PHOEBE tutorial


Adopted for PHOEBE version 0.29











March 2007


Contents

Introduction

This document is a part of the ongoing effort to bring PHOEBE closer to its users. Here you will find step-by-step instructions for using PHOEBE for the first time. When you get acquainted with the main ideas and design, you may find the PHOEBE Reference Manual (which doesn't exist at the time) more instructive.

PHOEBE Licence

PHOEBE is released under a GNU General Public Licence. A copy of this licence is distributed along PHOEBE and you may find it in a file COPYING in a base PHOEBE directory. To quote the most important part:

The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software-to make sure the software is free for all its users. This General Public License applies to most of the Free Software Foundation's software and to any other program whose authors commit to using it. (Some other Free Software Foundation software is covered by the GNU Library General Public License instead.) You can apply it to your programs, too.

When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for this service if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs; and that you know you can do these things.

To protect your rights, we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights. These restrictions translate to certain responsibilities for you if you distribute copies of the software, or if you modify it.

For example, if you distribute copies of such a program, whether gratis or for a fee, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.

We protect your rights with two steps: (1) copyright the software, and (2) offer you this license which gives you legal permission to copy, distribute and/or modify the software.

Also, for each author's protection and ours, we want to make certain that everyone understands that there is no warranty for this free software. If the software is modified by someone else and passed on, we want its recipients to know that what they have is not the original, so that any problems introduced by others will not reflect on the original authors' reputations.

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The precise terms and conditions for copying, distribution and modification may be found in a file COPYING.

Some general remarks about PHOEBE

Although this document is a tutorial and as such doesn't even tend to cover all PHOEBE aspects, a few introductory lines are in order. Let us start by explaining the name: PHOEBE stands for PHysics Of Eclipsing BinariEs.

Probably the most important thing for any new PHOEBE user is to understand the limitations and capabilities of this software. Although PHOEBE aims to be as useful as possible, it still tends to follow the basic principle of simplicity. In other words: although PHOEBE may assist you in solving your specific problem, it won't write an article for you or even make you a good cup of coffee; that is still up to you.

To describe PHOEBE in one sentence: for a set of user-supplied experimental light curves and radial velocity curves PHOEBE estimates physical parameters of an eclipsing binary that theoretically best matches the data.

The basic goal of PHOEBE is to aid you with intuitive interface through the pain of solving this problem. Since the dimension of the parameter phase-space is tremendous and the inverse problem for eclipsing binaries itself highly non-linear, this task is hard to live up to and PHOEBE is thus subject to constant changes and improvements. Hopefully each new version converges to a global optimum.

First steps with PHOEBE

In this section we will take PHOEBE for a spin to get a feeling about what it can do for us. We shall start with proper configuration, continue with inspecting the overall structure and finally, demonstrate how the data is supplied to the program.

Configuring PHOEBE

PHOEBE may be installed in two ways: system-wide, meaning all users may use it, or user-oriented, designated only for an individual user. These two differ only in the location on the hard disk: the system-wide installation usually resides in /usr/local/share/phoebe directory and the user-oriented installation in /home/username/phoebe.

We shall presume that your PHOEBE installation was successful; if not, please turn to installation instructions found in the top-level PHOEBE directory. For troubleshooting please refer to the appendices.

As most computer programs, PHOEBE must be set up before the first use. When you first start it, the following warning will appear:

Figure 1: PHOEBE configuration warning.
phoebe_configuration_warning

This warning lets you know that PHOEBE was unable to find its configuration directory; this is a hidden directory in your personal HOME directory1 (usually /home/username/.phoebe). By clicking on OK button this directory will be created and PHOEBE will take you to the configuration window (Fig. 2). If you decide not to create this directory, you may click on Cancel button, which will terminate PHOEBE.

Figure 2: PHOEBE configuration window.
phoebe_configuration_window

Configuration window is composed of two tabs: Files and Options. The first tab contains system-dependent file locations and the other contains options that reflect on the overall PHOEBE performance.

The default settings in the Files tab of the configuration window are appropriate for the system-wide installation; let us analyse the contents of these directories in turn:

The Base directory is the directory of your current distribution; it contains generic distribution files such as the README file, ChangeLog file etc. It doesn't contain any execution-specific files.
The Source directory is a Base subdirectory and it contains PHOEBE sources (.c files and header files). This directory is optional, the generic make install script won't install it for you. Sources are needed if you want to build a binary package for your Linux distribution or if you want to change the code.
The Defaults directory contains a single PHOEBE defaults file. This file, called default.phoebe, governs PHOEBE's startup values: values of parameters with which PHOEBE starts. Changing this file (either by hand or overwriting it with any other keyword file) will change the default values permanently.
The Working directory is a temporary directory where PHOEBE stores all its I/O files. These files are intermediate WD files, logs, images etc. This directory obviously must have write permissions set, otherwise PHOEBE will issue a warning.
The Data directory is where PHOEBE expects to find user data (PHOEBE keyword files and LC/RV data files). For this directory also write permissions must be set.
The WD LC and WD DC directories set the location of Wilson-Devinney executables lc and dc. These are core files for PHOEBE and thus must be present.
The WD Flux Files directory is meant for future implementation. Since 2003, WD algorithm changed the model from effective wavelength treatment to integrated flux treatment, so to each experimental LC/RV curve there will be a flux file assigned. This directory will contain these flux files.

Other options include the choice of the plotting package2, limb darkening tables etc. We will talk about them later on, when we discuss their application.

To change a specific directory, you may enter new pathnames either by typing in the new values directly to the entry box, or by clicking on the Browse button and then selecting the right directory through a file selector. The latter is preferable, since it is easy to make typing errors otherwise.

The directory you will most probably want to change is the Data directory; it is advisable to create a dedicated directory somewhere in your home directory and supply it to PHOEBE, e.g. /home/username/data.

When finished, click on the Save button. The settings will be saved, so you need to set it up only this one time. If you want to review or modify your settings later, you may find them in Settings->Configuration menu. If you click on the OK button instead, the configuration will be kept for that PHOEBE session only. Clicking on Cancel will leave PHOEBE unconfigured. Strange things might happen...

When you are finished setting up, it's time to get some real work done!

PHOEBE structure

The main PHOEBE window is divided in 9 tabs - screen pages with selected content that fits into a specific category (Fig. 3). Data tab contains the information on experimental data, which is supplied by the user. System-related tab holds all parameters that characterize the binary system as a whole - such are the epoch parameters, semi-major axis, mass ratio and so on. Component-related tab, in contrast, contains curve-independent parameters that are characteristic of each star individually, such as the effective temperature and gravity potentials. Orbit tab contains parameters that describe the orbit, like orbital ellipticity. There is planned support for Roche geometry graphs. Next tab, Luminosities, contains curve-dependent parameters: they have different values for different filters. Typical parameters that fit into this category are flux levelers, $3^{\mathrm{rd}}$ light and weighting functions. Limb darkening and Surface tabs contain parameters that contribute in perturbative orders of corrections to curves. Finally, the Fitting tab contains functions and parameters that define and support minimization algorithms. Utilities tab holds functions that are not directly connected to PHOEBE and prototypes for functions to be implemented into PHOEBE shortly.

Figure 3: PHOEBE window.
phoebe_luminosities_tab

Beside the main window there are three other types of PHOEBE windows. Calculation windows help the user to quickly estimate the values of some specific parameters, such as limb darkening coefficients and gravity potentials. Plotting windows are meant for quick and simple eye inspections of both experimental and synthetic light and radial velocity curves. A Fitting window guides the user through the minimization process and iterative solution convergence. Finally, a Scripter window enables the user to write and execute powerful PHOEBE scripts.

All windows may be open at the same time because of the modular design of PHOEBE. It is up to you to decide the layout that suits you best, PHOEBE poses no limitations in this respect. You may not, however, have more copies of the same window, since the contents of one window might not be the same as the contents of another instance of the same window and this might lead to ambiguity problems.

The very top of the PHOEBE window is occupied by a drop-down menu. Here you will find other actions PHOEBE can perform.

On the bottom of the main PHOEBE window there's a status bar (labelled Last Event), where PHOEBE notifies you of all events when they are initiated. There is planned support for history browsing and undoing unwanted changes, which will be closely related to events in the status bar.

Finally, the lowermost part of the main PHOEBE window is occupied with a button box. The buttons in this box let you quickly perform some specific operations. Open and Save buttons let you manipulate PHOEBE keyword files - files containing all the information about a certain star and are usually identified by the suffix .phoebe. The Defaults button restores all parameter values and entries to a state they were at startup. LC Plot and RV Plot buttons open the plotting windows, Fitting takes you to the fitting window, Help button will eventually bring up the online help screen, but this is not implemented yet. Finally, the Exit button closes all PHOEBE windows and exits the program.

Getting started

During this tutorial, we shall try and find the preliminary solution of the UV Leonis binary system. UV Leonis is a detached close binary with coordinates $\alpha_{\mathrm{2000}} = 10^h 38^m 20.7699^s$ and $\delta_{\mathrm{2000}} = +14^o 16' 03.670''$. Data files for this system are acompanying every PHOEBE release and reside in data/UVLeo directory. There are two photometric light curves3 in B and V filters (both from Bessel 1990 series): UVLeo.B and UVLeo.V and two radial velocity curves4: UVLeo.rv1 and UVLeo.rv2 - one for each star.

Supplying experimental data

First and foremost, we have to tell PHOEBE where to find our experimental data.

On top of the Data tab (Fig. 4) you may see a Star Name entry field. This field is for your convenience only and carries no modeling significance. Click on it and enter UV Leonis. Note that the main PHOEBE window title has changed accordingly.

Figure 4: PHOEBE data tab.
phoebe_data_tab

Now we shall face our first physical decision: we must assign a model to our star by choosing the most appropriate from the given choices (the Model field). The choice of a model is very important, since it poses additional constraints on our solution. Take a minute to browse through available models, but since our star in fact is a detached system, leave it on default5.

Next, we will supply PHOEBE with two light curves and two radial velocity curves; enter the number 2 in fields labelled Light Curves and RV Curves. You may change these values directly, by clicking on the entry and typing 2 instead of 0, or use the arrows next to the entry. As you change the number of experimental data curves, empty records are added to light curves list and to radial velocity curves list. Note that the used abreviation for light curves is LC and for radial velocity curves is RV. A major break-through will be accomplished by implementing complete spectra as experimental input data and this is planned for the intermediate future.

Assign the first light curve to PHOEBE. Edit the first LC data entry by either double-clicking on it, or by clicking on the Edit Data Entry button on the lower right part of the list while that entry is highlighted. An assignment window (Fig. 5) will appear.

Figure 5: PHOEBE light curve assignment.
phoebe_assign_lc_record

Input files should contain two or three columns. The first column contains the independent variable, which may be either phase or heliocentric Julian date (HJD). The second column contains the dependent variable, which may be in flux or magnitude units. The third column is optional; if present, it contains individual weights given either as weights or as standard deviation. The normalization is arbitrary; higher the number, stronger the weight.

Now inspect your input files from the terminal; simply write something like6:

   [andrej@gemma andrej]$ head phoebe/data/UVLeo/UVLeo.B
   # UV Leo Bessell B
   # Data from the Crni Vrh Observatory, Slovenia
   #
   # Standard star T0845-00073 used to measure nightly sigma.
   # Weights calculated as 1/sigma^2, renormalized to [0.1, 9.9].
   #
   #     HJD          Mag      w
   2451602.395411   10.130   2.53
   2451602.397101   10.129   2.53
   2451602.398802   10.157   2.53
   [andrej@gemma andrej]$
Immediately you see that a # delimeter is used for commenting; whenever PHOEBE reaches it, the rest of the line is discarded. It is common practice to put a small header (such as the one you see above) to your input files. Spaces, tabs and empty lines in input files are allowed.

In our case the independent variable is heliocentric Julian date (HJD, PHOEBE refers to it simply as Time) and the dependent variable is magnitude. The third column contains individual weights. These weights are usually based on some sound time-dependent insight, such as nightly variation of a standard deviation $\sigma$ of a comparison star. If these $\sigma$s are to be trusted, the weights are transformed as $1/\sigma^2$. Higher the weight, stronger is the influence of a specific data point on the solution.

Now that we know about the input file structure, set all three column definitions accordingly. Assign a filename by clicking on the Browse button and selecting UVLeo.B from the data/UVLeo directory.

The dispersion (standard deviation) of our data is changing from night to night, which we already compensate with individual weighting. Thus a simple average over all nights is a good initial approximation. Set Sigma to 0.026. We shall correct this approximation when the time comes, during the course of solution seeking.

Finally, choose apropriate filter: click on the filter button and select 440nm (B) from the Bessell subgroup.

If you wish, PHOEBE may compensate automatically for the reddening effect given that the reddening correlation parameter $R$ and the color excess E(B-V) are known. The measured magnitude decreases due to interstellar extinction by a factor $A_\lambda = 0.68 \, R \, \textrm{E(B-V)} (1/\lambda \mathrm{[\mu m]} - 0.35)$. We consider UV Leo's reddening effect to be negligible, so leave the reddening option off.

When you are content with the settings, click the OK button. This will close the assignment window and update the list of light curves. Repeat the above assignment for the other light curve file; this time set the filter to 550nm (V) from the Bessell subgroup and Sigma to 0.016.

All internal calculations in PHOEBE are performed in fluxes, not magnitudes. Therefore PHOEBE automatically transforms magnitudes to fluxes, if magnitudes are dependent variables, which in our case they are. The conversion is done by using the user-supplied zero magnitude (the Zero Magnitude for Flux Normalization option). If the zero magnitude corresponds to the magnitude at quarter phase, then the flux will be normalized at quarter phase. This is what the user should tend to, but not rigorously. By this we mean the following: don't spend your time seeking the right zero magnitude, leave this to PHOEBE. Just be sure you are close enough: $\pm$ 2 magnitudes or so. In our case leaving zero magnitude at 10.0 will do just fine.

Next, assign radial velocity curves to PHOEBE. In much the same manner as above, inspect the contents of a radial velocity file:

   [andrej@gemma andrej]$ head phoebe/data/UVLeo/UVLeo.rv1
   # UV Leo Primary Star Radial Velocities
   # Data from the Asiago Observatory, Italy
   # 
   # All data reduced with IRAF from acquired Echelle spectra
   #
   #    HJD          RV
   2451209.51624  -151.4
   2451210.43364   166.6
   2451216.49289   116.0
   2451217.46653    18.9
   [andrej@gemma andrej]$
Independent variable for radial velocities is much the same as for light curves: either phase or HJD. The dependent variable is different; it is a measure of radial velocities, which is given either in km/s or in 100km/s. The latter is a WD legacy, because numerous input files from the past had these units as input. You are thus advised to use km/s if possible, because PHOEBE does all the renormalization internally. Weighting is done in exactly the same manner as for light curves.

Figure 6: PHOEBE radial velocity curve assignment.
phoebe_assign_rv_record

From inspecting the radial velocities experimental data, we conclude that the independent variable is Time and that the dependent variable is radial velocity in km/s. This time the weights are missing, so the third column of an input file should be set to Unavailable. Set both Sigmas to 10 km/s (the initial approximation) and choose a filter 861nm (RVIJ) from RV filters subgroup. Radial velocities should theoretically be independent of the filter choice and in most cases they are; this specific filter tells PHOEBE to use a straight flux file and not think of radial velocities as wavelength dependent values.

After assigning a second radial velocity curve, note that filter names in RV columned list have changed: two extra numbering delimeters #1 and #2 have been added to make a distinction in curve dependent assignments.

If you want to use bins instead of complete data arrays, you may select data binning next. This is advisable only when you deal with considerable number of data points (5000 or more), and that only when seeking a preliminary solution. Be aware that you are degrading light curves heavily this way.

If you followed the described procedure correctly, you should end up with a Data tab as shown in Fig. 7.

Figure 7: This is how your Data tab should look after experimental data assignment.
phoebe_data_tab_filled

System-related parameters

When we are finished with specifying experimental data for PHOEBE in the Data tab, let's proceed to the System-related tab (Fig. 8).

Figure 8: PHOEBE system related tab.
phoebe_system_related_tab

The System-related tab contains 8 physical parameters that apply to the binary system as a whole (contrary to component-related parameters, which apply to each component of the binary system individually). They are:

HJD0: the reference time epoch of the zero phase. It is in units of HJD and it determines the initial point (the origin) for the epoch of the binary.
PERIOD: the period of the binary. It is given in days.
DPDT: the first time derivative of the binary. It gives the period change rate in days per day, so if different than 0, it is a very small quantity.
PSHIFT: a constant phase shift of ephemeris time. By applying the shift 0.5, one easily exchanges the roles of both components.
SMA: the semi-major axis of the binary. It is the maximum distance between both binary components given in solar radii, $R_\odot$.
RM: the mass ratio. It is defined as the ratio of a secondary component mass over primary component mass, primary star being the one eclipsed at phase 0.
INCL: the inclination of the system. It is measured in degrees: inclinations smaller than $90^o$ mean that the binary rotates counter-clockwise as projected onto a plane of sky; inclinations larger than $90^o$ mean that the binary rotates clockwise as projected onto a plane of sky.
VGA: the radial velocity of the binary system center of mass expressed in km/s.

First four parameters, HJD0, PERIOD, DPDT and PSHIFT are called the epoch parameters, since they determine the ephemeris epoch of the binary. With these parameters the mapping to phase-space is readily possible. The following four parameters, SMA, RM, XINCL and VGA determine the global binary shape.

Let us focus for one second at parameter input form (Fig. 9); each parameter is given by its keyword, a concise description and a value.

Figure 9: A typical PHOEBE parameter entry explained.
phoebe_parameter_entry_explained

PHOEBE keyword is the symbol by which a particular parameter is identified in PHOEBE . Keyword names follow a well established WD standard, but since PHOEBE does much more, new keyword names had to be introduced. In appendix D a complete list of all PHOEBE keywords is given. A file containing all parameter values in form KEYWORD = VALUE is called a keyword file. These files usually have the extension .phoebe, but there are no limitations in this respect.

A concise description is meant to elaborate on sometimes cryptic parameter keywords and give additional explanation on some conventions that are not obvious (e.g. the mass ratio being defined as the mass of the secondary star divided by the mass of the primary star - see Fig. 8).

The parameter value depends on parameter type. Usually the value is a number (integer or real), but it may also be boolean (TRUE and FALSE) or a string (text). A typical example of a boolean value is the adjustment switch (Fig. 9), where a string value is, in example, the experimental data filter name.

The Adjust switch is used to mark the parameter for adjustment. If turned on, this switch causes the value of the corresponding parameter to be fitted during the minimization process7. Otherwise the value is kept.

If the value is to be adjusted, the minimization step size tells PHOEBE how crudely to change the values during the minimization process. These values should be carefully chosen, because a value too large will skip the minimum and the value too small will take unnecessarily long computing time.

Finally, on the right-most part there are parameter boundaries for the simplex minimization routine. This routine scans through the parametric hyperspace in search for a global minimum. This is yet to be implemented, so all these boundaries are currently shaded out.

$\vdots$

Now try and change the value for period and the origin of time! Change the period to 0.600086 and HJD0 to 2448500.562. These two numbers you should estimate yourself independently of using PHOEBE. Since we have no physical indications about the values of other parameters, leave them be as they are.

What else may we presume?

Since we know virtually nothing in advance of our star, it makes no solid sense to make any further presumptions. We cannot safely say that the orbit is either circular or elliptical, until we examine the shape of radial velocity curves. We cannot be sure about the temperatures until we examine light curves and measure the color index. We have no idea about the parameters that influence the shape of synthetic curves, such as the inclination, mass ratio or the semi-major axis; we definitely have no clues about the secular changes (time derivatives of the period and the periastron); these are of such perturbative order that it is rarely obvious how to change these values. As the fit brings you closer to the solution, more of the star's nature will be revealed; take it one step at the time.

Inspecting data files closely

The next logical step would usually be eye inspection of light curves and radial velocity curves. PHOEBE enables you to plot experimental curves in a separate window, which you access by clicking on the Plot LC button on the button box for light curves and Plot RV button for radial velocity curves.

Let us start with light curves. Click on the Plot LC button on the button-box.

Figure 10: The light curves plotting window.
phoebe_lc_plotting_window

The top left part of the LC plotting window (Fig. 10) is used to select what it is we want plotted. Synthetic light curves are computed from theory, based on the current palette of physical parameters. Experimental light curves are taken from our input files. You may choose which experimental curve to plot by selecting its photometric filter; all PHOEBE operations are in fact identified with their filter names rather than filenames.

The resolution of a synthetic light curve is defined by the number of vertexes that we use for plotting. This value is held by an entry labelled Vertexes. If we have steep and narrow eclipses, we need finer resolution, especially in eclipse regions. The number of vertexes defaults to 100, but sometimes this does not suffice for accurate inspections. Try setting it to 300 for the time being.

The top right part of the LC plotting window is used to assign the variables of our plot. The independent variable is plotted on x-axis and the dependent variable is plotted on y-axis.

Independent variable may be time (HJD) or phase. If the independent variable for our plot is phase and what we have in our input files is heliocentric Julian date, then PHOEBE automatically transforms HJD to phase according to the origin of time and period that we set in the main PHOEBE window. This automatic procedure transforms all data points to have phases between $-0.5$ and $0.5$. Thus, if we use a plotting range wider than this, the regions outside this interval will contain no data points. That is why PHOEBE allows you to map existing points to the whole region of the plot. This means that a single data point will occur multiple times, wherever the phase of that point is the same. This procedure is referred to as aliasing8. Aliasing is turned on through a Turn on data aliasing switch.

Dependent variable is a bit tricky. It may be:

Let's first see whether our period and the origin of time make sense; thus we do not want to plot synthetic light curve, so turn the button Synthetic LC off. The data filter is already set to 440nm (B) in the drop-down list. Note that this list contains only filters that correspond to experimental data curves as defined in the Data tab. Now turn the Experimental light curve switch on. We shall plot the total flux against phase (which is the default). The phase range lets you select what phases you want plotted. This too we shall at this time leave at their current values.

To actually plot the diagram, click on the Plot button. The plot in Fig. 11 appears.

Figure 11: Plotted 440nm (B) light curve.
phoebe_lc_plotting_window_experimental_data

You may see that the values for period and the origin of time are pretty much what one would expect, since the minima are at phases 0.0 and 0.5 for all times.

Whenever you click on the Plot button, the plot is displayed; underneath it you may find the number of experimental data points, in our case 1563. The values of weighted and unweighted $\sigma$s (Sigma) are calculated between experimental data and synthetic light curve, so they are not calculated now that we don't plot a synthetic lightcurve. On the right of the plot you may see the Modeling parameters part of the plotting window; these parameters are calculated from synthetic light curve parameters, so again they are not displayed for our first plot.

Now select the other light curve by choosing 550nm (V) filter instead of 440nm (B).

Let us check how big is the difference between the initial synthetic light curve and experimental data! Turn on synthetic light curve (Synthetic LC switch) and click on Plot again! You are presented with the picture as it Fig. 12.

Figure 12: Synthetic light curve with plotted 440nm (B) light curve.
phoebe_lc_plotting_window_both_data

Synthetic curve is denoted with a solid red line and experimental data with blue filled circles.

As you may see, the result is disasterous. Standard deviations $\sigma$ are now calculated. If individual weights are included in the computation, we obtain the value of $\sigma_W = 0.288071$ and if they are not included in the computation, we obtain $\sigma = 0.272182$. It is $\sigma_W$ that drives our fit during the minimization scheme, more precisely a convolution of all lightcurve-dependent $\sigma_W$s.

Synthetic light curve computation now supplies values to the Modeling Parameters window (on the right of the plot itself). Browse through parameters to see what is contained in the output. Abreviations P and S stand for primary and secondary star, respectively. The masses are expressed in units of solar mass, mean radii in units of solar radius. Bolometric magnitude Mbol is in magnitude units and surface gravity potential has standard cgs units g cm/s${}^2$. PSB stands for polar surface brightness, Surface is in units of the Roche lobe surface size and all radii informations are given with respect to the semi-major axis. The values you see make no sense, but this is of course to be expected, since parameter values are way wrong. Don't worry about it, they will converge eventually.

The fit between experimental points and the synthetic curve is, for the lack of a better word, terrible. Inspect the other light curve. To get the feeling of discrepancy, you may use the Plot Residuals switch in the mid-upper part of the window. This switch tells PHOEBE to calculate the residuals between synthetic and experimental light curve and to plot them against independent variable. One would hope to get a pure scatter around 0, but this of course cannot yet be the case.

Just above the plot there are plot control arrows and zooming buttons. These controls enable you to move your plot according to your needs. Try zooming in on the eclipse minimum, move the plot around and, all in all, get a feeling of the plotting window as a whole. A button in between the direction arrows is the reset button; clicking on it will restore original plot settings. On the far right of that part of the plotting window you see the grid options and the coordinate options for your plot. The grid (both coarse and fine) is used to easily read out the values from the plot directly. Some users might prefer axes to a boxed coordinate system and this option is supported too.

When you had your share of playing with the LC plotting window and have examined the results, click on the Done button to close the window. Note that you may keep the window open if you prefer - this is usually sound to do when your monitor resolution is very high (1600$\times$1200 or so).





We shall now do much the same for radial velocity curves; open the RV plotting window by clicking on the RV Plot button on the button box.

Figure 13: The radial velocities plotting window.
phoebe_rv_plotting_window

The shape and feel of the RV plotting window is much the same as the LC plotting window, although few differences should be stressed. Dependent plot variables are as follows:

This time leave both experimental data and synthetic RV switches on; if you select the first filter (carrying #1 in its name), set Primary Normalized RV as the dependent variable. If you select the other RV curve, set Secondary Normalized RV as the dependent variable. Try plotting both radial velocity curves on on plot.

Figure 14: Primary RV curve in km/s; the discrepancy consists of two contributions: one is the amplitude misfit ($K_1$ value is wrong) and the other is vertical offset (center-of-mass $v_\gamma $ velocity is wrong, although hardly noticeable).
phoebe_rv_plotting_window_both_data

Feel free to examine other options of the plotting window. See what is accomplished when different choices are made. Move the plot around, turn the grid on and off, change the coordinate system shape to axes or boxes. Make your PHOEBE session as comfortable as possible. Again, if you wish, you may leave your RV plotting window open or close it by clicking on the Done button.

As you have seen, it doesn't make much sense to plot synthetic solution over experimental data at first. The fit is of course bad and the only thing you should really check at this early stage is that ephemerides for your binary are correct: the period ($P_0$) and the the origin of time ( $\mathrm{HJD}_0$). If your binary exhibits a period change (typically seen by smearing you cannot fix by changing $P_0$ or $\mathrm{HJD}_0$), you should set that parameter value appropriately.

Component-related parameters

We will now proceed with looking for clues about our binary that would help us set a decent starting point for automatic minimization algorithm.

Figure 15: PHOEBE component-related tab.
phoebe_component_related_tab

UV Leonis is a detached binary system and as such we may estimate the gravity potential. PHOEBE uses modified Kopal potentials as defined in (Wilson 1979). Their WD names are PHSV for a primary star and PCSV for a secondary star. In Component-related tab (Fig. 15) you may find the calculation buttons next to initial parameter values. Clicking these brings out the calculation window (Fig. 16).

Figure 16: The calculation window of the primary star's gravitational potential.
phoebe_phsv_calculation

You may quickly calculate the value of the potential by supplying the instantaneous separation between both stars, normalized to the semi-major axis (thus always 1.0 for circular orbits), the mass ratio and the radius-vector to the desired point in space, again in units of semi-major axis. To calculate the modified Kopal potential, set this radius-vector to the star radius towards the companion star (if tidal forces are strong, the star is deformed into a dumbell shape, so stellar radius is direction-dependent). In case of elliptical orbit the potential will depend on synchronicity parameter and direction cosines, but this is out of scope of this tutorial. Please refer to PHOEBE Reference Manual for further assistance.

On the lowermost part of the calculation window there is a button box. The Calculate button calculates the value of potential. PHOEBE lets the user inspect it and decide whether the calculated value is acceptable or not. If not, you may close the calculation window by clicking on the OK button and there will have been no changes made. If so, clicking on the Update button will copy the calculated value to the main PHOEBE window.

UV Leonis seems to have a pretty circular orbit and in the first approximation it seems that the default values in the calculation window are as might be expected (we will be proven wrong shortly, but this in fact is the realistic presumption). Therefore click on the Calculate button. The value 11.00504 will show up in PHSV entry. Since we are happy with it, click on Update button. The value will be copied to the Advanced tab of the main PHOEBE window. Close the calculation window by clicking on the OK button.

Repeat the same procedure for the secondary potential. The value will be the same.

At this time we cannot safely presume anything about the values of gravity brightening, albedo and synchronicity parameters, so we continue without modifying anything.

First steps of fitting

There is a preferable order of which parameters should be fitted when. Patience is advisory; as you gain experience, you will get the feeling of how to approach to certain types of problems. But the usual recipe would be this: start with luminosities. These quantities, although carrying physical name, are actually "normalizers" of your experimental data; original minimization routines work with fluxes, not with magnitudes, so PHOEBE internally does all the necessary conversions. These conversions are based on the normalizing magnitude that is set in the Data tab. It defaults to 10, but should be set to the magnitude of the quarter phase; as this cannot (and shouldn't) be done too accurately, luminosities jump in and normalize the data so that the normalization is actually done internally and you don't have to worry about it. These luminosities are referred to as HLA and CLA in in WD.

Go to the Luminosities tab (Fig. 17) and turn on the Adjust HLAs switch. Leave the step as it is, since we are going to use Levenberg-Marquardt's $\lambda$ multiplicator routine, which automatically chooses a proper step size when close to the solution.

Figure 17: PHOEBE luminosities tab.
phoebe_luminosities_tab

Now go to the Fitting tab (Fig. 18); the upper part of the tab is dedicated to the method of multiple subsets, which we will not use in this tutorial. The lower left part belongs to the two-dimensional projection of the parameter phase-space, which helps us analyse parameter correlations. We shall demonstrate its use in the course of this tutorial. On the right, there are fitting parameters, which are generally a good choice. Change grid sizes only if stellar shapes deviate significantly from spheroids, change Marquardt's $\lambda$ only if convergence isn't satisfactory. Choosing symmetrical derivatives is a better approximation, but at the expence of computing time. Since the time isn't that important at the moment, turn on symmetrical derivatives! After you have gotten yourself comfortable, click on the Fit Parameters Set For Adjustment switch!

Figure 18: PHOEBE fitting tab.
phoebe_fitting_tab

After you have clicked on that button, a fitting window (Fig. 19) will show up. This window is composed of three lists that contain important information for monitoring the progress of your soulution trough iterations.

Figure 19: PHOEBE differential corrections window.
phoebe_dc_corrections

On the top of the fitting window is a columned list that contains all parameters selected for adjustment. In case of curve-independent parameters there is only one entry and in case of curve-dependent parameters there are as many entries as there are affected curves. Light curves and radial velocity curves differ in this respect and have each their own adjustment switches.

Parameter names are followed by their current values. The fitting algorithm will calculate the correction of one iteration. Be very careful about these values, for they are the best marker to show if the fit converges or diverges. Luminosities essentially always converge, but you can never be too sure. This list also shows the corrected value and standard deviation of the fit of the particular parameter.

Next is a list of all experimantal curves detailing the number of points and the $\chi^2$ values before and after calculation for each curve. The $\chi^2$ values change after each iteration to reflect the similarity of the newly calculated curves to experimental data, and thus are another way to monitor the convergence of your soulution.

Finally, in the lower part of the fitting window there is a list of passband luminosities for the primary and secondary stars, as well as for the $3^\mathrm{rd}$ body.

Now click on the Calculate button! After some seconds of calculating (depending on the speed of your computer), the window as in Fig. 20 will show up.

Figure 20: PHOEBE differential corrections window with calculated parameters.
phoebe_dc_corrections_calculated

You may see that both corrections (and their corrected values) are in agreement with what we expect (recall what offsets we saw in plotting windows). Since we are happy with the fit results, we want to keep them. So click on Update all corrections button to replace the original values in both the fitting window and in the main PHOEBE window with corrected values. The fitting function is now ready for the second iteration, so click on Calculate button again! The corrections will now be even smaller and the convergence obvious. Update the corrected values and recalculate the values as many times as you see fit. As long as you are far away from the solution, be sure to let the values converge on a decimal or two, but as the fit gets better and better, try to be as accurate as possible. If you persisted completely through the convergence, you should have come up with these values:

Figure 21: PHOEBE differential corrections window with calculated parameters after several itterations.
phoebe_dc_corrections_final

Before you close the fitting window, take notice of the Show QuickBar button. It opens a window where you can mark any system parameter for adjustment and change its adjustment step. This is a time-saving shortcut worth of keeping in mind.

When you've finished (and are happy) with fitting, close the window by clicking on the Close button. Go to the Luminosities tab (or enter the Fitting QuickBar), turn off the adjustment switch of the luminosities and inspect the results, as you have before, for both light curves. Notice any difference? Still not perfect, but much better.

I feel comfortable, what next?

What's next basically depends on your particular problem. In our case, it seems that the shape of the light curve is way off. The parameters which influence the light curve shape the most are Kopal potentials, so let us try and fit those. Go to the System-related tab and mark both primary and secondary star potentials (PHSV, PCSV) to be adjusted. Go to the Fitting tab and do as before: correct the values of potentials one step at the time. Note that the convergence is slower than for the luminosities, but be patient and keep going until you have reached satisfactory values. A piece of advice: keep a plotting window open and inspect what each iteration does to your synthetic light curve!

Figure 22: The 550nm (B) light curve after the fit of potentials.
phoebe_after_phsv_fit_2

If you have followed the above scheme, you should have landed with the primary and secondary potential values of 4.878 and 4.686, respectively. We now see how way off we have been with our calculated values. You may play with the values in the potential calculation window to see what presumption was wrong. The corrected values produce the following for both light curves:

This is much better already!




... TO BE CONTINUED ...

Physical parameters' conventions

Semi-major axis: measured in solar radii, $R_\odot = 696\,000$km. It is a sum the two absolute semi-major axes: $a = a_1 + a_2$.
Orbital inclination: measured in degrees, with respect to the plane of the sky. If the inclination is in range $0^o$ to $90^o$, the binary orbits counter-clockwise as projected onto the plane of the sky, while above $90^o$ it orbits clockwise.

Common wisdom

Exclusive RV fitting: to correctly determine mass ratio, it is wise to fit only the radial velocity data. There is a very important caveat you must be aware of while doing this: if you try to fit only the mass ratio $q$ while you keep inclination and semi-major axis fixed, you most probably won't obtain a proper solution! The sound policy is to keep the inclination at $90^o$ and release the semi-major axis, where one must recognise the obtained solution for $a$ is in fact $a \sin i$ and not merely $a$, since the RV amplitude (e.g. $K_1$) is:

\begin{displaymath}
K_1 = \frac{2 \pi a \sin i}{t_0} \left( \frac{q}{q+1} \right).
\end{displaymath}

Transformation rules

Transforming HJD to phases: the transformation of heliocentric julian date (t) to phase is done by solving the differential equation:

\begin{displaymath}
d\Phi\, (t) = \mathrm{mod} \left( \frac{dt}{P(t)} \right),
\end{displaymath}

where $P(t)$ is the period, which might or might not be a function of $t$. If $\dot P = dP/dt$ is 0, then integrating that equation yields:

\begin{displaymath}
\Phi\, (t) = \mathrm{mod} \left( \frac{t-t_0}{P} \right).
\end{displaymath}

If, on the other hand, $\dot P \neq 0$, we obtain:

\begin{displaymath}
\Phi\, (t) = \mathrm{mod} \left( \frac {1}{\dot P} \log \left[ P_0 + \dot P (t - t_0) \right] \right).
\end{displaymath}

In C, the division moduo is obtained by function fmod, which returns the value on the interval $(-1, 1)$, so to obtain desired phases set in plotting windows, simple interval mapping with optional aliasing is applied.


Keywords supported in PHOEBE

This appendix lists the most important PHOEBE keywords and their corresponding types. Keywords are symbolic identifiers for parameters, and are used to describe the model in the keyword file that PHOEBE creates when you save your work.

Model parameters

These are the parameters which determine morphological and conditional constraints of the model.



PHOEBE keyword Type Short description
NAME string Star common name
MODEL string Model morphological constraint
LCNO integer Number of observed light curves
RVNO integer Number of observed radial-velocity curves
SPECNO integer Number of spectra
LD string Adopted limb darkening model

Data parameters

The following are parameters which determine observational data (light curves and radial velocity curves), and transformation factors.



PHOEBE keyword Type Short description
MNORM real Quarter phase magnitude
BINNING boolean Should data be binned
BINVAL integer Number of bins
REDDENING boolean Should data be de-reddened
REDDENING_R real De-reddening factor
REDDENING_E real De-reddening color excess
LCFN string LC data filename
LCCOL1 string LC data independent variable type
LCCOL2 string LC data dependent variable type
LCCOL3 string LC data individual weight type
LCSIG string LC data standard deviation
LCFLT string LC data passband
RVFN string RV data filename
RVCOL1 string RV data independent variable type
RVCOL2 string RV data dependent variable type
RVCOL3 string RV data individual weight type
RVSIG string RV data standard deviation
RVFLT string RV data passband

Physical parameters

These are physical and geometrical parameters that determine the shape and the structure of the modeled binary.

System-related parameters

PHOEBE keyword Type Short description
HJD0 adjustable Zero epoch time in HJD
PERIOD adjustable Pperiod value in days
DPDT adjustable Period time derivative
PSHIFT adjustable Phase shift
SMA adjustable Semi-major axis in $R_\odot$
RM adjustable Mass ratio ($M_2/M_1$)
INCL adjustable System inclination in degrees
VGA adjustable Systemic radial velocity

Component-related parameters

PHOEBE keyword Type Short description
TAVH adjustable $T_{eff}$ of primary star in $K$
TAVC adjustable $T_{eff}$ of secondary star in $K$
PHSV adjustable Surface potential of primary star
PCSV adjustable Surface potential of seconary star
LOGG1 adjustable Surface gravity of primary star
LOGG2 adjustable Surface gravity of secondary star
MET1 adjustable Metallicity of primary star
MET2 adjustable Metallicity of secondary star

Luminosity parameters

PHOEBE keyword Type Short description
HLALC adjustable Passband luminosity of primary star
CLALC adjustable Passband luminosity of secondary star
EL3 adjustable Third light
EL3_FLUX boolean Is the $3^\mathrm{rd}$ light in flux units
OPSF adjustable Opacity function
IPB_ON boolean Include proximity effects?
NOISE_ON boolean Add synthetic scatter to LCs
NOISE_TYPE string Synthetic scatter level weighting scheme
NOISE_VAL real Synthetic scatter $\sigma$
SEED_VAL real Synthetic scatter RNG seed
IFAT1_ON boolean Model atmosphere for primary star
IFAT2_ON boolean Model atmosphere for secondary star
MODELLGG_ON boolean Use gravity acceleration from the model
NREF_ON boolean Detailed reflection effect
NREF_VAL integer Number of reflections
ICOR1_ON boolean luminosities icor1 switch ???
ICOR2_ON boolean luminosities icor1 switch ???

Surface parameters

PHOEBE keyword Type Short description
ALB1 adjustable Albedo of primary star
ALB2 adjustable Albedo of secondary star
GR1 adjustable Gravity darkening of primary star
GR2 adjustable Gravity darkening of secondary star

Orbit parameters

PHOEBE keyword Type Short description
E adjustable Orbital eccentricity
PERR0 adjustable Argument of periastron
DPERDT adjustable Periastron time derivative
F1 adjustable Synchronicity of primary star
F2 adjustable Synchronicity of secondary star

Limb darkening parameters

PHOEBE keyword Type Short description
XBOL1 adjustable Bolometric LD coefficient $x_1$
YBOL1 adjustable Bolometric LD coefficient $y_1$
XBOL2 adjustable Bolometric LD coefficient $x_2$
YBOL2 adjustable Bolometric LD coefficient $y_2$
X1ALC adjustable Passband LD coefficient $x_1^{LC}$
Y1ALC adjustable Passband LD coefficient $y_1^{LC}$
X2ALC adjustable Passband LD coefficient $x_2^{LC}$
Y2ALC adjustable Passband LD coefficient $x_2^{LC}$

Spots parameters

PHOEBE keyword Type Short description
NSPOTSPRIM integer Number of spots on primary star
NSPOTSSEC integer Number of spots on secondary star
XLAT1 real Spot latitude on primary star
XLONG1 real Spot latitude on primary star
RADSP1 real Spot latitude on primary star
TEMSP1 real Spot latitude on primary star
XLAT2 real Spot latitude on secondary star
XLONG2 real Spot latitude on secondary star
RADSP2 real Spot latitude on secondary star
TEMSP2 real Spot latitude on secondary star


Footnotes

... directory1
If you do not have your home directory set, PHOEBE will display a notice window with instructions how to add your home directory to the environment and exit.
... package2
Currently, the supported packages are SuperMongo and GNUPlot.
... curves3
Photometric measurements have been acquired and are the property of the Cni Vrh observatory, Slovenia.
... curves4
Spectroscopic measurements have been acquired and are the property of the Asiago observatory, Italy.
... default5
If in doubt what to choose, you may start with a general binary system and estimate the type of your binary as you get closer to the solution; then set the appropriate model.
... like6
Change the path to your distribution archive; note that data files aren't installed to any system-wide directory.
... process7
Currently two minimization schemes are supported: differential corrections and Levenberg-Marquardt least squares.
...aliasing8
Alias stands for "also listed as".