UNIVERSITY COLLEGE LONDON Department of Physics and Astronomy University Of London Observatory PHAS2130 2013 14.3 Analysis of Eclipsing Binaries 1 Introduction By modelling the light-curves of eclipsing binary stars, we can infer a number of astrophysical parameters. Assuming, for simplicity, that we know the orbital period and that the orbit is circular, those parameters are (i) the sizes of the two stars relative to the orbital separation (R 1 /a, R 2 /a), (ii) the stars surface-brightness ratio (usually expressed in terms of temperature, T 1, T 2 ), and (iii) the orbital inclination (i, where i = 90, means that the observer is in the plane of the orbit). The spectroscopic orbit of each star depends on the stellar masses, the binary separation, and the orbital inclination. If the spectra of both stars can be measured that is, if the system is a double-lined spectroscopic binary then the actual masses and separations can be determined, modulo factors involving the inclination. In the (relatively uncommon) situation where a double-lined spectroscopic binary is also an eclipsing binary, we can combine all the information in order to determine the masses and radii of each star. This methodology is by far the most most important technique for determining stellar masses (with few special-case exceptions, there are no other direct methods available), and yields the most precise radii. The purpose of this practical is to analyse data for an eclipsing, double-lined system to derive fundamental parameters. You will be using phoebe ( PHysics Of Eclipsing BinariEs ), a graphical front-end to the so-called Wilson-Devinney (WD) program, which is the standard research-level tool for this type of work. First, though, it s instructive to play with simpler, less quantitative, but more interactive tool, Starlight Pro (slp). Your write-up should explicitly addressed points in the script flagged as a Question, and/or with a symbol. It should not reproduce large chunks of this script, but should include fully annotated tables of measurements, and sufficient details of any calculations for your working to be readily followed. Of course, any numerical results you give should include a statement of the units, and some indication of the likely uncertainties. 2 Binary-star classification There are several schemes for classifying binary systems, but the most widely used scheme relies on the concept of the Roche lobe (Fig. 1). Imagine each star in a binary is compressed down to a point mass. A test particle will feel the gravitational pull of both stars, but if it is close enough
2 Binary-star analysis Figure 1: Types of binary system: (clockwise from top left) detached, semi-detached, overcontact, and contact. to the centre of mass of one of them, it will orbit just that one star star. If it is much further from both stars than their mutual separation, it will orbit the common centre of mass. At intermediate distances, it may perform a figure-of-eight orbit around both stars (or more complex motions). The Roche lobe is the volume of space within which a test particle belongs to just one star. It is, in effect, the maximum size allowed for that star if the star becomes bigger, then particles on the suurface will be influenced by the gravity of the companion to the extent that they may be pulled away from the parent star. The sizes of the Roche lobes depend on the masses of the stars in the system. If the surfaces of both stars are inside their respective Roche lobes, the binary is said to be detached. The more massive star the primary evolves faster. As it evolves away from the main sequence, it grows in size, and it may then come to fill its Roche lobe. The system is now semi-detached; as the star attempts to grow larger still, the material closest to the companion is captured by the secondary, preventing the primary from growing larger. (This mass transfer also changes the mass ratio, and the sizes of each Roche lobe.) In principle, two stars of equal mass can each exactly fill their Roche lobes at the same time, forming a double-contact system. In practice, such system are short-lived statistical flukes, and few (if any) are known. It is, however, possible for both stars to grow larger than their individual Roche lobes at the same time; they then share a common envelope of material that remains bound to the binary, but which no longer belongs uniquely to one or other star. These overcontact systems are quite common. 3 Starlight Pro Open up the slp program. If the display is animated, uncheck the Animate tick-box; set the viewing angle (just to the right of the Animate box) to close to 0. (The exact value is unimportant, but you want an edge-one view at this stage. Note that slp is idiosnycratic; any sane astronomer would call say that the resulting view corresponded to phase 0.25.)
PHAS2130 2013 14 3 To speed things up a bit, change the Photometry option from RGB to just R. You can work through a few of the pre-loaded stellar parameters in the list of Pictorial Atlas Stars to get some sense of the diversity in binary stars. You will see that most of the systems illustrated are rather close (meaning that the orbital separation isn t much greater than the sum of the radii). Question: Why do you think this is? Select 44i Boo, the first entry in the list of Pictorial Atlas Stars. With the default parameters, the system is just detached. Check the Animate tick-box to generate a light-curve. Question: The light-curve shows substantial variations outside of eclipses why? You should next vary parameters one at a time as directed below (with all other parameters at their default values). Tips: You can restore the base set of parameters at any time by double-clicking on the star name in the Pictorial Atlas Stars list. You will find the program to be more stable if you turn off animation while changing parameter values. Note that the light-curve plot autoscales - it fills the y range of the plot irrespective of the actual amplitude of variability (and there are no units plotted). Therefore, with slp, you can t make judgements about the absolute range of variation one good reason for progressing to phoebe later on. Change the temperature of Star 1 to 10 000 K Change the temperature of Star 2 to 10 000 K Change the inclination to 90, 80, and 35. Change the radius of Star 1 to 0.2. Change the radius of Star 2 to 0.2. Record, in your script, the effect on the light-curve of each adjustment, with a brief explanation. (Pay particular attention to the widths, and relative depths, of the eclipses.) This should give you some idea of how the light-curve varies in response to each parameter. 4 PHOEBE You ll now use phoebe to estimate the physical parameters of an eclipsing binary. 4.1 Load data Fire up phoebe. Choose the Data tab if it isn t already open. In the LC data panel, click Add, and select UVLeo.B from the Filename drop-down menu in the new window. As you might guess, this loads photometry in the Johnson B band for the system UV Leonis. (You might select the Filter manually.) Close the Add LC Data window.
4 Binary-star analysis You should add the additional light-curve (LC) and radial-velocity (RV) data provided. Make sure that you load each dataset in the right subwindow light-curve data in the LC data frame, radial-velocity data in the RV data frame. Also, make sure that each dataset is properly identified; e.g., Primary RV or Secondary RV, etc. You should specify the Filter for the photometry, but not for the radial velocities. Go to the Parameters tab, and select the Ephemeris sub-tab. Set the period to 0.600086d and HJD0 (the time of eclipse minimum) to 2448500.562. Select the Component sub-tab. Set the Primary star effective temperature to 5900K (this value is estimated from the spectral type, and should not be changed). Go to the Plotting tab and inspect the observed light-curve and radial-velocity curves. (Note that you have to click the Plot button, at bottom-left, whenever you want to refresh the display it doesn t do this automatically.) Turn on the plot of the Synthetic light-curve, and plot. Almost certainly, the fit will be awful. Your task is to improve that fit, as described in the following sections. There are several ways in which this improvement can be attempted, but one convenient way to work is to first estimate the mass ratio, on which the shapes of the stars depend. Question: Why does the mass ratio matter? (Why aren t the stars spherical?) 4.2 Modelling: radial velocities Open a separate window for the radial-velocity plot (click the RV Plot option from the menu at the top). Again, the modelled ( Synthetic ) curve will probably look fairly awful. In the main window, open the Parameters tab. Try varying the semi-major axis and the mass ratio until the observed and modelled curves agree as well as you can manage, for both stars (simultaneously). You will need to plot both primary and secondary radial-velocity curves together (in the RV Plot tab, choose the Primary+Secondary RV option for Y) you can t determine the mass ratio from a single radial-velocity curve! Record your adopted mass ratio. 4.3 Modelling: light-curves Once you have a good match, close the RV Plot, and open a light-curve plot. Then, in the main window, open the Parameters tab. First, adjust the Luminosities until the observed and modelled brightnesses agree around phases 0.25 and 0.75 ( 0.25). Keep the primary and secondary levels equal in each passband. Next, start varying the inclination (in the System sub-tab), the primary & secondary potentials, 1 and the secondary effective temperature (in the Component sub-tab). By replotting the observed and modelled light-curve as you go along, you should be able to identify how each of the 4 parameters separately affects the light-curve. Adjust each until you can get as good a match as reasonably 1 The surface potential is a characterization of the size of the star (see the Appendix); a smaller potential corresponds to a bigger star. Be careful not to make the star too big you can keep an eye on this by plotting the star shape, or by using the built-in potential calculator.
PHAS2130 2013 14 5 possible, not worrying about any remaining zero-point offset (you may find it useful to plot the residuals). (This is the major part of the effort required for this practical. You should be prepared to spend a significant amount of time getting a good fit, so that the synthetic light-curve passes through the midst of the observed data points, paying particular attention to getting the widths and the depths of the model eclipses in agreement with observations.) Record your adopted light-curve parameters. These should include, as a minimum, the two stellar radii, the two stellar temperatures, and the orbital inclination. 4.4 Modelling: radial velocities revisited Once you have a good light-curve fit, go back and inspect your radial-velocity curves. These may well have changed since you initially set the mass ratio even though you haven t changed the parameters that are critical to the radial-velocity curves (the mass ratio and semi-major axis). You may note, in particular, that the radial-velocity curves are not the sine waves you might expect for two stars in circular orbits. Question: Why do you think this is? binaries.) (Hint: the major effect is absent from non-eclipsing 4.5 Recording your results If you ve followed the instructions given above, your write-up should include a tabular summary of the values you ve adopted for the main parameters. You should try to include a rough estimate of the uncertainty of each parameter (e.g., by how much can you change the inclination before the light-curve fit becomes significantly worse 0.1? 1? 10?), as well as plots of your final observed and modelled radial-velocity and light curves. [As there is no built-in print option, you might do this with a screen grab, using (for example) the gimp program.] 5 Analysis 5.1 Evolution The results of your work so far should include estimates of the sizes and masses of the two components of the binary (with some indication of their uncertainties). Use the EZ-Web interface 2 to compute evolutionary tracks for stars of the masses you have obtained. Accept defaults for everything except Initial Mass. (You may assume that the initial ZAMS masses match your observed masses, as any mass loss should be negligible.) It may take a few minutes for your results to be returned. Meanwhile, compute the luminosities of both stars (in solar units), from the radii and temperatures. You should plot the computed evolutionary tracks, and your observed stellar parameters, in a theoretical HR diagram (i.e., log(l/l ) vs. log T eff ; take care to ensure that the axes correspond 2 If you re reading this in a suitable browser, such as Google Chrome, you can just click on the EZ-Web link; otherwise, the URL is http://www.astro.wisc.edu/ townsend/static.php?ref=ez-web
6 Binary-star analysis to astronomical conventions), using the plotting package of your choice, and plot everything in a single diagram. If your observations and analysis were perfect, and if the evolutionary calculations (for isolated single stars) were completely reliable, then your observed points should each lie somewhere along the relevant evolutionary track. You should be able to say something about the ages and evolutionary status of the stars, and to draw some conclusions about how well the evolutionary and dynamical (observed) masses agree. 5.2 Distance [Show all your working, and give the units of your calculations at every stage.] Assume that each star has the same temperature everywhere on its surface, and radiates as a black body (see Appendix). The effective wavelength of the Johnson V passband is 550 nm. For each star, calculate the V -band spectral irradiance (or flux, or monochromatic luminosity), which has units of energy emitted per unit area, per unit time, per unit wavelength interval. 3 Hence, assuming that each star is spherical, calculate its total emitted V -band flux, Outside of eclipse, UV Leo has V = 8.9. Observed flux and V magnitudes are related through V = 2.5 log f V 26.1 where f V is measured in J m 2 s 1 nm 1. Calculate the distance (in parsecs) to UV Leo. Apply a sanity check to test whether your answer is vaguely plausible (e.g., is your inferred distance larger than the distance to Proxima Centauri, and less than the radius of the Galaxy?). 3 Per unit area, per unit wavelength interval implies dimensionality of (length) 2 (length) 1 ; i.e., (length) 3, or per unit volume. Physically, however, it makes better sense to keep the two terms separate (and in practice, it s often convenient to consider them in different units; e.g., m 2 and nm 1 ).
PHAS2130 2013 14 7 Appendix: Useful Formulae Black-body flux: The spectral irradiance for a black body is πb λ = 2πhc2 λ 5 { e hc/kt λ 1} 1. Pay close attention to units! Projected orbital semi-major axis (for star 1,2): a 1,2 sin i = (1 e2 ) 1/2 K 1,2 P 2π = (1.976 10 2 )(1 e 2 ) 1/2 K 1,2 P R where the numerical values in the second line follow if K is in km s 1 and P is in days. Masses: M 1,2 sin 3 i = 1 2πG (1 e2 ) 3/2 (K 1 + K 2 ) 2 K 2,1 P = (1.036 10 7 )(1 e 2 ) 3/2 (K 1 + K 2 ) 2 K 2,1 P M If a radial-velocity curve is available for only one star, then we can only find a 1 sin i and the so-called mass function : f(m) = M 3 2 sin 3 i (M 1 + M 2 ) 2 = (1.036 10 7 )(1 e 2 ) 3/2 K 3 1P M The shapes of stars in close binary systems are conventionally described in terms of equipotential surfaces (i.e., surfaces of constant gravitational potential ψ, taking into account the gravitational forces of both stars [computed on the basis of point masses], and centripetal forces introduced by orbital and stellar rotation. These surface can be cast in a form that is independent of the absolute sizes and masses of the stars, by using the binary separation as the unit of distance. This normalized potential may be written as ψ = 2 (1 + q)r 1 + ( 2q + x (1 + q)r 2 ) 2 q + y 2 (1 + q) in an (xyz) co-ordinate system where the centre of mass of star 1 is at (0, 0, 0), the centre of mass of star 2 is at (1, 0, 0), and the stars equators are in the xy plane. The mass ratio is q = M 2 /M 1, and r 1,2 are the distances from the centres of mass, i.e., r 2 1 = x 2 + y 2 + z 2 r 2 2 = (x 1) 2 + y 2 + z 2 Initial author: IDH, 2/13 Other contributors: Latest revision: 4/2013