \times \exp\left\{-\frac{1}{2} \frac{1-\cos \theta}{1-\cos \sigma}\right\}\,, where :math:`\theta` is the sky separation to the model center. In this case, the Gaussian is normalized to 1 on the sphere: .. math:: N = \frac{1}{4\pi a\left[1-\exp(-1/a)\right]}\,,\,\,\,\, a = 1-\cos \sigma\,. In the limit of small :math:`\theta` and :math:`\sigma`, this definition reduces to the usual form: .. math:: \phi(\text{lon}, \text{lat}) = \frac{1}{2\pi\sigma^2} \exp{\left(-\frac{1}{2} \frac{\theta^2}{\sigma^2}\right)}\,. In case an eccentricity (:math:`e`) and rotation angle (:math:`\phi`) are passed, then the model is an elongated Gaussian, whose evaluation is performed as in the symmetric case but using the effective radius of the Gaussian: .. math:: \sigma_{eff}(\text{lon}, \text{lat}) = \sqrt{ (\sigma_M \sin(\Delta \phi))^2 + (\sigma_m \cos(\Delta \phi))^2 }. Here, :math:`\sigma_M` (:math:`\sigma_m`) is the major (minor) semiaxis of the Gaussian, and :math:`\Delta \phi` is the difference between `phi`, the position angle of the Gaussian, and the position angle of the evaluation point. **Caveat:** For the asymmetric Gaussian, the model is normalized to 1 on the plane, i.e. in small angle approximation: :math:`N = 1/(2 \pi \sigma_M \sigma_m)`. This means that for huge elongated Gaussians on the sky this model is not correctly normalized. However, this approximation is perfectly acceptable for the more common case of models with modest dimensions: indeed, the error introduced by normalizing on the plane rather than on the sphere is below 0.1\% for Gaussians with radii smaller than ~ 5 deg. .. GENERATED FROM PYTHON SOURCE LINES 51-54 Example plot ------------ Here is an example plot of the model: .. GENERATED FROM PYTHON SOURCE LINES 54-92 .. code-block:: Python import numpy as np from astropy.coordinates import Angle from gammapy.maps import WcsGeom from gammapy.modeling.models import ( GaussianSpatialModel, Models, PowerLawSpectralModel, SkyModel, ) phi = Angle("30 deg") model = GaussianSpatialModel( lon_0="2 deg", lat_0="2 deg", sigma="1 deg", e=0.7, phi=phi, frame="galactic", ) geom = WcsGeom.create( skydir=model.position, frame=model.frame, width=(4, 4), binsz=0.02 ) ax = model.plot(geom=geom, add_cbar=True) # illustrate size parameter region = model.to_region().to_pixel(ax.wcs) artist = region.as_artist(facecolor="none", edgecolor="red") ax.add_artist(artist) transform = ax.get_transform("galactic") ax.scatter(2, 2, transform=transform, s=20, edgecolor="red", facecolor="red") ax.text(1.5, 1.85, r"$(l_0, b_0)$", transform=transform, ha="center") ax.plot([2, 2 + np.sin(phi)], [2, 2 + np.cos(phi)], color="r", transform=transform) ax.vlines(x=2, color="r", linestyle="--", transform=transform, ymin=-5, ymax=5) ax.text(2.25, 2.45, r"$\phi$", transform=transform) .. image-sg:: /user-guide/model-gallery/spatial/images/sphx_glr_plot_gauss_001.png :alt: plot gauss :srcset: /user-guide/model-gallery/spatial/images/sphx_glr_plot_gauss_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(2.25, 2.45, '$\\phi$') .. GENERATED FROM PYTHON SOURCE LINES 93-96 YAML representation ------------------- Here is an example YAML file using the model: .. GENERATED FROM PYTHON SOURCE LINES 96-104 .. code-block:: Python pwl = PowerLawSpectralModel() gauss = GaussianSpatialModel() model = SkyModel(spectral_model=pwl, spatial_model=gauss, name="pwl-gauss-model") models = Models([model]) print(models.to_yaml()) .. rst-class:: sphx-glr-script-out .. code-block:: none components: - name: pwl-gauss-model type: SkyModel spectral: type: PowerLawSpectralModel parameters: - name: index value: 2.0 - name: amplitude value: 1.0e-12 unit: cm-2 s-1 TeV-1 - name: reference value: 1.0 unit: TeV spatial: type: GaussianSpatialModel frame: icrs parameters: - name: lon_0 value: 0.0 unit: deg - name: lat_0 value: 0.0 unit: deg - name: sigma value: 1.0 unit: deg - name: e value: 0.0 - name: phi value: 0.0 unit: deg metadata: creator: Gammapy 1.3.dev222+g8af9dbfc9 date: '2024-05-01T14:12:20.264986' origin: null .. _sphx_glr_download_user-guide_model-gallery_spatial_plot_gauss.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/gammapy/gammapy-webpage/main?urlpath=lab/tree/notebooks/dev/user-guide/model-gallery/spatial/plot_gauss.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_gauss.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_gauss.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_