.. _measurements_page: ======================== Measurements description ======================== .. image:: ../0_resources/images/measurements_table.PNG :align: center This section describes the parameters measured by :math:`\mathrm{LiMe}`. Unless otherwise noted, these parameters have the same notation in the output measurements log as the attributes in the programing objects generated with the ``lime.Spectrum`` class. These parameter references are also the column names of the ``pandas.DataFrame`` lines log (``lime.Spectrum.log``). Inputs ++++++ This section includes 3 parameters which are actually provided by the user inputs. However, they are also included in the output log for consistency. * **line** (``.line``, ``str``): This attribute is name of the line the measurements belong to. It has the :math:`\mathrm{LiMe}` line notation: format. * **band** (``.band``, ``np.array()``): This attribute consists in a six-value vector with the line bands: In the ``lime.Spectrum`` object, the mask is stored as a vector under the ``lime.Spectrum.mask`` attribute. In the ``.log`` the wavelengths are stored in individual columns with the headers: ``w1``, ``w2``, ``w3``, ``w4``, ``w5`` and ``w6``. * **profile_label** (``.profile_label``, ``str``): This attribute consists in a string with the line components separated by dashes (-). The individual components labels have the :math:`\mathrm{LiMe}` line notation and they may also have a suffix for the kinematic component. In single lines, the default value for this attribute is ``None`` (string variable). As an example, two profile labels are included below: .. code-block:: H1_6563A_b = H1_6563A-H1_6563A_b1-N2_6584A-N2_6548A O2_3727A_m = O2_3727A-O2_3729A Identification ++++++++++++++ These parameters are not attributes of the ``lime.Spectrum`` class. Nonetheless, they are stored in the ``lime.Spectrum.log`` ``pandas.DataFrame`` and the output measuring logs for their convenience in posterior treatments. * **wave**: This parameter contains the theoretical, rest-frame, wavelength for the emission line. This value is derived from the line label provided by the user. * **ion**: This parameter contains the ion responsible for the emission line photons. This value is derived from the line label provided by the user. * **latex_label**: This parameter contains the transition classical notation in latex format. This string includes the profile components if they were provided during the fitting. .. _intgreatedProperties: Integrated properties +++++++++++++++++++++ These attributes are calculated by the ``lime.Spectrum.line_properties`` function. In these calculations, there is no assumption on the emission line profile shape. .. attention:: In the output measurements log and the ``lime.Spectrum.log``, these parameters have the same flux units as the input spectrum. However, the attributes of the ``lime.Spectrum`` are normalized by the ``.norm_flux`` constant provided by the user at the ``lime.Spectrum`` definition. * **peak_wave** (``.peak_wave``, ``float``): This variable is the wavelength of the highest pixel value in the line region. * **peak_flux** (``.peak_flux``, ``float``): This variable is the flux of the highest pixel value in the line region. * **m_cont** (``.m_cont``, ``float``): Using the line adjacent continua regions :math:`\mathrm{LiMe}` fits a linear continuum. This variable represents is the gradient. :code:`y = m*x + n` * **n_cont** (``.n_cont``, ``float``): Using the line adjacent continua regions :math:`\mathrm{LiMe}` fits a linear continuum. This variable represents is the interception. :code:`y = m*x + n` * **cont** (``.cont``, ``float``): This variable is the flux of the linear continuum at the ``.peak_wave``. * **cont_err** (``.cont_err``, ``float``): This variable is standard deviation of the adjacent continua flux. It is calculated from the observed continuum minus the linear model for both continua masks. * **intg_flux** (``.intg_flux``, ``float``): This variable contains measurement of the integrated flux. This value is calculated via a Monte Carlo algorithm: * If the pixel error spectrum is not provided by the user, the algorithm uses the line ``.cont_err`` as a uniform uncertainty for all the line pixels. * The pixel error is added stochastically to each pixel in the line region mask. * The flux in the line region is summed up taking into consideration the line region averaged pixel width and removing the contribution of the linear continuum. * The previous two steps are repeated in a 1000 loop. The mean flux value from the resulting array is taken as the integrated flux value. * **intg_err** (``.intg_err``, ``float``): This attribute contains the integrated flux uncertainty. This value is derived from the standard deviation of the Monte Carlo flux calculation described above. .. attention:: Blended components have the same ``.intg_flux`` and ``.intg_err`` values. * **eqw** (``.eqw``, ``float`` or ``np.array()``): This parameter is the equivalent of the emission line. It is calculated using the expression below: .. math:: Eqw = \int_{\lambda_{1}}^{\lambda_{2}}\frac{F_{c}-F_{\lambda}}{F_{c}}d\lambda = \int_{\lambda_{1}}^{\lambda_{2}}\frac{F_{line}}{F_{c}}d\lambda where :math:`F_c` is the integrated flux of the linear continuum in the line region (``.cont``) and :math:`F_\lambda` is the spectrum flux. In single lines, :math:`F_{line}` is the integrated flux (``.intg_flux``) while in blended lines, the corresponding gaussian flux (``.gauss_flux``) is used. The integration limits for the line region are ``w3`` and ``w4`` from the input user mask. * **eqw_err** (``.eqw``, ``float`` or ``np.array()``): This parameter is the uncertainty in the equivalent width. It is calculated from a Monte Carlo propagation of the ``.cont`` and its ``.cont_err`` and the uncertainty of the line flux. * **z_line** (``.z_line``, ``float``): This variable is the emission line redshift: .. math:: z_{\lambda} = \frac{\lambda_{obs}}{\lambda_{theo}} - 1 where :math:`\lambda_{obs}` is the ``.peak_wave``. In blended lines, this variable is computed using the same ``.peak_wave`` for all transitions (this is the most intense pixel in the line band). * **FWHM_int** (``.FWHM_int``, ``float``): This variable is the Full Width Half-Measure in :math:`km/s` computed from the integrated profile: The algorithm finds the pixel coordinates which are above half the line peak flux. The blue and and red edge :math:`km/s` are subtracted (blue is negative). .. attention:: This operation is only available for lines whose width is above 15 pixels. * **snr_line** (``.FWHM_int``, ``float``): This variable is the signal to noise ratio of the emission line using the definition by `Rola et al. 1994 `_: .. math:: \frac{S}{N}_{line}\approx\frac{\sqrt{2\pi}}{6}\frac{A_{line}}{\sigma_{cont}}\sqrt{N}\approx\frac{F_{line}}{\sigma_{cont}\cdot\sqrt{N}} where :math:`A_{line}` is the amplitude of the line, :math:`F_{line}` is the integrated flux of the line (``.intg_flux``) :math:`\sigma_{cont}` is the continuum flux standard deviation (``.cont_err``) and :math:`N` is the number of pixels in the input line band. The later parameter approximates to :math:`N=6\sigma` in single lines, where :math:`\sigma` is the gaussian profile standard deviation. * **snr_cont** (``.snr_cont``, ``float``): This variable is the signal to noise ratio of the emission line region using the formula: .. math:: \frac{S}{N}_{cont} =\frac{F_{cont}}{\sigma_{cont}} where :math:`\sigma_{cont}` is the continuum flux at the peak wavelength and :math:`\sigma_{cont}` is the continuum flux standard deviation. * **v_med** (``.v_med``, ``float``): This variable is the median velocity of the emission line. The emission line wavelength is converted to velocity units using the formula: .. math:: V (Km/s) = c \cdot \frac{\lambda_{obs}}{\lambda_{peak}} - 1 where :math:`c = 299792.458 km/s` is the speed of light, :math:`\lambda_{obs}` is the wavelength mask array selection between :math:`w3` and :math:`w4` points and :math:`\lambda_{peak}` is the ``.peak_wave`` of the emission line. * **v_50** (``.v_50``, ``float``): This variable is velocity corresponding to the 50th percentile of the emission line spectrum where the wavelength array is in :math:`km/s`. A cumulative sum is performed in the line flux array. Afterwards, this array is multiplied by the ``.pixelWidth`` and divided by the ``.intg_flux``. The resulting vector quantifies the flux percentage corresponding to each pixel in the :math:`w3` and :math:`w4` mask selection. Afterwards, this vector is interpolated with respect to the velocity array (whose calculation can be found above). .. attention:: This operation is only available for lines whose width is above 15 pixels. * **v_5** (``.v_5``, ``float``): This variable is the velocity corresponding to the 5th percentile of the emission line flux. The calculation procedure is described at the ``.v_50`` entry. * **v_10** (``.v_10``, ``float``): This variable is the velocity corresponding to the 10th percentile of the emission line flux. The calculation procedure is described at the ``.v_50`` entry. * **v_90** (``.v_90``, ``float``): This variable is the velocity corresponding to the 90th percentile of the emission line flux. The calculation procedure is described at the ``.v_50`` entry. * **v_95** (``.v_95``, ``float``): This variable is the velocity corresponding to the 95th percentile of the emission line flux. The calculation procedure is described at the ``.v_50`` entry. Gaussian properties +++++++++++++++++++ These attributes are calculated by the ``lime.Spectrum.gauss_lmfit`` function. These calculations assume a Gaussian or multi-Gaussian profile: .. math:: F_{\lambda}=\sum_{i}A_{i}e^{-\left(\frac{\lambda-\mu_{i}}{2\sigma_{i}}\right)^{2}} where :math:`F_{\lambda}` is the combined flux profile of the emission line for the line wavelength range :math:`\lambda`. :math:`A_{i}` is the height of a gaussian component with respect to the line continuum (``.cont``), :math:`\mu_{i}` is the center of the of gaussian component and :math:`\sigma_{i}` is the standard deviation. The first parameters has the input flux units (``lime.Spectrum.flux``), while the latter two have the input wavelength units (``lime.Spectrum.wave``). The output uncertainty in these parameters corresponds to the `1σ error `_: This is the standard error which increases the magnitude of the :math:`\chi^2` calculated by the least squares algorithm. .. note:: The Gaussian built-in model in `LmFit `__ defines the amplitude :math:`(A_{i})` as the flux under the gaussian profile. :math:`\mathrm{LiMe}` defines its own model where the amplitude is defined as the height of the line with respect to the adjacent continuum. * **amp** (``.amp``, ``np.array()``): This array contains the amplitude of the Gaussian profiles. The parameter units are those of the input spectrum flux (``lime.Spectrum.flux``). * **amp_err** (``.amp_err``, ``np.array()``): This array contains the uncertainty on the Gaussian profiles amplitude. The parameter units are those of the input flux (``lime.Spectrum.flux``). * **center** (``.center``, ``np.array()``): This array contains the Gaussian components central wavelength. The parameter units are those of the input spectrum wavelength (``lime.Spectrum.wave``). * **center_err** (``.center_err``, ``np.array()``): This array contains the uncertainty on the Gaussian profiles central wavelength. * **sigma** (``.sigma``, ``np.array()``): This array contains the Gaussian components standard deviation. The parameter units are those of the input spectrum wavelength. * **sigma_err** (``.sigma_err``, ``np.array()``): This array contains the uncertainty on the Gaussian profiles standard deviation. * **v_r** (``.v_r``, ``np.array()``): This array contains the Gaussian components radial velocity in :math:`km/s`. This parameter is calculated using the expression: .. math:: v_{r} = c \cdot \frac{\lambda_{center}}{\lambda_{ref}} - 1 where :math:`c = 299792.458 km/s` is the speed of light, :math:`\lambda_{center}` is the Gaussian profile central wavelength (``.center``) and :math:`\lambda_{ref}` is the reference wavelength. In non-blended lines :math:`\lambda_{ref}` is the observed peak wavelength (``.peak_wave``). In blended lines, :math:`\lambda_{ref}` is the component transition wavelength (``.wave``) shifted to the observed frame using the redshif provided by the user at the ``lime.Spectrum``. * **v_r_err** (``.v_r_err``, ``np.array()``): This array contains the uncertainty of the Gaussian components radial velocity in :math:`km/s`. * **sigma_vel** (``.sigma_vel``, ``np.array()``): This array contains the Gaussian components standard deviation in :math:`km/s`. This parameter is calculated using the expression: .. math:: \sigma_{v} (km/s) = c \cdot \frac{\sigma}{\lambda_{ref}} where c :math:`c = 299792.458 km/s` is the speed of light, :math:`\sigma` is the Gaussian profile standard deviation (``.sigma``) and :math:`\lambda_{ref}` is the reference wavelength. In non-blended lines :math:`\lambda_{ref}` is the observed peak wavelength (``.peak_wave``). In blended lines, :math:`\lambda_{ref}` is the component transition wavelength (``.wave``) shifted to the observed frame using the redshif provided by the user at the ``lime.Spectrum``. * **sigma_vel_err** (``sigma_vel_err``, ``float`` or ``np.array()``) This array contains the uncertainty of the Gaussian components standard deviation in :math:`km/s`. * **FWHM_g** (``.FWHM_g``, ``np.array()``): This array contains the Full Width Half Maximum of the Gaussian components in in :math:`km/s`. This parameter is calculated as: .. math:: FWHM_{g}=2\sqrt{2\,ln2}\sigma_{v} where :math:`\sigma` is the velocity dispersion of the Gaussian components (``.sigma_vel``). * **gauss_flux** (``.gauss_flux``, ``np.array()``): This array contains the flux of the Gaussian components. It is calculated using the expression: .. math:: F_{i, g} = A_i \cdot \sqrt{2\pi} \cdot \sigma_i where :math:`A_i` is Gaussian component amplitude (``.amp``) and :math:`\sigma_{i}` gaussian component standard deviation (``.sigma``) * **gauss_flux_err** (``.gauss_flux_err``, ``np.array()``): This array contains the uncertainty of the Gaussian components flux. Diagnostics +++++++++++ These section contains the parameters which provide a qualitative or quantitative diagnostic on the line measurement. * **chisqr** (``.chisqr``, ``float``): This variable contains the :math:`\chi^2` diagnostic `calculated by LmFit `_ * **redchi** (``.redchi``, ``float``): This variable contains the reduced :math:`\chi^2` diagnostic `calculated by LmFit `_: .. math:: \chi_{\nu}^2 = \frac{\chi^2}{N-N_{varys}} where the :math:`\chi^2` diagnostic is divided by the number of data points, :math:`N`, minus the number of dimensions :math:`N_{varys}` * **aic** (``.aic``, ``float``): This variable contains the `Akaike information criteria `_ calculated by `LmFit `_ * **bic** (``.bic``, ``float``): This variable contains the `Bayesian information criteria `_ calculated by `LmFit `_ * **observation** (``.observation``, ``str``): This variable contains errors or warnings generated during the fitting of the line (not implemented). * **comments** (``.comments``, ``str``): This variable is left empty for the user to store comments.