Lab 7 Eclipsing Binary Stars (1)

Eclipsing Binary Stars Remember to use blue text for your answers Background Material Complete the following section after reviewing all the Background Material. Question 1: Describe where the center of mass is located relative to the two stars that make up a binary system. Question 2: Complete the following table related to the location of the center of mass. The distance between the stars is always measured from the center of Star 1, so the separation between the stars (x) is the location of Star 2. The center of mass (x CM ) of the system is the distance from Star 1 to the center of mass. Star Star 1 Star 2 x CM Mass (M ⊙ ) x Mass (M ⊙ ) x System A 5 0 5 10 6 System B 2 0 4 6 2 System C 3 0 2 12 8 Question 3: If a small, very hot star is completely obscured by a large, cool star, the drop in total luminosity of the binary system would be best described as … (highlight your answer in blue) a) very small b) around 50% c) very large Question 4: Complete the following table related to stellar luminosities. Star Luminosity (L ⊙ ) Radius (R ⊙ ) Temperature(T ⊙ ) Star A 8 4 1 Star B 9 3 1 Star C 2 4 1/2 NAAP – Eclipsing Binary Simulator 1/10 The center mass is known as the balance point between two stars. It will always be closer to the more massive star, and it will be in the middle if the stars have the same mass.

Figure : Light Curve Visualization Panel for Example 1 at phase = 0.5. Part 1: Exploration with Presets Open up the Eclipsing Binary simulator. You should note that there are four distinct panels:  a Perspective from Earth panel in the upper left where you can see the binary system in motion.  a Light Curve panel in the upper right where you can see the light variations of the system graphed in either flux or magnitude.  a Presets panel in the lower right which contains settings to configure the simulator to Student Guide Examples, Datasets with Complete Parameters (real eclipsing binary light curve data will be displayed and the simulator will be configured so as to mimic this system), and Datasets with Incomplete Parameters (real eclipsing binary light curve data will be displayed and students will be asked to configure the system to match). There is also a System Properties sub-panel where you can set the semi-major axis and eccentricity.  a System Orientation panel in the lower left that allows you to adjust the inclination at which the system is viewed, as well as the viewing longitude. The Animation and Visualization control allows you to start and stop the animation, and view it from different phases of the orbit. This panel also contains a button that allows you to see the stars’ location on the HR Diagram. Spend some time experimenting with the simulator, and make sure that you are experimenting in a systematic matter. Make adjustments to one variable exclusively until you develop a good grasp of how it affects the light curve. It is difficult to develop physical intuition if you hop from one variable to another in a haphazard fashion. Set the system to Example 1 . This preset has stars of the same size, radius, and temperature in circular orbits. Click on the start animation button to show the stars in simulated orbit around their common center of mass. After you have watched the system through several orbits click the pause animation button when the system is near a phase of 0.5 (if the light curve is not exactly at 0.5 you can drag the vertical red line cursor in the light curve display to the appropriate phase, or use the phase slider under the Animations and Visualization Controls sub panel). This phase represents the instant when one star completely covers the other from the Earth’s perspective. A “pointed” eclipse on a light curve typically reflects a partial eclipse when one star doesn’t completely NAAP – Eclipsing Binary Simulator 2/10

Figure : 3D Visualization panel for Example #4 looking down onto the plane of the binary system. In the 3D Visualization panel deselect lock on perspective from Earth . You may now click and drag anywhere in the 3D Visualization Panel to change the viewer’s perspective. Note that the Lightcurve Visualization panel still illustrates the light curve as detected from the Earth’s perspective. Thus, the 3D Visualization panel is allowing you to leave the Earth and view the system from any vantage point. The red arrow indicates the direction to Earth. Manipulate the panel until you are looking directly down onto the plane of the binary system as is shown in the figure to the right. The green cross represents the center of mass of the system. You are encouraged to open up the Star 1 Properties sub-panel and play with the mass of Star 1. Note that as the stars’ distances from the center of mass change the mean separation of the stars is still 6R ⊙ and the light curve shape is also unchanged. NAAP – Eclipsing Binary Simulator 4/10

Figure 4: 3D Visualization panel for Example #5 looking down onto the plane of the binary system Figure 3: 3D Light curve for Example #5. Set the system to Example 5 . This example once again has stars of the same size, radius, and temperature but they are now in an elliptical orbit with eccentricity (e) = 0.4. You are still looking down on the plane of the system. Click show the system in motion and note that the eclipses no longer occur symmetrically in time. Note that the eclipses on the light curve correspond with the positions when both stars are aligned with the red arrow pointing toward the Earth. Select lock on perspective from Earth and note how the asymmetric occurrence of eclipses appears from this perspective. NAAP – Eclipsing Binary Simulator 5/10

Figure 5: Light curve illustrating only 1 eclipse. Return to Example 2 and click start animation . Note that the two eclipses are “symmetric” in that when Star 1 is closest to the earth the amount of star area obscured is the same as when Star 2 is closest to the earth. The two stars have the same temperature, thus the depths of the two eclipses are identical with this geometry. Return to Example 3 and click start animation . Note that these two eclipses are also symmetric in that the same amount of star area is obscured in each. However, since the surface temperatures are different the eclipse depths are different as well in this geometry. Examples 2 and 3 seem to suggest that the eclipses are always symmetric in obscured area. However, this only occurs in the simple cases where eccentricity is zero. When eccentricity is non-zero, the perspective described by the inclination and longitude of the system become important. The longitude of the system is the angle specifying the orientation of the semi-major axes of the orbits relative to your perspective. Set the system to Example 5 (where the eccentricity is 0.4). This example presently has two eclipses with equal depths. Go to the Systems Properties sub-panel and decrease the inclination to 75  . Note that with longitude equal to 0  (meaning that our line-of-sight is perpendicular to the semi-major axes of the orbits), the eclipse depths are still equal. Now experiment with the longitude parameter and note how it affects the light curve. You should be able to identify a range of longitude values for which only one eclipse occurs! Thus, the interplay of inclination, eccentricity, and longitude allow for some very complicated light curves (even in our simplified model). Part II: Exploring with Real Data In the previous section we looked at how various aspects of the system affected the light curve. In this section we will reverse the process by exploring real datasets of photometric data and NAAP – Eclipsing Binary Simulator 7/10

using the simulator to learn about the system from the light curve. However, it should be remembered that there are many parameters involved. Typically, light curve information is incorporated with spectra (giving radial velocity curves and spectral type) to give a more complete understanding of system. Without this additional information the system parameters cannot be uniquely determined from the light curve data. Thus, in this lab we will focus on discussing some of the possibilities involved rather than reaching any definitive conclusions on the nature of the system. It should also be remembered that many approximations are used in this simulator. It assumes that the stellar discs are round and of uniform brightness. This approximation works better for some systems than others. Read the information about each real binary star light curve listed below, and investigate each light curve with the simulator. Look at the photometric data for KP Aql . We can draw a number of conclusions about the system by looking at the light curve:  Since the two eclipses are evenly spaced in phase (at 0.0 and 0.5) we can conclude that it is likely that the orbit is circular and we can ignore “longitude” effects.  Since the two eclipses have very similar depths and we are assuming circular geometry (i.e. symmetric eclipses), we can conclude that the two stars have very similar surface temperatures (here both temperatures are 7400 K).  Since only a small percentage of the light curve is made up of eclipses – most of the light curve consists of the “flat top” – we know that the stars are relatively far apart (KP Aql has a separation 13.61 R ⊙ , compared to the sizes of the stars.) This is the main reason that our simulator can fit the data so well!  The fact that the flux falls to very nearly 50% during eclipse is especially significant. For stars of the same temperature and symmetric eclipses, a 50% drop is the largest possible and can only occurs when the two stars are the same size and the inclination must be near 90º. Thus, we are really seeing a total eclipse for a short time. (It is known that KP Aql has an inclination very near 90  and the two stars are very close in size, 1.7 R ⊙ and 1.8 R ⊙ ). The system EW Ori has stars that are widely spaced compared to their size since the eclipses are deep and narrow. And since the secondary eclipse doesn’t occur exactly at phase 0.50 the orbit must be slightly eccentric (e = 0.07). It is difficult to make any firm conclusion related to the uneven depths of the eclipses. If the system has an inclination of exactly 90º, the uneven depths must be due to slight temperature differences in the stars (which is the known case for EW Ori, T 1 = 5970 K and T 2 = 5780 K). However, it is also possible that if inclination is appreciably less than 90º that longitude could be affecting the eclipses and the stars could have the same temperatures. Select preset FL Lyr . Once again, the two eclipses are evenly spaced in phase (at 0.0 and 0.5) suggesting a circular orbit. This light curve has eclipses of uneven depths, which suggests that the stars are not the same temperature (known values of 6150 K and 5300 K). The EK Cep system is slightly eccentric from the spacing of the eclipses in phase. However, since the eclipses have very different depths the stars must have very different temperatures. NAAP – Eclipsing Binary Simulator 8/10

Question 17: DI Her – Adjust the eccentricity. a) Best value of eccentricity = Question 18: Ag Phe – Adjust the temperature of Star 2. a) Best value of Star 2 Temperature = Summary/Conclusion (5 points): The masses of stars in an eclipsing binary system can be determined very accurately, which is important for our understanding of stellar evolution. To determine the masses of stars in a binary system, we need to know how long they take to orbit one another (period), and the average separation between the stars (semi-major axis). Newton’s version of Kepler’s third law can then be used to find their masses. Explain how the period and semi-major axis can be determined for an eclipsing binary system. Be specific and provide detail. (You will likely need to use your book for help in determining how the semi-major axis is found since this question is not addressed in the lab. Post a question to the weekly discussion board if you need help). NAAP – Eclipsing Binary Simulator 10/10 0.49 5300 k A period and a semi major axis can be determined from an eclipsing binary system if you can solve the orbit motion of an eclipsing binary. You can find the mass and diameter of each star and compare them with the spectral lines in the solar system. The times of a super-giant star’s spectrum are more narrow.