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Chapter 11 Measuring the Masses of Binary Stars

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Chapter 11 Measuring the Masses of Binary Stars Chapter 11 Measuring the masses of binary stars Goal-of-the-Day To use the radial velocity variations of a double-lined spectroscopic binary to derive the masses of the stars in the binary system. 11.1 Binary systems as astrophysical tools A large fraction of stars are found to be part of multiple systems. These binary systems are important astrophysical tools, as they allow us to deduce the properties of the individual stars more accurately than can be done for single stars. A very important stellar parameter that is very difficult to measure for single stars is the stellar mass. Binary systems can however be used to estimate stellar masses. The physics that govern the stars’ orbits in a binary system (or a planet’s orbit in a planetary system) were developed by Newton and Kepler. It can be summarised in the relatively simple equation: 4π2 P 2 = r3, (11.1) G(m1 + m2) where P is the orbital period of the system, r is the separation between the two stars and (m1 + m2) is the added mass of the two stars in the system. The more massive star in the system is usually called the primary and the less massive star the secondary. For simplicity in this lab session we are going to assume that the two stars orbit around their centre of mass in circular orbits and that the observer (i.e. us on Earth) is viewing the system along the orbital plane. We will make use of a few other concepts from first year mechanics. From the definition of the centre of mass: r1 m1 = r2 m2, (11.2) × × where ri is the distance of star i from the centre of mass. This means that the centre of mass sits closer to the more massive star, i.e. the more massive star has a shorter orbit. Remember that stars stay on the opposite side of the centre of mass from each other! You should also recall that the orbital period and the orbital velocity are easily related in circular orbits. 59 60 CHAPTER 11. MEASURING THE MASSES OF BINARY STARS Exercise 11.1 (a) In a circular orbit, how are the angular velocity, radius, period and velocity related? (b) Since the two binary components are in circular orbits around the centre of mass, how does the mass ratio of the components relate to the ratio of orbital velocities? 11.2 Spectroscopic binaries Binaries that can be detected spectroscopically (spectroscopic binaries) are particularly useful. If the spectral lines due to both components can be observed in the spectrum (double-lined spectroscopic binaries), we are able to measure the periodic changes in Doppler shifts of the spectral lines, as the binary components move about their centre of mass. Remember that only motions along the line-of-sight cause a Doppler shift: to longer wavelengths (moving away from the observer, i.e. redshifted) or to shorter wavelengths (moving towards the observer, i.e. blueshifted). The radial (or line-of-sight) velocity of each binary component can be determined from the observed wavelength shift: ∆λ ∆v = , (11.3) λ0 c where λ0 is the rest wavelength of the spectral line being considered and c is the speed of light. The Doppler shifts of spectral lines are used to construct a radial velocity curve, a plot of the radial velocity (line-of-sight velocity) versus time. As you will see, for each star 11.3. CONSTRUCTING RADIAL VELOCITY CURVES 61 the radial velocity follows a sinusoidal variation, with the maximum velocity amplitude giving us the velocity of the star with respect to the centre of mass. The binary as a system is also moving with respect to the observer, i.e. the centre of mass of the binary system also has a radial velocity of its own. Exercise 11.2 (a) Why is it important that the observer is viewing the system along the orbital plane? Think how inclination affects the radial component of the velocity vector. Double-lined spectroscopic binaries can be used to obtain the individual velocities of the stars in the system. As we have seen in Section 11.1, if the period P is also known we can determine the binary separation r = r1 + r2. Armed with the binary components’ velocities, the period and the separation, the individual masses of both stars can be derived. This is the goal of today’s lab session. 11.3 Constructing radial velocity curves We are going to construct the radial velocity curve of a binary called V615 Per that has an orbital period of 13.7 days. We will make use of a timeseries of spectra that includes the Mg ii transition, at λ0 = 4481.2 A,˚ that we will use to measure the Doppler shift. Exercise 11.3 (a) Create and change to a new directory called project5, and make Iraf — do not forget to update the login.cl file! (b) Retrieve the data from ftp.astro.keele.ac.uk, directory /pub/astrolab/project5; (c) Make a data file listing for each spectrum the time of observation. Remember the Iraf task hselect that was introduced in session 2. The time information we are looking for is the HJD (Heliocentric Julian Date) date, which is given in days and fractions of days. In Iraf, the command splot — which you already used in the previous experiment to measure the spectral line strengths — can be used also to measure the Doppler shift of a spectral line. Better still, it can be used to simultaneously measure the Doppler shifts of two spectral lines belonging to the same electronic transition (but arising in the spectra of each of the two stars in the binary system). To do this, we will follow the procedure outlined below: splot [spectrum], where [spectrum] is the name of the relevant FITS file. You • might find it useful to add to the command xmin=λ1 and xmax=λ2 to constrain the wavelength range plotted; place the cursor just left of the set of two spectral lines, and press d (for “multiple • profile deblending”); place the cursor just right of the set of two spectral lines, and press d again; • place the cursor near the centre of the spectral line on the left, and press g (to fit • a Gaussian function to the spectral line profile); place the cursor near the centre of the spectral line on the right, and press g again; • press q to finish this step; 62 CHAPTER 11. MEASURING THE MASSES OF BINARY STARS press a in response to the question whether to fit the positions (or centroids) of the • two spectral lines independently (which is indeed what you want); press a again in response to the question whether to fit the widths of the two spectral • lines independently (which again is what you want); press y in response to the question whether to fit the background together with the • two spectral line profiles; use - to toggle between the solutions of the fits to the two spectral lines; • press q a couple of times to terminate the splot task. • Exercise 11.3 (d) Display the first spectrum and identify the Mg ii transition; (e) Using the procedure outlined above, measure the wavelengths of the two spectral lines belonging to the Mg ii transition, and record them in the data file with the times of observation that you prepared previously; (f) Repeat (d)–(e) for the remaining spectra. You now have a data file with 3 columns. For each of the 23 spectra you should have the time of observation (HJD date), and the wavelength position of each of the binary components. We will use gnuplot to plot and characterise the radial velocity curve. Exercise 11.3 (g) How do you obtain the radial velocities from your measurements? (h) Plot the radial velocities versus time of observation; (i) Identify which data belong to the primary, and which data belong to the secondary. Re-organise the table such that all the radial velocities belonging to the primary are grouped in one column, and those belonging to the secondary are grouped in another. We are going to use the gnuplot fit procedure outlined in session 6 to fit a sine function to the velocity variations of each component. We will however keep the period fixed: f(x)= a1 + a2 sin(2 pi x/13.7 + a3), with ai the parameters to be fitted. ∗ ∗ ∗ Exercise 11.3 (j) Fit a sine function to the radial velocity curve of the primary; (k) Fit a sine function to the radial velocity curve of the secondary; (l) How are the three fitting parameters of the sine function related, between the primary and secondary? What is the meaning of parameter a1 in terms of binary properties? Which parameters give you the radial velocities of the binary components? (m) what is the mass ratio of the binary? (n) Determine the mass of the primary and the mass of the secondary, and compare these values with the mass of the Sun. (Hint: start by calculating the binary separation r = r1 + r2, and then make use of equation 11.1.).
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