Measurements description

This section describes the parameters measured by $\textsc{LiMe}$ . 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 $\textsc{LiMe}$ 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 $\textsc{LiMe}$ 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:
```
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.

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 $\textsc{LiMe}$ fits a linear continuum. This variable represents is the gradient. y = m*x + n
n_cont (.n_cont, float): Using the line adjacent continua regions $\textsc{LiMe}$ fits a linear continuum. This variable represents is the interception. 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:

$Eqw = \int_{\lambda_{1}}^{\lambda_{2}}\frac{F_{c}-F_{\lambda}}{F_{c}}d\lambda = \int_{\lambda_{1}}^{\lambda_{2}}\frac{F_{line}}{F_{c}}d\lambda$

where $F_c$ is the integrated flux of the linear continuum in the line region (.cont) and $F_\lambda$ is the spectrum flux. In single lines, $F_{line}$ 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:

$z_{\lambda} = \frac{\lambda_{obs}}{\lambda_{theo}} - 1$

where $\lambda_{obs}$ 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 $km/s$ 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 $km/s$ 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:

$\frac{S}{N}_{line}\approx\frac{\sqrt{2\pi}}{6}\frac{A_{line}}{\sigma_{cont}}\sqrt{N}\approx\frac{F_{line}}{\sigma_{cont}\cdot\sqrt{N}}$

where $A_{line}$ is the amplitude of the line, $F_{line}$ is the integrated flux of the line (.intg_flux) $\sigma_{cont}$ is the continuum flux standard deviation (.cont_err) and $N$ is the number of pixels in the input line band. The later parameter approximates to $N=6\sigma$ in single lines, where $\sigma$ 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:

$\frac{S}{N}_{cont} =\frac{F_{cont}}{\sigma_{cont}}$

where $\sigma_{cont}$ is the continuum flux at the peak wavelength and $\sigma_{cont}$ 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:

$V (Km/s) = c \cdot \frac{\lambda_{obs}}{\lambda_{peak}} - 1$

where $c = 299792.458 km/s$ is the speed of light, $\lambda_{obs}$ is the wavelength mask array selection between $w3$ and $w4$ points and $\lambda_{peak}$ 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 $km/s$ . 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 $w3$ and $w4$ 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:

$F_{\lambda}=\sum_{i}A_{i}e^{-\left(\frac{\lambda-\mu_{i}}{2\sigma_{i}}\right)^{2}}$

where $F_{\lambda}$ is the combined flux profile of the emission line for the line wavelength range $\lambda$ . $A_{i}$ is the height of a gaussian component with respect to the line continuum (.cont), $\mu_{i}$ is the center of the of gaussian component and $\sigma_{i}$ 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 $\chi^2$ calculated by the least squares algorithm.

Note

The Gaussian built-in model in LmFit defines the amplitude $(A_{i})$ as the flux under the gaussian profile. $\textsc{LiMe}$ 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 $km/s$ . This parameter is calculated using the expression:

$v_{r} = c \cdot \frac{\lambda_{center}}{\lambda_{ref}} - 1$

where $c = 299792.458 km/s$ is the speed of light, $\lambda_{center}$ is the Gaussian profile central wavelength (.center) and $\lambda_{ref}$ is the reference wavelength. In non-blended lines $\lambda_{ref}$ is the observed peak wavelength (.peak_wave). In blended lines, $\lambda_{ref}$ 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 $km/s$ .
sigma_vel (.sigma_vel, np.array()): This array contains the Gaussian components standard deviation in $km/s$ . This parameter is calculated using the expression:

$\sigma_{v} (km/s) = c \cdot \frac{\sigma}{\lambda_{ref}}$

where c $c = 299792.458 km/s$ is the speed of light, $\sigma$ is the Gaussian profile standard deviation (.sigma) and $\lambda_{ref}$ is the reference wavelength. In non-blended lines $\lambda_{ref}$ is the observed peak wavelength (.peak_wave). In blended lines, $\lambda_{ref}$ 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 $km/s$ .
FWHM_g (.FWHM_g, np.array()): This array contains the Full Width Half Maximum of the Gaussian components in in $km/s$ . This parameter is calculated as:

$FWHM_{g}=2\sqrt{2\,ln2}\sigma_{v}$

where $\sigma$ 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:

$F_{i, g} = A_i \cdot \sqrt{2\pi} \cdot \sigma_i$

where $A_i$ is Gaussian component amplitude (.amp) and $\sigma_{i}$ 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 $\chi^2$ diagnostic calculated by LmFit
redchi (.redchi, float): This variable contains the reduced $\chi^2$ diagnostic calculated by LmFit:

$\chi_{\nu}^2 = \frac{\chi^2}{N-N_{varys}}$

where the $\chi^2$ diagnostic is divided by the number of data points, $N$ , minus the number of dimensions $N_{varys}$
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.