Relation between inner and outer eccentricities in a hierarchical triple star system

Relation between inner and outer eccentricities in a hierarchical triple star system

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According to Table 3 of Sterzik & Tokovinin (2002), the inner eccentricity of a triple star system (i.e. the eccentricity of the inner binary) is, most of the times (70%), larger than the outer star eccentricity. This seems counterintuitive, since I've always thought the outer eccentricity was usually larger than the inner one. Any reason for this? Or is this data wrong?

The statistics of the angle Φ between orbital angular momenta in hierarchical triple systems with known inner visual or astrometric orbits are studied. A correlation between apparent revolution directions proves the partial orbit alignment known from earlier works. The alignment is strong in triples with outer projected separation less than

50 au, where the average Φ is about . In contrast, outer orbits wider than 1000 au are not aligned with the inner orbits. It is established that the orbit alignment decreases with the increasing mass of the primary component. The average eccentricity of inner orbits in well-aligned triples is smaller than in randomly aligned ones. These findings highlight the role of dissipative interactions with gas in defining the orbital architecture of low-mass triple systems. On the other hand, chaotic dynamics apparently played a role in shaping more massive hierarchies. The analysis of projected configurations and triples with known inner and outer orbits indicates that the distribution of Φ is likely bimodal, where 80% of triples have and the remaining ones are randomly aligned.

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Title: Eccentricity growth and orbit flip in near-coplanar hierarchical three-body systems

The secular dynamical evolution of a hierarchical three-body system in which a distant third object orbits around a binary has been studied extensively, demonstrating that the inner orbit can undergo large eccentricity and inclination oscillations. It was shown before that starting with a circular inner orbit, large mutual inclination (40°-140°) can produce long timescale modulations that drive the eccentricity to extremely large values and can flip the orbit. Here, we demonstrate that starting with an almost coplanar configuration, for eccentric inner and outer orbits, the eccentricity of the inner orbit can still be excited to high values, and the orbit can flip by ∼180°, rolling over its major axis. The ∼180° flip criterion and the flip timescale are described by simple analytic expressions that depend on the initial orbital parameters. With tidal dissipation, this mechanism can produce counter-orbiting exoplanetary systems. In addition, we also show that this mechanism has the potential to enhance the tidal disruption or collision rates for different systems. Furthermore, we explore the entire e and i parameter space that can produce flips.

Hierarchical index¶

The following is the nomenclature used in setup_disc.f90 in order to refer to different hierarchical levels and stars in a general hierarchical system. In particular the subst parameter in the .setup file is the hierarchical index of the star that has to be substituted.

The hierarchical indexes (1, 11, 12, 112, …) are used to identify the hierarchical position of the star in the system. In the code they are used to specify the sink to be substituted with a binary when calling the set_multiple routine in order to build an hierarchical system.

A Study on the Kinematics of Hierarchical Triple Stars ☆,☆☆

Our statistical analysis shows that, because of the three-body effects, the classical double two-body model cannot be used to describe the kinematics of most hierarchical triple star systems with the precision of present day observations (1 mas). Even for the usual practical requirements, this model is not suitable for figuring those systems suffering significant three-body effects. Because it is not practical to use numerical ephemerides with a large volume of data to describe the kinematics of component stars in a star catalogue, a kinematical model as simple and practical as possible for the hierarchical triple star systems is needed. Based on the existing observations and their associated fitting results, we obtain consistent mass parameters and initial conditions of six hierarchical triple star systems. Then the description of the kinematics of these systems on the basis of practical requirements is discussed.

The dependence of the stability of hierarchical triple systems on the orbital inclination

In this paper we study numerically the effect of the initial mutual orbital inclination on the stability of hierarchical triple systems with initially circular orbits. Our aim is to investigate the possibility that the stability boundary may be independent of the orbital inclination for certain mass ratios. We integrate numerically the equations of motion of hierarchical triple systems with initially circular orbits and different orbital configurations. The mass ratios cover the range from 10 - 6 to 10 6 and the initial mutual inclination angle varies from 0° to 180°. The results from the numerical simulations show that for hierarchical triple systems with initially circular orbits and for the mass ratios we used, the initial mutual inclination angle does affect the stability boundary.


► We study the effect of the mutual inclination on the stability of triple systems. ► The orbits are initially circular. ► We investigate whether the stability boundary depends on the orbital inclination. ► We conclude that the mutual inclination angle does affect the stability boundary.

Scientists Test Einstein’s Theory of Gravity on Unique Triple-Star System

In a new test of Einstein’s theory of gravity, a group of astronomers from the Netherlands, the United States, Australia and Canada has demonstrated that the theory holds up, even for a stellar triple system. Their work is published in the journal Nature.

An artist’s impression of the triple star system PSR J0337+1715, which is located about 4,200 light-years from Earth. Image credit: NRAO / AUI / NSF / S. Dagnello.

Einstein’s theory of gravity states that all objects fall the same way despite their mass or composition, like a cannonball and apple falling off the Leaning Tower of Pisa and hitting the ground at the same time.

But alternatives to his theory predict that compact objects with extremely strong gravity, like neutron stars, fall a little differently than objects of lesser mass. That difference would be due to a compact object’s so-called gravitational binding energy — the gravitational energy that holds it together.

In 2011, astronomers discovered a natural laboratory to test Einstein’s theory in extreme conditions — PSR J0337+1715, a hierarchical system of three stars in which a binary consisting of a millisecond radio pulsar and a white dwarf in a 1.6-day orbit is itself in a 327-day orbit with another white dwarf.

“PSR J0337+1715 is a unique star system. We don’t know of any others quite like it. That makes it a one-of-a-kind laboratory for putting Einstein’s theories to the test,” said team member Dr. Ryan Lynch, an astronomer at the Green Bank Observatory in West Virginia.

White dwarfs are very dense stars while their size is comparable to the Earth, their mass is similar to that of our Sun.

Neutron stars are even smaller and denser than white dwarfs. They are made from collapsed cores of stars that have undergone supernova explosions and are the densest stars in the Universe.

Many spinning neutron stars are pulsars, sending regular lighthouse-like electromagnetic signals out through space that can be captured by radio telescopes here on Earth.

“We can account for every single pulse of the neutron star in PSR J0337+1715 since we began our observations,” said team leader Dr. Anne Archibald, from the University of Amsterdam and the Netherlands Institute for Radio Astronomy.

“We can tell its location to within a few hundred meters. That is a really precise track of where the neutron star has been and where it is going.”

If alternatives to Einstein’s picture of gravity were correct, then the neutron star and the inner white dwarf in PSR J0337+1715 would each fall differently toward the outer white dwarf.

“The inner white dwarf is not as massive or compact as the neutron star, and thus has less gravitational binding energy,” said team member Dr. Scott Ransom, an astronomer with the National Radio Astronomy Observatory.

Through careful observations and calculations, the researchers were able to test the system’s gravity using the pulses of the neutron star alone.

They found that any acceleration difference between the neutron star and inner white dwarf is too small to detect.

“If there is a difference, it is no more than three parts in a million,” said team member Dr. Nina Gusinskaia, from the University of Amsterdam.

“Now, anyone with an alternative theory of gravity has an even narrower range of possibilities that their theory has to fit into, in order to match what we have seen.”

“Every single time we’ve tested Einstein’s theory of relativity so far, the results have been consistent,” said team member Professor Ingrid Stairs, from the University of British Columbia.

“But we keep looking for departures from relativity because that might help us understand how to describe gravity and quantum mechanics with the same mathematical language.”

The team’s result is 10 times more precise that the previous best test of gravity, making the evidence for Einstein’s Strong Equivalence Principle that much stronger.

“We’re always looking for better measurements in new places, so our quest to learn about new frontiers in our Universe is going to continue,” Dr. Ransom said.

Anne M. Archibald et al. 2018. Universality of free fall from the orbital motion of a pulsar in a stellar triple system. Nature 559: 73-76 doi: 10.1038/s41586-018-0265-1

Relation between inner and outer eccentricities in a hierarchical triple star system - Astronomy

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Binary systems (systems of 2 stars) orbit a common centre of mass, known as the barycentre. The stars, as well as any planets in the system (only in the case of close binary systems), will orbit this barycentre.

Given the average separation of the stars (a, in AU) and their masses (MA and MB), we can calculate the average distance between the primary star and the barycentre, using the following formula ΐ] Α] :

Therefore, the average distance between the secondary star and the barycentre is:

The orbits of the stars are not usually circular, but are instead elliptical. Eccentricity is a measure of just how "un-circular" an orbit is, where 0 is a perfect circle and 1 is a parabola. Both stars orbiting a single barycentre will have equal orbital eccentricity, which should ideally be between 0.4 and 0.7 for a close orbiting pair.

Given the eccentricity (e) of the orbit and the average distance between a star and the barycentre, the maximum and minimum distances between the star and the barycentre can be calculated as:

Using the above, we can calculate the maximum and minimum separation between the stars themselves as:

When determining the habitable zone, planetary limits and frost line of a close-orbiting pair (see Planetary Systems), all one needs to do is to add the masses or luminosities of the stars together. Far-orbiting systems follow the usual rules, as each star in the system may have its own planetary system.

In addition to the above, there are also limits on planetary formation due to the gravitational effects of the stars. No planets can form within these "forbidden zones", whose limits are given as:

Habitability [ edit | edit source ]

A binary star system, as seen from a planet's surface

For close binary systems, we would like the outer planetary boundary to be less than the inner planetary limit if we wish the system to be habitable, so that the maximum area for planet formation is available to us. For a planet to be habitable, it must also orbit more than 4 times the outer forbidden zone from the barycentre.

For far binary systems, we would like the inner planetary boundary to be greater than the outer planetary limit of both stars, in order to have the maximum area for planet formation available to us.

The placement of planets in a binary system is identical to placing them in a single-star system, except that the above restrictions must be observed.

Recent Main Research Projects

A figure from Stephan et al. 2020 showcasing the parameter space for which mass loss enhances WD pollution and short-period planet formation.

Forming Short-Period Planets around White Dwarfs

Two recently discovered short-period planets around white dwarfs, WD J0914 b and WD 1856 b, have reinforced the notion that planets exist around stars of all stellar evolutionary stages. However, the planets were most likely not formed at their current orbits, but had to migrate there through dynamical interactions. In this recent project, we show that the interplay of stellar evolution and the EKL mechanism can naturally explain the formation of such systems, similar to work we had done in 2017. Crucially, we show that the EKL oscillations can lead to tidal stripping or migration of a giant planet within the first few million years after white dwarf formation, important for explaining WD J0914 b. The paper has been submitted for peer review and is available for pre-publication view on the arXiv as A.P. Stephan, S. Naoz, B.S. Gaudi, 2020, Giant Planets, Tiny Stars: Producing Short-Period Planets around White Dwarfs with the Eccentric Kozai-Lidov Mechanism, arXiv:2010.10534.

Effects of Planet Consumption on Evolving Stars

A common outcome of the dynamical evolution of planets in binary star systems is the collision of planets with their host stars due to extreme eccentricity excitations or the engulfment of planets as their host stars evolve into red giants and inflate in radius. These collisions and engulfments can cause a variety of effects on the stars themselves, including stripping of the stellar envelope, changes in the stars' angular momenta and spin rates, luminosity and X-ray radiation spikes, or chemical composition anomalies. As such, if stars that have engulfed their planets can be reliably identified, it would be possible to gain a more complete understanding of the formation and

Parameter space showing when surface grazing planets can cause surface eruptions from their host stars, from Stephan et al. 2020.

evolution histories of planetary systems. In my recent paper A.P. Stephan, S. Naoz, B.S. Gaudi, J.M. Salas, 2020, Eating Planets for Lunch and Dinner: Signatures of Planet Consumption by Evolving Stars, ApJ, Volume 889, Issue 1, article id.45, 11 pp., doi: 10.3847/2041-8213/ab21d3 we investigated several of these effects and defined a number of observable features that can help future surveys to identify stars that have recently engulfed their planets. This work was also featured for an article in the New Scientist magazine.

A flowchart from Stephan et al. 2019 outlining the outcomes of binary stellar evolution in the galactic center.

Binary Stellar Evolution in the Galactic Center

My previous work on galactic center binaries from 2016 was mostly focused on the dynamical evolution of such objects up until they either merged or separated into single stars. The exact nature of the resulting "merger products", however, was not well defined in that work. I therefore revisited the topic in 2019 by investigating the binary stellar evolution of merger candidates with a binary stellar evolution code. In this project we showed that binary stars in galactic centers can lead to the formation of a large variety of interesting objects, such as X-ray binaries, type Ia supernovae, cataclysmic variables, as well as black hole and neutron star binaries, and much more. The work shows that galactic centers can be an extremely important location for black hole-neutron star mergers, potentially explaining recent observations by LIGO of such objects. The work was published in A.P. Stephan, S. Naoz, A.M. Ghez, M.R. Morris, A. Ciurlo, T. Do, K. Breivik, S. Coughlin, C.L. Rodriguez, 2019, The Fate of Binaries in the Galactic Center: The Mundane and the Exotic, ApJ, Vol. 878, Issue 1, article id. 58, 7 pp., doi: 10.3847/1538-4357/ab1e4d and has been featured by AAS Nova.

Temporary Hot Jupiters

Hierarchical three-body dynamics has been used as a possible explanation for so-called "Hot Jupiters", giant gas planets that orbit their host stars on short-period orbits (P
A plot from Stephan et al. 2018 showing the outcomes of gas giant planet dynamical evolution calculation for different stellar lifetimes, and comparing these calculations to observed planets orbiting evolved A-type stars (marked as black stars).

much faster than sunlike stars. They also nearly always possess stellar companions, leading to EKL effects for any planets orbiting either planet. We found that the enhanced tidal effects during the Red Giant phase of the stars can lead to the formation of "Temporary Hot Jupiters", Hot Jupiters that only exist for a few hundred thousand years before they become engulfed by their evolving host star. Overall we found that A-type stars would destroy about 70 % of all gas giant planets orbiting them, by turning them into Hot Jupiters or Temporary Hot Jupiters and subsequently engulfing them. Our work was published in A.P. Stephan, S. Naoz, B.S. Gaudi, 2018, A-type Stars, the Destroyers of Worlds: The Lives and Deaths of Jupiters in Evolving Stellar Binaries, AJ, Vol. 156, Issue 3, article id. 128, 12 pp., doi: 10.3847/1538-3881/aad6e5 and has been featured as a Nature Research Highlight.

Dynamical evolution of a Neptune-like planet (a) and a Kuiper belt analog object (b) in wide binary star systems. Large eccentricity excitations during the white dwarf phase lead to pollution. [Stephan et al. 2017]

White Dwarf Pollution in Wide Binaries

Some of my previous work has been concerned with the pollution of white dwarfs in evolving wide binary star systems. Over recent decades many studies have revealed that about a quarter to a half of all white dwarf stars, the last life-stage of most stars, contain heavy elements in their atmosphere. However, these elements are expected to sink to the core of the star over short time-scales, indicating that they have been recently polluted by planetary or asteroidal material. In our work, published in A.P. Stephan, S. Naoz, B. Zuckerman, 2017, Throwing Icebergs at White Dwarfs, ApJL, Vol. 844, L16, doi: 10.3847/2041-8213/aa7cf3, we show that some of these white dwarfs can be polluted by planets that orbited far from the star during the main sequence, if they have a stellar companion on a very distant orbit. This is

due to the mass loss that a star undergoes during its evolution, which changes the orbital configuration of the systems and allows the EKL mechanism to become stronger. Thus, objects that did not get destroyed during the main sequence might instead end up coming close to the white dwarf, potentially polluting it. This work perfectly explains recent observations by Xu et al. 2017 that have for the first time found nitrogen pollution in a white dwarf. In our own solar system nitrogen is only abundant in the Kuiper belt, as it is a volatile element. In fact, Earth's atmospheric nitrogen probably came from a Kuiper belt asteroid. The white dwarf in Xu et al. is also part of an extremely wide binary star system, and was therefore most likely polluted by such a Kuiper belt analog object through the EKL mechanism. Our work has also been featured by AAS Nova and Physics Today.

Galactic Center Stellar Binary Dynamics

My Master Thesis project, in collaboration with the UCLA Galactic Center Group, studied the EKL mechanism in the context of binary stars that are orbiting the massive black hole at the center of the Milky Way. For these systems we included the effects of general relativity, tides, and stellar evolution into our calculations, in order to accurately model the possibility of binary mergers in the galactic center. We see a binary merger as a good candidate to explain the G2 object that has been discovered by Gillessen et al. (2012). Our findings were published in:
A.P. Stephan, S. Naoz, A.M. Ghez et al., 2016, Merging Binaries in the Galactic Center: The eccentric Kozai-Lidov mechanism with stellar evolution, MNRAS, Vol. 460, 3494-3504, doi: 10.1093/mnras/stw1220

Binary merger scenarios from Stephan et al. 2016

Other Significant Contributions

The EKL mechanism, including tides and stellar evolution, can be used to study a large variety of different astronomical systems. A project I have been part of was a study by my adviser, Prof. Smadar Naoz, investigating the formation of low-mass X-ray (LMXR) binaries through the EKL perturbations by a third companion. For this study the inclusion of stellar evolution effects, which I assisted with, was of crucial importance. The study was published by ApJ Letters:
S. Naoz, T. Fragos, A. Geller, A.P. Stephan, and F.A. Rasio, 2016, Formation of Black Hole Low-Mass X-Ray Binaries in Hierarchical Triple Systems, ApJ, 822, L24, doi: 10.3847/2041-8205/822/2/L24

Another study I have been involved with was concerned with the stability of multi-planet systems in the presence of an outer companion. In particular, the project considered two planets orbiting a star in a co-planar configuration that are being perturbed by an outer, inclined fourth body, and showed for what configurations such an ensemble would be stable or unstable. The project was lead by a former undergraduate mentee of my adviser and myself, Paul Denham, and was published by MNRAS: P. Denham, S. Naoz, B.-M. Hoang, A.P. Stephan, W.M. Farr, 2018, Hidden Planetary Friends: On the Stability Of 2-Planet Systems in the Presence of a Distant, Inclined Companion, MNRAS, 482, 3, 4146-4154, doi: 10.1093/mnras/sty2830

Sometimes the EKL mechanism can also be used to explain observed exotic stellar phenomena. V Hydrae is a carbon star that shows periodic ejections of gas as highly collimated "bullets" every 8.5 years. It has been suggested that these ejections are caused by the close passage of an eccentric companion during its periastron. However, for such a companion tidal effects are expected to quickly shrink and circularize the orbit, long before the star reaches its current evolutionary stage. In a project by my UCLA graduate student colleague Jesus Salas, on which we collaborated, we show that the continued high eccentricity of the companion can be explained by the existence of a third companion forming a hierarchical triple. We furthermore can set constrains on the mass of the bullet-causing companion (probably a brown dwarf). The project was published by MNRAS: J.M. Salas, S. Naoz, M.R. Morris, A.P. Stephan, 2019, Unseen companions of V Hya inferred from periodic ejections, MNRAS, 487, 3, 3029-3036, doi: 10.1093/mnras/stz1515

Several of my research projects have investigated the evolution of binary stars in the galactic center, to a large part motivated by the observations of dusty objects orbiting Sagittarius A*, such as the so-called G2 object. Past work by the UCLA Galactic Center Group have strongly suggested that such objects are held together by the gravity of a central star, potentially indicating that they are the result of stellar mergers. My dynamical studies have supported that hypothesis, and made predictions that a whole population of such objects should exist in the galactic center, with a large variety of orbital parameters. In a large study led by Dr. Anna Ciurlo, this prediction has been confirmed as a population of six G2-like sources on very different orbits has been identified. The results have been published as a Nature article: A. Ciurlo, R.D. Campbell, M.R. Morris, T. Do, A.M. Ghez, A. Hees, B.N. Sitarski, K.K. O’Neil, D.S. Chu, G.D. Martinez, S. Naoz, A.P. Stephan, 2020, A population of dust-enshrouded objects orbiting the Galactic black hole, Nature, 577, 337–340, doi: 10.1038/s41586-019-1883-y

Current and Future Projects

In general I continue to be interested of the various applications of the EKL mechanism to various astrophysical settings, including exoplanets, stellar remnants, and the galactic center. Future projects will also focus more on the effects of supernova kicks, which introduce short-term changes to a system's orbital configuration that can radically change the dynamical behavior of such systems. These effects are especially relevant for the study of black hole and neutron star binary progenitor systems, which can produce gravitational wave sources relevant for LIGO and LISA. I plan to study the effects on such phenomena in greater detail in the future.

Past Work and Other Research Interests

During my time at the University of Chicago I had the great opportunity to work with Prof. James W. Truran and Michael Florian on problems concerning the chemical evolution of galaxies and dwarf galaxies. While I am not currently working on chemical evolution problems, I continue to be interested in these systems. In particular I wish to further explore the role of galaxy dynamics in these circumstances, at some point.

I also had the great fortune to have had Prof. Michael D. Gladders as my bachelor thesis adviser. The project for my bachelor thesis was concerned with strong gravitationally lensing galaxy clusters and the relation between the brightest cluster galaxy (BCG) mass and the strength of the lens. This was an extraordinarily fascinating project, exploring the structure of galaxy clusters with general relativity.