BINARYrep

BINARYrep

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  THE STABILITY OFPLANETS IN BINARY STARS SYSTEMS Brian Lonergan blonergan@trentu.ca Cristina Chifor cchifor@trentu.ca ABSTRACT This paper investigates regions of instability for infinitesimal planets in binary star systems with a direct application to the Alpha Centauri and Sirius systems. Mass-less test planets are placed in circular orbits about the primary and secondary stars for several hundred binary periods. Planets are also placed in circular orbits about the binary’s center of mass at radii from 30 AU to 100 AU. Planets orbiting Alpha Centauri’s primary and secondary are found to be highly unstable past 2.5 AU while those in exterior orbits are found only to be unstable for all regions smaller than 50 AU. Inside the Sirius binary, planetary orbits were found to be unstable for distances greater than 1 AU and 2 AU respectively, with high instability for high inclinations. In the region exterior to Sirius, orbits are unstable up to 40 AU. Results are presented in charts of ranges of distances and inclinations that were tested for both the interior and exterior orbits. The main purpose of the project is to find regions of instability for binary systems which can aid searches for extrasolar planets in our galaxy. Subject headings: Exoplanets, binary stars, stability 1. Introduction The search for Earth-like planets in other sys- tems of our galaxy now plays an important role in observational astronomy. After the discovery of a Jupiter-like planet orbiting the solar like star 51 Pegasi by Mayor and Queloz in 1995, the search for extrasolar planets has expanded greatly. There have been many discoveries made since then, but all observed planets were Jupiter- like. Finding Earth-like planets is incredibly difficult due to their small masses and sizes, as they have almost no effect on the motion of the star or the luminos- ity of stars during eclipses. Knowing where and where not to look becomes a big problem when dealing with small objects such as terrestrial plan- ets. The question of what sort of stars to look at is one that needs answering before time and ef- fort are spent on observations. It is thought that slightly more than 50% of all the stars in our own Milky Way Galaxy are in multiple or binary sys- tems. If we were to find that planets cannot form in such systems, we would drastically reduce the sample space of which to observe. It turns out however, that planets are likely to exist and be sta- ble in binary systems, as we uncovered instability regions that leave room for stable zones. We dis- cuss possible planetary formation regions in such systems, by determining regions around the stars that are unstable after tens of thousands of years. This timescale cannot predict stability over the ages of the stars, but with this approach we can find regions where the planets are certain not to have formed. We can therefore narrow the search space for planets in these common systems and help to speed up the process of finding the first Earth-like planet. The Alpha Centauri system is a triple system with two stars forming a close binary and a third member, Proxima, more than 12,000 AU away from the inner binary. The masses of the two 1
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  stars are similarto each other (Table 1) and con- veniently close to our Sun (1.3 pc). The closest approach of these stars is 11.2 AU so we should ex- pect to find stability only near the two stars. The brightest star in the Northern Hemisphere, Sirius, was also discovered to be in a binary or multiple system. In 1862, Alvan Clarr sighted a companion to Sirius, now called Sirius B; Sirius was then re- named Sirius A. The companion is much too small to see with the unaided eye, and even with large expensive telescopes. Sirius B is a white dwarf with a mass of about 1 solar mass MJ (Table 2)., meaning that it must have been the biggest and brightest of the two stars. It is very interesting to think about the possibility of planets orbiting white dwarf. It should be noted, however, that Sirius A is presently more than 20 times as lumi- nous as our Sun, and if Sirius B were even larger and more luminous than Sirius A in the past, any planet that was close near Sirius B would have been burned over the star’s evolution off the Main Sequence. The Sirius example is nevertheless an interesting experiment, which we considered in or- der to examine a more extreme case than Alpha Centauri. Using a mathematical model of the 3-body problem, we intended to map the position of both stars and a planet orbiting about a common center of mass The mass of the planets was assumed to be zero, an assumption which is plausible since the masses of Earth-like objects are insignificant when compared with the masses of the stars. The test particles were integrated in the binary system for timescales of the tens of thousands of years. Any planet placed in an initially circular orbit quickly evolves into its own seemingly unpredictable or- bit with an eccentricity that evolves over time. As measured by planetary observations so far, the semi-major axis of stable planets does not fluctu- ate beyond 5% . Hence, in our model, we rendered as unstable any test particle whose time -averaged semi-major axes differed by more than 5% from their initial value. Moreover, planets that suffered close encounters with the stars they were elimi- nated. A close encounter with the primary was assumed to be within 5 stellar radii. Also, plan- ets which came within half the minimum separa- tion of the stars were also assumed unstable. Our assumptions are based on observations and theo- retical considerations of planets in our own solar system as well as observed extrasolar planets. 2. The Model Planetary masses was assumed to be zero due to the tiny fraction of mass Earth-like objects pos- sess when compared with stars. We inserted only one planet at a time since we are not concerned yet with finding the dynamics of entire systems of planets. Each was placed at a distance ranging from 0.2 AU to 10 AU in steps of 0.2 AU, and at an angle from 0 to 180 degrees in 15 degree incre- ments. A detailed derivation of the equations used in the model is given in Appendix A. The differential equations were solved by nu- merical integration, using a FORTRAN program (Appendix B) that implements Euler’s integration method. Euler’s method assumes a constant ac- celeration for calculating velocity, and a constant velocity for calculating position over a given time step. We therefore had to determine an appropri- ate time step to ensure accuracy. This was done by incorporating a 4th order Runge-Kutta numerical integrator in calculating the orbit of one star (Ap- pendix C). We found that a maximum time step of 5∗10−3 years was suitable and we used 36,140,000 time steps of 5∗10−4 years ,in order to ensure accu- racy for a complete integration. It should be noted that given an equal time step, Euler’s method is faster than the Runge-Kutta method, which es- sentially is a cubic approximation of a curve and requires the calculation of four points inside a time step. In the case of the Alpha Centauri system, we assumed that Proxima, the third member of the system, is small enough, and at a large enough distance from the binary. Therefore, it was con- sidered to have no gravitational effect on the bi- nary or any orbiting planets. Hence, the orbit of the central pair, is taken to be a fixed Kepler el- lipse that determines a binary plane. The physi- cal properties of the binary are outlined in Table 2. The binary has a semi-major axis (average separa- tion) of 23.4 AU and a highly eccentric orbit with e=0.52. It should be noted that this is a specific example of a well known binary star system. The model is generalized, as any binary system with known mass, semi-major axis and eccentricity may be implemented and analyzed by using this ap- proach. In order to emphasize this, we considered 2
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  Table 1 Physical characteristicsof α Cen binary, including star masses, sizes and luminosities (Wiegert & Holman (1997), Kamper & Wesselink (1978)). The semi-major axis is 23.4 AU and eccentricity is 0.52. Star Mass MJ Radius RJ Luminosity LJ α Cen A 1.1 1.227 1.6 α Cen B 0.91 0.865 0.45 Table 2 Physical characteristics of Sirius binary, including star masses, sizes and luminosities (Gatewood & Gatewood (1978), Holberg et al. (1998)). The semi-major axis is 19.74 AU and eccentricity is 0.5292. Star Mass MJ Radius RJ Luminosity LJ Sirius A 2.14 1.7 26 Sirius B 1.03 0.022 0.002 another binary star, namely Sirius, which an even higher eccentricity of 0.5292 and semi-major axis of 19.74 AU. The physical properties of Sirius are detailed in Table 2. In the binary field, we integrated test parti- cles, representing point masses, ie. planets with insignificant mass compared to the masses of the stars. We considered each test planet individually, as we do not address stability of multiple plane- tary systems, ad hence we need not consider any interactions between planets. We adopt a sim- ple, empirical, observationally motivated criterion for stability (Wiegert & Holman (1997)). The possible candidates for stable planets were inte- grated for 18000 years and were eliminated from the system in two cases: they either escape or- bits or they suffer close encounters with one of the stars. A close encounter with the primary was as- sumed to be within 5 stellar radii. This would be enough for a planet to be disintegrated, at this close distance from a star. . As measured by plan- etary observations so far, the semi-major axis of stable planets does not fluctuate beyond 5% . In our model, we consider unstable any test planets whose time -averaged semi-major axes differ by more than 5% from their calculated initial value. It should also be noted that our definition of stabil- ity excludes planets that remain close to a binary, but migrate to smaller or larger orbits, encompass- ing only planets that remain near their initial or- bits. The deviation in the semi-major axis is hence only a measure of the available phase space. 3. Initial Conditions Initial information required for the binary sys- tem itself includes only the masses of the stars, the average separation (semi-major axis), and the ec- centricity. We start the stars at closest approach to each other and to the center of mass. At this point (periastron) the initial velocity of the pri- mary (arbitrarily chosen to be the more massive star) can be derived as shown in equation 7 in Ap- pendix A. The condition that the secondary star must, at all times, be exactly opposite the center of mass from the primary, allowed us to skip the numerical integration for the secondary (speed- ing up the numerical integration) by directly as- signing a position to it, with respect to the pri- mary.(Equation 2 and Fig. 1 in Appendix A). Test planets are placed in circular orbits about each star and about the center of mass in a wide range of radii and inclinations. In both systems, around each star, planets were placed at a distance 3
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  ranging from 0.2AU to 10 AU in steps of 0.2 AU, and at an angle from 0 to 180 degrees in 15 de- gree increments. In order to determine stability for the outer regions, planets were placed at dis- tances from the center of mass, ranging from 30 AU to 100 AU every 5 AU. Again, the inclination from the binary plane was varied every 15 degrees , from 0 to 180 degrees. The initial velocity of the planet can be found by knowing its radius from the star and the mass of the star it orbits (Equation 6 in Appendix A). It is important to note that the velocity calculated for the planet has to be added to the initial velocity of the star that it orbits, since we are working in the center of mass frame at all times. Being given an initial velocity, the planets and stars are therefore set in motion and their trajectories over a period of 18,000 years are calculated. If a planet is found unstable by either suffer- ing a close encounter with either of the stars, or by deviating from its inital orbit (as detailed in the model section) , d from the system into outer space, the integration is stopped and the region of instability is recorded. The program itself had conditions that gov- erned its accuracy and speed. We used Euler’s method for numerical integration, which it is rel- atively straightforward and fast, but can be inac- curate if used with an inappropriate time step. A 4th order Runge-Kutta numerical integrator was used to define an accurate time step. In the in- tegration of the primary star, Runge-Kutta was used for a step size of 0.1 years. This data file was then compared with data files created for the primary star by using Euler’s method. We found that a time step of 5∗10−3 years was suitable and the corresponding data files were virtually indis- tinguishable. In our final integrations we used a time step of 5 ∗ 10−4 years. This ensured that we were getting accurate results for our time period. 4. Simulations and Results Our results refer to three main regions of the bi- nary systems; near the primary star, near the sec- ondary star and exterior to the binary. The inner regions were found to be highly unstable in both the Alpha Centauri and Sirius systems. There is only stability very close the star that the planet orbits. The Sirius system is more unstable than Alpha Centauri, as we have expected, due to its greater eccentricity and masses, and smaller aver- age separation. In the Alpha Centauri system, the primary’s potentially stable regions extend from 0.2 AU up to a maximum of about 2.5 AU. Ret- rograde orbits seem to be more stable than pro- grade ones, but they suffer more close encounters, as shown in Fig. 1 and Fig. 3. Alpha Centauri B has potential stable regions extending from 0.2 AU all the way out to 2.6 AU for retrograde or- bits although these seem to suffer close encoun- ters. Prograde orbits and slightly inclined orbits average stability up to 2 AU, while highly inclined orbits are less stable (Fig. 1 and Fig. 3). The exterior region of the Alpha Centauri system was found to be potentially stable from 45 AU and above for retrograde orbits and from 60 AU and above for prograde orbits. Highly inclined orbits were also found to be potentially stable for dis- tances from 50 AU and above, as shown in Fig. 5. The further a planet is placed from the center of mass, the more the two stars can be regarded as a point mass, which explains why stability can be found at such great distances. Sirius A has potentially stable regions for distances less than about 1.2 AU and for inclinations less than 120 degrees. Planets orbiting this star do suffer close encounters but there are not noticeably preferred conditions for this to happen (Fig. 6 and Fig. 8). Sirius B can have stable regions only very close to the star with a maximum radius of about 1.2 AU for prograde and inclined orbits. Retrograde or- bits were found to be unstable below 1.4 AU and above 2 AU but not for lesser or greater radii. All close encounters suffered by planets orbiting Sir- ius B occurred at close to retrograde orbits. (Fig. 6 and Fig. 8). We found potential stability on the the exterior region of the binary at distances greater than 40 AU for inclined orbits and greater than 60 AU from the center of mass for prograde and retrograde orbits.(Fig. 10). In order to test our model, we designed a simple experiment, by considering known orbiting bodies in our own solar system: we assumed that the Sun and Jupiter represent the binary system , while the Earth is orbiting the ’primary’star, which in this case it is the Sun. As expected, the orbits of the Earth and Jupiter are nearly circular, with the Earth found ’stable’ at exactly 1 AU. The trajec- tories of the Earth and Jupiter were plotted and 4
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  are given inAppendix D. 5. Discussion and Conclusions Considering the amount of data processed, the FORTRAN program used in simulations has proven to be time efficient. For each run of approx- imately 23 minutes, the program we were able to asses the stability of approximately 600 separate planets orbiting a each star. For the external re- gions we calculated almost 200 separate planetary orbits and determined their stability. Although the integrations cannot assure stability of planets on time scales greater than 20,000 years, they do identify important instability regions. It should also be noted that that if we were using this pro- gram to determine stability over the probable 5 Gyr life span of the stars, we would not be able to use the Euler integration method. The slightest propagating deviations would propagate to erro- neous results, and the time required to finish the integration would result in the computer running out of memory. For a longer integration, a Runge- Kutta method and a more performant computer would be desired. The regions of potential stability were found to be relatively small, but not nearly small enough to rule out the possibility of finding Earth-like plan- ets in such a system. In fact, due to the high frequency of binary star systems in our galaxy it is plausible that such planets will be observed in practice. The most likely place to find a terres- trial planet is in the outer regions of the binary, circling at a large radius. This region is not likely the region where such a planet would form, how- ever, during the formation of the binary system, planets could have been captured. Regarding the inner regions of a binary system, it seems that there are many factors which constrain the pos- sible regions for planets to form. Greater binary semi-major axes and smaller eccentricities would allow for much larger regions of stability. These systems would then be the best candidates for fu- ture observations, while the highly eccentric, close binaries should be given a lower priority. REFERENCES Gatewood, G. D., & Gatewood, C. V., 1978, ApJ, 225, 191 Fig. 1.— Regions of potential stability near Alpha Centauri A. Axes are initial distance from star in AU vs. inclinations angle from the plane of the binary in degrees. Black squares indicate plan- ets that have survived the integration, and not deemed unstable. Fig. 2.— Regions of potential stability near Alpha Centauri B. Axes are initial distance from star in AU vs. inclinations angle from the plane of the binary in degrees. Black squares indicate plan- ets that have survived the integration, and not deemed unstable. 5
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  Fig. 3.— Initialdistance vs. inclination angle for planets orbiting Alpha Centauri A which suffer close encounters with either Alpha Centauri A or B. Axes and legend as in Fig. 1. Fig. 4.— Initial distance vs. inclination angle for planets orbiting Alpha Centauri B which suffer close encounters with either Alpha Centauri A or B. Axes and legend as in Fig. 1. Fig. 5.— Stability of planets orbiting in the region exterior to the Alpha Centauri system. Axes and legend are as in Fig. 1. Fig. 6.— Regions of potential stability near Sirius A. Axes are initial distance from star in AU vs. inclinations angle from the plane of the binary in degrees. Black squares indicate planets that have survived the integration, and not deemed unsta- ble. 6
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  Fig. 7.— Regionsof potential stable planets near Sirius B. Axes are initial distance from star in AU vs. inclination angle from plane of binary in de- grees. Fig. 8.— Values of initial distance vs. inclina- tion angle for planets orbiting Sirius A which suf- fer close encounters with either Sirius A or B. Axes and legend as in Fig. 1. Fig. 9.— Values of initial distance vs. inclina- tion angle for planets orbiting Sirius B which suf- fer close encounters with either Sirius A or B. Axes and legend as in Fig. 1. Fig. 10.— Stability of planets orbiting in the re- gion exterior to the Sirius binary system. Axes and legend as in Fig. 1. 7
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  Fig. 11.— Anunstable planet orbiting Alpha Centauri A at an initial distance, 2.4 AU, that is ejected outside the system. This is a result of strong perturbations from Alpha Centauri B. Axes are in Astronomical Units. Fig. 12.— 8
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  Kamper, K. W.,& Wesselink, A. J. 1978, AJ, 83,1653 Kov´acs, T., 2004 Holberg, J.B. et al.,1998, ApJ, 497, 942 Wiegert, P. A., & Holman, M. J., 1997, ApJ, 113, 4 This 2-column preprint was prepared with the AAS LATEX macros v5.0. 9