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  1. 1. Orbital Period of the Pre-Cataclysmic Variable: NN Serpentis Carlos Osorio∗ (Physics 134L) (Dated: June 3, 2016) Abstract. A pre-cataclysmic variable is a binary star system consisting of a white dwarf and a less massive star, whose separation is not small enough to allow mass transfer between them. These systems possess some interesting characteristics, for which they are commonly a subject of study. On this lab a pre-cataclysmic variable by the name of NN Serpentis (NN Ser) is observed using a CCD sensor located at Las Cumbres Observatory Global Telescope Network in Goleta, California. After the extraction of the data from FIT files into catalogues, and the isolation of the NN Ser data points from these catalogues, the light curve for this binary system in the V-band filter was successfully plotted. By then plotting a waveform to fit the data, the period of the orbit of such system was calculated to be 3.206±0.023 hrs. This value, along with some others acquired from the scientific literature, were used to find a total of 7 parameters defining the binary, including the mass and radius of the red dwarf, as well as the separation between the two stars. The calculated parameters were within the scope of the previously published values, thus rendering the observation as successful. CONTENTS I. INTRODUCTION 1 A. Roche Geometry 1 B. NN Ser 2 II. METHOD 2 A. Data Acquisition 2 B. Data Extraction and Calibration 2 C. Graph Generation and Curve-Fitting 3 III. RESULTS 4 A. Orbital Angular Frequency and Eclipse Duration 6 B. The NN Ser Orbit and the Reflection Effect 8 IV. CALCULATIONS 9 V. CONCLUSION 10 VI. CURRENT RESEARCH 10 VII. REFERENCES 11 I. INTRODUCTION A. Roche Geometry A closed binary system is a system of two stellar objects (such as stars, or dwarfs) that orbit around ∗ Also at Physics Department, University of California Santa Bar- bara; each other due to a gravitational attraction. Closed binary systems also possess the trait of having other significant, non-gravitational interactions taking place between the two stellar objects (Warner et al., 1995). Among the most diverse and commonly studied types of closed binary systems are the cataclysmic variables. These are binary systems composed of a white dwarf and some other, less massive, type of star. These two stellar objects are usually referred to as the primary and secondary stars, respectively. Due to the small separation between the two stars, and the high density of the white dwarf, the secondary star of a cataclysmic variable becomes highly distorted and is no longer spherical, but instead has a tear-like shape pointing towards the primary star. The geometry that dominates this type of system is called Roche geometry. If the stars are close enough so that the gravita- tional attraction of the primary star at the radius of the secondary is greater than that of the secondary itself, the mass of the secondary star begins flowing into the primary, creating an accretion disk around it. This mass flow is the defining characteristic of a cataclysmic variable. The maximum region a secondary star could occupy, with still complete gravitational binding over its material, is called the Roche lobe, and it is bounded by the critical gravitational potential at which mass transfer can happen.
  2. 2. 2 B. NN Ser Now, if the secondary star of this type of binary system does not fill its Roche lobe completely, no mass transfer can happen between the stars, and the system is said to be detached. This binary is what is commonly referred to as a pre-cataclysmic variable. An example of such a binary system is the NN Serpentis, located about 500 parsecs away from earth in the Serpens constellation. First referenced in the Palomar-Green Survey in 1980, and originally catalogued as PG 1550+131, the NN Ser system has a Right Ascension of 15h 53m 31.051s and a declination of +120◦ 57’ 40.13” in FK5 coordinates (Simbad, 2016). It is a pre-cataclyismic binary system composed of a red dwarf of low mass and a white dwarf of about half the mass of the sun. With an inclination of almost 90◦ , and a primary star much smaller than its companion, the NN Ser undergoes through deep periodic eclipses (Parsons et al., 2010). The purpose of this lab is to observe the NN Serpentis system through a CCD sensor to obtain the period at which the stars orbit each other. This result will then be used, along with some published parameters, to calculate the radius and mass of the stars, as well as the separation between the dwarfs and the orbital velocity of the red dwarf. II. METHOD A. Data Acquisition The NN Ser binary system was observed using a CCD sensor located at Las Cumbres Observatory Global Tele- scope Network in Goleta, California. The observation took place on May 8th , 2014 at 6:05am and lasted for about 4 hours until 10:04am of the same day. The data acquired consisted of 331 FIT images, 303 of which were taken with a V band pass filter, while the rest were taken with a B-filter. All 331 images were obtained with an exposure time of 30 seconds, and had an average of 47 seconds between recordings. Due to the small amount of data collected with the B-filter, this data was not ren- dered useful for any of the calculations involved, and was thus disregarded. B. Data Extraction and Calibration After the data was collected, the following step was to extract from the FIT images the information that was needed to obtain the orbital period of the binary system. For this we used SExtractor to get a catalogue of each FIT file. These newly created cat files contained the time of observation, the aperture magnitude and isophotal flux of each star in the image with their associated errors, and each stars’ right ascension and declination. Now, in order to isolate the information of the NN Ser system from the rest of the data points in the cat files, a python code was written that would recognize and extract the data on this binary system from within each of the files. To do this an algorithm was implemented in which the program would select from each file the data point with the closest right ascension and declination to that of the NN Ser system. This was done by finding the data point within each file that had the minimum square difference with the NN Ser’s coordinates, as given by the following formula; d = (ra − RA)2 + (δ − D)2 (1) In here d represents the total distance of each data point to the NN Ser’s coordinates, ra and δ are the right ascension and declination of each data point respectively, and RA and D the constant coordinate values for the NN Ser, as given in section 1B. After secluding the data for the NN Ser from the cat files, the next step was to calibrate the aperture magnitudes obtained from the CCD sensor to represent the apparent magnitude of the stars. To do so we chose two stars from the fit files and compared their aperture magnitudes to the apparent V-Tycho magnitudes given to them in the catalogue. Before this, however, the Tycho magnitudes given by Sky-Map were converted to the standard Johnson-Cousins passband magnitudes by using the conversion formulas provided by Mamajek et al. (2002). The function used to signal out these two stars from each cat file and compare them to their Sky-Map values was the same as that used above to extract the NN Ser data points. The average of the differences between the data and Sky-Map
  3. 3. 3 V-magnitude values, for each cat file, was then added to all the aperture magnitudes of the NN Ser data points in their corresponding cat file in order to convert them into calibrated apparent magnitudes. C. Graph Generation and Curve-Fitting The last stage was to plot the graphs of the data col- lected, and, using an existing curve-fitting function from Python, draw a line of best fit for the data. Matplotlib’s pyplot object was used to create four graphs in total. The first (Fig.1) was a plot of the NN Ser’s uncalibrated V-magnitudes against time, with the second plot (Fig.2) being the same but with calibrated magnitudes instead. The third graph (Fig.3) was a plot of the isophotal flux as a function of time. Finally, the fourth plot (Fig.4) was a scaled version of the second graph, with a sine wave plotted to fit the data points. For all of these plots, the time of each data point was specified to be the mid-point of the exposure time, that is, 15 seconds after the expo- sure started. To get a curve to fit the third plot, scipy’s curve fit function was used. For the function to work, a sine wave function was created and given as argument, as well as an initial guess for the sine wave’s parameters. This function would then return the fitted parameters of the curve, as well as the errors in calculating such pa- rameters. 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 Time of Observation [hr.] 8.5 8.0 7.5 7.0 6.5 6.0 UncalibratedApertureMagnitude Uncalibrated V-Magnitude Light Curve FIG. 1. Plot of the aperture magnitude of NN Serpentis, taken from Las Cumbres Observatory Global Telescope Network, as a function of time. This plot shows some evidence for a fluctuation of NN Ser’s apparent magnitude with time, as well as a dip in magnitude around 8.7 hours that is characteristic of a deep eclipse.
  4. 4. 4 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 Time of Observation [hr.] 16.0 16.5 17.0 17.5 18.0 18.5 CalibratedApparentMagnitude V-Magnitude Light Curve FIG. 2. Plot of the calibrated apparent magnitude of NN Serpentis as a function of time. This plot shows a significant increase in resolution from the uncalibrated plot in Fig.1. We can now see more clearly the periodic fluctuation of the binary system’s apparent magnitude. The dip in magnitude caused by the red dwarf eclipsing the white dwarf around 8.7 hours is also more evident. There exist a number of outliers between 8 and 8.5 hours that indicate the visibility of the sky around that time might have been affected due to weather. III. RESULTS The graphs discussed in the previous section are plotted below. The results given by these graphs turned out to be very interesting, and it is valuable to run them down one by one. As stated before, Fig.1 shows the uncalibrated V-magnitude light curve. At first glance this graph does not seem to have a concrete periodic oscillation. Although this plot does show sign of a rela- tion between the V-magnitude of NN Ser with time, the seemingly rough ends and uneven spacings between the minimums give little hope for finding a fitting model to calculate the period of the binary’s orbit. Nevertheless, by calibrating the aperture magnitudes into apparent ones, in the process mentioned in section 2.B above, this apparent chaotic result turns into a smooth waveform, whose sinusoidal appearance is hard to overlook (Fig.2). Looking at this graph we don’t only see with clarity a periodic oscillation of the apparent magnitude, but there also appears to be a hole in the plot once the data points get close to the waveform’s minimum. This gap on the outline of the graph is not due to lack of measurements, but is instead the outcome of what is called a primary eclipse. Since the white dwarf of this binary system is almost half the size of its companion, and the inclination of the system relative to us is close to 90◦ , when the red dwarf’s orbit passes in front of its primary star, the light emitted by the latter is completely blocked.
  5. 5. 5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 Time of Observation [hr.] 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 IsophotalFlux Time dependece of Isophotal Flux FIG. 3. Graph of the Isophotal Flux of NN Serpentis, captured with an exposure time of 30 seconds by a CCD sensor, and plotted as a function of time. This plot strongly correlates with the plot of apparent magnitude in Fig.2 as is expected since the magnitude of a system is just the integral of the flux density over the filter’s banpass range. We see once again sinusoidal fluctuations of the signal, as well as strong evidence of a deep eclipse around 8.7 hours, and signs for bad weather conditions between 8 and 8.5 hours. Therefore, for those brief minutes, we receive only the light from the red dwarf, whose low temperature makes its apparent magnitude so high (and its brightness so low) that the system becomes no longer visible in the V-bandpass region for the CCD sensor used. Such a drastic change of magnitude during the eclipse is why this phenomenon is also termed as a deep eclipse. This effect can also be witnessed in Fig.6, in which the binary star appears to completely fade away from the FITs image, as compared to a few moments before, in Fig.5, in which it ws completely visible. The error bars of both these diagrams were taken to be the aperture magnitude errors recorded for the binary in the cat files. The next graph (Fig.3) serves as a verification of the effects seen in the light curve of Fig.2. In it, the sinusoidal aspect of the binary’s signal, as well as the primary eclipse’s dip in signal around 8.7 hrs can be recognized almost immediately. Notice also the set of outliers between 8 and 8.5 hours present in both of these graphs. The appearance of this outliers is thought to be due to bad weather conditions during this time that may have clouded the sky, thus making some of the measurements produced in this range appear as if they were dimmer. It is no surprise that the flux diagram seems so closely related to the the light curve of Fig.2,
  6. 6. 6 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 Time of Observation [hr.] 16.0 16.2 16.4 16.6 16.8 17.0 17.2 CalibratedApparentMagnitude V-Magnitude Light Curve Best Fit Curve FIG. 4. A scaled up version of the light curve shown in Fig.2. This plot also has a sine wave of the form A sin(ω×t+φ)+b plotted to fit the data. The sine wave seems to fit the data very closely, and thus serves as a good model for the periodic oscillation of the binary star’s magnitude. With this fit, the angular frequency of the orbit was determined to be ω = 1.960 ± 0.014 rad−1 , and thus the period to be P = 3.21 ± 0.11 hrs. The duration of the eclipse was also calculated to be ∆t = 0.176 ± 0.025 hrs. after all, the magnitude of a system is just the integral of its flux density over the specified band range. A. Orbital Angular Frequency and Eclipse Duration The last graph presented (Fig.4) is a zoomed in version of Fig.2 with a sine wave plotted to fit the data. As explained in the methods section, this was possible by using scipy’s curve fit function in python. The curve of best fit generated through this method coincides very accurately with the data points, as we see that the curve passes through most of the data points’ error bars. It is thus safe to say that this curve can serve as an accurate model of the NN Ser’s light curve, at least in the short-term range. The form of this sinusoidal curve was A sin(ωt + φ) + b, and through curve fit the best-fit parameters were found to be those shown below in Table 1. More specifically, the final wave plotted on this graph was V = 0.293 sin(1.960t + 3.21) + 16.538. There are three important aspects of these cal- culated parameters. First of all, the angular frequency of the magnitude, and thus the angular velocity of the system’s orbit was calculated to be 1.960±0.014 rads/hr. This value will later be used to find the orbital period of the NN Ser binary. The amplitude of the wave described above also tells us the change in magnitude of
  7. 7. 7 -500- 500 FIG. 5. FITs image of the NN Serpentis system. In this image we can clearly see the appearance of the binary system, along with some neighbor stars. the system as it completes the orbit. The analysis of this result will be shown in the next section (3.B). Finally, we see that the waveform is centered at a magnitude of 16.538±0.004 which relates very closely to the calcu- lated mean V-magnitude of 16.8 by Haefner et al. (1989). From the curve fit function it was also possible to obtain the variance of the best fit parameters calcu- lated. These values were then square rooted to give the standard deviation of each parameter, and thus used as their respective errors. From this graph it was also possible to obtain the duration of the secondary eclipse. This was calculated as being the time difference between the midpoints of both the last data point before the magnitude dip and the first recording after it, and the last exposure of this event and the first data point back near the fitted waveform. This way, the duration of the eclipse (∆t) was found to be around 10.56 ± 1.5 min, which is very close to the value found by Haefner et al. (1989) to be around 12 min. The error in the calculation of this value takes into account the amount of time between the observations of the transition from the waveform magnitude to the eclipsed value.
  8. 8. 8 -500- 500 FIG. 6. FITs image of the NN Serpentis system during a primary eclipse. This image is centered around the same location in space as Fig.5, but NN Serpentis does not appear to be visible. This is a clear indication of the large change in magnitude this binary undergoes through whenever the red dwarf passes in front of its primary, in what is generally called a deep eclipse. More over, this total eclipse gives evidence for the inclination of the system being close to 90◦ . B. The NN Ser Orbit and the Reflection Effect One last thing to make sense of from the light curves plotted in Fig.2 and Fig.4 is the respective orbital mo- tion of the stellar objects in the system, along with the reasoning behind this periodic motion of the apparent magnitudes. As stated a couple of times before, the minimum point in the light curves, where the dip in magnitude occurs, corresponds to the eclipse of the white dwarf by the red dwarf. It is known that the orbit of these two stars are circular and not eccentric since the tidal interactions of a pre-cataclysmic variable eliminate any initial eccentricity in the system’s orbit (Warner et al., 1989). This effect is also visible in the light curves plotted, as any eccentricity in the orbit would cause there to be different time displacements in between extrema. With this being said, it is quite obvious that the moment in which the red dwarf is situated behind the white dwarf, in what is called a secondary eclipse, is exactly one half-period away from the primary eclipse.
  9. 9. 9 This result is seemingly contradicting at first due to the fact that the brightness of the binary system seems to increase when the red dwarf is covered. The reason behind this increase in brightness when the white dwarf is in front of the red dwarf is what is usually referred to as the reflection effect. This a phenomenon caused by the red dwarf’s absorption and re-emission of the white dwarf’s radiation (Hellier et al., 2001). Now, since the red dwarf is tidally locked to the white dwarf due to tidal interactions (J. Horner et al., 2012), there is only one side of the red dwarf the absorbs all the incoming radiation from the white dwarf, making that side significantly hotter than the rest of its mass. Since in stellar objects higher temperatures mean higher brightnesses (and thus lower magnitudes), the side of the red dwarf facing the white dwarf has a substantially brighter surface than the rest of its body. That being said, when the white dwarf is in front of the red dwarf, the secondary’s brightest surface is also pointing directly at Earth, and so we receive both the light of the white dwarf, and it’s companion’s highest brightness, thus reaching a maximum in the light curve. With this known, we return to the amplitude calculated for the light curve’s fitted waveform in Fig.4. This amplitude tells us the change in magnitude of the system as it completes the orbit. The only thing changing throughout these stars’ orbit, in terms of the brightness emitted towards the earth, is the angle at which we face the hottest and brightest region of the red dwarf. Therefore, the amplitude of the fitted sine wave (0.293 ± 0.006) gives us a relation between the brightnesses of the red dwarfs’ faces, and thus a relation between each sides temperature. Specifically, this result tells us that the brightest and dimmest faces of the red dwarf have a magnitude difference of twice the derived amplitude (0.586 ± 0.012). Although this result will not be fully explored in this paper, it should pose as an interesting calculation for any future observations on the system’s temperature distribution. IV. CALCULATIONS Now that the sinusoidal aspect of the NN Ser system was modeled and used to find the angular frequency of the orbit, it is time to use these results to find some defining parameters of the system. Unfortunately there are very few things that can be done with only the data that was used for this lab. One of the few parameters that can be calculated on its own is the period of the orbit. For this, the following equation was used, with P being the period in hrs: P = 2π ω (2) The result of this calculation, along with all the ones that will follow, are displayed in Table 1.1 below. With a value of approximately 3.21 hrs, the period calculated closely approaches its published value of 3.12 hrs (Brinkworth et al., 2006). Notice also that all of the derived values below take into account the errors of their calculations. The propagation of this error was done by using the un- certainties’ ufloat function in python, which was checked throughout to show accurate error values. Parameters Other sources Derived % Error ω(rads/hr) - 1.960±0.014 - A(magnitude) - 0.293±0.006 - b(magnitude) ∗ 16.8 16.538±0.004 1.6 φ(rads) - 3.21±0.011 - ∆t(min) 12 10.56±1.5 12 M1(M ) 0.535±0.012 - - M2(M ) 0.111±0.004 - - R1(R ) 0.0211±0.0002 - - R2(R ) 0.149±0.002 0.184±0.023 23.5 P(hrs) 3.12 3.206±0.023 2.9 i(◦ ) 0.149±0.002 - q(M2/M1) 0.207±0.006 - - a(R ) 0.934±0.009 0.947±0.016 1.4 Vt(km/s) - 359±6 - TABLE I. List of all the parameter values acquired from other sources, as well as the ones derived in this lab. The percent- age error of the derived values with the accepted ones is also shown. All the published values were taken from Parson et al. (2010) and ∗ Haefner et al. (1989) For the following calculations, three parameter values were extracted from the literature, to compute a total of six parameters. The first two of the parameters taken from Parsons et al. (2010) are the mass ratio of the stars and the mass of the primary star. The former of these two has a value of about 0.207, as shown in the table above as q. This ratio, along with the white dwarfs mass, also given above, was later on used to calculate the separation of the two stars, a, by using Keppler’s third law in the
  10. 10. 10 following form: a3 = GP2 orbitM1(1 + q) 4π2 (3) This equation gave a separation between stars of 0.947 ± 0.016R , whose accepted value of 0.934R falls within its errors. The next step taken was to use the newly ac- quired values of the period and star separation in an equation to obtain the tangential velocity of the red dwarf, Vt, modeled as if it travelled around the whit dwarf in a perfect circle. As explained before, due to tidal interactions, the orbit of this pre-cataclysmic variable is circular and this model renders as an accurate description of the system’s orbit. Vt = 2πa P (4) The tangential velocity of the red dwarf taken this way was found to be 359 ± 6km/s. Now, assuming that the white dwarf is small enough, as compared to the red dwarf and the separation between them, that it can be taken to be as a dot, the equation to find the radius of the red dwarf becomes: R2 = Vt × ∆t 2 (5) From which we see that the secondary star has a radius a little less than a fifth of that of the sun. Notice how this is the calculation with the biggest percentage error with 23.5%. The big errors involved in this calculation are thought to be a result of all the approximations that were used to acquire this value. By checking once again at the parameter values in Table 1, we can see how close the calculated values were to the already published ones, with each of them having a percentage error of less than 3%, with the exception of the red dwarf’s radius. V. CONCLUSION In this lab we were able to use FIT data files of the NN Serpentis pre-cataclysmic variable, taken by Las Cumbres Observatory Global Telescope Network, to plot the V-band light curve of this binary system. After calibration of the magnitudes, and curve fitting using Python, the sinusoidal aspect of this binary’s light curve was exposed. From here, the orbital period of the two stars, P, was calculated to be 3.206±0.023 hrs, differing only by 2.9% from the published value. Through this process, the mean V-magnitude of the system, b, and the amplitude of the magnitude change for each periodic oscillation, A, were able to be computed. The light curves plotted were also used to determine the length of the periodic deep eclipse caused by the red dwarf on the white dwarf primary, found to be 10.56 ± 1.5 min. Finally, these results, along with some parameter values taken from the scientific literature, were used to calculate the radius (R2) and tangential velocity (Vt) of the white dwarf, as well as the separation (a) between the two stars in the system to be 0.184 ± 0.023 R , 359 ± 6 km/s, and 0.947 ± 0.016 R , respectively. All of the calculated parameters were within the scope of the accepted values, and, with the exception of R2, they all had percentage errors of less than 3%. Due to the general low percentage error of the derived values in this observation, this experiment can be termed successful. Nonetheless, the lack of data in other bandpass filters (apart from V), and the short amount of time of the recordings, exerted a constraint on the potential of this observation’s findings. This should be taken into account when taking future observations. Also, from the amplitude change (A) of the system, a more detailed analysis can be done to calculate the temeprature distribution of the red dwarf. VI. CURRENT RESEARCH Recent studies surrounding the NN Serpentis binary system concern about the existence of a planetary sys- tem in its orbit. This theory was proposed due to some small variations in the binary’s orbital period, which could be explained by the gravitational interaction with planets orbiting it. The latest studies suggest that there are two gas planets orbiting the pre-cataclysmic variable, each with an orbit of 15.5 and 7.7 years, and a mass of 6.9MJup and 2.2MJup, respectively, where MJup repre- sents Jupiter masses (Beuerman et al., 2010). Studying such a system offers the opportunity to understand bet-
  11. 11. 11 ter the evolution and formation of planets and stars in a pre-cataclysmic variable. VII. REFERENCES -Haefner. ”PG 1550 131 - A Short Periodic Precataclysmic Binary with Very Deep Eclipses.” Astronomy and Astrophysics 213.1-2 (1989): L15-18. Web. 20 May 2016. -Hellier, Coel. Cataclysmic Variable Stars: How and Why They Vary. London: Springer, 2001. Print. - Horner, J., R. A. Wittenmyer, T. C. Hinse, and C. G. Tinney. ”A Detailed Investigation of the Proposed NN Serpentis Planetary System.” Monthly Notices of the Royal Astronomical Society 425.1 (2012): 749-56. Web. - Parsons, S. G., T. R. Marsh, C. M. Copperwheat, V. S. Dhillon, S. P. Littlefair, B. T. Gnsicke, and R. Hickman. ”Precise Mass and Radius Values for the White Dwarf and Low Mass M Dwarf in the Pre-cataclysmic Binary NN Serpentis.” Monthly Notices of the Royal Astronomical Society 402.4 (2010): 2591-608. Web. - Warner, Brian. Cataclysmic Variable Stars. Cambridge: Cambridge UP, 1995. Print. - Wilson, Miller, Africano, Goodrich, and Mahaffey. ”Photoelectric Photometry of Six Cataclysmic Variable Stars.” Astronomy and Astrophysics Supplement Series 66.3 (Dec. 1986): 323-30. Web. 20 May 2016.. - Beuermann, K., F. V. Hessman, S. Dreizler, T. R. Marsh, S. G. Parsons, D. E. Winget, G. F. Miller, M. R. Schreiber, W. Kley, V. S. Dhillon, S. P. Littlefair, C. M. Copperwheat, and J. J. Hermes. ”Two Planets Orbiting the Recently Formed Post-common Envelope Binary NN Serpentis.” Astronomy and Astrophysics AA 521 (2010): n. pag. Web. - Mamajek, Eric E., Michael R. Meyer, and James Liebert. ”PostT Tauri Stars in the Nearest OB Association.” The Astronomical Journal 124.3 (2002): 1670-694. Web.