FAMIAS User Manual Wolfgang Zima Instituut voor Sterrenkunde, K.U. Leuven B-3001 Leuven, Belgium e-mail: zima@ster.kuleuven.be http://www.ster.kuleuven.be/∼zima/famias 1. Introduction famias (Frequency Analysis and Mode Identiﬁcation for AsteroSeismology) is a collection of state-of-the-art software tools for the analysis of photometric and spectroscopic time series data. It is one of the deliverables of the Work Package NA5: Asteroseismology of the European Coordination Action in Helioand Asteroseismology (HELAS). Two main sets of tools are incorporated in famias. The ﬁrst set allows to search for periodicities in the data using Fourier and non-linear least-squares ﬁtting algorithms. The other set allows to carry out a mode identiﬁcation for the detected pulsation frequencies to determine their pulsational quantum numbers, the harmonic degree, , and the azimuthal order, m. The types of stars to which famias is applicable are main-sequence pulsators hotter than the Sun. This includes the Gamma Dor stars, Delta Sct stars, the slowly pulsating B stars and the Beta Cep stars - basically all pulsating main-sequence stars, for which empirical mode identiﬁcation is required to successfully carry out asteroseismology. This user manual describes how to use the diﬀerent features of famias and provides two tutorials that demonstrate the usage of famias for spectroscopic and photometric mode identiﬁcation. 1.1 Overview The following key features are provided by famias: • Search for periodicities in photometric/spectroscopic time series using Fourier analysis and multi-periodic least-squares ﬁtting techniques. • Spectroscopic mode identiﬁcation using the moment method (Briquet & Aerts 2003) and Fourier parameter ﬁt method (Zima 2006) • Photometric mode identiﬁcation using the method of amplitude ratios and phase diﬀerences based on pre-computed model grids (Balona & Stobie 1979; Watson 1988; Cugier et al. 1994; Daszyńska-Daszkiewicz et al. 2002). • Eﬃcient usage of multi-core processors with parallel computing. 20 Overview The user interface of famias is structured into diﬀerent tabs that contain the modules dedicated to the diﬀerent tools. The tabs can be selected by clicking on their descriptive name. The two main modules are for the spectroscopic and photometric analysis. Each of these modules is subdivided into tools for data management, frequency searching and mode identiﬁcation. In the following, the diﬀerent available tabs are brieﬂy described. Spectroscopy Tabs • Data Manager Edit the time series of spectra or moments, perform statistics, compute line moments, examine the spectra, extract spectral lines, etc. • Fourier Analysis Compute a Fourier analysis (Discrete Fourier Transformation) for each pixel of a spectrum (pixel-by-pixel) or for the diﬀerent line moments in order to detect periodicities and their statistical signiﬁcance. • Least-Squares Fitting Compute a non-linear multi-periodic least-squares ﬁt across a line proﬁle (pixel-by-pixel) or for the diﬀerent line moments and pre-whiten the data. • Line Proﬁle Synthesis Compute a time series of theoretical line proﬁles of a radially or nonradially pulsating star. • Mode Identiﬁcation Identify pulsation modes by means of the Fourier parameter ﬁt method or the moment method. • Results The results of the mode identiﬁcations are displayed and logged in this tab. • Logbook Log of all actions that were carried out in the spectroscopy module. Photometry Tabs • Data Manager Edit and modify the photometric time series. • Fourier Analysis Compute a Fourier analysis (DFT) and determine the statistical signiﬁcance of detected frequency peaks. Introduction 21 • Least-Squares Fitting Compute a non-linear multi-periodic least-squares ﬁt and pre-whiten the data. • Mode Identiﬁcation Carry out a photometric mode identiﬁcation with the method of amplitude ratios and phase diﬀerences in diﬀerent photometric passbands. • Results The results of the mode identiﬁcation are displayed and logged in this tab. • Logbook Log of all actions of the photometry module. Figure 1: Screenshot of famias. 22 Requirements 1.2 What Data Can Be Used? The spectroscopic as well as the photometric data that can be analysed with famias must fulﬁll speciﬁc quality criteria and must have been fully reduced. More speciﬁcally, this implies the following requirements: • Requirements for spectroscopic data – Time series of fully reduced and normalised spectra, including barycentric time and velocity correction – Dispersion better than 40000 – Signal-to-noise ratio higher than 200 – Unblended absorption line • Requirements for photometric data – Time series of fully reduced diﬀerential photometric data, including barycentric time correction – Multi-colour data in Strömgren, Johnson/Cousins, or Geneva ﬁlters for mode identiﬁcation – Milli-mag precision 1.3 Requirements famias has been written in the programming language C++. For the graphical user interface, the open source version of the Qt 4 library (from Trolltech1 ) has been adopted. This combination enabled the development of a software tool that requires high computational speed in combination with the ability to create cross platform versions for Linux and Mac OS X. famias also features a builtin help system with an extended manual describing the tools and providing introductory tutorials. The homepage of famias2 provides the possibility to download the software, read the on-line documentation, and to submit bug reports. 1 http://www.trolltech.com 2 http://www.ster.kuleuven.be/∼zima/famias 2. The Main Window After the start-up of famias, the Data Manager Tab of the Spectroscopy Module is shown. If you wish to work with photometric data, you have to switch to the Photometry Module. In this chapter, the main menu entries of famias are described. 2.1 The File Menu This menu contains entries for opening and saving project ﬁles and importing time series of spectra or photometric measurements. A session with famias can be saved as a project ﬁle. Such a project ﬁle contains all the data included in the current session of famias. The following entries are available in this menu. • New Project Creates a new, empty project. All entries of the current famias session will be cleared. • Open Project Opens an existing project. All current entries in famias will be cleared and replaced by the opened project. • Recent Projects Shows a list of previously opened project ﬁles. • Save Project Saves the current session of famias as a project ﬁle with the current ﬁle name (if existent). In a project the complete content of all modules of famias is saved. This includes time series data, diagrams, results from the analyses and the logbook. • Save Project as Saves the current session of famias as a project ﬁle with a new ﬁle name. • Import Set of Spectra Opens a dialogue to import a time series of spectra. The selected ﬁle must be in ASCII format and list the ﬁlenames of the spectra, the observation 24 The File Menu times (Heliocentric Julian Date), and optionally the weights of the spectra and their signal-to-noise ratio. The separate spectrum ﬁles must also have ASCII format and require the following two column structure: wavelength or Doppler velocity in km s−1 and normalised intensity. File headers can be skipped during the import of the ﬁles. Figure 2: Screenshot of the dialogue for importing a time series of spectra. Figure 2 shows a screenshot of the dialogue for importing spectra. The left column shows the raw data ﬁle. You can indicate the number of header lines to be skipped (Skip ﬁrst X lines). At the top of the right part of the window you can select the column number which contains the data type selected in the box below. If your spectrum ﬁles have headers, you can skip them by choosing the number of lines to skip. The time must be in units of days (Heliocentric Julian Date), and the weights are point weights per spectrum. Click on Ignore if no column with weights is available. All weights are then automatically set to 1. Optionally, a column with the signal-to-noise ratio per spectrum can be imported. These values are used to estimate the uncertainties of the line moments in famias. The SNR can also be estimated within famias (see p. 32). If your selections are valid and the ﬁle structure is acceptable for importing, the text on the lower right will read File OK. Otherwise, you have to check the structure of your data ﬁle. Still, if there is a problem with the structure of the spectrum ﬁles, famias might give an error message during importing, indicating in which ﬁle the read-in error occurred. After you click on OK, you must select the dispersion scale of your spectra (units of Ångstrom or Doppler velocity (km s−1 )). If the import was successful, you will see the imported data in the Data Manager Tab of the spectroscopy module. The Main Window 25 • Import Light Curve(s) Opens a dialogue to import a time series of photometric data. You can import several ﬁles simultaneously by making a multi-selection in the ﬁle import window (by pressing the Shift or Ctrl-key when selecting ﬁles). Imported ﬁles are required to be in ASCII format and must consist of two or three columns separated by spaces or tabulators. The following two columns are mandatory: time in days (Heliocentric Julian Date) and diﬀerential magnitude. A third column can consist of weights for the single measurements. A ﬁle header can be skipped during import. Figure 3: Screenshot of the dialogue for importing light curves. Figure 3 shows a screenshot of the dialogue for importing light curves. The left column shows the raw data ﬁle. You can indicate the number of header lines to be skipped (Skip ﬁrst X lines). At the top of the right part of the window you can select the column number which contains the data type selected in the box below. The time must be in units of days. You must select a passband for the magnitude, and the weights are point weights per data point. Click Ignore if no column with weights is available. All weights are then automatically set to a value of 1. If your selections are valid and the ﬁle structure is acceptable for importing, the text box on the lower right will display File OK. Otherwise, you will have to check the structure of your data ﬁle. If the import was successful, you will see the imported data in the Data Manager of the photometry module. • Quit Exit famias. 26 The Help Menu 2.2 The Edit Menu The items in this menu provide the possibility to clear the input ﬁelds, plots, and stored data of selected tabs of famias. • Clear Spectroscopy Tabs Clear all data in the selected tabs of the spectroscopy module. • Clear Photometry Tabs Clear all data in the selected tabs of the photometry module. 2.3 The Tools Menu This menu provides some useful tools that are related to asteroseismology and mode identiﬁcation. • Stellar Rotation Compute the equatorial rotational velocity, rotation period or rotation frequency, theoretical critical Keplerian break-up velocity or critical v sin i, and the critical minimum inclination for a given stellar mass, radius, v sin i, and inclination angle. • Pulsation Parameters Compute the horizontal-to-vertical amplitude ratio, the frequency in the stellar frame of reference, and the rotation frequency and the ratio of the rotation to the pulsation frequency for a given pulsation mode, stellar mass, radius, v sin i, and inclination angle. 2.4 The Help Menu The Help Menu provides access to the famias-manual, enables to submit bug reports and provides general information about the software. • famias Help This opens the built-in user manual of famias. The manual is regularly updated with new versions of famias. • Update Information This shows a list containing update information about the current and previous versions of famias. • Report a Bug Provides a link to the webpage of famias, where bug reports can be submitted on-line. The Main Window 27 • Copyright and User Agreement View general copyright information for famias and the user agreement. • About FAMIAS Provides some general information about famias. • About Qt Provides an information box about the version of Qt that was used for the current version of famias. The graphical user interface of famias has been programmed with the Trolltech Qt-library. 3. The Plot Window A plot can be zoomed in by pressing the left mouse button while moving the mouse to draw a zoom box. Pressing the right mouse button zooms out. Keep the middle mouse button pressed to pan the plot. The following commands are available in the menu Plot: • Refresh Plot/Show All Refresh the contents of the current plot. • Set Viewport Set the viewport of the current plot. • Detach Plot Open current plot in a new window. • Print Plot Print the current plot. • Export Plot To PDF Write the current plot into a PDF ﬁle. If this is a multi-plot (e.g., zeropoint, amplitude and phase from least-squares ﬁtting), the sub-plots will be written into separate ﬁles. The following commands are available in the menu Data: • Overplot If this option is checked, the plot window is not cleared when a new plot is drawn. • Show Original and Fit If a least-squares ﬁt has been computed for these data, this option shows the original data (spectrum, line moments or light curve) and the multiperiodic least-squares ﬁt. • Show Residuals Only available if the current data set consists of residuals (pre-whitened in the Least-Squares Fitting Tab). The original data minus the leastsquares ﬁt are shown. • Show Phase Plot Plot the data phased with the indicated frequency. 4. The Spectroscopy Modules After the start-up of famias, the Data Manager Tab of the Spectroscopy Module is shown. The Spectroscopy Module contains the tools that are required to search for frequencies in time series of spectra and to carry out a spectroscopic mode identiﬁcation. Additionally, synthetic line proﬁle variations of a multi-periodic pulsating star can be computed. The tools are located in tabs that have the following denominations: Data Manager, Fourier, LeastSquares Fitting, Line Proﬁle Synthesis, Mode Identiﬁcation, Results, and Logbook. These tools are described in the following sections. 4.1 Data Manager The Data Manager Tab provides information about the data that have been imported and permits to edit the data, calculate statistics, compute moments of a spectral line, set the weights of individual spectra, or extract a line using sigma-clipping. The window is structured into three data boxes and one plot window. A menu is located above each box. In the Data Sets Box you can select the time series of spectra you want to work with. The Time Series Box shows the time, number of dispersion bins, weight, and optionally the signal-tonoise ratio of all spectra of the selected data set. The Spectrum Box shows the dispersion and intensity of the spectrum currently selected in the Time Series Box. The Plot Window shows the currently selected spectrum, statistics of a spectrum (mean or standard deviation), or a time series of moments (if selected in the Data Sets Box). A screenshot of the Data Manager Tab is shown in Figure 1. Once you have successfully imported a set of spectra, its name will be added to the list of data sets (Data Sets Box). The times of measurements, number of wavelength bins, and the weight of each spectrum will be listed in the Time Series Box. The Plot Window will remain empty until you click on one of the spectra in the Time Series Box. In this case, the dispersion (in Ångstrom or km s−1 , dependent on your selection) and intensity of the selected spectrum will be listed in the Spectrum Box and the spectrum will be plotted as a blue line in the Plot Window. You can select multiple wavelength bins in the Spectrum Box. They will be displayed as red crosses in the Plot Window. 30 Data Manager 4.1.1 Data Sets Box This box shows a list of the diﬀerent data sets that have been imported or created. The data can consist of a time series of spectra (green background) or of line moments (yellow background). To select a data set, click on it or select it in the combo box at the top right of the information bar. The selected data set is used for all operations of famias. The following commands can be selected in the Data Menu: • Remove Data Set Removes the currently selected data set from the list. • Rename Data Set Renames the currently selected data set. • Export Data Set The currently selected data set will be exported as ASCII-ﬁle(s) to the disk. The suﬃx of the ﬁles has to be entered by the user. For a time series of spectra the exported ﬁles will have the following structure: One ﬁle, called times.suffix consisting of a list of three columns, namely spectra ﬁlenames, times and weights. Each spectrum of the time series will be written into a separate ASCII ﬁle and called number.suffix, where number is a running counter. If a data set of line moments is exported, a single ASCII ﬁle having the following four columns is created: time, moment value, uncertainty, and weight. • Combine Data Sets Combines the selected data sets to a new single time series. The data sets to be combined must have the same units of dispersion. Moreover, all times of measurement have to diﬀer. • Change Dispersion Scale Select wavelength in Ångstrom or Doppler velocity in km s−1 as dispersion scale of the current data set of spectra. A conversion between the two scales can be carried out in the Modify Menu of the Time Series Box. 4.1.2 Time Series Box The content of this list depends on the selected data set. If a time series of spectra is selected in the Data Sets Box, the list will consist of three (or four) columns: times of measurement, number of wavelength bins, and weight (and optionally signal-to-noise ratio). If a moment time series is selected in the Data Sets Box, the list will consist of times of measurement, moment value, and weight. The Spectroscopy Modules 31 Click on an item of the list to display dispersion and intensity of the selected spectrum in the Spectrum Box. The selected spectrum will also be displayed in the Plot Window. Multiple spectra can be selected by clicking with the left mouse button on several items in the list while pressing the Ctrl-key or the Shift-key. All items can be selected by pressing Select All. Only items that have been selected in this list (with blue background) are taken into account for the data analysis (e.g., Fourier analysis or least-squares ﬁtting). The following commands are available in the Data Menu: • Edit Data Opens a table of times and weights in a new window with the possibility to edit these values. Modiﬁcations can be written to the current data set. • Copy Selection to New Set A new data set with currently selected spectra is created and written to the Data Sets Box. Use this option to create subsets of your data. • Remove Selection The currently selected spectra are removed from the time series/data set. • Extract Dispersion Range A new data set with the currently selected spectra and the indicated dispersion range is created and written to the Data Sets Box. Use this option, e.g., to cut out certain spectral lines from your data set. The following commands are available in the Calculate Menu: • Mean Spectrum The weighted temporal mean for each pixel of the selected spectra is computed and displayed in the Plot Window. Important: All spectra must have the same dispersion scale, i.e., they must be interpolated on a common scale (use the tool Interpolate Dispersion in the Modify Menu). • Median Spectrum The weighted temporal median for each pixel of the selected spectra is computed and displayed in the Plot Window. Important: All spectra must have the same dispersion scale, i.e., must be interpolated on a common scale (use the tool Interpolate Dispersion in the Modify Menu). • Std. Deviation Spectrum The weighted temporal standard deviation (σ) for each pixel of the selected spectra is computed and displayed in the Plot Window. Important: 32 Data Manager all spectra must have the same dispersion scale, i.e., must be interpolated on a common scale (use the tool Interpolate Dispersion in the Modify Menu). • Compute Signal-to-Noise Ratio Opens a dialogue for computing the signal-to-noise ratio (SNR) of the selected spectra by making use of sigma-clipping to determine the continuum range. The calculated SNR of each spectrum can be used to set the weights of the data. In order to compute the SNR of the spectra, a suﬃciently large range of continuum must be present in the selected spectra. Description of the SNR-dialogue (see Figure 4): Initially, the Time and SNR Box displays only the list of times of the selected spectra. Once a SNR computation has been performed, it also displays the SNR of each spectrum. Clicking on a time will show the according spectrum in the Current Spectrum Box. Figure 4: Screenshot of the dialogue for computing the SNR of the spectra. In the Settings Box, the sigma-clipping factor and the number of iterations can be indicated and must be adapted for diﬀerent data sets. Sigma clipping iteratively removes outliers of a Gaussian distribution. In this case, the sigma clipping algorithm tries to ﬁnd the continuum and to exclude the spectral lines. The Spectroscopy Modules 33 After clicking on Calculate, the position of the pixels detected as continuum is marked as red crosses in the Current Spectrum Box. The three plots at the right side show information about all spectra of the time series in order to check the overall results. The top plot shows the number of bins detected as continuum of each spectrum. The middle plot shows the SNR of all spectra and the overall mean SNR. The bottom plot shows the mean intensity value of the pixels detected as continuum. The latter values should be around 1. Outliers in this plot can indicate that the continuum has not been detected properly in some spectra. In this case, the settings for the sigma-clipping must be adapted. If Write SNR as normalised weights is clicked, the time series is written into a new data set having normalised weights W =(SNR)2 . Also, each spectrum of the time series is assigned its SNR-value (fourth column in the Time Series Box). • Compute Weights from SNR This function computes the weights of each spectrum according to its SNR. The values of the SNR must already have been imported together with the spectra. The weights are calculated from (SNR)2 and normalised such that the mean value is 1. • Compute Moments Opens a dialogue for computing line moments of the selected spectra. The dispersion scale of the spectra must be in Doppler velocity expressed in km s−1 . The moments time series can either be written into a new data set to the Data Sets Box or directly to the disk as ASCII ﬁles (if the option Write Moments 0 to 6 in a ﬁle has been checked). The nth normalised moment <v n > of a line proﬁle I(v, t), corrected for the velocity of the star with respect to the sun, at the time t is deﬁned by ∞ v n I(v, t) dv −∞ n <v >I (t) = , (1) ∞ I(v, t) dv −∞ where v denotes the line-of-sight Doppler velocity of a point on the stellar surface and the denominator of this expression is equal to the equivalent width of the line. The 1st moment is the radial velocity placed at average zero, the 2nd moment describes the line variance, and the 3rd moment describes the skewness of the proﬁle. If the shape of the line proﬁle changes periodically due to stellar pulsations, the line moments also vary with the period of pulsation (or a sub-multiple thereof). For more details, we refer to Aerts et al. (1992). 34 Data Manager famias can compute the uncertainties of the diﬀerent line moments if the SNR of the spectra is known. The uncertainty is used to derive the χ2 -value of the theoretical moments when applying the moment method and consequently, to determine the statistical signiﬁcance of obtained solutions of the mode identiﬁcation. We provide here the formalism to calculate the uncertainties of the moments. The formal uncertainty of each wavelength bin of a line proﬁle σI(v,t) can be derived from the signal-to-noise ratio SNR of the spectrum by σI(v,t) = If Δ<v 0 >(t) and Δ<v n >(t) 2 2 = = SNR I(v, t) ∞ −∞ ∞ −∞ . (2) σI(v,t) dv 2 v n σI(v,t) dv (3) 2 (4) v n I(v, t) dv 2 then the variance σ 2 of the moment <v n > is 2 σ<v n> = Δ<v n >(t) ∞ I(v, t) dv −∞ 2 + Δ<v 0 >(t) ∞ −∞ ∞ −∞ I(v, t) dv 2 . (5) Description of the line moments dialogue (see Figure 5): Select the dispersion range for the computation of the line moments. The range must be large enough to include the complete line proﬁle, i.e., from continuum to continuum. Optionally, the complete dispersion range can be selected for the computations by checking Complete range. The mean SNR of all spectra or the individual SNR of each spectrum is required to compute the statistical uncertainties of the moments. The mean SNR of all selected spectra at the continuum can be estimated by computing the standard deviation spectrum and taking the inverse of the standard deviation at the position of the continuum. When selecting Individual SNR, each spectrum must be assigned a speciﬁc SNR (column 4 in the Time Series Box). The following procedure is strongly recommended for the calculation of line moments: compute the SNR of each spectrum with the function Compute signal-to-noise ratio in the Calculate Menu. Then extract the line proﬁle with the function Extract line in the Modify Menu. Use the resulting spectra for computing the moments by selecting the complete The Spectroscopy Modules 35 Figure 5: Screenshot of the dialogue for computing the line moments. dispersion range and the option Individual signal-to-noise ratio. Note that the function Extract line determines integration boundaries for the moments that are diﬀerent from one spectral line to another in order to avoid the noisy continuum in the moment computations. The use of this function is thus indispensable when the line proﬁle is moving a lot in time. If the user wants the nth moment to be written to the Data Sets Box, the moment index n must be indicated in the combo box below. The moments 0 to 6 can be exported as ASCII ﬁles, by selecting the check box Write Moments 0 to 6 in a ﬁle and indicating a ﬁle suﬃx. The output ﬁles will be written into the directory selected in the following dialogue and called Moment*.suffix. The following commands are available in the Modify menu: • Interpolate Dispersion Linear interpolation of all selected spectra onto a common grid of dispersion values. This is necessary for most data operations such as computing line statistics, Fourier analysis, least-squares ﬁtting and mode identiﬁcation with the FPF method. In the dialogue window, three diﬀerent options for the interpolation can be selected. In all three cases, the interpolated spectra will be written to a new data set. – Interpolate onto scale of ﬁrst spectrum: all spectra will be interpolated onto the dispersion scale of the ﬁrst spectrum of the currently selected spectra. – Choose a ﬁle to interpolate: interpolate onto the scale read from an ASCII ﬁle. In the dialogue window you can select in which column the dispersion values are listed. – Compute grid for interpolation: interpolate on a grid of equidistant dispersion values. The minimum, maximum and step values must be indicated by the user. 36 Data Manager • Convert Dispersion Convert the dispersion scale of the selected spectra from Ångstrom to km s−1 or vice versa, dependent on the dispersion scale of the current spectra. The value of the zero-point for the conversion must be indicated in the dialogue window. The converted spectra will be written into a new data set. • Extract Line Opens a dialogue for determining the position of a line proﬁle using sigma clipping. This tool is especially useful when line moments have to be calculated for a time series where the wavelength position line proﬁle shifts signiﬁcantly due to pulsation. Since ideally, for the computation of the moments, the continuum should not be included, the line has to be extracted. This tool determines the position of the left and right line limits through a sigma clipping algorithm, which detects the continuum. Description of the extract line dialogue (see Figure 6): The Times of Observations Box shows the list of times of the selected spectra. Clicking on a time will show the according spectrum in the Current Spectrum Box. Figure 6: Screenshot of the dialogue for extracting a spectral line. The Spectroscopy Modules 37 In the Settings Box, the sigma-clipping factor and the number of iterations can be indicated and must be adapted for diﬀerent data sets. Sigma clipping iteratively removes outliers of a Gaussian distribution. In this case, the sigma clipping algorithm tries to ﬁnd the pixels belonging to the continuum and thus to determine the limits of the line. These limits can be expanded with a number of pixels indicated with Expand limits. Alternatively, the limits can be set at certain dispersion values indicated in the spin boxes. The limits can be changed for each spectrum individually or applied to all spectra when clicking on Set current limits for all spectra. After clicking on Calculate, the position of the pixels detected as continuum are marked as red crosses in the Current Spectrum Box. The two plots on the right-hand side show information about all spectra of the time series in order to check the overall results. The top plot shows the number of bins detected as line of each spectrum. The bottom plot shows the left and right dispersion limits of each spectrum. Outliers in this plot (marked red in the plots and in the Times of Observation Box) can indicate that the line position has not been detected properly in some spectra. When clicking on OK, the extracted line will be written into a new data set. • Shift Dispersion Shift the zero-point of the dispersion of the selected spectra with a ﬁxed value (positive or negative). The shifted spectra will be written into a new data set. • Subtract Mean Subtract the temporal mean from all selected spectra, i.e., compute the diﬀerence of each spectrum from the mean spectrum. The dispersion scale of all spectra must be interpolated on each other to use this function. The new spectra will be written into a new data set. • Add Noise Add white Gaussian noise to the selected spectra. The continuum SNR must be indicated in the dialogue window. The new spectra will be written into a new data set. 4.1.3 Spectrum Box This box shows a list of the currently selected spectrum in the Time Series Box. It consists of two columns, dispersion and intensity. The dispersion can be in 38 Data Manager units of Ångstrom or km s−1 , dependent on the selected data set. Multiple bins can be selected by clicking with the left mouse button on several items in the list while pressing the Ctrl-key or the Shift-key. All items can be selected by pressing Select All. Selected items are displayed in the plot window as red crosses. The following commands are available in the menu Data: • Edit Data Opens a table of dispersion values and intensity in a new window with the possibility to edit these values. Modiﬁcations can be written to the current spectrum. • Remove Selection Remove the selected bins from the spectrum/data set. Use this function to remove bad pixels with deviating intensities from the spectra. After removal of a pixel, interpolation of the spectra onto a common velocity grid might be necessary. 4.1.4 Plot Window The plot window shows the currently selected spectrum with selected wavelength bins, a time series of moments, or the statistics of a time series of spectra (mean, standard deviation). For more information about the plot window, we refer to p. 28. The Spectroscopy Modules 39 4.2 Fourier Analysis With this module, a discrete Fourier transform (DFT) can be computed to search for periodicities in the data set selected in the Data Sets Box of the Data Manager Tab. The data can consist of a time series of spectra (twodimensional) or of a time series of moments (one-dimensional). For the latter, we refer to the photometry manual (see p. 96). To compute a Fourier analysis for a time series of spectra, you must indicate the dispersion range (in Ångstrom or km s−1 ) that should be taken into account, the frequency range, and what the calculations are based on (pixel-by-pixel line proﬁle or moments). The Fourier spectrum is displayed in the plot window and saved as data set in the List of calculations. A screenshot of the Fourier Tab is displayed in Figure 7. Figure 7: Screenshot of the Fourier Tab. 40 Fourier Analysis 4.2.1 Settings Box In this box, the settings for the Fourier analysis are deﬁned. • Dispersion range Minimum/Maximum values of the dispersion range in Ångstrom or km s−1 , dependent on the input data. The range speciﬁes which wavelength bins of the spectrum will be taken into account for the computation of the Fourier spectrum. The Complete range is selected if the corresponding box is checked. • Frequency range Minimum/maximum values of the frequency range. The Fourier spectrum will be computed from the minimum to the maximum value. • Nyquist frequency Estimate of the Nyquist frequency (mean sampling frequency). For nonequidistant time series, a Nyquist frequency is not uniquely deﬁned. In this case, the Nyquist frequency is approximated by the inverse mean of the time-diﬀerence of consecutive measurements by neglecting large gaps. • Frequency step Step size (resolution) of the Fourier spectrum. Three presets are available: Fine (≡ (20ΔT )−1 ), Medium (≡ (10ΔT )−1 ), and Coarse (≡ (5ΔT )−1 ). The corresponding step size depends on the temporal distribution of the measurements, i.e., the time diﬀerence ΔT of the last and ﬁrst measurement. It is recommended to select the ﬁne step size to ensure that no frequency is missed. The step value can be edited if desired. • Use weights If the box is checked, the weight indicated for each spectrum is taken into account in the Fourier computations. Otherwise, all weights are assumed to have equal values. • Compute spectral window If the box is checked, a spectral window of the current data set is computed. A spectral window shows the eﬀects of the sampling of the data on the Fourier analysis and thus permits to estimate aliasing eﬀects. The spectral window is computed from a Fourier analysis of the data taking the times of measurements and setting all measurement intensities to the value 1. The shape of the spectral window does not depend on the selected dispersion range and should be plotted for a frequency range that is symmetric around 0 for visual inspection. The Spectroscopy Modules 41 • Compute signiﬁcance level If the box is checked, the signiﬁcance level at a certain frequency value is computed and shown in the plot window as a red line. The following parameters can be set: – Frequency Frequency value of the peak of interest. The data will be prewhitened with this frequency and the signiﬁcance level will be computed from the pre-whitened Fourier spectrum. – S/N level Multiplicity factor of the signal-to-noise level. The displayed significance level will be multiplied by this factor. – Box size Box size b for the computation of the noise-level in units of the frequency. The displayed signiﬁcance level is computed from the running mean of the pre-whitened Fourier spectrum. For each frequency value F , the noise level is calculated from the mean of the range [F − b/2, F + b/2]. You must choose what data the calculations are based on and then press Calculate Fourier. The signiﬁcance level will be shown as a red line in the plot window together with the Fourier spectrum of the data (blue line). This option cannot be selected when computing a Fourier spectrum across the line proﬁle. In this case, no signal-to-noise criterion (e.g., signiﬁcance of a peak when SNR ≥ 4) can be applied, because the computed Fourier spectrum is an average of all Fourier spectra across the line proﬁle. To determine the signiﬁcance of a frequency peak across the line proﬁle, one should use the function Pixel with highest amplitude at f= in combination with Compute signiﬁcance level. By doing so, only the dispersion bin having the largest amplitude of the indicated frequency is taken into account for the computation of its SNR. • Calculations based on Deﬁnes what the calculation of the Fourier analysis is based on. The following settings are possible: – Pixel-by-pixel (1D, mean Fourier spectrum) Computes a Fourier spectrum which is the mean of all Fourier spectra across the selected dispersion range. The resulting Fourier spectrum is therefore one-dimensional with frequency on the x-axis and mean amplitude on the y-axis. The signal-to-noise ratio of a peak 42 Fourier Analysis cannot be determined since a frequency can have diﬀerent amplitudes across the line proﬁle. For this, use the option Pixel with highest amplitude at f= in combination with Compute signiﬁcance level. – Pixel-by-pixel (2D, only export) Computes a Fourier spectrum for each pixel (= bin) across the selected dispersion range. The output is a two-dimensional Fourier spectrum where the amplitude is a function of frequency and dispersion. Due to the generally large data size of such a Fourier spectrum (some megabytes), it can only be exported to an ASCII ﬁle. A contour plot of these data can easily be created by the user, e.g., with the program gnuplot with the commands set pm3d map and splot. – Pixel with highest amplitude at f = Computes a Fourier spectrum at the pixel where the given frequency has the highest amplitude. The purpose of this task is to determine the signiﬁcance of a frequency peak in a line proﬁle. You must indicate a frequency value to carry out this operation. This task computes for each pixel across the selected dispersion range a Fourier spectrum and determines at which position in the proﬁle the given frequency has the highest amplitude. – Equivalent width Computes the equivalent width of the line proﬁle (inside the indicated dispersion range) and calculates its Fourier spectrum. – 1st moment (radial velocity) Computes the ﬁrst moment <v 1 > of the line proﬁle (inside the indicated dispersion range) and calculates its Fourier spectrum. – 2nd moment (variance) Computes the second moment <v 2 > of the line proﬁle (inside the indicated dispersion range) and calculates its Fourier spectrum. – 3rd moment (skewness) Computes the third moment <v 3 > of the line proﬁle (inside the indicated dispersion range) and calculates its Fourier spectrum. – 4th moment Computes the fourth moment <v 4 > of the line proﬁle (inside the indicated dispersion range) and calculates its Fourier spectrum. – 5th moment Computes the ﬁfth moment <v 5 > of the line proﬁle (inside the indicated dispersion range) and calculates its Fourier spectrum. The Spectroscopy Modules 43 – 6th moment Computes the sixth moment <v 6 > of the line proﬁle (inside the indicated dispersion range) and calculates its Fourier spectrum. • Calculate Fourier Computes the discrete Fourier transform (DFT) according to your settings and displays it in the plot window as a blue line. The mean of the time series is automatically shifted to zero before the Fourier transform is computed. The peak having highest amplitude in the given range is marked in the plot window. A dialogue window reports the frequency having the highest amplitude in the selected frequency range and asks if it should be added to the frequency list of the Least-Squares Fitting Tab. 4.2.2 List of Calculations Previous Fourier calculations can be selected from the list. Each computed Fourier spectrum is saved and listed here. If a project is saved, the list of computed Fourier spectra is also saved but compressed to decrease the project ﬁle size (only extrema are saved). The following operations are possible via the Data Menu: • Remove Data Set Removes the currently selected data set from the list. • Rename Data Set Renames the currently selected data set. • Export Data Set Exports the currently selected data set to an ASCII ﬁle having the following three-column format: frequency, amplitude, power. 4.2.3 Fourier Spectrum Plot Shows the most recently computed Fourier analysis or the selection from the list of calculations. The Fourier spectrum is shown as a blue line, the signiﬁcance level, if included, is shown as a red line. The frequency and amplitude of the peak having the highest frequency are indicated. For more information about the plot window, we refer to p. 28. 44 Least-Squares Fitting 4.3 Least-Squares Fitting This module provides tools to compute a non-linear multi-periodic least-squares ﬁt of a sum of sinusoidals to your data. The ﬁtting can be applied for every bin of the spectrum separately (pixel-by-pixel) or for the diﬀerent line moments. The ﬁtting formula is Ai sin 2π(Fi t + φi ) . Z+ (6) i Here, Z is the zero-point, and Ai , Fi , and φi are respectively amplitude, frequency and phase (in units of 2π) of the i-th frequency. The least-squares ﬁt is carried out with the Levenberg-Marquardt algorithm. For a given set of frequencies, either their zero-point, amplitude and phase can be optimised (Calculate Amplitude & Phase), or additionally also the frequency value itself (Calculate All). The latter is only available for onedimensional time series (i.e., the line moments). The data can be pre-whitened with the computed ﬁt and written to the Data Sets Box of the Data Manager Tab. A screenshot of the Least-Square Fitting Tab is displayed in Figure 8. Before a mode identiﬁcation can be carried out, a least-squares ﬁt to the data must be calculated. In order to apply the Fourier parameter ﬁt method, the ﬁt must be based on the pixel-by-pixel values. To apply the moment method, the ﬁt must be based on the ﬁrst moment. 4.3.1 Settings Deﬁnes the settings for the calculation of the least-squares ﬁt. • Dispersion range Minimum/Maximum values of the dispersion range in Ångstrom or km s−1 (dependent on the input data). The range speciﬁes which wavelength bins of the spectrum will be taken into account for the computation of the least-squares ﬁt. • Use weights If this box is checked, the weight indicated for each spectrum is taken into account in the least-squares ﬁt. Otherwise, all weights are assumed to have equal values. • Pre-whiten data If this box is checked, the data will be pre-whitened with the computed least-squares ﬁt and written into a new data set. If the calculations are based on pixel-by-pixel, a new time series of spectra will be created. In The Spectroscopy Modules 45 Figure 8: Screenshot of the Least-Squares Fitting Tab. this case, the zero-point proﬁle will not be taken into account for the pre-whitening to preserve the mean shape of the line proﬁle. When a least-squares ﬁt of a line moment is computed, the pre-whitened time series of moments is written into a new data set (one-dimensional timeseries). • Calculations based on This drop-down box deﬁnes what the computation of the least-squares ﬁt is based on. The following settings are possible: – Pixel-by-pixel (MI: FPF) For each pixel (= dispersion bin) across the selected dispersion range, a separate least-squares ﬁt is computed by improving zeropoint, amplitude, and phase. For this option, the frequency value cannot be improved. The results of the ﬁt for each frequency are displayed in the plot window. The integral of the amplitude across the line in the indicated dispersion range is written to the frequency list. 46 Least-Squares Fitting This option has to be chosen if the Fourier parameter ﬁt mode identiﬁcation method should be applied. The computed least-squares ﬁts can be imported from the Mode Identiﬁcation Tab. – Equivalent width Computes the equivalent width of the line proﬁle (inside the indicated dispersion range) and calculates a least-squares ﬁt. The results are written to the frequency list. – 1st moment (radial velocity, MI: moment) Computes the ﬁrst moment <v 1 > of the line proﬁle (inside the indicated dispersion range) and calculates a least-squares ﬁt. This option has to be chosen if the moment method should be applied for the mode identiﬁcation. The computed least-squares ﬁt and time series of moments can be imported from the Mode Identiﬁcation Tab. – 2nd moment (variance) Computes the second moment <v 2 > of the line proﬁle (inside the indicated dispersion range) and calculates a least-squares ﬁt. – 3rd moment (skewness) Computes the third moment <v 3 > of the line proﬁle (inside the indicated dispersion range) and calculates a least-squares ﬁt. – 4th moment Computes the fourth moment <v 4 > of the line proﬁle (inside the indicated dispersion range) and calculates a least-squares ﬁt. – 5th moment Computes the ﬁfth moment <v 5 > of the line proﬁle (inside the indicated dispersion range) and calculates a least-squares ﬁt. – 6th moment Computes the sixth moment <v 6 > of the line proﬁle (inside the indicated dispersion range) and calculates a least-squares ﬁt. • Compute signal-to-noise ratio Computes the amplitude SNR of each selected frequency and displays it in the list of frequencies. The noise is computed from the Fourier spectrum of the pre-whitened data. The Box size is the width of the frequency range which is taken into account for the calculation of the noise. For a box width of b, the noise of a given frequency F is the mean value of the Fourier spectrum of the residuals in the range [F − b/2, F + b/2]. How the SNR is computed depends on the selected calculation basis. In the case of pixel-by-pixel, for each frequency, the dispersion bin where this The Spectroscopy Modules 47 frequency has the highest amplitude is determined. The SNR is derived from this bin alone. For the moments, the SNR is computed from the ratio of AF and the noise of pre-whitened Fourier spectrum at the position of F . • Calculate Amplitude + Phase Computes a least-squares ﬁt with the Levenberg-Marquardt algorithm using the above mentioned ﬁtting formula. The zero-point, amplitude and phase are calculated, whereas the frequency is kept ﬁxed. If the computations are based on pixel-by-pixel, the determined (improved) values of zero-point, amplitude, and phase are plotted for each frequency in the plot window. The uncertainties are derived from the error matrix of the least-squares ﬁtting algorithm. The residuals (=mean standard deviation of the residuals) and the integral of the amplitude across the selected dispersion range are written to the frequency list. For the moments, the following optimised values are written into the frequency list: the zero-point and its uncertainty, the standard deviation of the residuals, for each selected frequency its amplitude and phase, and their formal uncertainties derived from the error matrix of the leastsquares ﬁtting algorithm. • Calculate All Computes a least-squares ﬁt with the Levenberg-Marquardt algorithm using the above mentioned ﬁtting formula. The zero-point, amplitude, phase and frequency are improved. This option cannot be selected for computing a least-squares ﬁt across the proﬁle (pixel-by-pixel). For the moments, the following optimised values are written to the frequency list: the zero-point and its uncertainty, the standard deviation of the residuals, the frequency, amplitude and phase and their formal uncertainties derived from the error matrix of the least-squares ﬁtting algorithm. 4.3.2 List of Frequencies The List of Frequencies Box displays the results of the least-squares ﬁt. Frequencies that should be included in a least-squares ﬁt can be entered in the column Frequency and selected by clicking on the check box in the column Use. The following values are shown in this box after a least-squares ﬁt has been calculated: 48 Least-Squares Fitting • Least-squares ﬁt across the proﬁle with the option Pixel-bypixel The mean standard deviation of the residuals (pre-whitened spectra) across the selected dispersion range (Residuals) and, for each selected frequency, the integral of the amplitude across the dispersion range and its uncertainty are shown. If Compute signal-to-noise ratio has been selected, the amplitude SNR of each frequency is shown in the column SNR. This value refers to the SNR of the dispersion bin where the particular frequency has its highest amplitude. • Least-squares ﬁt with the option Moments (Equivalent width and 1st through 6th moment) The zero-point, its formal uncertainty and the standard deviation of the residuals are shown at the top. The improved values of frequency, amplitude and phase and their formal statistical uncertainties are shown in the list. The phase and its uncertainty is in units of 2π. The last column lists the SNR for each frequency (only shown when the box Calculate signal-to-noise ratio has been checked). The SNR is computed from the Fourier spectrum after pre-whitening with all selected frequencies. For each frequency, the assumed noise-level is computed from the mean amplitude around the frequency value with the box size indicated in the ﬁeld Calculate signal-to-noise ratio. • Export frequencies Exports all frequency, amplitude and phase values of the List of frequencies to an ASCII ﬁle. The ﬁle format is compatible with the program Period04 (Lenz & Breger 20051 ). • Import frequencies Imports an ASCII list of frequencies having the following four-column format separated by tabulators: frequency counter, frequency value, amplitude, phase (see example below). Bracketed values are unchecked frequencies. This format is compatible with the program Period04 (Lenz & Breger 20051 ). F1 F2 F3 F4 (5.2861 6.2566 5.885284 (10.583572 0.029179815 0.017759398 0.029203887 0.022958049 1 http://www.univie.ac.at/tops 0.534 ) 0.7461502 0.47617591 0.55097456 ) The Spectroscopy Modules 49 4.3.3 Least-Squares Fit plot The plot window displays zero-point, amplitude and phase and their uncertainties of the current least-squares ﬁt across the line proﬁle (only active when pixel-by-pixel was selected for the calculations). The frequency can be selected in the combo box at the top. • Export current LSF Export the current least-squares ﬁt across the line proﬁle to ASCII-ﬁles. You must indicate a ﬁle name with an extension (like name.ext). For each frequency x, a separate output ﬁle, called name Fx.zap is created. The ﬁles consist of the following columns: dispersion value, zero-point, standard deviation of the zero-point, amplitude, standard deviation of the amplitude, phase (in units of 2π), standard deviation of the phase. For more information about the plot window, we refer to p. 28. 50 Line Proﬁle Synthesis 4.4 Line Proﬁle Synthesis This module can be used to compute a time series of synthetic line proﬁles of a multi-periodically radially or non-radially pulsating star. The synthetic line proﬁles are written as a new data set to the Data Manager Tab. The Fourier parameter ﬁt method in famias uses the same implementation for the computation of the synthetic line proﬁles. A screenshot of the Line Proﬁle Synthesis Tab is displayed in Figure 9. 4.4.1 Theoretical background We now brieﬂy describe the approach for computing the line proﬁles. For a more detailed description, we refer to Zima (2006). The following is slightly modiﬁed from this publication. We assume that the displacement ﬁeld of a pulsating star can be described by a superposition of spherical harmonics. Our description of the Lagrangian displacement ﬁeld is valid in the limit of slow rotation taking the eﬀects of the Coriolis force to the ﬁrst order into account (Schrijvers et al. 1997). Since deviations from spherical symmetry due to centrifugal forces are ignored, our formalism is reliable only for pulsation modes whose ratio of the rotation to the angular oscillation frequencies Ω/ω < 0.5 (Aerts & Waelkens 1993). This limitation excludes realistic modeling of rapidly rotating stars and low-frequency g-modes. For higher frequency p-modes, such as observed in many δ Scuti and β Cephei stars, the given criterion is fulﬁlled and a correct treatment is provided. The intrinsic line proﬁle is assumed to be a Gaussian. This is a good approximation for strong spectral lines of metals where the rotational broadening dominates over other line-broadening mechanisms. A distorted proﬁle is computed from a weighted summation of Doppler shifted proﬁles over the visible stellar surface. Additionally, we take into account a parametrised variable equivalent width due to temperature and brightness variations across the stellar surface. We assume an unperturbed stellar model to be spherically symmetric, in hydrostatic equilibrium, and unaﬀected by a magnetic ﬁeld or rotation. The position of a mass element of such a star can be written in spherical coordinates (r, θ, φ) deﬁned by the radial distance to the stellar centre r, the co-latitude θ ∈ [0, π], i.e., the angular distance from the pole, and the azimuth angle φ ∈ [0, 2π]. Any shift of a mass element from its equilibrium position is given by the Lagrangian displacement vector ξ = (ξr , ξθ , ξφ ). This displacement modiﬁes the initial pressure p0 , the density 0 , and the gravitational potential Φ0 as a function of r, θ, φ, and the time t. The linear, adiabatic perturbations of these parameters are governed by the four equations of hydrodynamics, i.e., Poisson’s equation, the equation of motion, the equation of continuity, and 51 The Spectroscopy Modules the energy equation, which translates into the condition for adiabacity in the absence of non-adiabatic eﬀects in the stellar envelope. This set of diﬀerential equations is solved by assuming that all perturbed quantities depend on Y m (θ, φ) eiωt , where Y m (θ, φ) denotes the spherical harmonic of degree and of azimuthal order m, ω is the angular pulsation frequency, and t the time. The spherical harmonic can be written as Y m (θ, φ) ≡ N m P |m| (cos θ) eimφ . (7) |m| denotes the associated Legendre function of degree Here, P imuthal order m, given by P m (x) ≡ +m m d (−1)m (1 − x2 ) 2 (x2 − 1) , 2 ! dx +m and az- (8) and N m = (−1) m+|m| 2 (2 + 1) ( − |m|)! 4π ( + |m|)! (9) is a normalisation constant. The deﬁnition of N m changes from author to author, which must be taken into account when comparing derived amplitudes. In our formalism a positive value of m denotes a pro-grade mode, i.e., a wave propagating in the direction of the stellar rotation around the star. We model uniform stellar rotation, including ﬁrst-order corrections due to the Coriolis force, which gives rise to toroidal motion. The resulting displacement ﬁeld in the case of a slowly rotating non-radially pulsating star cannot be described by a single spherical harmonic anymore. It consists of one spheroidal and two toroidal terms, which only have a horizontal component, and is given for an angular frequency ω in the stellar frame of reference and a time t by ξ= √ 4π as, 1, k ∂ 1 ∂ ,k ∂θ sin θ ∂φ Y m (θ, φ) e−iωt + at, +1 0, 1 ∂ ∂ ,− , sin θ ∂φ ∂θ −i(ωt+ 2 ) Ym +1 (θ, φ) e + at, −1 0, 1 ∂ ∂ ,− , sin θ ∂φ ∂θ −i(ωt− 2 ) Ym −1 (θ, φ) e π (10) π (Martens & Smeyers 1982, Aerts & Waelkens 1993, Schrijvers et al. 1997). Note that the term proportional to Y m −1 is not deﬁned for radial and sectoral modes. Here, as, denotes the amplitude of the spheroidal component of the displacement ﬁeld, whereas at, +1 and at, −1 are the corresponding amplitudes 52 Line Proﬁle Synthesis of the toroidal components. We neglect the ﬁrst order correction of the amplitude as due to rotation, whereby the amplitudes of the toroidal terms can be approximated by the following relations: at, +1 = as, at, −1 = as, Ω − |m| + 1 2 (1 − k), ω +1 2 +1 Ω + |m| 2 1 + ( + 1)k . ω 2 +1 (11) √ The 4π in Eq. (10) is introduced in order to scale the normalisation √ factor 4πN00 = 1, such that as represents the fractional radius variation for radial pulsation. The ratio of the horizontal to vertical amplitude, which attains quite diﬀerent values for p- and g-modes, can be approximated by the following relation in the limit of no rotation GM ah (12) = 2 3 k≡ as ω R where ah and as are the horizontal and vertical amplitude, G is the gravitational constant, M is the stellar mass, and R is the stellar radius. We assume that the intrinsic line proﬁle is a Gaussian, which may undergo equivalent width changes due to temperature variations. The distorted line proﬁle is calculated from an integration of an intrinsic proﬁle over the whole visible stellar surface, which - for computational purposes - numerically results in a weighted summation over the surface grid. We deﬁne the intrinsic Gaussian proﬁle in a surface point having the lineof-sight velocity V as I(v, Teﬀ , log g) = 1+ δF F 1− Wint (Teﬀ ) −( V −v )2 σ √ . e σ π (13) Here, v is the velocity across the line proﬁle, δF/F takes the surface ﬂux of the emitting segment into account, Wint (Teﬀ ) is the equivalent width as a function of the eﬀective temperature (see Eq. (14)); and σ is the width of the intrinsic proﬁle. The distorted line proﬁle is calculated by summation over all visible segments on the surface grid of (θ, φ) weighted over the projected surface. The response of a line’s equivalent width to local temperature changes is dependent on the involved element, its excitation, and the temperature in the zone where the line originates. In order to take this eﬀect into account, a variable equivalent width of the intrinsic line proﬁles must be considered for calculating the distorted proﬁle. Since there is no phase shift between δWint (T ) and δT , we can write, following Schrijvers & Telting (1999), Wint (Teﬀ ) = W0 (1 + αW δTeﬀ ), (14) 53 The Spectroscopy Modules where αW is a parameter denoting the equivalent width’s linear dependence on δTeﬀ , which can be approximated for δTeﬀ 1. In famias, this parameter is denotes as d(EW)/d(Teﬀ). For calculating the local temperature, surface gravity, and ﬂux variations, we closely followed Balona (2000) and Daszyńska-Daszkiewicz et al. (2002). Since the ﬂux variation δF/F is mainly a function of Teﬀ and log g, we can write in the limit of linear pulsation theory δTeﬀ δg δF = αT + αg = F Teﬀ g δR 1 3ω 2 = αT f eiψf − αg 2 + R0 4 4πG<ρ> , (15) where αT and αg given by αT = ∂ log F ∂ log Teﬀ g and αg = ∂ log F ∂ log g Teﬀ (16) are partial derivatives of the ﬂux, which can be calculated from static model atmospheres for diﬀerent passbands. Here, R0 is the unperturbed radius, G denotes the gravitational constant, <ρ> is the mean density of the star, f the absolute value of the complex fR +ifI , and ψf the phase lag of the displacement between the radius and temperature eigenfunctions. Then f describes the ratio of ﬂux to radius variations, which can be transformed into the ratio of temperature to radius variations due to the fact that the ﬂux is proportional to T 4. 4.4.2 Stellar Parameters In this box, the global stellar parameters that should be used for the computation of the synthetic line proﬁles are deﬁned. • Radius Stellar radius in solar units. In combination with the stellar mass, this parameter determines the k-value of the pulsation mode, i.e., the ratio of the horizontal to vertical displacement amplitude. • Mass Stellar mass in solar units. In combination with the stellar radius, this parameter determines the k-value of the pulsation mode, i.e., the ratio of the horizontal to vertical displacement amplitude. 54 Line Proﬁle Synthesis Figure 9: Screenshot of the Line Proﬁle Synthesis Tab. In famias, the following non-linear limb darkening law, described by Claret et al. (2000), is used to determine the brightness of the surface elements as a function of the line-of-sight angle α: I(µ) =1− I(1) 4 k ak (1 − µ 2 ). (17) k=1 Here, I(µ) is the speciﬁc intensity on the stellar disk at a certain line-of-sight angle θ with µ = cos θ and ak is the k-th limb darkening coeﬃcient. The limb darkening coeﬃcients are determined through the values of Teﬀ , log g, and metallicity by bi-linear interpolation in a precomputed grid (Claret et al. 2000). • Teﬀ Eﬀective temperature of the stellar surface in Kelvin. The Spectroscopy Modules 55 • log g Value of the logarithm of the gravity at the stellar surface in c.g.s. units. • Metallicity Stellar metallicity [m/H] relative to the sun. • Inclination Angle between the line of sight and the stellar rotation axis, which is assumed to be the symmetry axis for pulsation, in degrees. • v sin i Projected equatorial rotational velocity in km s−1 . The model assumes rigid rotation. 4.4.3 Line Proﬁle Parameters In this box, parameters of the synthetic line proﬁle are deﬁned. • Central wavelength Central wavelength of the line proﬁle in units of Ångstrom. This parameter determines the limb darkening coeﬃcients, which are linearly interpolated in precomputed grids using the formalism by Claret (2000). • Equivalent width Equivalent width of the line proﬁle in km s−1 . • d(EW)/d(Teﬀ ) Ratio between the equivalent width variations of the local intrinsic Gaussian line proﬁle and the local temperature variations. This parameter can have positive as well as negative values (in the latter case the equivalent width decreases with increasing temperature). In combination with the non-adiabatic parameter f , this parameter controls the temporal equivalent width variations of the line proﬁle. • Intrinsic width Width of the intrinsic Gaussian line proﬁle in km s−1 . This is the width of the line proﬁle, unbroadened by stellar rotation and pulsation. • Zero-point shift Shift of the line proﬁle with respect to zero Doppler velocity in km s−1 . The synthetic line proﬁles are computed for the assumption that the barycentre of the line proﬁle is at zero Doppler velocity. 56 Line Proﬁle Synthesis 4.4.4 Pulsation Mode Parameters In this list, the parameters of the pulsation modes are deﬁned. • Use If a box is checked, the corresponding pulsation mode is taken into account for the computation of the synthetic line proﬁles. • Frequency Value of the pulsation frequency in d−1 in the observer’s frame of reference. • Degree Spherical degree of the pulsation mode ( ≥ 0). • Order m Azimuthal order m of the pulsation mode (|m| ≤ ). A positive value of m denotes a pro-grade pulsation mode. • Vel. Amp. Velocity amplitude of the pulsation mode in km s−1 . The amplitude is normalised in such a way that it represents the intrinsic velocity for a radial pulsation mode. • P Phase φ of the pulsation mode in units of 2π. • ||f || Absolute value of the complex non-adiabatic parameter f . For a deﬁnition, we refer to Eq. (30) on p.103. In combination with the parameter d(EW)/d(Teﬀ) this parameter controls the equivalent width variations of the line proﬁle. • P (f ) Phase lag ψf between the radius and temperature eigenfunctions, in units of radians. 4.4.5 General Settings In this box, some general parameters for the computation of the synthetic line proﬁles can be deﬁned. • No. of segments Total number of segments (visible and invisible) on the stellar surface. The Spectroscopy Modules 57 The segments are uniformly distributed across the surface, i.e., each segment covers approximately the same surface area. The segments lie on a spiral that has its endpoints at the poles of the sphere. At each segment, a local intrinsic Gaussian proﬁle is deﬁned and shifted by the local Doppler velocity. The overall synthetic line proﬁle is computed by summing up over all visible local proﬁles. The higher the number of segments is, the better the precision of the computation gets at the cost of computational speed (linear increase). • Dispersion range These values deﬁne the dispersion grid in Doppler velocity (km s−1 ) for the computation of the synthetic line proﬁles. A minimum, maximum, and a step value must be indicated. Internally, a ﬁxed step size of 1 km s−1 is taken, and the minimum and maximum limits expanded by 20 km s−1 to ensure that the proﬁle is computed correctly at the limits. These synthetic proﬁles are then linearly interpolated onto the grid deﬁned by the dispersion range. • Time range Deﬁnes the minimum, maximum, and step values of the grid of times that should be used for the computation of the line proﬁles. • Import times from ﬁle This allows the user to import a ﬁle that contains a list of time values that should be used for the line proﬁle computation. • Data set name Name of the data set of synthetic line proﬁles that is written to the Data Manager Tab. • Save parameters Saves the parameters you entered in this tab to a ﬁle. • Load parameters Loads the parameters for computing synthetic line proﬁles from a ﬁle. • Compute line proﬁles Computes the synthetic line proﬁles and writes them into a new data set to the Data Manager Tab. 58 Mode Identiﬁcation 4.5 Mode Identiﬁcation This module is dedicated to the spectroscopic mode identiﬁcation with the Fourier parameter ﬁt (FPF) method and the moment method. Its main functions are: importing the observational data for the mode identiﬁcation (leastsquares ﬁt across the line proﬁle or line moments), setting stellar and pulsational parameters, deﬁning the free parameters for the optimisation (=mode identiﬁcation), and setting the parameters of the optimisation procedure. The results of the mode identiﬁcation are written to the Results Tab and to log-ﬁles on the disk. A screenshot of the Mode Identiﬁcation Tab is displayed in Figure 10. Observational data can be imported and the parameters to be optimised can be chosen. Two diﬀerent approaches for the identiﬁcation of pulsation modes are available: the FPF method (Zima 2006) and the moment method (Balona 1986a,b, 1987; Briquet & Aerts 2003). Both methods assume the following: oscillations in the limit of linearity (sinusoidal variations), slow rotation (neglecting second order rotational eﬀects), a limb darkening law according to Claret (2000), a symmetric intrinsic line proﬁle, which is a Gaussian for the FPF method, and a displacement ﬁeld that can be described by a sum of spherical harmonics. The FPF method furthermore permits to model a variable equivalent line width caused by local temperature variations on the stellar surface. Both methods rely on the fact that the bin-intensities across an absorption line proﬁle vary with the period of the associated non-radial pulsation mode. Whereas the FPF method makes use of the intensity information of each dispersion bin across the line proﬁle, the moment method uses integrated values across the proﬁle. This is the main diﬀerence between the two methods and leads to a diﬀerence in the capability of identifying pulsation modes. For the FPF method there is in principle no upper limit for the identiﬁcation of ( , m), but a very small value of v sin i as well as a large pulsation velocity relative to the projected rotational velocity can make mode identiﬁcation impossible. In the latter two cases, the moment method is better suited, but this method is only sensitive for low-degree pulsation modes with ≤ 4. In the way as they are implemented in FAMIAS, both methods take into account the uncertainties of the observations and the goodness of the ﬁt (=mode identiﬁcation) is expressed as a chi-square value. The optimisation is carried out using genetic optimisation. Such an approach permits to search for local minima, and consequently the global minimum, in a complex large multi-parameter space. For excellent reviews of spectroscopic mode identiﬁcation techniques we refer to Aerts & Eyer (2000), Balona (2000), and Mantegazza (2000). The Spectroscopy Modules 59 Figure 10: Screenshot of the Mode Identiﬁcation Tab. 4.5.1 The FPF method This method relies on the rotational broadening of a line proﬁle and thus delivers good and reliable results for v sin i > 20 km s−1 . The main assumptions of the models have been described above. For a more detailed description of this method, we refer to Zima (2006). For each detected pulsation frequency and each dispersion bin across the line proﬁle, a multi-periodic non-linear least-squares ﬁt of sinusoids is computed (use the Least-Squares Fitting Tab of famias). This delivers the observational values of zero-point (Zo ), amplitude (Ao ) and phase (Po ) as a function of the position in the line proﬁle. These observational values are ﬁtted with theoretical values derived from synthetic line proﬁles. The FPF method comes in diﬀerent ﬂavours in famias, the main diﬀerences concerning the temporal distribution of the synthetic line proﬁles and the number of pulsation modes taken into account simultaneously. The FPF method makes use of the fact that the zero-point, amplitude and phase (ZAP) across 60 Mode Identiﬁcation the line proﬁle depend on the ( , m)-values of the associated pulsation modes. By comparing the theoretical values of ZAP with the observed ZAP-values, one can, in principle, determine the degree and azimuthal order of a pulsation mode. The reduced χ2ν , which is regarded as goodness of the ﬁt, is calculated from complex amplitudes in order to combine amplitude and phase information as follows χ2ν = 1 2nλ − N nλ i=1 (AoR,i − AtR,i )2 (AoI,i − AtI,i )2 . + 2 2 σR,i σI,i (18) Here, nλ is the number of pixels across the proﬁle, N is the number of free parameters, Ao and At denote observationally and theoretically determined values, respectively, AR = Aλ cos φλ and AI = Aλ sin φλ are the real and imaginary part of the complex amplitude, and σ is the observational error. Since the amplitude and phase of a given wavelength bin are treated as independent variables, the variances are calculated from 2 = σ(Aλ )2 cos2 φλ + σ(φλ )2 A2λ sin2 φλ , σR,λ (19) 2 = σ(Aλ )2 sin2 φλ + σ(φλ )2 A2λ cos2 φλ . σI,λ (20) 4.5.2 Optimisation settings for the FPF method In the drop-down menu Select MI method, the following selections are possible as optimisation settings: • Compute vsini, EW, intrinsic width, and velocity oﬀset (ﬁt Z) With this setting, the pulsationally independent parameters v sin i, the equivalent line width, the intrinsic width σ, and the Doppler velocity oﬀset are determined from a ﬁt of a rotationally broadened synthetic line proﬁle to the observational zero-point proﬁle. This method only provides reliable results if the line proﬁle is not signiﬁcantly broadened by pulsation. The determined values can be used as starting values for the mode identiﬁcation. • FPF Method: ﬁt AP For each selected pulsation frequency, a single-mode displacement ﬁeld and the corresponding line proﬁles are computed for 10 phase bins evenly distributed over one pulsation cycle. The theoretical values for AP are computed from a mono-periodic least-squares ﬁt to these synthetic line proﬁles. A chi-square value is computed by taking into account the observed and theoretical Fourier parameters and their observational uncertainties (for details see Zima, 2006). The zero-point across the line proﬁle, The Spectroscopy Modules 61 which gives a strong constraint on v sin i, the intrinsic line width and the equivalent width, is ignored in this case. Therefore, this option should only be chosen if already good constraints on these global parameters are known. This method assumes that the diﬀerent pulsation modes do not have a signiﬁcant inﬂuence on each other’s ZAP values. Such an assumption is valid if the ratio of the radial velocity amplitude to the projected rotational velocity for all frequencies is < 0.2. For higher values, the ZAP-values across the line proﬁle might be distorted and impossible to model with a single-mode displacement ﬁeld. In this case, the approach FPF Method (complete time series): ﬁt AP (see below) or the moment method are better suited. • FPF Method: ﬁt ZAP This option is identical to FPF Method: ﬁt AP with the exception that also the observed and theoretical zero-points are taken into account for computing the ﬁts. • FPF Method (complete time series): ﬁt AP With this option, multi-periodicity and the complete series of observational times are considered for applying the FPF method. Synthetic line proﬁles are computed from a multi-mode displacement ﬁeld, taking all selected pulsation modes into account. For each time step of the observed time series, one proﬁle is computed. The theoretical AP across the proﬁle are derived from a multi-periodic least-squares ﬁt. Since multi-periodicity is considered for this method, it can in principle be applied to stars for which the radial velocity amplitude to the projected rotational velocity is of the order of 1. This method is computationally much slower since not only 10 synthetic proﬁles but the complete time series have to be modelled. The zero-point across the line proﬁle, which gives a strong constraint on v sin i, on the intrinsic line width and on the equivalent width, is ignored in this case. Therefore, this option should only be chosen if already good constraints on these global parameters are available. • FPF Method (complete time series): ﬁt ZAP This option is identical to FPF Method (complete time series): ﬁt AP with the exception that also the observed and theoretical zero-points are taken into account for computing the ﬁts. 62 Mode Identiﬁcation 4.5.3 Practical information for applying the FPF method • The dispersion range of the least-squares ﬁt (ZAP across the line proﬁle) that is imported to the Mode Identiﬁcation Tab should cover the range where the amplitude reaches signiﬁcant values. In general, the continuum should therefore be excluded and the line wings can in many cases be excluded. The least-squares ﬁt should include all signiﬁcant frequencies also combination and harmonic frequencies since they can have a signiﬁcant eﬀect on the Fourier parameters of the other pulsation frequencies. During the mode identiﬁcation, combination and harmonic frequencies should not be set as free parameters unless one has a reason to assume that they are pulsation modes intrinsic to the star. • The stellar parameters radius, mass, Teﬀ , log g, and metallicity should be quite well-known. Radius and mass can be set as variable during the ﬁt and have an inﬂuence on the k-value (ratio of horizontal to vertical displacement amplitude) of the pulsation modes. The three other parameters determine the limb darkening coeﬃcients and slightly aﬀect the ﬁtted v sin i and intrinsic line width. • Before starting the mode identiﬁcation, one should determine starting values for v sin i, the intrinsic width, the equivalent width, and the velocity zero-point oﬀset. This can be done by selecting Compute vsini, EW, intrinsic width, and velocity oﬀset (ﬁt Z) in the ﬁeld Optimisation Settings. For this optimisation no pulsation mode should be selected and v sin i, the equivalent width, the intrinsic width and the zero-point shift should be set as a variable in a reasonable range. This mode of optimisation ﬁts a theoretical rotationally broadened line proﬁle to the observational zero-point proﬁle. Any pulsational broadening of the latter is neglected during the ﬁt and can lead to an overestimation of v sin i or the intrinsic width. • The most reliable results for the mode identiﬁcation will be obtained when following an iterative scheme. In general, one can ﬁx the equivalent width and the zero-point shift, once they have been determined with suﬃcient precision in the previous step. The values of v sin i and the intrinsic width should be set as variable in a range that is determined by the ﬁt in the previous step (taking the chi-square values into account). The inclination should be set as variable in the complete possible (realistic) range between about 5 and 90 degrees taking a step of about 10 degrees. It does, in general, not make sense to set the lower range to 0 since this would imply inﬁnitive equatorial velocity if v sin i > 0 km s−1 . For each pulsation mode, a separate mode identiﬁcation should be acquired ﬁrst The Spectroscopy Modules 63 (using FPF Method: ﬁt AP or FPF Method: ﬁt ZAP). The degree, the azimuthal order and amplitude should be set to reasonable ranges. • If the pulsation frequency has a signiﬁcant amplitude in the least-squares ﬁt of the ﬁrst moment, its phase value can be used for the mode identiﬁcation. In this case, the phase can be set as variable in the range φ<v1 > , φ<v1 > +0.5 with a step of 0.5. If the frequency is not detected in the ﬁrst moment, the phase value has to be set as free in the range between 0 and 1 with a step-size of ≤ 0.01. After a ﬁrst run of the mode identiﬁcation with and m as free parameters (see ﬁeld Optimisation Settings), one normally has a constraint on the phase value φ and it should be set to φ, φ + 0.5 with a step of 0.5 (diﬀerent pulsation modes have their best ﬁt at phase values that diﬀer by half a period - this is due to the fact that we limit the inclination angle to a range between 0 and 90 degrees and not between 0 and 180 degrees). The further mode identiﬁcation should be carried out by setting the search method to and m: grid search. • Equivalent width variations of the line proﬁle due to local temperature variations at the stellar surface can be taken into account by considering the parameters ||f || and Phase (f) (see Eq. (15)) in combination with the parameter d(EW )/d(T ef f ) which can be positive or negative (see Eq. (14)). The parameter space is signiﬁcantly enlarged by setting these parameters as variable, so it is important to already have some constraints on and m before attempting to ﬁt the equivalent width variations. • Multi-mode identiﬁcation gives only good results if already some constraint about and m of the pulsation modes has been obtained. Otherwise the genetic optimisation algorithm may end up in a local minimum due to the large parameter space. • The number of segments on the stellar surface (see ﬁeld General Settings) should have a value of at least 1000. The lower this number, the lower the precision of the computations. For slowly rotating stars having low-degree modes ( < 4), a value between 1000 and 3000 in general is suﬃcient. For more rapidly rotating stars (v sin i > 50 km s−1 ) and high-degree modes, this value should be between 3000 and 10000. 4.5.4 The moment method This method uses the ﬁrst (radial velocity) and second (line width) moments of a line proﬁle as a discriminator for mode identiﬁcation. The version of the moment method we adopted, has been described in detail by Briquet & Aerts (2003) and has been slightly modiﬁed in famias. The complete time series of 64 Mode Identiﬁcation observed moments is ﬁtted with theoretical moments to determine and m. The main assumptions for the computation of the theoretical moments have been described above. We take into account the uncertainties of the observed moments that can be computed numerically if the signal-to-noise ratio of the spectra is known. The observational uncertainties are used to compute a chisquare value which provides a statistical criterion for the signiﬁcance of the mode identiﬁcation. The formalism for the calculation of the statistical uncertainty of the moments has been described on p. 34. The reduced χ2ν goodness-of-ﬁt value is computed from χ2ν 1 = 2N N i=1 <v 1 >o −<v 1 >t σ<v1 >o 2 + <v 2 >o −<v 2 >t σ<v2 >o 2 , (21) where N is the number of measurements of the time series, and the indices o and t denote observed and theoretical values, respectively. To speed up the computations of the theoretical moments, a grid of integrals is precomputed for all possible and m-combinations for 0 ≤ ≤ 4 and all inclinations between 0 and 90◦ . This computation is performed once the mode identiﬁcation has been started and may take a few minutes. The precomputed integrals depend on the limb darkening coeﬃcients and the number of segments on the stellar surface. They are thus only recomputed in a subsequent mode identiﬁcation if the latter parameter or the parameters Teﬀ, log g, Metallicity, or Central wavelength have been modiﬁed. 4.5.5 Practical information for applying the moment method • The dispersion range that is imported to the Mode Identiﬁcation Tab should cover the complete range of the line proﬁle (from continuum to continuum). The best approach would be to extract the line proﬁle (see p. 36) and select the complete dispersion range when computing the leastsquares ﬁt. The least-squares ﬁt should include all signiﬁcant frequencies as well as combination and harmonic frequencies. During the mode identiﬁcation, the latter should not be set as free parameters unless one has a reason to assume that they are pulsation modes intrinsic to the star. • The stellar parameters radius, mass, Teﬀ , log g, and metallicity should be quite well known. Radius and mass can be set as variable during the ﬁt and have an inﬂuence on the k-value (ratio of horizontal to vertical displacement amplitude) of the pulsation modes. The three other parameters determine the limb darkening coeﬃcients and thus mainly aﬀect the ﬁtted v sin i and intrinsic line width. The Spectroscopy Modules 65 • The inclination should be set as variable in the complete possible (realistic) range between about 5 and 90 degrees taking a step of about 5 degrees. It does in general not make sense to set the lower range to 0 since this would imply inﬁnite equatorial velocity if v sin i > 0 km s−1 . • After importing the line moments, the phase of each pulsation mode is set as free parameter between φ<v1 > , and φ<v1 > +0.5 with a step of 0.5 (diﬀerent pulsation modes have their best ﬁt at phase values that diﬀer by half a period. This is due to the fact that we limit the inclination angle to a range between 0 and 90 degrees and not between 0 and 180 degrees). • To obtain the most reliable results with the moment method, one should set all detected pulsation frequencies (except combinations/harmonics) as free parameters during the ﬁt. This is due to the fact that the complete time series of observed moments is ﬁtted with theoretical moments. The number of segments on the stellar surface (see General Settings Box) should have a value of at least 1000. The lower this number, the lower the precision of the computations becomes. For slowly rotating stars having low-degree modes ( ≤ 4), a value between 1000 and 3000 is in general suﬃcient. For more rapidly rotating stars (v sin i ≥ 50 km s−1 ) this value should be between 3000 and 5000. • The chi-square values of the ﬁts are derived numerically from the computation of the line moments taking into account the SNR of the spectra. If the SNR is not known for the single spectra, one can provide a mean SNR (in the box Line Proﬁle Parameters). This mean value can be obtained by computing the inverse of the standard deviation at the continuum close to the line proﬁle. In this case, the chi-square value is based on the assumption that all spectra have the same SNR and may thus not be reliable. If the SNR of each spectrum is known and listed in the Data Manager, one should select the option Individual signal-to-noise ratio in the box Line Proﬁle Parameters. 4.5.6 Setting of parameters During the mode identiﬁcation several values can be set as ﬁxed or free parameters. These values are listed in the boxes Stellar Parameters, Pulsation Mode Parameters, and Line Proﬁle Parameters. If the check box associated with the parameter is unchecked, the parameter is ﬁxed at a constant value during the optimisation. In this case a value must be entered in the input box of the column Min/Const. A parameter can be set to be variable (free) during the optimisation if the check box has been checked. In this case, two additional input 66 Mode Identiﬁcation boxes appear and values for the search range (Min, Max, Step) have to be entered. Some parameters, such as Teﬀ or log g cannot be set as free parameters since they determine the limb darkening coeﬃcient. The more parameters are set as variable simultaneously and the ﬁner the step, the larger the parameter space becomes. This must be taken into account when setting the optimisation parameters to avoid ending up in a local minimum (see Section 4.5.10 for more details). 4.5.7 Stellar Parameters This box deﬁnes the global stellar parameters that should be used for the optimisation. • Radius Stellar radius in solar units. In combination with the stellar mass, this parameter determines the k-value of the pulsation mode, i.e., the ratio of the horizontal to vertical displacement amplitude. • Mass Stellar mass in solar units. In combination with the stellar radius, this parameter determines the k-value of the pulsation mode, i.e., the ratio of the horizontal to vertical displacement amplitude. • Teﬀ Eﬀective temperature of the star in Kelvin. Together with the parameters log g and Metallicity, this parameter deﬁnes the limb darkening coeﬃcients by linear interpolation in a precomputed grid (Claret et al. 2000). • log g Value of logarithm of the gravity at the stellar surface. Together with the parameters Teﬀ and Metallicity, this parameter deﬁnes the limb darkening coeﬃcients by linear interpolation in a precomputed grid (Claret et al. 2000). • Metallicity Stellar metallicity [m/H] relative to the Sun in logarithmic units. Together with the parameters log g and Teﬀ, this parameter deﬁnes the limb darkening coeﬃcients by linear interpolation in a precomputed grid (Claret et al. 2000). • Inclination Angle between the line of sight and the stellar rotation axis in degrees. The Spectroscopy Modules 67 The models assume that the axis of rotation is aligned with the symmetry axis of the pulsational displacement ﬁeld. • v sin i Projected equatorial rotational velocity v sin i in km s−1 . 4.5.8 Pulsation Mode Parameters This box deﬁnes the parameters of the pulsation modes that should be identiﬁed. The observed data can be imported by clicking on the button. The imported frequencies can be selected with the combo box. Each frequency that should be taken into account for the mode identiﬁcation must be selected by clicking on the check box next to the frequency value. • Button Import Data Import the observational data for the mode identiﬁcation. You must ﬁrst compute a least-squares ﬁt across the line proﬁle (for FPF method) or the ﬁrst moment (for moment method) in the Least-Squares Fitting Tab. – For the moment method Select in the combo box Calculations based on of the Least-Squares Fitting Tab the option 1st moment and compute the least-squares ﬁt by clicking on Calculate Amplitude + Phase or Calculate All. The import button in the Pulsation Mode Parameters Box of the Mode Identiﬁcation Tab will now display Import data for moment method. After clicking, the spectra on which the least-squares ﬁt was based, and the selected frequencies and their phases are imported and displayed in the Pulsation Mode Parameters Box. The frequencies can be selected from the combo box next to the import button. – For the FPF method Select in the combo box Calculations based on of the Least-Squares Fitting Tab the option Pixel-by-pixel and compute the least-squares ﬁt by clicking on Calculate Amplitude + Phase. The import button in the Pulsation Mode Parameters box of the Mode Identiﬁcation Tab will now display Import data for FPF method. After clicking, the parameters zero-point, amplitude, and phase across the line proﬁle and the selected frequencies are imported and displayed in the Pulsation Mode Parameters Box. The frequencies can be selected from the combo box next to the import button. 68 Mode Identiﬁcation • Frequency Value of the pulsation frequency as it was imported from the LeastSquares Fitting Tab. This value cannot be modiﬁed (only by importing a new least-squares ﬁt). • Degree Spherical degree of the pulsation mode ( ≥ 0). Deﬁnes the search parameter space for the associated pulsation mode. The step size must have a value of ≥ 1. • Order m Azimuthal order m of the pulsation mode (|m| ≤ ). Deﬁnes the search parameter space for the associated pulsation mode. The step size must have a value of ≥ 1. • Vel. Amp. Velocity amplitude of the pulsation mode in km s−1 . The amplitude is normalised in such a way that it represents the intrinsic velocity for a radial pulsation mode. • Phase Phase φ of the pulsation mode in units of 2π. • ||f || Absolute value of the complex non-adiabatic parameter f . For a deﬁnition, we refer to Eq. (30) on p.103. In combination with the parameter d(EW)/d(Teﬀ) this parameter controls the equivalent width variations of the line proﬁle. • P (f ) Phase lag ψf between the radius and temperature eigenfunctions, in units of radians. 4.5.9 Line Proﬁle Parameters In this box, the parameters of the line proﬁle are deﬁned. • Central wavelength Central wavelength of the line proﬁle in units of Ångstrom. This parameter determines the limb darkening coeﬃcients, which are linearly interpolated in precomputed grids using the formalism by Claret et al. (2000). The limb darkening coeﬃcients slightly inﬂuence the derived values of v sin i and the intrinsic width, but generally have negligible eﬀect on the mode identiﬁcation. The Spectroscopy Modules 69 • Equivalent width Equivalent width of the line proﬁle in km s−1 . • d(EW)/d(Teﬀ ) Ratio between the equivalent width variations of the local intrinsic Gaussian line proﬁle and the local temperature variations. This parameter can have positive as well as negative values (in the latter case the equivalent width decreases with increasing temperature). For a deﬁnition see Eq. (14). • Intrinsic width Width of the intrinsic Gaussian line proﬁle in km s−1 . • Velocity oﬀset Oﬀset of the line proﬁle with respect to zero Doppler velocity in km s−1 . The synthetic line proﬁles are computed for the assumption that the barycentre of the line proﬁle is at zero Doppler velocity. In general, this is not the case for the observed line proﬁles. The following parameters are only available for the moment method. • Centroid velocity Centroid velocity of the line proﬁle. In ideal cases, this is the mean radial velocity (<v 1 >) of the star. It is in any case best to use the zero-point of the least-squares ﬁt to the ﬁrst moment (which is automatically done in famias), especially if the time series consists only of few measurements or the radial velocity amplitude is large. • Mean signal-to-noise ratio This value is used for the computation of the statistical uncertainties of the line moments if the SNR of the individual spectra is not known. In this case, the determined χ2ν -values might not be reliable if some individual spectra deviate strongly from this value. • Individual signal-to-noise ratio If the SNR is known for each spectrum, this option should be chosen to determine the statistical uncertainties of the line moments. The values of the SNR can be imported with the spectra (additional column in the list of times) or computed in famias in the Data Manager → Calculate → Compute Signal-To-Noise Ratio (see p. 32). 4.5.10 Optimisation Settings In this box, the settings for the optimisation procedure are deﬁned. The optimisation is carried out with a genetic algorithm (Michalewicz 1996). These 70 Mode Identiﬁcation settings are crucial for the mode identiﬁcation and must be chosen very carefully. The most important aspect is to avoid ending up in a local minimum. Since the computations of theoretical line proﬁles and moments is generally very time consuming, one must ﬁnd a compromise between the coverage of the parameter space and CPU time eﬃciency. Although famias provides default values for diﬀerent optimisation problems, the best way to proceed is trial-and-error, i.e., to test diﬀerent optimisation settings and to proceed iteratively. • Select MI method Selection of the mode identiﬁcation method. See above for a description of the diﬀerent possibilities. In the case of the moment method, only the option Moment method is available. • No. of starting models Generation size during the genetic optimisation. Larger values for a larger parameter space. • Max. number of iterations Stop criterion for the genetic optimisation. This number deﬁnes after how many iterations (=generations) the optimisation will stop. • Max. iterations w/o improvement Stop criterion for genetic optimisation. The optimisation stops if no improvement of the best found model has been achieved after n iterations. • Convergence speed Deﬁnes how quickly the algorithm is forced to converge. Value must be between 0 and 1. Higher values cause quicker convergence at the cost of parameter space exploration and thus precision. • No. of elite models This number deﬁnes how many of the best models will be copied unaltered to the following generation. This parameter ensures quicker convergence. • & m: free parameters/grid search Deﬁnes if the and m values are free parameters in the given range, or if they are subsequently ﬁxed (grid search) while the other parameters are being optimised. • Number of CPUs to use Number of processors that are used in parallel during the optimisation. The Spectroscopy Modules 71 4.5.11 General Settings • No. of segments Total number (visible + invisible) of segments on the stellar surface to compute the line proﬁles. Higher numbers provide higher accuracy but slower computational speed (linear dependence). • Extension Extension of output and log ﬁles. The output directory can be chosen in the Settings menu (see below). The default output directory is the directory of the project ﬁle. During the mode identiﬁcation, a log-ﬁle, called logMI.extension, is written to the disk. It contains a list of all computed models, their χ2ν -values and parameter values. After a mode identiﬁcation has stopped, the results are written to the ﬁle bestFitsLog.extension. The best 20 ﬁts (ASCII ﬁles and plots containing theoretical and observed zero-point, amplitude, and phase or moments) are written to ﬁles in the directory deﬁned in Settings (see below). • Set ﬁelds to default Set the optimisation, line and stellar parameters to default values dependent on the selected mode identiﬁcation method. The proposed values are just a guidance and have to be adapted for many optimisation problems. • Settings Opens a menu that provides the following functions: silent mode, save/load parameters, set output path for the logﬁles and clear all ﬁelds of the Mode Identiﬁcation Tab. The silent mode toggles the updating of the progress bars. For some optimisations, the computational performace decreases signiﬁcantly, if the progress bars are updated. • Reset Resets the previous optimisation. Must be pressed if a new optimisation procedure should be started. • Start mode identiﬁcation Starts the mode identiﬁcation. The optimisation process can be stopped at any time by clicking again on this button. The results of the mode identiﬁcation will be written into the Results Tab. After an optimisation process has stopped, again clicking this button will continue the optimisation at the stage where it stopped. To begin a new process, click the button Reset. 72 Mode Identiﬁcation • Progress bars and counter The progress bars show the total progress and the progress in the current iteration. Below the bars, a counter gives the total number of computed models. The Spectroscopy Modules 73 4.6 Results This window shows the results of the current and previously derived spectroscopic mode identiﬁcations. A list of the parameters of the best ﬁtting models, the ﬁts of the theoretical models to the observations, and diagrams where the free parameters are plotted against the corresponding χ2ν -values are displayed. The results of previously performed mode identiﬁcation process are logged. Once a mode identiﬁcation has been started, this window is updated regularly with the actual status of the optimisation. A screenshot of the Results Tab is displayed in Figure 11. Figure 11: Screenshot of the Results Tab. Press the button Update to update the list of best models and the plots with the current status of the mode identiﬁcation. 74 Results 4.6.1 Best models This table lists the parameters of the 20 best ﬁtting models. The ﬁrst column always shows the χ2ν -value. The other columns contain the parameters that have been set as free for the mode identiﬁcation. Two display options are possible and can be selected in the combo box above: • Best models List the free parameters of the models having the lowest χ2ν -values. • Best (l,m)-combinations List the free parameters of the models having the lowest χ2ν -values and diﬀerent ( ,m)-combinations. For each possible combination of and m, the best model is shown. 4.6.2 Chi-square plots This plot-window displays the χ2ν -values (log-scale) of all models that have been computed in the current optimisation as a function of the free parameters. The uncertainty of the ﬁt for the diﬀerent parameters can thus be estimated. The free parameter can be selected in the combo box above. By selecting Model # in the combo box, the temporal evolution of the χ2ν -values during the optimisation is plotted. 4.6.3 Comparison between ﬁt and observation This box shows the ﬁt of the theoretical values to the observations. The content depends on the selected mode identiﬁcation method and is described below. The model can be selected by clicking on the corresponding row in the table of best models. The observed values are shown as blue line or symbols, the statistical uncertainty as a green line, and the modelled values as a red line. • Compute vsini, EW, intrinsic width, and velocity oﬀset (ﬁt Z) The observed (blue line with uncertainty range as green line) and synthetic (red line) zero-point proﬁle are displayed. • FPF methods Three panels are displayed: zero-point (top panel), amplitude (middle panel), and phase (bottom panel) in units of 2π are shown as a function of Doppler velocity (km s−1 ). The ﬁt is shown as a red line, whereas the observed values are shown as a blue line with the uncertainty range indicated by green lines. The ﬁt for a certain frequency can be selected in the combo box above. The Spectroscopy Modules 75 • Moment method The complete time series of observed and modeled moments is shown. The two panels show the ﬁrst and second moment, respectively. 4.6.4 List of calculations This box lists all previously performed optimisations. By clicking on an item in the list, the corresponding parameters are shown in the other windows of this tab. 4.7 Logbook The logbook provides the list of actions that have been performed with famias and corresponding information. Each time an operation is carried out in famias, a new log-entry is written to the List of actions. Clicking on an entry of this list shows the corresponding information in the text box. Entries of the List of actions can be renamed or deleted by using the menu Data. The text box can be modiﬁed and saved in famias by clicking on the button Save. 76 Tutorial: Spectroscopic mode identiﬁcation 4.8 Tutorial: Spectroscopic mode identiﬁcation This tutorial demonstrates how to perform a mode identiﬁcation of a time series of synthetic spectra with famias. The synthetic spectra can be found in the installation directory of famias in the directory tutorial/*.coasttutorial. The synthetic data simulate spectroscopic observations of a multi-periodic δ Scuti star which consist of one absorption line having realistic observation times and signal-to-noise ratio. The time-series contains 490 spectra that consist of 91 and 77 pixels, respectively, and cover a wavelength range between 5381 and 5385 Ångstrom. These spectra have been computed using the tool Line Proﬁle Synthesis of famias. The input parameters of the model can be found in the ﬁle tutorial/coasttutorial.star. References to functions of famias are written in the following manner: Main Window → File → Import Set of Spectra, which could be translated as: Select in the Main Window the function Import Set of Spectra in the menu File. In each tab, there are named boxes, which can also be referred to. For instance, Fourier Tab → Settings → Calculations based on: 1st moment implies that you have to select the Fourier Tab and choose the option 1st moment in the combo box denominated Calculations based on in the box called Settings. 4.8.1 Import spectra Follow the following procedure to import the spectra to famias. 1. Import the spectra by selecting Main Window → File → Import Set of Spectra. In the ﬁle manager that opens, select the directory tutorial located in the installation directory of famias and double-click on the ﬁle times.coasttutorial. This ﬁle contains the observation times and ﬁle names of all spectra of this time series. Figure 12 shows a screenshot of famias after importing the tutorial time series of spectra. 2. The Import ﬁle dialogue that opens shows the contents of this ﬁle. Click OK to import this ﬁle. In the following dialogue that opens, select Ångstrom as dispersion scale and click OK. 3. After successful import, the spectra are displayed as data set in the Data Manager Tab. Click into the Time series list to display speciﬁc spectra in the plot window. The Spectroscopy Modules 77 Figure 12: Data Manager after importing the tutorial time series of spectra. 4.8.2 Select dispersion range The synthetic data consist of one absorption line. In general, one has to select a suited spectral line for analysis. Such a line should be an unblended metallic line. Balmer and He-lines are not well suited for the mode identiﬁcation, since they cannot be well approximated with an intrinsic Gaussian line proﬁle. To study a speciﬁc line with famias, follow the following procedure: Click on one spectrum of the time series. Zoom in on the line in the plot window. Click on Time Series → Select All and then select Time series → Data → Extract Dispersion Range. You can modify the dispersion range in the dialogue window that opens. After you clicked OK, a new data set has been written that only contains the selected dispersion range. 4.8.3 Convert from wavelength to Doppler velocity To compute moments and to carry out a mode identiﬁcation, the dispersion scale of the spectra has to be converted from Ångstrom to km s−1 . To do so, select all spectra, click on Time series → Modify → Convert Dispersion 78 Tutorial: Spectroscopic mode identiﬁcation and enter the value of the central wavelength in the dialogue window, which is 5383.369 Ångstrom in this case. The converted spectra are written as a new data set. 4.8.4 Compute signal-to-noise ratio and weights Computing the SNR of the spectra is important for weighting the spectra, for calculating the statistical uncertainty of the moments, and for enabling the calculation of chi-square with the moment method. There are two ways to estimate the SNR of your spectra. The simple way is to compute the mean SNR of all spectra. To do so, compute the standard deviation of your spectra (see Section 4.8.7). The mean SNR is the inverse value of the standard deviation at a dispersion position of the continuum. A more sophisticated and better approach is of course to calculate the SNR of each spectrum separately. If these values have been determined with an external program, they can be imported with the list of times and ﬁle names. The weight of each spectrum can then be computed with the function Time series → Calculate → Compute weights from SNR. To compute the SNR with famias, select the function Time series → Calculate → Compute Signal-toNoise Ratio. Adapt the parameters Factor for sigma clipping and Number of iterations in such a way that only continuum is selected in all spectra. Click on Write signal-to-noise ratio as normalised weights to write a new weighted data set. 4.8.5 Compute moments 1. Compute the SNR and weights as described in Section 4.8.4. Before computing the moments, the spectral line should be extracted by excluding the continuum. We refer to Section 4.1.2 for a detailed description how to extract a spectral line with famias. If the line borders do not move signiﬁcantly due to the pulsation (=low radial velocity), one can cut out the line, assuming ﬁxed left and right limits. The position of these limits can be determined by interpolating the dispersion scale of the spectra onto a common scale (see Section 4.8.6), computing the mean spectrum with Time series → Calculate → Mean Spectrum, and noting the Doppler velocity of the left and right borders of the line (transition to the continuum). Select the original data set (non-interpolated) and extract the line with Time series → Data → Extract dispersion range. The extracted spectra are written into a new data set. Select this data set and press Select All. The Spectroscopy Modules 79 2. Call the dialogue for computing the moments by pressing Time series → Calculate → Compute Moments. Select Individual signal-to-noise ratio and the moment that you want to compute in the combo box below. Press OK and leave the centroid velocity at the proposed value (=mean barycentre of the line). 3. The time series of moments is written as a new data set. It is advisable to check for systematic trends, especially in the equivalent width and the ﬁrst moment. 4.8.6 Interpolate on common dispersion scale The tutorial spectra have diﬀerent dispersion scales. Therefore, they have to be interpolated onto a common dispersion scale to carry out several tasks, such as to compute a two-dimensional Fourier transform or a least-squares ﬁt across the line proﬁle (pixel-by-pixel), and to apply the FPF method. This is not mandatory when only line moments are used. To carry out a linear interpolation on a common dispersion scale, select all spectra and click on Time series → Modify → Interpolate Dispersion. It is advisable to interpolate onto the spectrum having the highest resolution in order not to lose information. In our case, the ﬁrst spectrum of the time series has the highest resolution. Therefore, select the function Interpolate onto scale of ﬁrst spectrum and click OK. The following dialogue window shows the dispersion values of the mask. To start the interpolation, click on OK. 4.8.7 Compute line statistics The temporal weighted mean of the spectra can be computed with the function Time series → Calculate → Mean Spectrum. To check for line proﬁle variability and estimate the SNR, the standard deviation at each pixel of the spectrum can be computed with the function Time series → Calculate → Std. Deviation Spectrum. 4.8.8 Searching for periodicities It is advisable to search for periodicities in the data in the pixels across the line proﬁle as well as in the line moments. For the ﬁrst approach, a data set should have the following properties: interpolated on a common dispersion scale, converted to km s−1 , and weighted. For the analysis of the line moments, the data should be converted to km s−1 , weighted, and the SNR should be computed for each spectrum. The two approaches can reveal pulsation modes having diﬀerent characteristics. 80 Tutorial: Spectroscopic mode identiﬁcation The search for periodicities should be carried out in the following iterative schematic way: 1. Compute a Fourier spectrum in a frequency range where you expect pulsation. 2. Compute the signiﬁcance level at the frequency having the highest amplitude and include this frequency in the least-squares ﬁtting if it is signiﬁcant. 3. Compute a multi-periodic least-squares ﬁt of the original data with all detected frequencies. In case that no unique frequency solution exists due to aliasing, compute least-squares ﬁts for diﬀerent possible frequency sets. The solution resulting in the smallest residuals should be regarded as best solution. 4. Exclude frequencies from the ﬁt that do not have a SNR above 4 (3.5 for harmonics/combination terms). 5. Pre-whiten the data with all signiﬁcant frequencies. 6. Continue with the ﬁrst point using the pre-whitened data until no significant frequency can be found. Line moments 1. Select data set Select the spectra that were prepared for the analysis of the line moments and go to the Fourier Tab. 2. Calculate Fourier spectrum of equivalent width Select the option Fourier Tab → Settings → Calculations based on → Equivalent width and click on Calculate Fourier. The plot window now displays the Fourier spectrum of the equivalent line width. A dialogue opens, indicating the highest frequency peak at F1 = 3.148 d−1 and asking if this frequency should be added to the frequency list of the Least-Squares Fitting Tab. Since we ﬁrst want to check the signiﬁcance of this frequency, click on No. 3. Compute signiﬁcance level Select the option Settings → Compute signiﬁcance level. The ﬁeld at frequency should now contain the value 3.148038. Compute the Fourier spectrum once more by clicking on Calculate Fourier. The plot window now also displays the signiﬁcance level as a red curve and the dialogue window indicates the SNR of the highest peak. Since it has a SNR of 4.1, click on Yes to include it in the frequency list. The Spectroscopy Modules 81 Figure 13: Fourier spectrum of the ﬁrst moment after pre-whitening with F2 = 11.53 d−1 . 4. Compute least-squares ﬁt Go to the Least-Squares Fitting Tab and select the ﬁeld Settings → Compute signal-to-noise ratio and the frequency F1. Compute a leastsquares ﬁt by pressing Settings → Calculate Amplitude + Phase. Improve the frequency solution by clicking Settings → Calculate All. According to the List of frequencies, this frequency has a SNR of 3.96, which is just below the signiﬁcance limit. The diﬀerence with the SNR determined in the Fourier transform is due to the fact that, in this case, the amplitude determined from the least-squares ﬁt is taken as signal. We can conclude now that there are no signiﬁcant periodic equivalent width variations in the line proﬁle. 5. Calculate Fourier spectrum of ﬁrst moment Select the option Fourier Tab → Settings → Calculations based on → 1st moment and click on Calculate Fourier. The highest peak is at the frequency F2 = 11.53 d−1 . Check for signiﬁcance as described in the previous point. Since this peak is highly signiﬁcant, it should be included 82 Tutorial: Spectroscopic mode identiﬁcation in the List of frequencies. A screenshot of famias showing the Fourier spectrum of the ﬁrst moment is displayed in Figure 13. 6. Compute least-squares ﬁt and pre-whiten data Select the detected frequency F2 in the Least-Squares Fitting Tab, compute a least-squares ﬁt and pre-whiten the data (Settings → Pre-whiten data). The residuals are written as a new data set in the Data Manager Tab. The List of frequencies shows the results of the computed ﬁt and the derived uncertainties of the parameters. The value of the ﬁeld Residuals is computed from the standard deviation of the residuals. The frequency is always indicated in units of the inverse of the input timestring. The units of the amplitude depend on the selected calculation basis. The equivalent width is in units of km s−1 . The n-th moments is in units of (km s−1 )n . The phase is in units of 2π. 7. In the Data Manager Tab, select the time series of residuals and check the computed ﬁt (red line). 8. If you want to compute a Fourier spectrum or a least-squares ﬁt of line moments, you have two possibilities. The ﬁrst option is to compute the moments of the line proﬁle in the Data Manager Tab (see Section 4.8.5) and then analyse this one-dimensional time series. The other possibility is to choose a time series of spectra and then to select the option Settings → Calculations based on → n-th moment in the Fourier Tab and the Least-Squares Fitting Tab. In this case, the corresponding time-series of moments is computed automatically for the selected dispersion range. If you then pre-whiten your data, the residuals are written as time series of moments to the Data Manager Tab (yellow background in the list of Data sets. You can calculate a Fourier transform of these residuals to search for further peaks. If you want to compute another least-squares ﬁt with an additional frequency, you have to select the original time-series of spectra (green background in the list of Data sets). 9. Compute Fourier spectrum of residuals Compute a Fourier spectrum of the residuals. A frequency at F3 = 17.5 d−1 is signiﬁcant and should also be included in the least-squares ﬁt. Also select this frequency in the Least-Squares Fitting Tab → List of frequencies and compute a least-squares ﬁt for both frequencies simultaneously. You have to select the original data set that was prepared for computing the moments. Pre-whiten the data and compute a Fourier spectrum of the residuals. No signiﬁcant peaks are left. 10. Analyse the ﬁrst three moments Analyse also the second and third moments since modes of higher degree The Spectroscopy Modules 83 Figure 14: Results of the least-squares ﬁt to the ﬁrst moment. might only have signiﬁcant amplitudes for these diagnostics. The analysis of the second moment should reveal F2 , 2F2 , 2F3 , and an additional frequency at F4 = 23.998 d−1 . The third moment only has F2 as signiﬁcant peak. 11. We can conclude that three signiﬁcant independent frequencies are present in the ﬁrst three moments of the tutorial data. Only two of them are visible in the ﬁrst moment and thus analysable with the moment method. Figure 14 displays a screenshot of famias showing the results of the least-squares ﬁt to the ﬁrst moment. Pixel-by-pixel across the line proﬁle 1. Select data set Select the data set that was prepared for the frequency analysis across the line proﬁle (pixel-by-pixel) and go to the Fourier Tab. 2. Calculate Fourier spectrum Select the option Fourier Tab → Settings → Calculations based on 84 Tutorial: Spectroscopic mode identiﬁcation → Pixel-by-pixel (1D, mean Fourier spectrum) and click on Calculate Fourier. 3. Compute signiﬁcance level The plot window now shows the mean of all Fourier spectra across the line proﬁle. A dialogue opens, indicating the highest frequency peak at F1 = 11.53 d−1 . Since we ﬁrst want to check the signiﬁcance of this frequency, click on No. To determine the signiﬁcance of F1 , check the ﬁeld Settings → Compute signiﬁcance level and select the option Settings → Calculations based on → Pixel with highest amplitude at f=. The latter option is necessary, since the signiﬁcance level cannot be determined from the mean Fourier spectrum across the line proﬁle. Click Calculate Fourier to compute the Fourier spectrum and its signiﬁcance level at the dispersion position, where the given frequency has its highest amplitude. Since this frequency is highly signiﬁcant, add it to the List of frequencies in the Least-Squares Fitting Tab. 4. Compute least-squares ﬁt In the Least-Squares Fitting Tab, select the option Settings → Calculations based on → Pixel-by-pixel (MI:FPF), check the box next to the detected frequency, and press Calculate Amplitude + Phase to compute the least-squares ﬁt. Zero-point, amplitude, and phase will be displayed in the plot panel at the right-hand side. The blue lines denote the derived ﬁt, whereas the green lines indicate the statistical uncertainty range of the ﬁt. 5. The List of frequencies shows the results of the computed ﬁt. The ﬁeld Results shows the mean standard deviation of the residual spectra. The frequency is indicated in inverse units of the input time-string. The IAD is the integrated amplitude distribution, and is calculated from the integral of the amplitude across the line proﬁle inside the selected dispersion range. 6. Pre-whiten spectra Pre-whiten the data with the determined least-squares ﬁt by checking the box Settings → Pre-whiten data and clicking Calculate Amplitude + Phase. The pre-whitened spectra are written as a new data set to the Data Manager Tab. 7. Select the time series of residual spectra in the Data Manager Tab and click on several spectra to check the quality of the ﬁt (red line). 8. Compute Fourier spectrum of residuals Compute a Fourier spectrum of the residuals by selecting the time series The Spectroscopy Modules 85 of residual spectra in the Data Manager Tab and proceeding as described in point 2. 9. When computing further multi-periodic least-squares ﬁts, the original time series of spectra has to be selected. Figure 15: Results of the least-squares ﬁt across the line proﬁle. 10. The period analysis of the tutorial data set reveals three frequencies, F1 = 11.53 d−1 , F2 = 17.50 d−1 , and F3 = 2F1 = 23.06 d−1 . The frequency F3 is a harmonic of F1 . 4.8.9 Mode identiﬁcation famias provides two diﬀerent approaches for the spectroscopic mode identiﬁcation, the moment method and the Fourier parameter ﬁt method. In the following, we will describe in detail the approach for each method separately. • Setting the parameters Parameters on the Mode Identiﬁcation Tab that have a check box next to the parameter name can be set as variable during the optimisation. In 86 Tutorial: Spectroscopic mode identiﬁcation this case, a minimum, maximum, and step value have to be indicated. If the box is unchecked, the parameter is set as constant during the optimisation with the value indicated. • Stellar parameters You need to provide estimates for the stellar radius, mass, Teﬀ , log g, and metallicity in the ﬁeld Stellar Parameters. The indicated radius and mass mainly aﬀect the numerical calculation of the horizontal to vertical pulsation amplitude and can be set as variable during the optimisation. The three other parameters determine the limb darkening coeﬃcient, which is interpolated linearly in a pre-computed grid (Claret et al. 2000). The inclination and v sin i can be ﬁxed, when they are known. Otherwise, they can be estimated during the mode identiﬁcation and should be set as variable in a reasonably large range (see Figure 16). • Line Proﬁle Parameters The only parameter which has to be known a priori is the Central wavelength of the considered line proﬁle. This parameter determines the adopted limb darkening coeﬃcient. If one deals with a cross correlated proﬁle, this value of course does not have a physical meaning. In this case, it is best to enter the mean value of the cross correlated range into this ﬁeld. The other parameters in this ﬁeld can be determined during the mode identiﬁcation. In the case of the moment method, also the centroid velocity and the signal-to-noise ratio have to be set. See Section 4.5.9 for details about these parameters. • Pulsation Mode Parameters This ﬁeld controls the settings for the parameters of each imported pulsation frequency. A frequency will be taken into account for the optimisation if the check box next to the ﬁeld Frequency [c/d] is checked. • Optimisation Settings This ﬁeld controls how the mode identiﬁcation is applied as well as the settings for the genetic optimisation. For a detailed explanation of the settings, we refer to Section 4.5.10. • General Settings For the tutorial spectra, you can leave the number of segments that are taken into account for the computation of the line proﬁle at the value of 1000. For rapidly rotating stars and high-degree pulsation modes, a higher value is required. For details, we refer to Section 4.5.11. famias The Spectroscopy Modules 87 proposes default parameter settings for the optimisation if you press Set ﬁelds to default. Figure 16: Settings of the Mode Identiﬁcation Tab. Fourier parameter ﬁt method 1. Determine pulsation frequencies Determine all pulsation frequencies, including harmonics and combinations, that have signiﬁcant amplitude across the line proﬁle (=pixel-bypixel) as described in the previous section. Select all frequencies in the List of frequencies and compute a least-squares ﬁt across the line proﬁle with the option Settings → Calculations based on → Pixel-by-pixel (MI:FPF). 2. Selection of dispersion range After you have imported the current least-squares ﬁt to the mode identiﬁcation tab, the dispersion range that is taken into account for the mode identiﬁcation can no longer be modiﬁed. Therefore, you have to deﬁne the dispersion range already when you compute the least-squares ﬁt. An optimal range excludes the continuum and the line wings. Only the range 88 Tutorial: Spectroscopic mode identiﬁcation Figure 17: Results of the ﬁt to the zero-point proﬁle. where the amplitude across the proﬁle reaches signiﬁcant values should be selected. You can either modify the dispersion range in the ﬁeld Settings or zoom into the selected region in the plot window. In the latter case, the left and right dispersion values of the zoomed range are automatically written to the Settings-ﬁeld. Uncheck the box Settings → Complete range and compute a least-squares ﬁt. In the tutorial example a range between −70 and 45 km s−1 would be optimal. 3. Import frequencies to Mode Identiﬁcation Tab Switch to the Mode Identiﬁcation Tab and import the current multiperiodic least-squares ﬁt by clicking on Pulsation Mode Parameters → Import data for FPF method (from current LSF). In the ﬁeld Pulsation Mode Parameters you can now switch between the diﬀerent imported pulsation frequencies. 4. Determine pulsationally independent parameters For the tutorial spectra, the stellar parameters have been saved in a ﬁle called coasttutorial.star. You can import this ﬁle by selecting General Settings → Settings → Import stellar parameters. We will ﬁrst de- The Spectroscopy Modules 89 Figure 18: Results of the mode identiﬁcation for F1 = 11.53 d−1 . termine starting values for the pulsationally independent parameters, i.e., v sin i, the equivalent width, the intrinsic width, and the velocity zeropoint shift of the proﬁle. The search range of these parameters should be suﬃciently large with a reasonable step-size. For the tutorial example, good starting values would be [min;max;step]: v sin i ∈ [1; 100; 1], equivalent width ∈ [1; 20; 0.1], intrinsic width ∈ [1; 20; 1], and zero-point shift ∈ [−20; 20; 0.1]. The step width should generally not be smaller than the precision to which a parameter can be determined. The best approach is to begin with a relatively large search range and step size, and to iteratively narrow the range. See Figure 16 for a screenshot of the Mode Identiﬁcation Tab with the settings before the ﬁrst optimisation. Select the option Select MI method → Compute vsini, EW, intrinsic width, and velocity oﬀset (ﬁt Z) and press on General Settings → Set ﬁelds to default to set default parameters for the genetic optimisation and to let famias propose free parameters for the optimisation. In this case, v sin i, the equivalent width, the intrinsic width, and the zero-point shift are set as free. 90 Tutorial: Spectroscopic mode identiﬁcation Press General Settings → Start mode identiﬁcation to start the optimisation. The results are written to the Results Tab. Figure 17 shows the results of the ﬁt to the zero-point proﬁle. The ﬁeld Best Models shows the 20 best solutions. You can click into the table to display the ﬁt in the plotting window Comparison between Fit and Observation. Here, the observational zero-point proﬁle is displayed as blue line, its statistical uncertainty as green lines, and the modeled proﬁle as red line. The chi-square plots can be used to estimate the uncertainty of the ﬁt. It is evident in Figure 17 that the best solution can still be improved. It is a good idea to note the parameter values for the best solutions and reﬁne the search range of these parameters in the Mode Identiﬁcation Tab to the following values [min;max;step]: v sin i∈ [30; 50; 1], equivalent width ∈ [7; 9; 0.01], intrinsic width ∈ [7; 15; 1], and zero-point shift ∈ [−13; −11; 0.01]. Reset the optimisation procedure by pressing General Settings → Reset and start another optimisation. This optimisation should result in a much lower chi-square value and thus a better constraint on the free parameters. We will take the obtained solution as a starting point for the mode identiﬁcation. Generally, the equivalent width and the zero-point shift are quite well constrained and can be set as constant during the optimisation. Figure 18 shows the parameter range we selected for the mode identiﬁcation. 5. Identify pulsation modes Select the Fourier parameter ﬁt method with the option to ﬁt zero-point, amplitude, and phase across the line proﬁle through the combo box Optimisation Settings → Select MI method → FPF Method: ﬁt ZAP. Select the option Optimisation Settings → l & m: grid search to obtain more reliable results of the optimisation procedure. Click on General Settings → Set ﬁelds to default. The inclination i is now also set as free parameter. Enter the following values as range: i ∈ [5; 90; 10]. Select the frequency 11.53 d−1 in the ﬁeld Pulsation Mode Parameters and mark the check box next to the frequency value. The parameter ranges should be as follows: degree ∈ [0; 3; 1], order m ∈ [−3; 3; 1], vel.amp v ∈ [0; 30; 1], and phase ψ ∈ [0.4715; 0.9715; 0.5]. The value of the phase is taken from the least-squares solution of the ﬁrst moment of this frequency (see Section 4.5.4 for details). The maximum value of the velocity amplitude should be set at least an order of magnitude higher than the amplitude of the ﬁrst moment. In general, you should extend the range of a parameter, if the lowest chi-square value was found at one of the search border (minimum or maximum of the range). Start the optimisation by pressing Start mode identiﬁcation. The other pulsation mode at The Spectroscopy Modules 91 17.5 d−1 can be analysed in the same manner. You can compare your results with the input values by loading the ﬁle coasttutorial.star into the Line Proﬁle Synthesis Tab. All computed mode identiﬁcations are saved in the Results Tab → List of Calculations and are logged in the Logbook of famias. Figure 19: Settings of the Mode Identiﬁcation Tab for the moment method. Moment method 1. Determine pulsation frequencies Determine all frequencies that have signiﬁcant amplitude in the ﬁrst moment, including harmonics and combination frequencies (see previous section). Compute a multi-periodic least-squares ﬁt using the option Least-Squares Fitting Tab → Settings → Calculations based on → 1st moment (radial velocity, MI: moment). 2. Import frequencies to Mode Identiﬁcation Tab Switch to the Mode Identiﬁcation Tab and click on Pulsation Mode Parameters → Import data for moment method (from current LSF). 92 Tutorial: Spectroscopic mode identiﬁcation After the import, you can switch between the two pulsation frequencies in the ﬁeld Pulsation Mode Parameters with the top left combo box. Mark the check box next to the frequency value for both imported frequencies. 3. Identify pulsation modes The starting parameters and settings should be adopted as displayed in Figure 19. Start the mode identiﬁcation by clicking on General Settings → Start mode identiﬁcation. The results are written to the Results Tab. 5. The Photometry Modules The Photometry Module contains tools that are required to search for frequencies in photometric time series and to perform a photometric mode identiﬁcation. The tools are located in tabs that have the following denominations: Data Manager, Fourier, Least-Squares Fitting, Mode Identiﬁcation, Results, and Logbook. These tools are described in the following sections. 5.1 Data Manager The Data Manager Tab gives information about light curves that have been imported, allows to edit them, and select the data sets for analysis. The window is divided into two data boxes and one plot window. A menu is located above each box. In the Data Sets Box you can select the light curve you want to analyse. The Time Series Box displays the time of measurement, magnitude, and weight of the selected data set. The Plot Window displays the currently selected light curve and data points that have been selected in the Time Series Box. A screenshot of the Data Manager is displayed in Figure 20. 5.1.1 Data Sets Box This box contains a list of the diﬀerent data sets that have been imported or created. To select a data set, click on it or select it in the combo box at the top right of the information bar. The following commands can be selected in the Data Menu: • Remove Data Set Removes the currently selected data set from the list. • Rename Data Set Renames the currently selected data set. • Export Data Set Exports the currently selected light curve as an ASCII-ﬁle to the disk. The suﬃx of the ﬁles has to be entered by the user. The exported ﬁles have the following three columns: time, magnitude, and weights. 94 Data Manager Figure 20: Screenshot of the Data Manager Tab. • Combine Data Sets Combines the selected data sets to a new single time series. The data sets to be combined must have the same units of the dispersion. Moreover, all times of measurement have to diﬀer. • Change Assigned Filter Change the ﬁlter that is assigned to the current time string. The correct ﬁlter has to be assigned to assure correct working of the mode identiﬁcation. 5.1.2 Time Series Box This list shows the measurements of the currently selected light curve. It consists of three columns: times of measurement, magnitude, and weight. Multiple measurements can be selected by clicking with the left mouse button on several items in the list while pressing the Ctrl-key or the Shift-key. All items can be selected by pressing Select All. Only items that have been selected in this list The Photometry Modules 95 (with blue background) are taken into account for the data analysis (e.g., Fourier analysis or least-squares ﬁtting). Selected items are marked with a red cross in the Plot Window. The following commands are available in the Data Menu: • Edit Data Opens a table of times and weights in a new window with the possibility to edit these values. Modiﬁcations can be written to the current data set. • Copy Selection to New Set A new data set with currently selected measurements is created and written to the Data Sets Box. Use this option to create subsets of your data. • Remove Selection The currently selected measurements are removed from the time series/data set. 5.1.3 Plot window The plot window shows the currently selected light curve as blue symbols. Selected measurements are marked with a red cross. For more information about the plot window, we refer to p. 28. 96 Fourier Analysis 5.2 Fourier Analysis With this module, a discrete Fourier transform (DFT) can be computed to search for periodicities in the data set selected in the Data Sets Box of the Data Manager Tab. The Fourier spectrum is displayed in the plot window and saved as data set in the List of calculations. A screenshot of the Fourier Tab is displayed in Figure 21. 5.2.1 Settings Box In this box, the settings for the Fourier analysis are deﬁned. • Frequency range Minimum/Maximum values of the frequency range. The Fourier spectrum will be computed from the minimum to the maximum value. • Nyquist frequency Estimation of the Nyquist frequency (mean sampling frequency). For non-equidistant time series, a Nyquist frequency is not uniquely deﬁned. In this case, the Nyquist frequency is approximated by the inverse mean of the time-diﬀerence of consecutive measurements by neglecting large gaps. • Frequency step Step size (resolution) of the Fourier spectrum. Three presets are available: Fine (≡ (20ΔT )−1 ), Medium (≡ (10ΔT )−1 ), and Coarse (≡ (5ΔT )−1 ). The corresponding step size depends on the temporal distribution of the measurements, i.e., the time diﬀerence ΔT of the last and ﬁrst measurement. It is recommended to select the ﬁne step size to ensure that no frequency is missed. The step value can be edited if desired. • Use weights If the box is checked, the weight indicated for each data point is taken into account in the Fourier computations. Otherwise, all weights are assumed to have equal values. • Compute spectral window If the box is checked, a spectral window of the current data set is computed. A spectral window shows the eﬀects of the sampling of the data on the Fourier analysis and thus permits to estimate aliasing eﬀects. The spectral window is computed from a Fourier spectrum of the data taking the times of measurements and setting all intensities to the value 1. The shape of the spectral window should be plotted for a frequency range that is symmetric around 0 for visual inspection. The Photometry Modules 97 • Compute signiﬁcance level If the box is checked, the signiﬁcance level at a certain frequency value is computed and displayed in the plot window as a red line. The following parameters can be set: – Frequency Frequency value of the peak of interest. The data will be prewhitened with this frequency and the noise level will be computed from the pre-whitened Fourier spectrum. – S/N level Multiplicity factor of the signal-to-noise level. The displayed noise level will be multiplied by this factor. – Box size Box size b for the computation of the noise-level in units of the frequency. The signiﬁcance level is computed from the running mean of the pre-whitened Fourier spectrum. For each frequency value F , the noise level is calculated from the mean of the range [F −b/2, F + b/2]. • Calculate Fourier Computes the discrete Fourier transform (DFT) according the user’s settings and displays it in the plot window as a blue line. The mean intensity value of the time series is automatically shifted to zero before the Fourier analysis is computed. The peak having highest amplitude in the given range is marked in the plot window. A dialogue window reports the frequency having the highest amplitude in the selected frequency range and asks if it should be added to the frequency list of the Least-Squares Fitting Tab. 5.2.2 List of Calculations Previous Fourier calculations can be selected from the list and viewed. Each computed Fourier spectrum is saved and listed here. If a project is saved, the list of computed Fourier spectra is also saved but compressed to decrease the project ﬁle size (only extrema are saved). The following operations are possible via the Data Menu: • Remove Data Set Removes the currently selected data set from the list. • Rename Data Set Renames the currently selected data set. 98 Fourier Analysis Figure 21: Screenshot of the Fourier Tab. • Export Data Set Exports the currently selected data set to an ASCII ﬁle having the following three-column format: frequency, amplitude, power. 5.2.3 Fourier Spectrum Plot Shows the most recently computed Fourier analysis or the selection from the list of calculations. The Fourier spectrum is shown as a blue line, the signiﬁcance level is shown as a red line. The frequency and amplitude of the peak having the highest frequency are indicated. For more information about the plot window, we refer to p. 28. The Photometry Modules 99 5.3 Least-Squares Fitting This modules provides tools for the computation of a non-linear multi-periodic least-squares ﬁt of a sum of sinusoidals to your data. The ﬁtting formula is Z+ Ai sin 2π(Fi t + φi ) (22) i Here, Z is the zero-point, and Ai , Fi , and φi are amplitude, frequency and phase (in units of 2π) of the i-th frequency, respectively. The least-squares ﬁt is carried out with the Levenberg-Marquardt algorithm. For a given set of frequencies, either their zero-point, amplitude and phase can be optimized (Calculate Amplitude & Phase), or additionally also the frequency value itself (Calculate All). The data can be pre-whitened with the computed ﬁt and written to the Data Sets Box of the Data Manager Tab. Before a mode identiﬁcation can be carried out, a least-squares ﬁt to the data must be calculated. To carry out a photometric mode identiﬁcation, light curves from diﬀerent ﬁlters must be imported to famias, and amplitudes and phases of the pulsation frequencies must be determined by least-squares ﬁtting. These values can then be copied to the Mode Identiﬁcation Tab to carry out the mode identiﬁcation method using amplitude ratios and phase diﬀerences. 5.3.1 Settings Deﬁnes the settings for the calculation of the least-squares ﬁt. • Use weights If this box is checked, the weight indicated for each data point is taken into account in the least-squares ﬁt. Otherwise, all weights are assumed to have equal values. • Pre-whiten data If this box is checked, the data will be pre-whitened with the computed least-squares ﬁt and written as a new data set to the Data Manager Tab. • Compute signal-to-noise ratio Computes the amplitude signal-to-noise ratio (SNR) of each selected frequency and displays it in the List of Frequencies. The noise is computed from the Fourier spectrum of the pre-whitened data. The Box size is the width of the frequency range which is taken into account for the calculation of the noise. For a box width of b, the noise of frequency F is the mean value of the Fourier spectrum of the residuals in the range [F − b/2, F + b/2]. The SNR is the ratio of Af and the noise level of the pre-whitened Fourier spectrum at the position of f . 100 Least-Squares Fitting Figure 22: Screenshot of the Least-Squares Fitting Tab. • Calculate Amplitude + Phase Computes a least-squares ﬁt with the Levenberg-Marquardt algorithm using the above mentioned ﬁtting formula. The zero-point, amplitude and phase are calculated, whereas the frequency is kept ﬁxed. The following optimized values are written to the frequency list: zeropoint and its uncertainty, the standard deviation of the residuals, for each selected frequency its amplitude and phase and their formal uncertainties derived from the error matrix of the least-squares ﬁtting algorithm. • Calculate All Computes a least-squares ﬁt with the Levenberg-Marquardt algorithm using the above mentioned ﬁtting formula. The zero-point, amplitude, phase and frequency are improved. The following optimized values are written to the frequency list: zero-point and its uncertainty, the standard deviation of the residuals, for each selected frequency its frequency value, amplitude and phase and their formal uncertainties derived from the error matrix of the least-squares ﬁtting algorithm. The Photometry Modules 101 • Copy values to MI Computes a least-squares ﬁt by improving amplitude and phase (equivalent to Calculate Amplitude + Phase) and copies the derived values (frequencies, amplitudes, and phases and their uncertainties) to the Mode Idenitiﬁcation Tab. For diﬀerent ﬁlters, a least-squares solution with identical frequency values must be computed to ensure that the phases in the diﬀerent ﬁlters can be compared. 5.3.2 List of Frequencies The List of Frequencies Box shows the results of the least-squares ﬁt. Frequencies that should be included in a least-squares ﬁt can be entered in the column Frequency and selected by clicking on the check box in column Use. The following values are shown in this box after a least-squares ﬁt has been calculated: The zero-point, its formal uncertainty and the standard deviation of the residuals are shown at the top. The improved values of frequency, amplitude and phase and their formal statistical uncertainties are shown in the list. The phase and its uncertainty, in units of 2π. The last column lists the SNR of each frequency (only shown when option Calculate signal-to-noise ratio has been checked). The SNR is computed from the Fourier spectrum, pre-whitened with all selected frequencies. For each frequency, the assumed noise-level is computed from the mean amplitude around the frequency value with the box size indicated at the option Calculate signal-to-noise ratio. • Export frequencies Exports all frequency, amplitude and phase values of the List of frequencies to an ASCII ﬁle. The ﬁle format is compatible with the program Period04 (see example on p. 48). • Import frequencies Imports an ASCII list of frequencies having the following four-column format separated with tabulators: frequency counter, frequency value, amplitude, phase (see example on p. 48). 102 Mode Identiﬁcation 5.4 Mode Identiﬁcation This module can be used to perform a photometric mode identiﬁcation based on the method of amplitude ratios and phase diﬀerences of pulsation modes in diﬀerent photometric passbands (Balona & Stobie 1979; Watson 1988; Cugier et al. 1994). This method permits to determine the harmonic degree of pulsation modes in general up to = 6. This upper limit is due to partial geometric cancelation of the observable pulsation amplitude over the stellar disc. The determination of the -degrees is based on static plane-parallel models of stellar atmospheres and on linear non-adiabatic computations of stellar pulsation. In the present version of famias, these are provided in the form of pre-computed grids and interpolated linearly to obtain values appropriate for the observed parameters. The theoretical values of the amplitude ratio and phase diﬀerence in a certain ﬁlter depend strongly on pulsational input. This points out a very important diﬀerence between spectroscopic and photometric mode identiﬁcation: the former is model independent, the latter is not. To be able to compare the results, famias incorporates grids computed from diﬀerent pulsational codes and from diﬀerent atmosphere models. The present version of famias includes grids from two diﬀerent scientiﬁc institutions (see details below). It is planned to include model grids from more groups in the future, whenever they are provided. 5.4.1 Theoretical background In famias, we apply the approach proposed by Daszyńska-Daszkiewicz et al. (2002) to compute the theoretical photometric amplitudes and phases due to pulsation. For more details see instruction on the Wroclaw HELAS Webpage1 . In the limit of linear pulsation, zero-rotation approximation and assuming static plane-parallel atmospheres, we can write the ﬂux variations in the passband λ caused by a oscillation mode having a frequency ω and a degree as ΔFλ λ λ = εY m (i, 0)bλ Re{[D1, f + D2, + D3, ]e−iωt }, (23) Fλ0 where 1 ∂ log(Fλ |bλ |) , 4 ∂ log Teﬀ = (2 + )(1 − ), λ = D1, D2, λ =− D3, 2 3 (24) λ ∂ log(Fλ |b |) ω R +2 . 0 GM ∂ log geﬀ 1 http://helas.astro.uni.wroc.pl/deliverables.php 103 The Photometry Modules or equivalently λ = D1, 1 βT,λ 1 αT,λ + , 4 ln 10 bλ D2, = (2 + )(1 − ), 2 3 ω R +2 GM λ =− D3, (25) αg,λ + 1 βg,λ . ln 10 bλ Here, ε is the pulsation mode amplitude expressed as a fraction of the equilibrium radius of the star, Y m (i, 0) describes the mode visibility with the inclination angle, i, and ( , m) being the spherical harmonic degree and the azimuthal order, respectively, G is the gravitational constant, M is the stellar mass, and bλ is the disc averaging factor deﬁned as bλ = 1 0 h0λ (µ)µP (µ)dµ. (26) λ λ and D3, terms describe temperature and gravity eﬀects, respectively, The D1, and both include the perturbation of the limb-darkening. The D2, term stands for geometrical eﬀects. For computing bλ , we use a non-linear limb darkening law, deﬁned by Claret et al. (2000) as I(µ) =1− I(1) 4 k ak (1 − µ 2 ), (27) k=1 where I(µ) is the speciﬁc intensity on the stellar disk at a certain line-of-sight angle θ with µ = cos θ and ak is the k-th limb darkening coeﬃcient. The parameters αT,λ and αg,λ are the partial ﬂux derivatives over eﬀective temperature and gravity, respectively, that are calculated from static model atmospheres for diﬀerent passbands αT,λ = ∂ log Fλ ∂ log Fλ and αg,λ = , ∂ log Teﬀ ∂ log g (28) whereas, the parameters βT,λ and βg,λ are partial derivatives of the bλ factor βT,λ = ∂ log bλ ∂ log bλ and βg,λ = . ∂ log Teﬀ ∂ log g (29) The f parameter is a complex value which results from linear non-adiabatic computations of stellar pulsation and describes the relative ﬂux perturbation at the level of the photosphere δTeﬀ 1 m −iωt }. 0 = ε 4 Re{f Y e Teﬀ (30) 104 Mode Identiﬁcation According to Eq. (23), the complex amplitude of the light variations is expressed as (Daszyńska-Daszkiewicz et al. 2002) λ λ Aλ (i) = −1.086εY m (i, 0)bλ (D1, f + D2, + D3, ), (31) and the amplitudes and phases of the light variation are given by Aλ = |Aλ | = and where A2λ,R + A2λ,I ϕλ = arg(Aλ ) = arctan(AI /AR ) λ λ fR + D2, + D3, ), Aλ,R = −1.086εY m (i, 0)bλ (D1, λ fI . Aλ,I = −1.086εY m (i, 0)bλ D1, Calculating amplitude ratio and phase diﬀerences the εY m (i, 0) term goes away making these observables independent of the inclination angle, i, and the azimuthal order, m, in the case of zero-rotation approximation. 5.4.2 Approach for mode identiﬁcation in famias famias computes the theoretical amplitude ratios and phase diﬀerence according to the above described scheme in diﬀerent photometric passbands. To identify the spherical harmonic degree, , the user must provide its frequency, amplitude and phase in diﬀerent ﬁlters, ranges for Teﬀ and log g, a number of stellar model parameters such as mass and metallicity, and the source of the stellar models. famias then derives the theoretical values from pre-computed model grids and displays the results of the mode identiﬁcation on the Results Tab. At the time this manual was written, we had pre-computed grids of the parameters αT,λ , αg,λ , βT,λ , βg,λ , bλ at our disposal. These atmospheric parameters have been computed by Leszek Kowalczuk and Jadwiga DaszyńskaDaszkiewicz using Kurucz and NEMO atmospheres. The grids are available for the following photometric systems: Johnson/Cousins U BV RI, Strömgren uvby, and Geneva, and for diﬀerent values of metallicity parameter [m/H] and microturbulence velocity, ξt . All these results can be found on the Wroclaw HELAS Webpage2 . These model grids contain stellar evolution tracks for diﬀerent masses computed by the Warsaw-New Jersey code (Paczyński 1969, 1970) and pulsational 2 http://helas.astro.uni.wroc.pl/deliverables.php The Photometry Modules 105 models from ZAMS to TAMS with some step in time (or eﬀective temperature) computed for mode degree from 0 to 6. The full description of the evolutionary and pulsational models is given at the Wroclaw HELAS Webpage. Furthermore, we used pulsational models from two diﬀerent sources available. First, a grid for main-sequence stars with masses from 1.8 to 12 M computed by Jadwiga Daszyńska-Daszkiewicz, Alosha Pamyatnykh, and Tomasz Zdravkov using the non-adiabatic Dziembowski code (Dziembowski 1971, 1977), which can be downloaded also from the above mentioned web site. Second, a grid for δ Sct stars computed with ATON (Ventura et al. 2007) and MAD by Montalban & Dupret (2007). The grids included in the present version of famias cover the following range: • 1.6 ≤ M ≤ 12 • 3500 ≤Teﬀ ≤ 47500 K • 1 ≤log g≤ 5 • −5 ≤ [m/H] ≤ 1 • 0 ≤ vmicro ≤ 8 km s−1 (for some metallicities). In more detail, the mode identiﬁcation is carried out in the following way: • The user must provide the pulsation frequency F , its amplitude Aλ , the uncertainty of the amplitude σAλ , the phase φ, and the uncertainty of the phase σφ . • The following values and options for the stellar models must be indicated: ranges for Teﬀ and log g, stellar mass, metallicity, micro turbulence, source of the atmosphere grid, and the source of the non-adiabatic observables. • The evolutionary stellar model grid for the indicated mass is searched for models that lie in the given range of Teﬀ and log g. • For each found model, the atmospheric parameters αT,λ , αg,λ , βT,λ , βg,λ , and bλ, are determined by bi-linear interpolation in the grid of the indicated ﬁlter set, metallicity, and micro turbulence. • The program searches in the lists of the found non-adiabatic pulsation models for diﬀerent -values. For each , the frequency that is closest to the observed value is searched for. The values of the real and complex non-adiabatic parameters, fR and fI , respectively, are taken from this frequency value. 106 Mode Identiﬁcation • The theoretical amplitudes and phases are computed from Eq. (31) for each selected ﬁlter. • The amplitude ratios and phase diﬀerences are computed with respect to a selected ﬁlter (ideally the one with the largest observed amplitude). famias creates an error message if no atmospheric or evolutionary models have been found in the grids for the indicated parameters. Figure 23: Screenshot of the Mode Identiﬁcation Tab. 5.4.3 Observed values This box contains the frequencies, amplitudes, and phases of the observed pulsation frequencies in diﬀerent photometric ﬁlters. These values can be imported from the Least-Squares Fitting Tab or entered manually. • Frequency selection Diﬀerent frequencies can be selected with this combo box. Each item in the combo box is related to a frequency value and its amplitudes and phases in diﬀerent ﬁlters. Frequencies can be added by importing from The Photometry Modules 107 the Least-Squares Fitting Tab or by selecting in the Action menu the option Add frequency. • Action menu This menu allows to add or remove frequency sets. • Frequency value Frequency value in d−1 . • Filter system Select the ﬁlter system for which the mode identiﬁcation should be carried out. Three diﬀerent systems are available: Johnson/Cousins U BV RI, Strömgren uvby, and Geneva. • Table of amplitudes and phases This table contains for each ﬁlter the observed values of Aλ and σAλ in mmag, and of φ, and σφ in units of 2π. You do not have to ﬁll out all ﬁelds. Empty ﬁelds (or = 0) are not used for the computation of the amplitude ratios and phase diﬀerences. Theoretical values are anyway computed for all ﬁlters. 5.4.4 Stellar model parameters • Teﬀ Observational value of the eﬀective temperature in Kelvin and its uncertainty. • log g Observational value of the logarithm of the gravity in c.g.s and its uncertainty. • Mass Stellar mass in solar units. The available values depend on the selected non-adiabatic model source. You can only obtain a mode identiﬁcation for one selected mass-value at a time. • Atmosphere grid Model source of the grid of the atmospheric parameters αT,λ , αg,λ , βT,λ , βg,λ , and bλ . • Overshooting This box indicates if models with core overshooting should be taken into account. 108 Mode Identiﬁcation • Metallicity Stellar metallicity value [m/H]. The available range depends on the selected non-adiabatic model source. • Micro turbulence Micro turbulence value of the stellar atmosphere models. • Non-adiabatic obs. source Select here the source for the grid of non-adiabatic observables. • Identify mode Start the mode identiﬁcation. famias computes the observed amplitude ratios and phase diﬀerences as well as the corresponding values for all found pulsation models. The results are written to the Results Tab. The Photometry Modules 109 5.5 Results This module contains the results of the photometric mode identiﬁcations. It gives the observed and theoretical values of the amplitude ratio and phase diﬀerence in diﬀerent ﬁlters in a text window as well as in diagrams. 5.5.1 List of Calculations Each time a mode identiﬁcation is carried out, its results are saved as a new data set in this list. Click on an item to display the results in the ﬁeld Mode identiﬁcation and the corresponding diagrams in the ﬁeld Mode identiﬁcation plots. Figure 24: Screenshot of the Results Tab. 5.5.2 Settings This box can be used to set the reference ﬁlter and to set which -values should be displayed in the plot window. 110 Results • Reference ﬁlter This is the reference ﬁlter r for the amplitude ratio and phase diﬀerence with respect to ﬁlter x. The amplitude ratios are computed as Ax /Ar . The phase diﬀerence is calculated as φx − φr . Exceptions are the mode identiﬁcation plots, where the phase diﬀerence is plotted against the amplitude ratio. There, the indices r and x are exchanged. • Box of -values You can select here which -values should be displayed in the plot window. • Update This updates the Mode identiﬁcation box and plot window with the current settings. 5.5.3 Mode Identiﬁcation Report This ﬁeld displays the main information about the observed and theoretical parameters for the obtained mode identiﬁcation. It lists the input values and settings for the models, the observed amplitude ratios and phase diﬀerences, and for each pulsation model that matches the search criteria, its degree and corresponding amplitude ratio and phase diﬀerences. Amplitude ratios in the ﬁlters x and y are denoted as A(x)/A(y). Phase diﬀerences are indicated as P (x − y) and in units of degrees. For the observed values, the 1σ standard deviation is indicated. 5.5.4 Mode Identiﬁcation Plots Three kinds of plots, that can be selected via the combo box above, are available in this ﬁeld. They are described in detail below. In each plot, the observed values are displayed as black crosses with error bars. The theoretical values are displayed as lines. Each colour represents another value of the degree and coincides with the colour-scheme in the ﬁeld Settings. Generally, more than one theoretical pulsation model is found that matches the input criteria (e.g., Teﬀ and log g). All these models are displayed as apart lines in the plots (as well as listed in the ﬁeld Mode Identiﬁcation). • Amplitude ratio x This plot displays the amplitude ratio A Ar relative to the reference ﬁlter r (see Settings) as a function of the central wavelength of the corresponding ﬁlter x. The values of each ﬁtting theoretical model are drawn as apart lines. Plots of this kind are suited to identify modes in SP B or β Cep stars, i.e., stars where the amplitude ratio depends strongly on the degree of the mode. The Photometry Modules 111 • Phase diﬀerence This plot displays the phase diﬀerence P (x − r) relative to the reference ﬁlter r (see Settings) as a function of the central wavelength of the corresponding ﬁlter x. Values of diﬀerent ﬁtting theoretical models are plotted as apart lines. • Phase diﬀ / Ampl. ratio Ar as a function of the phase These plots display the amplitude ratio A x diﬀerence P (r − x) for each ﬁlter x of the selected ﬁlter system relative to the reference ﬁlter r. Each plot displays the results for another combination of r and x. In these plots, the lines of a certain colour represent the range of all found models that fulﬁl the search criteria. 5.6 Logbook The logbook shows the list of actions that have been performed with the photometric set of tools of famias and corresponding information. Each time an operation is carried out in famias, a new log-entry is written to the List of actions. Clicking on an entry of this list shows the corresponding information in the text box. Entries of the List of actions can be renamed or deleted by using the menu Data. The text box can be modiﬁed and saved in famias by clicking on the button Save. 112 Tutorial: Photometric mode identiﬁcation 5.7 Tutorial: Photometric mode identiﬁcation This tutorial demonstrates the use of famias for the photometric mode identiﬁcation based on multi-colour light curves. 5.7.1 Importing and preparing data 1. Select the Photometry page and click on File → Import Light curve. 2. Select one or several ﬁles that contain the photometric data. A data ﬁle must be in ASCII format and consist of at least two columns, separated by a space or tabulator. Columns of observation time in d−1 and magnitude are required. An additional column listing the weights of the measurements is optional. Once you have selected your ﬁles, click on Open. Figure 25: Data manager after importing Geneva light curves of an SPB star. 3. For each ﬁle that you import, an import-dialogue will open. Select the columns that you want to import and specify the photometric passband of The Photometry Modules 113 the observations. For a more detailed description of the import-dialogue see Section 2.1. The successfully imported data sets will be listed in the Data Manager. 4. You can use the tools in the Data Manager to edit your data (delete data points, change weights, etc.). See Section 5.1.2 for further details. 5.7.2 Searching for periodicities The search for periodicities should be carried out in the following iterative schematic way: 1. Compute a Fourier spectrum in a frequency range where you expect pulsation. 2. Compute the signiﬁcance level at the frequency having the highest amplitude and include this frequency in the least-squares ﬁtting if it is signiﬁcant. 3. Compute a multi-periodic least-squares ﬁt of the original data with all detected frequencies. In case that no unique frequency solution exists due to aliasing, compute least-squares ﬁts for diﬀerent possible frequencies. The solution resulting in the lowest residuals should be regarded as best solution. 4. Exclude frequencies from the ﬁt that do not have a SNR above 4 (3.5 for harmonics and combination terms). 5. Pre-whiten the data with all signiﬁcant frequencies. 6. Continue with the ﬁrst point using the pre-whitened data until no significant frequency can be found. In famias follow the following procedure: 1. Select data set In the Data Manager or the use combo box (top right), select the data set you want to analyse. 2. Calculate Fourier spectrum Switch to the Fourier Tab. Select a reasonable frequency range and click on Settings → Calculate Fourier to compute a Fourier spectrum. A dialogue box will pop up and ask you, if you would like to include the highest frequency peak in the Least-Squares Fitting Tab. Before doing so, it is a good idea to check for the statistical signiﬁcance of this peak. 114 Tutorial: Photometric mode identiﬁcation 3. Compute signiﬁcance level Mark the check box Settings → Compute signiﬁcance level. The frequency value of the highest peak is automatically written to the corresponding text ﬁeld. Modify this value, if you are interested in the signiﬁcance of another frequency peak. Click on Calculate Fourier to compute another Fourier spectrum. The signiﬁcance level will be shown in the plot as a red line. If the examined frequency peak is signiﬁcant, include it in the List of Frequencies of the Least-Squares Fitting Tab. Figure 26: Fourier spectrum of the Geneva U -band. 4. Compute least-squares ﬁt Switch to the Least-Squares Fitting Tab and mark the check boxes of all frequencies in the List of Frequencies that you want to include in the ﬁt. Also mark the check box Settings → Compute signal-to-noise ratio to determine the statistical signiﬁcance of the selected frequency peaks. Click on Calculate Amplitude + Phase to compute a least-squares ﬁt to the data by improving amplitude and phase values. Click on Calculate All to compute a ﬁt by improving frequency, amplitude, and phase. The List of Frequencies box shows the results of the ﬁt. The frequency is The Photometry Modules 115 displayed in inverse units of the input time and the phase is indicated in units of the period. 5. Pre-whiten light curve To pre-whiten the light curve with the obtained ﬁt, mark the check box Pre-whiten data and compute another ﬁt. The pre-whitened light curve is written to the Data Manager Tab as a new data set. 6. Compute Fourier spectrum of residuals Select the pre-whitened light curve and compute a Fourier spectrum thereof to search for further frequencies. If you want to compute another least-squares ﬁt with additionally found frequencies, you must select the original light curve. 5.7.3 Mode identiﬁcation The photometric mode identiﬁcation as it is implemented in famias uses the method of amplitude ratios and phase diﬀerences in diﬀerent photometric passbands. You therefore have to provide for each pulsation frequency that should be identiﬁed its amplitude and phase for diﬀerent ﬁlters. These values can be determined with a multi-periodic least-squares ﬁt of sinusoids to the data under the assumption that the variations are sinusoidal. To be able to compare the phase values determined for the diﬀerent ﬁlters, the same frequency values have to be taken into account in the least-squares ﬁts for all ﬁlters. Important note: the frequency has to be in units of d−1 . 1. Insert frequencies You can either enter input the observed values of frequency, amplitude, and phase manually or copy directly the results from the least-squares ﬁt. • Manual input Switch to the Mode Identiﬁcation Tab (see Figure 27). In the ﬁeld Observed Values, select the option User input in the top combo box. You can input the frequency value in the ﬁeld Frequency. Select the ﬁlter system of your observations and input the observed amplitude and phase in the corresponding ﬁelds. To add values for an additional frequency, use the function Action → Add frequency. • Automatic input To use this function, you need to import light curves from diﬀerent ﬁlters and assign to each light curve the correct ﬁlter name (during import or in the Data Manager). Mark the frequencies in the LeastSquares Fitting Tab → List of Frequencies that have signiﬁcant 116 Tutorial: Photometric mode identiﬁcation amplitude in all ﬁlters that you want to use for the mode identiﬁcation. Press Copy values to MI to compute a least-squares ﬁt and copy the frequency, amplitude, and phase to the Mode Identiﬁcation Tab. In this least-squares ﬁt, the frequency is kept constant whereas amplitude and phase are improved. Repeat this procedure for the light curves taken in other ﬁlters without modifying the frequency values or the number of marked frequencies. Figure 27: Mode Identiﬁcation Tab of famias. The observed amplitude and phase are listed in the left ﬁeld, whereas the options for the stellar models can be set in the right ﬁeld. For each imported frequency, a new item is added in the top combo box of the ﬁeld Mode Identiﬁcation Tab → Observed Values. To carry out a mode identiﬁcation, it is only obligatory to provide the observed frequency value. It is not necessary to input observed amplitude and phase values. The theoretical amplitude ratios and phase diﬀerences are in any case always computed for all ﬁlters of the selected ﬁlter system. The Photometry Modules Figure 28: Results of the tutorial data. The best identiﬁcation is achieved for (red lines). 117 =0 2. Set stellar model parameters In the Mode Identiﬁcation Tab, the parameters of the stellar models and the source of the non-adiabatic observables that should be used for the mode identiﬁcation have to be set. See Section 5.4.4 for detailed information about these parameters. 3. Start mode identiﬁcation Start the mode identiﬁcation by pressing the button Identify mode. The results will be written to the Results Tab. 4. Interpretation of results The Results Tab displays the results of the mode identiﬁcation in text form and in several plots. The text ﬁeld Mode Identiﬁcation lists the input parameters and the observed and theoretical amplitude ratios and phase diﬀerences. Its contents are described in detail in Section 5.5. As can be seen in Figure 28, the observed amplitude ratios are most consistent with theoretical models that have a degree = 0.

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