The Journey of Hypervelocity Stars

Transcription

1 Università degli Studi di Torino Dipartimento di Fisica Corso di Laurea Magistrale in Fisica Tesi di Laurea Magistrale The Journey of Hypervelocity Stars Candidato: Arianna Gallo Relatore: Prof.ssa Luisa Ostorero Controrelatore: Prof. Nicolao Fornengo Anno Accademico

2 Contents 1 Introduction to the Physics of Hypervelocity Stars Hypervelocity Stars The Hills Mechanism The fate of former companions to HVSs Variations on the Hills Mechanism Three-body and four-body interactions with a binary SMBH IMBH-SMBH in-spiral events Three-body interactions with stellar mass black holes Dwarf galaxy AGN Runaway Stars Runaway and hyper-runaway ejection rates Physical Properties of Stars HVS Ejection Rates Galactic Center SMBH loss cone Milky Way HVS ejection rate Galactic Potential Constraints on the Origin of HVSs Velocity distribution of HVSs Metallicity Stellar rotation ii

3 1.8.4 Flight time Arrival time Number and rate of unbound HVSs Spatial distribution Proper motion Summary of Observations Objectives: Hypervelocity Stars as a Probe of the Milky Way Mass Distribution 37 3 Numerical Simulations of the Journey of Hypervelocity Stars The Integration Method Derivation of the Verlet algorithm using Taylor s expansions Derivation of the Velocity Verlet algorithm Gravitational Potential of Point Sources Code Reliability Gravitational Potential of Extended, Spherical Sources Initial distribution of velocities of ejected stars Initial positions of ejected stars The survey volume Gravitational Potential of Extended, Axisymmetric Sources 70 4 Results Spherically symmetric potential: results Axisymmetric potential: results Conclusions and Future Prospects 99 Bibliography 103 Appendix 105 A B iii

4 Abstract Hypervelocity stars (HVSs) are stars that, according to Hills (1988), are ejected by the central super-massive black hole (SMBH) of the Milky Way (MW) at speeds that exceed the MW escape velocity. HVSs are now observed in the halo of the Milky Way and thus they link the outer parts of the MW to its very center. Their unbound trajectories are determined by the gravitational potential of the MW; thus, HVSs represent a unique probe of the MW mass distribution, from its very center out to the outer halo. In this Thesis, I study the time evolution of the positions and velocities for a sample of HVSs ejected by the Galactic SMBH, according to the Hills ejection mechanism. To this aim, I implement a numerical code to perform numerical simulations of the HVS trajectories for two different models of the MW potential. The aim is to investigate how the trajectories of HVSs and their velocities are affected by the MW mass distribution. After considering a simple, spherically symmetric potential for comparison with the literature, I move to a more complex and realistic axisymmetric potential. The code successfully computes the HVS trajectories and velocity distributions in all my case studies, yielding results broadly consistent with the literature, where the comparison can be performed. Minor inconsistencies are still under investigation. The code implemented in this Thesis, represents a useful tool for future work in this field. It is the starting point for the creation of mock catalogs of HVSs that include stars positions and velocities in different models of the MW potential. These mock catalogs will 1

5 be used both for comparing the simulated data with the available HVS data and for forecasts for future astrometric missions (as, e.g. Theia), with the aim to constrain the MW mass distribution for any given HVS generation mechanism. 2

6 Chapter 1 Introduction to the Physics of Hypervelocity Stars 1.1 Hypervelocity Stars Hypervelocity stars (HVSs) are stars ejected by the Galaxy s central super-massive black hole (SMBH) at speeds that exceed Galactic escape velocity. HVSs are now observed in the halo and thus they link the outer parts of the Milky Way (MW) to its very center and provide important tools for exploring a wide range of phenomena. In 1988 Hills [1] predicted the existence of such unbound stars and called them hypervelocity stars. In 2003 Yu & Tremaine [2] proposed a binary SMBH ejection scenario for HVSs and calculated rates for the single and the binary SMBH ejection scenarios. These papers represent the entire interest in HVSs for 17 years. Then, in 2005, the first HVS was discovered: a B-type star traveling out of the Milky Way with a velocity at least twice the Galactic escape velocity at its 50 kpc distance (Brown et al. 2005) [3]. The star s present motion exceeds the escape velocity from the surface of the star. To explain a motion in excess of the escape velocity from the surface of a star requires dynamical interaction with a massive compact object, as predicted by Hills. 3

7 1 Introduction to the Physics of Hypervelocity Stars Since 2005 there have been many more HVS discoveries, both targeted and serendipitous. In the 2014 HVS Survey, 21 unbound late B-type stars have been observed at kpc distance by the MMT Observatory (Brown et al. 2014) [4]. Assuming an isotropic and continuous ejection, around 100 HVSs are expected to be observed over the entire sky within 100 kpc. HVSs can be ejected from any galaxy with a SMBH; however, the intrinsic luminosity of stars limits the observers views to stars near the Milky Way. Identifying HVSs remains observationally difficult. To distinguish them from runaway stars (section 1.4) and other phenomena, Hills coined the term hypervelocity star to describe a star with: 1. SMBH ejection origin, 2. unbound velocity. With the growth of interest in HVSs, other scenarios able to produce HVSs were widely explored (section 1.3) and nowadays the word hypervelocity star defines a star ejected at the Galactic Center (not necessarily in a three-body encounter with a SMBH) with unbound velocity. On the other hand, the term bound HVS is used for stars ejected from the Galactic Center, but with a velocity smaller than the escape velocity from the Galaxy. 1.2 The Hills Mechanism SMBHs, i.e. black holes with masses > 10 5 M, are believed to be ubiquitous in the centers of galaxies. The orbital motions of stars in the center of the Milky Way provide compelling evidence for the presence of a M SMBH, referred to as Sagittarius A (Sgr A ). 4

8 1.2 The Hills Mechanism The environments of these SMBHs are characterized by extreme stellar densities, extreme velocities, and short dynamical timescales. Unusual dynamical phenomena are therefore frequent near a SMBH. A HVS is produced, according to Hills, by a 3-body exchange between a stellar binary and a SMBH. A SMBH tidally disrupts a stellar binary at the distance r bt in which the SMBH s gravitational tidal force exceeds the force that binds the binary, r bt = a bin ( 3M m b ) AU ( abin 0.1 AU ) ( M m b ) 1 3 ( M 10 6 M ) 1 3, (1.1) where a bin is the semi-major axis of the binary, m b is the total mass of the binary, and M is the SMBH mass. HVSs ejected by a 3-body exchange with a SMBH have a final velocity at infinite distance from the SMBH, ( ) ( ) abin 1/2 1/3 ( ) 1/6 mb M v ej = 1,370 km/s f 0.1 AU M R, M (1.2) where f R is a factor of order unity that depends on R min, the periapse distance to the SMBH. This is the equation in the case of equal mass binary. For an unequalmass binary, the ejection speeds of the primary and secondary stars are respectively [5]: v 1 = v ej ( 2 m2 m 1 + m 2 ) 1/2, v 2 = v ej ( 2 m1 m 1 + m 2 ) 1/2. (1.3) The probability P ej of an ejection depends on the periapse distance to the SMBH too: P ej = 1 D/175, (1.4) where D = ( Rmin a bin ) ( 10 6 ) 1/3 m b (1.5) 2M 5

9 1 Introduction to the Physics of Hypervelocity Stars and 0 D 175. For D 175, the binary does not approach close enough to the SMBH to produce an ejection. The Hills mechanism is illustrated in Figure 1.1: the super-massive black hole replaces one component of the stellar binary and ejects a HVS. The binary star with orbital velocity v b drops toward the SMBH. The velocity v at pericenter is orders of magnitude larger than v b. If the pericenter distance is less than the binary tidal disruption distance r bt, the binary is disrupted; one star becomes gravitationally bound to the SMBH and, by conservation of energy, the other star is ejected as a hypervelocity star. Figure 1.1: Schematic illustration of the Hills mechanism (Brown, W. R., 2015) [6] The fate of former companions to HVSs S-stars are stars orbiting in highly eccentric orbits around Sgr A. An intriguing open question is how did some of these stars obtain their highly eccentric orbits near the central SMBH. 6

10 1.2 The Hills Mechanism Adaptive optics assisted spectroscopy of the stars in the central arcsec (i.e pc) of the Galaxy established that the S-stars are normal main-sequence B-stars; yet the tidal forces of the SMBH are too strong to permit star formation in the central arcsec. A viable expla- Figure 1.2: S-stars S0-2 and S0-102 near Sgr A*. The image was taken at a wavelength of 2.12 µm (Meyer et al. 2012) [7]. nation for this paradox of youth is that the S-stars are captured by the SMBH in a 3-body exchange scenario (Zhang et al. 2013) [8]. So it is natural to ask if it is possible that some of these stars are former companions of HVSs. In their studies, Ginsburg & Loeb (2006) [9] simulated the orbits of the companion stars of HVSs. They found out that all of the orbits 7

11 1 Introduction to the Physics of Hypervelocity Stars have very high eccentricities, ranging from e = to e = This resulting eccentricity is similar to that observed for a number of S-stars, suggesting a possible binary origin for these stars. Both the number and the orbital distribution of S-stars are consistent with the Hills mechanism. 1.3 Variations on the Hills Mechanism Three-body and four-body interactions with a binary SMBH Binary SMBHs are an expected consequence of galaxy mergers. Threebody interactions with a binary SMBH can eject single stars as HVSs (Yu & Tremaine 2003) [2]. In this scenario, the most energetic ejections occur in the direction of SMBH orbital motion. Ejections become more energetic and isotropic as the binary SMBH hardens and then merges on 1-10 Myr timescales. An observational signature of a binary SMBH in-spiral event is thus a distinctive burst of HVSs in a ring or shell. The binary SMBH scenario differs from the Hills mechanism in that the ejection velocity of HVSs does not depend on the mass of the stars. In fact, a binary SMBH can eject also binary stars in a fourbody interaction with the binary SMBH (Wang et. al 2018) [16]. Binary HVSs may not be unique to binary SMBHs: encounters between hierarchical triples and a single SMBH can disrupt the outer star and eject the inner binary as a binary HVS. Simulations suggest that, in both scenarios, HVS binaries constitute a few percent of all HVS ejections IMBH-SMBH in-spiral events A variant of the binary SMBH is a binary system composed of an intermediate mass black hole (IMBH) with a mass of M and a SMBH, i.e. an IMBH-SMBH binary. Three-body interactions 8

12 1.3 Variations on the Hills Mechanism with an IMBH-SMBH binary can naturally eject HVSs. IMBHs are formed by collisional runaway of stars in massive young star clusters near the central SMBH. IMBH-SMBH in-spiral events may generate periodic bursts of HVSs which last a few Myr, until the IMBH is swallowed by the SMBH (Baumgardt et al. 2006) [11] Three-body interactions with stellar mass black holes Another mechanism for ejecting HVSs is the three-body interaction between a single star and a stellar mass black hole (SBH) orbiting a SMBH (O Leary & Loeb 2008) [12]. SBHs are the remnants of massive stars that should migrate, by dynamical friction, into the region surrounding the central SMBH. HVS ejections occur if the SBHs form a cusp and are fed by a relaxed distribution of stars. The SBH scenario differs from the Hills mechanism in two notable respects. First, SBH encounters are unlikely to generate ejection velocities exceeding 1000 km/s. Second, SBH encounters eject the lowest-mass stars with the highest ejection velocities Dwarf galaxy A tidally disrupting dwarf galaxy on a close pericenter passage can also eject stars (Abadi et al [13]), though a massive > M dwarf galaxy is necessary to eject unbound stars. The signature of the tidal debris scenario is a clump of unbound stars of all stellar types connected to a stream of bound stars AGN As an alternative to these stellar dynamical mechanisms, theorists propose that HVSs may be explained by stars formed inside gas clouds launched from an AGN jet (Silk et al [14]) or AGN spherical outflow (Zubovas et al [15] and Wang & Loeb

13 1 Introduction to the Physics of Hypervelocity Stars [16]). The challenge in this picture is whether a supersonic gas cloud collapses to form stars. Observed star formation occurs in clumps and cores inside subsonic giant molecular clouds. The uncertain physics of the AGN scenario has precluded any predictions of HVS velocities or ejection rates. If AGN-ejected gas clouds collapse to form stars, this scenario implies that HVSs should be found in discrete clumps with common velocities. 1.4 Runaway Stars Runaway stars are the most likely contaminant in HVS surveys. The Sun and all of the Milky Way s stars are moving in orbit around the center of our galaxy. It is more or less orderly, but there are local movements within this general stream of stars, too. In recent decades, astronomers have identified some Milky Way stars that are moving faster than expected, or in a direction that seems unusual. They use the term runaway star to describe these renegades. Thus, runaway stars are Galactic disk stars with very high peculiar motions (typically faster than 40 km/s). Runaways are a mix of objects: some are evolved stars that belong in the halo, but many are main-sequence B stars, short-lived stars that began their lives in the disk. In the Solar Neighborhood, approximately 30% of O-type stars and 10% of B-type stars are runaways based on their peculiar velocities. All runaway B stars in the Solar Neighborhood have orbits consistent with a disk origin. Because most O and B stars are born as binaries, runaways are naturally explained by binary disruption mechanisms. There are two proposed ejection mechanisms: 1. supernova ejections; 2. dynamical ejections. In the supernova ejection mechanism, a runaway is released from a binary when its former companion explodes as a supernova. The 10

14 1.4 Runaway Stars maximum possible ejection velocity in the supernova mechanism is the sum of the supernova kick velocity and the orbital velocity of the progenitor binary. Velocities up to 400 km/s may be possible. Most ejections occur with much lower velocity and in all cases are below the Galactic escape velocity. In the dynamical ejection mechanism, a runaway is ejected by dynamical 3-body or 4-body interactions in a young star cluster. The maximum possible ejection velocity in the dynamical ejection mechanism is set by the escape velocity of the most massive star. Ejection velocities can reach km/s for 4 M stars interacting with contact binaries 1 containing 100 M stars. However, high-velocity encounters frequently yield merger events instead of ejections owing to tidal dissipation effects. Recent simulations predict that the velocities from dynamical ejection form a power-law distribution; 99% of dynamical ejections have < 200 km/s velocities. Observations demonstrate that both the supernova and dynamical ejection mechanisms occur in nature. A runaway can possibly experience both a dynamical and a supernova ejection in a two-step process, but this phenomenon should apply to only 1% of O-type runaways. The most important observation in the context of HVSs is the discovery by Heber et al. (2008a) [17] of the first hyper-runaway B star: an unbound runaway star that was ejected from the outer disk. Simulations predict that the overall distribution of runaways in the Galaxy reflects their origin from the rotating disk. Runaways ejected from the Galactic disk naturally have a flattened spatial distribution with a scale height comparable to the thick disk. The kinematics of runaways also reflect their disk origin: the fastest runaways are those ejected at the lowest Galactic latitudes in the direction of rotation. 1 A contact binary is a binary star system whose component stars are so close that they touch each other or have merged to share their gaseous envelopes. 11

15 1 Introduction to the Physics of Hypervelocity Stars Runaway and hyper-runaway ejection rates If we assume that the Milky Way forms stars at 1M year 1, that the stars have a Kroupa mass function 2 ranging from 0.08 to 100 M, and that the 0.1% of stars become runaways, then runaways are launched from the disk at a rate of year 1. In the halo (see Section 1.7), runaways that move at > 400 km/s at R = 50 kpc are the only ones that can be confused with HVSs. Such hyper-runaways are rare. To have a 400 km/s velocity at R = 50 kpc requires a 600 km/s ejection from the Solar circle (8 kpc). A 600 km/s runaway can be explained only by the dynamical ejection mechanism. The rate of 600 km/s ejections is related to the joint probability of interacting contact binaries containing very massive stars within the lifetimes of the very massive stars. Quantitatively, the probability that runaways are ejected at 600 km/s is predicted to be The ejection rate of hyper-runaways with speeds comparable with HVSs is thus year 1. Galactic rotation and outer disk launch locations can reduce the velocity required to create a hyper-runaway; however, neither effect increases the overall ejection rate. 1.5 Physical Properties of Stars Stars have three fundamental properties that influence the detection of HVSs: their physical size, intrinsic luminosity and finite lifetime; all these properties are related to the stellar mass. 2 The Kroupa mass function is one possible functional form for the initial mass function and gives the number of stars with mass m per pc 3. It was proposed by Kroupa in 2001 and is proportional to m α, where α is a dimensionless exponent. Kroupa kept α = 2.3 above 0.5M, but introduced α = 1.3 between M and α = 0.3 below 0.08 M. 12

16 1.5 Physical Properties of Stars Physical size The physical size of stars imposes a speed limit on ejection velocity. For stellar encounters with a SMBH, the speed limit is set by the tidal disruption distance, r t, of a star by the SMBH: ( ) M 1 3 r t = r 0.5 AU m ( M 10 6 m ) 1 3 r R, (1.6) where M is the mass of the SMBH, m is the mass of the star, and r is the radius of the star. The fastest HVSs come from the closest encounters with a SMBH, but stars passing closer than r t produce tidal disruption events, not HVSs. For stellar encounters between individual stars, the speed limit of an ejection is set by the escape velocity from the surface of a star. To achieve higher speeds, stars would have to orbit inside each other, which is physically impossible. The escape velocity from the surface of a star is ( ) 1 ( ) 1 2Gm 2 m 2 R v esc = = 618 km s 1. (1.7) r M r Because of the quasi-linear relationship between the mass and radius of main-sequence stars, v esc 600 km/s to within 10% for M main-sequence stars. Stellar encounters between individual stars cannot produce speeds in excess of this velocity. HVSs are a natural explanation for main-sequence stars ejected at speeds greater than v esc. The typical v ej 1370 km/s ejection velocity of the Hills mechanism significantly exceeds the v esc of a typical main-sequence star. Intrinsic luminosity The intrinsic luminosity of stars determines the survey volume accessible for a given stellar type of HVS. Magnitude-limited surveys that reach to 20 th magnitude, such 13

17 1 Introduction to the Physics of Hypervelocity Stars as the Sloan Digital Sky Survey (SDSS 3 ) and Gaia 4, can sample solar metallicity 0.5M stars to heliocentric distance d = 1 kpc, 1 M stars to d = 10 kpc, and 3 M stars to d = 100 kpc. Metal-poor stars are systematically less luminous for the same effective temperature and thus probe smaller volumes. Finite lifetimes The finite lifetimes of stars dictate what types of HVSs are observed and where. In 1 Myr, a star moving at 1000 km/s travels 1 kpc. Thus short-lived O-type HVSs (that spend less than a million years on the main sequence) do not live long enough to travel out of the Galactic bulge (that extends out to 3 kpc). Long-lived HVSs can cross mega-parsec distances. Long-lived stars exist in every part of the Milky Way, which makes distinguishing longlived HVSs from normal Milky Way stars difficult. However, HVSs with short lifetimes can only originate from regions of ongoing star formation, such as the Galactic disk and Galactic Center. Stellar lifetime also affects the observed velocity distribution of HVSs. Stars with < 1 Gyr lifetimes ejected from the Galaxy on marginally bound orbits cannot be observed on inbound trajectories, as they do not live long enough to return. By contrast, stars 3 The Sloan Digital Sky Survey ( is a major multispectral imaging and spectroscopic redshift survey using a dedicated 2.5 m wideangle optical telescope at Apache Point Observatory in New Mexico, United States. Data collection began in 2000, and the final imaging data release covers over 35% of the sky. 4 Gaia ( is a space observatory of the European Space Agency (ESA) designed for astrometry: it measures the positions and distances of stars with unprecedented precision. The mission aims to construct the largest and most precise 3D space catalog ever made, totalling approximately 1 billion astronomical objects. 14

18 1.6 HVS Ejection Rates with > 1 Gyr lifetimes can fully populate bound orbits. Bound ejections accumulate and dominate the population of long-lived stars at all distances in the Galaxy. 1.6 HVS Ejection Rates HVSs are an inevitable consequence of SMBHs in the centers of galaxies, but the number of HVSs that can be observed depends on the ejection rate of HVSs. HVSs are most easily observed in the Milky Way. Thus, in order to know the rate of HVS production, it is important to study the environment of the SMBH in the Galactic Center and the observational constraints on binaries in this region. It is also fundamental to understand the dynamical processes that scatter these stars onto orbits that encounter the SMBH (the so-called loss cone ) Galactic Center The Galactic Center, i.e. the center of the Milky Way, presents the best-studied picture of a SMBH and its environment. The M SMBH sits at the dynamical center of the Milky Way surrounded by an immense crowd of stars. Within 1 arcsec (i.e pc) of the SMBH, several dozen main-sequence B-type stars (the so-called S stars) move on randomly oriented, eccentric orbits around the SMBH. Slightly further out, over 100 very young O-type stars exist between 1 arcsec and 12 arcsec (i.e. from 0.04 to 0.5 pc) from the SMBH. Half of these O stars orbit together in a clockwise thin disk. Spectroscopic analysis of the other stars in the central 1 pc indicates that the central region has experienced continuous but variable star formation over the past 12 Gyr. Surrounding the young stellar disk, but on a different orbital plane, there is a 10 6 M circumnuclear torus of molecular gas located at 2-8 pc from the SMBH. A nuclear star cluster, composed mostly of old stars with a half-light radius of 4.2 pc and a mass of M, 15

19 1 Introduction to the Physics of Hypervelocity Stars envelops the entire central region. Theoretically, the binary fraction of stars in the Galactic Center is expected to evolve as a result of dynamical interactions. In the environment near the SMBH, most binaries appear soft and will eventually be disrupted by encounters with other stars. The disruption timescale for an a bin = 1.0 AU binary, at R = 1 pc from the SMBH, is 10 8 years. Long-lived stars should have a very low binary fraction near the SMBH. Thus, low-mass binaries that encounter the SMBH must come from beyond R > 1 pc. Massive OB stars, in contrast, have lifetimes less than the binary disruption time; encounters between massive binaries and the SMBH are limited primarily by stellar evolution. Not all binary encounters with the SMBH necessarily make HVSs. Wide binaries are easily disrupted, but their disruption yields low ejection velocities (Equation (1.2)). Compact binaries, by contrast, have a lower probability of disruption (Equation (1.4)) but are more frequent than wide binaries. There is more energy in a compact binary. Thus the disruption of compact binaries yields the highest ejection velocities. Binaries with a bin < 1 AU are the most likely sources of unbound HVSs, and they are also the most likely to survive in the Galactic Center SMBH loss cone The rate at which stars encounter the SMBH depends on the timescale for scattering stars into the SMBH loss cone. The loss cone is the phase space of low angular momentum orbits that have close periapse distances to the SMBH. Assuming that stars on loss cone orbits are rapidly eliminated, the steady-state encounter rate of stars with the SMBH is controlled by dynamical processes that refill the empty loss cone. This process is complex, because different dynamical processes operate on different timescales and spatial scales in the Galactic Center. 16

20 1.6 HVS Ejection Rates Two-body relaxation, a gravitational encounter in which a star s orbit is altered due to the gravitational interaction with another star, is one of the dominant dynamical processes. In the spherical equilibrium assumption, there is a well-defined rate at which 2-body gravitational encounters between stars fill the loss cone. However observations show that the distribution of mass in the Galactic Center is not spherical but triaxial. Torquing of orbits by a triaxial potential causes much higher encounter rates than in a spherical potential. Thus, the assumption of a spherical, empty loss cone is very likely wrong, but its 2-body relaxation timescale remains a useful point of reference. Two-body gravitational encounters with massive objects, such as giant molecular clouds, greatly increase the rate of stellar encounters with the SMBH. Although the number density of massive perturbers is small compared with the density of stars, these objects are so much more massive than stars that they dominate the gravitational scattering inside any region that contains them. The observed distribution of massive perturbers in the Galactic Center enhances the SMBH encounter rate of stellar binaries by a factor of compared with spherical equilibrium 2-body relaxation. Binaries formed in a former gaseous disk around the SMBH may also add to the SMBH encounter rate. Binaries embedded in a massive gas disk migrate toward the SMBH as in planetary migration scenarios. The presence of a massive gas disk also acts to enlarge the SMBH loss cone. A cusp of SBHs further enhances the evolution of orbital eccentricities and thus the encounter rate of disk stars with the SMBH. Another possible process for quickly moving stars into the SMBH loss cone is the in-spiral of a massive star cluster hosting an IMBH. The in-spiral of a massive star cluster was originally proposed to explain the short-lived S-stars orbiting the SMBH, but it was later realized that any star cluster would be tidally destroyed before reaching the SMBH unless it hosts an IMBH. Models of a massive star cluster+imbh in-spiral can explain the observed distribution of S-stars, 17

21 1 Introduction to the Physics of Hypervelocity Stars but overpredict the number of short-lived stars outside the central 1-pc region. The Galactic Center clearly hosts a wealth of dynamical processes that cause binaries to encounter the SMBH. The binary encounter rate depends in part on where the binaries originate, but all the dynamical processes mentioned here act to increase the encounter rate over the spherical equilibrium, 2-body relaxation rate Milky Way HVS ejection rate Zhang et al. (2013) [8] provide the most sophisticated HVS ejection rate calculation to date. They consider models that inject binaries both from stellar disks and from infinity, which is appropriate for the massive perturber model (models that study the role of massive perturbers (MPs) in deflecting stars and binaries to almost radial ( loss cone ) orbits, where they pass near the central massive black hole and interact with it at periapse. They also consider different stellar mass functions to compare the models with the observed S-stars in the Galactic Center and the B- type HVSs in the Galactic halo. The models that best match the set of observations have HVS ejection rates of year 1. Theoretical HVS ejection rate estimates are also available for binary SMBH scenarios. In a spherical galaxy at equilibrium, a pair of SMBHs stalls at 1-pc separation and ejects HVSs at a rate of 10 4 year 1. For the case of an in-spiraling IMBH-SMBH binary, models predict comparable 10 4 year 1 HVS ejection rates. In my work, I choose 10 4 year 1 as the most reliable Milky Way HVS ejection rate (see Section 3.4.2). Tidal disruption events provide a consistency check. Tidal disruption events require closer encounters with a SMBH than the HVS scenario and thus should have comparable or lower rates. Indeed, the rate of observed tidal disruption events in nearby galaxies is 10 18

22 1.7 Galactic Potential times lower than the adopted Milky Way HVS ejection rate. Notably, the 10 4 year 1 Milky Way HVS ejection rate is 100 times larger than the ejection rate of disk runaways with speeds comparable with HVSs. Thus, contamination of HVS surveys with runaways should be negligible. 1.7 Galactic Potential HVSs are decelerated by the gravitational pull of the Galaxy s extended mass distribution as they escape the Milky Way. The Galaxy consists of several components: bulge; disk; stellar halo; dark matter halo. The bulge is a spheroidal stellar system with radius of 1 kpc, located at the center of the MW. This round structure is made primarily of old stars, gas, and dust and its properties are quite similar to that of an elliptical galaxy. Most of the MW s stars are in a thin disk, which represents about 70% of the total star light of the MW. The radius of this thin disk is roughly 10 kpc, and its thickness is roughly 0.5 kpc. The Sun is located near its mid-plane, about 8 kpc from the center. The disk also contains gas, mostly atomic and molecular hydrogen concentrated into clouds with a wide range of masses and sizes, as well as small solid particles, dust, which render interstellar gas opaque at visible wavelengths over distances of several kpc. Most of the hydrogen is neutral. Together, the gas and the dust are called interstellar medium (ISM). About 4% of the MW s stars belong to a thicker disk, which is aligned with the (thin) disk. Its existence has been 19

23 1 Introduction to the Physics of Hypervelocity Stars discovered only recently, in the 1980s. The difference with the thin disk is not just the kinematics of its stars: it turns out that thick disk stars are on average older than the stars of the thin disk, and have a lower metallicity. About 1% of the stellar mass in the Galaxy is contained in the stellar halo, which contains old stars of low metallicity (median about 0.02 Z 5 ). The stellar halo has little or no mean rotation, a density distribution that is approximately spherical, and a power-law function of radius, ρ r 3, out to at least 50 kpc. Also some metal-poor globular clusters are members of the stellar halo. The low metallicity of this population suggests that it was among the first components of the Galaxy to form. Much of the halo comprises the debris of disrupted stellar system, such as globular clusters and small satellite galaxies [18]. In addition to the stellar halo, there are evidences of a gaseous halo with a large amount of hot gas. The halo extends much further than the stellar halo and close to the distance of the Large and Small Magellanic Clouds. In addition to these components, the existence of a dark matter halo is hypothesized. In 1933 Fritz Zwicky began a systematic study of the Coma galaxy cluster, that contains over 1000 identified galaxies. Zwicky estimated the total mass of the cluster from the number of observable galaxies and from its total luminosity. On the other hand, 5 Z is the solar metallicity. Most of the physical matter in the Universe is in the form of hydrogen and helium, so astronomers use the word metals as a short term to refer to all elements except hydrogen and helium. Thus, the solar metallicity is the abundance of elements present in the Sun that are heavier than hydrogen or helium. Z is equal to and is calculated as: i>he m i M, where m i is the fractional mass of the metals contained in the Sun. 20

24 1.7 Galactic Potential he calculated the radial velocities of the galaxies by measuring the Doppler shift of the emitted light. From its measurements, it turned out that the velocities of the single galaxies were too high for the cluster to remain compact. With the observed velocities, the cluster would have to evaporate : the galaxies would not have been seen grouped together, but fleeing from one another. To overcome this problem, Zwicky proposed the idea that the cluster also contained invisible mass, i.e. matter able to exert gravitational attraction but which does not emit light and does not contribute, therefore, to the brightness of the galaxy. Thus, Zwicky coined the term dark matter (DM) to describe this invisible component [19]. In the forty years following Zwicky s observation, astronomers did not make any substantial progress in studying this anomaly. Then, in the early 70s, Vera Rubin, together with her collaborator, Kent Ford, began to study the rotation curves of spiral galaxies (i.e., the variation of the rotation speed of an astronomical system as a function of the distance from the center of the system itself) [20]. Rubin expected to find the trend predicted by Newton s law of gravity: for high distances from the galactic center, the rotation speed should be proportional to the inverse of the square root of the distance. However, she observed that far beyond the visible mass, the rotation speed at a great distance from the center is almost constant. There are two possible solutions to address this problem. The first is to consider the General Relativity on a galactic scale as unsuitable. This led to the birth of the modified gravity, which studies alternative theories of gravity with the aim of correctly reproducing the galactic rotation curves. On the other hand, in the context of the General Relativity, the second possible explanation to the observations is that the mass distribution extends far beyond that of visible stellar light. This means that dark matter must be present in the outermost regions of galaxies (the DM mass should be about 5 times that of the ordinary matter). The observed phenomenon is general: the rotation curve remains flat 21

25 1 Introduction to the Physics of Hypervelocity Stars beyond the visible disk in practically all the disk galaxies observed. In the context of the hypothesis of dark matter, this implies that each galaxy is immersed in a more extensive dark component, a sort of halo (dark matter halo). The dark matter halo is, by far, the larger component, both in size and mass, of the Galaxy. It has a radius of about 200 kpc and a mass of about M (both these values are quite uncertain). The dark matter halo is the least well understood of the Galaxy s components. Nowadays, there are only weak constraints on the composition, shape, size, mass, and local density. A wide variety of candidates have been suggested, most falling into one of two broad classes: (i) some unknown elementary particles; the preferred candidates are the weakly interacting massive particles (WIMPs); (ii) non-luminous macroscopic objects, such as neutron stars or black holes, which are usually called MACHOS (massive compact halo objects). Numerical simulations of the formation of dark matter halo suggest that it is triaxial, but there is little direct observational evidence of halo shape [18]. To understand the motion of HVSs requires a Galaxy model that accurately reproduces the gravitational potential from the very center out to the outer halo. Historically, potential models for the Milky Way developed for other applications were optimized to fit observations at distances R > 200 pc from the Galactic Center. However, for HVSs the largest deceleration occurs at R < 200 pc. To address this problem, Kenyon et al. (2008) [21] derive a three-component bulgedisk-halo potential model that fits observed mass measurements from the Galactic Center to the outer halo. The Galactic potential of this model is illustrated in Figure 1.3. The top panel shows the radial acceleration in the Galactic potential, decomposed into contributions from the central SMBH (purple line), bulge (blue line), disk (green line), and halo (red line). The black line is the total acceleration. The bottom panel shows the radial velocity profile of stars dropped from different radial starting locations (black lines). Acceleration and velocity profiles are calculated along the z-axis. 22

26 1.8 Constraints on the Origin of HVSs Figure 1.3: Radial acceleration (panel a) and radial velocity (panel b) as a function of the Galactocentric distance, in the Galactic potential proposed by Kenyon et al. in 2008 (Brown, W. R., 2015)[6]. 1.8 Constraints on the Origin of HVSs Comparing observations to theory can constrain the origin of HVSs. Different HVS ejection mechanisms predict different ejection velocity distributions, spatial distributions, and ejection rates for HVSs. No observation to date directly links an unbound star to the SMBH; however, there are many observations that provide indirect evidence for HVSs. The best evidence comes from the properties of the unbound 23

27 1 Introduction to the Physics of Hypervelocity Stars stars in the Galactic halo: their velocities, stellar nature, flight times, ages, angular positions, and overall numbers. In addition, the existence of HVSs requires that their former companions still orbit the SMBH. As illustrated in Section 1.2.1, the stellar nature and orbital distribution of the S-stars are well explained by the HVS ejection scenario. In this Section each observational constraint is taken into account. Although individual observations can be explained in different ways, only the HVS mechanism can self-consistently explain all the observations together Velocity distribution of HVSs Because HVSs are launched on radial trajectories from the Galactic Center, spectroscopic radial velocities measure the major component of motion. The measurement uncertainty depends on details like the spectral resolution and the spectral type of the star, but typical radial velocity errors are 0.1-1%. Tangential velocities can be measured for nearby distances ( 10 kpc). The measurement uncertainty depends on both proper motion errors and distance errors. At the >1 kpc distances of the nearest HVSs, the typical tangential velocity is 1-10% of HVS velocity. Radial velocities therefore provide the most accurate constraint on motion. The velocity distribution provides a second constraint. Different SMBH ejection mechanisms produce a different spectrum of ejection velocities, thus the velocity distribution of HVSs can distinguish the mechanisms. The HVS Survey (a well-defined spectroscopic survey of halo stars complete over deg 2 of sky) identified 21 unbound late B-type stars in the halo all moving outward, which is consistent with an ejection from the Galactic Center. The fastest unbound stars in the HVS Survey have speeds of 700 km/s at distances of kpc. A speed of 700 km/s significantly exceeds their stellar escape velocity and provides strong evidence for a SMBH ejection scenario. The present lack of a > 1000 km/s star is in slight tension with HVS ejection models; however, the overall 24

28 1.8 Constraints on the Origin of HVSs distribution of radial velocities and distances is consistent with HVS ejection models. Figure 1.4 shows the observed Galactic rest frame velocity, v rf, and Galactocentric distance, R, of late B-type stars in the Hypervelocity Star (HVS) Survey. The dashed line is the Galactic escape velocity from the updated Kenyon et al. (2008) model [21]. Dotted gray lines are the isochrones of flight time from the Galactic Center, i.e. the loci of (R, v rf ) pairs corresponding to the same travel time. For a given isochron: (i) the largest velocities correspond to the largest distances from the Galactic Center, R; (ii) points that lie below the dashed curve correspond to v rf values lower than the Galactic escape velocity at the corresponding R. The observed HVS values of (R, v rf ) that were ejected from the Galactic Center at the same epoch, i.e. HVSs that share the same flight time. Thus, Figure 1.4 clearly shows that the HVSs observed in the Hypervelocity Star Survey do not share all the same flight time. HVSs located below the dashed line (blue circles) will not be able to escape from the Galaxy, and are bound stars. Unbound HVSs are shown with magenta stars Metallicity Metallicity provides a possible constraint on the origin of HVSs. If HVSs are short-lived stars, their abundance patterns should reflect their place of origin. Stars formed in the Galactic Center are expected to have solar iron abundance and possibly enhanced alpha element abundance. Stars formed in the Galactic disk should have [M/H] = if they originate at R = 4 kpc and [M/H] = -0.4 if they originate at R = 14 kpc. Early B-type stars in the Solar Neighborhood have solar abundance [M/H] = 0. If HVSs are the tidal debris of a typical star-forming dwarf galaxy in the Local Group, the stars should have -2 < [M/H] < Unfortunately, metallicity does not always reflect the place of origin. There is observational and theoretical evidence that stars within a 25

29 1 Introduction to the Physics of Hypervelocity Stars Figure 1.4: Observed Galactic rest-frame velocity, v rf, and Galactocentric distance, R, of late B-type stars in the Hypervelocity Star (HVS) Survey (Brown, W. R., 2015) [6]. The dotted lines represent the isochrones of flight time from the Galactic Center. Flight times of 60 to 210 Myr. The dashed line indicates the Galactic escape velocity as a function of the Galactocentric distance. Blue circles indicate stars that will not be able to escape the Galaxy, while magenta stars indicate the unbound HVSs. single cluster have a statistically significant spread in abundances. Furthermore, hot stars with radiative atmospheres are frequently observed with peculiar abundance patterns. Gravitational settling and radiative levitation of elements in the atmospheres of these stars 26

30 1.8 Constraints on the Origin of HVSs systematically change their surface abundances over time. In other words, the surface abundance pattern of B-type HVSs may no longer reflect their original composition. Stellar rotation also puts a practical limitation on metallicity measurements. Young stars are fast rotators. Fast rotation smears weak metal lines and makes accurate abundance measurements difficult. The present abundance measurements of HVSs provide no meaningful constraint on their origin Stellar rotation Besides the velocity distribution and the metallicity, also the stellar rotation can constrain the origin of HVSs. Unless a star is being observed from the direction of its pole, sections of the surface have some amount of movement toward or away from the observer. The component of motion that is in the direction of the observer is called the radial velocity. For the portion of the surface with a radial velocity component toward the observer, the radiation is shifted to a higher frequency (blue-shift) because of Doppler shift. Likewise, the region that has a component moving away from the observer is shifted to a lower frequency (red-shift). Looking at the absorption lines of a rotating star, this shift is observed. The component of the radial velocity observed through line broadening depends on the inclination of the star s pole to the line of sight. The derived value is given as v e sin(i), where v e is the rotational velocity at the equator and i is the inclination. Figure 1.5 illustrates the concept. This projected stellar rotation constrains the origin of HVSs in two ways. First, v e sin(i) is the least ambiguous measure of the stellar nature of B spectral-type stars. B-type main-sequence stars and evolved horizontal branch stars can have similar effective temperatures and surface gravities, but very different rotations. Mainsequence B stars are young and have mean v e sin(i) 150km/s. Hot 27

31 1 Introduction to the Physics of Hypervelocity Stars Figure 1.5: Star with an inclination i to the line-of-sight of an observer on the Earth and rotational velocity v e at the equator. horizontal branch stars are old stars that have evolved off the giant branch and have v e sin(i) < 7 km/s. Fast rotation is thus the clear signature of a young main-sequence star. Second, the distribution of v e sin(i) may be linked to the nature of the HVS ejection mechanism. If HVSs are the product of disrupted compact binaries, then the mean v e sin(i) of late B-type HVSs should be relatively slow. The binary SMBH ejection mechanism, in contrast, can eject single fast-rotating stars. However present measurements do not significantly constrain the distribution of v e sin(i) Flight time The distribution of HVS flight times is another constraint on HVS origin. For example, a burst of HVSs is expected for an IMBH- SMBH in-spiral event. If HVSs are ejected in bursts, both long-lived and short-lived HVSs must share a common flight time. By contrast, a continuous distribution of HVS flight times is consistent with the single SMBH, continuous ejection scenario. The distribution of flight 28

32 1.8 Constraints on the Origin of HVSs times for the unbound stars in the HVS Survey is best described by a continuous ejection process. However, the uncertainties are large, and one cannot rule out that half of the unbound stars were ejected in a single burst. Multiple bursts would be required to explain the full span of observed flight times. In all cases, the unbound stars in the HVS Survey have flight times shorter than the main-sequence lifetimes of B stars, which is consistent with their identification as main-sequence stars ejected from the Milky Way Arrival time The time between the formation and the ejection of a HVS (arrival time) provides a strong test of origin. High-velocity runaway ejections require massive stars, stars that have 10 7 year lifetimes. The arrival time of unbound runaway ejections is therefore 10 7 years. For binaries formed in the Galactic Center, in contrast, there is no upper limit on arrival time. To estimate the arrival time, a HVS s age is subtracted from its flight time from the Galaxy. Age is determined by comparing a star s surface gravity and effective temperature against stellar evolution models. Statistically meaningful age estimates can only be obtained from high signal-to-noise, highresolution spectroscopy Number and rate of unbound HVSs Ejection rates provide a fundamental connection between observation and theory. Different ejection mechanisms produce different ejection rates of unbound stars. Thus, the observed number of unbound stars provides a constraint on their origin. The observed number of unbound B stars in the halo supports a Galactic Center origin. 29

33 1 Introduction to the Physics of Hypervelocity Stars Spatial distribution The spatial distribution of HVSs provides another constraint on their origin. In the single SMBH ejection scenario, the direction of HVS ejection corresponds to the direction of the encounters of the progenitors with the SMBH. If HVS progenitors come from a dynamically relaxed population of stars in the Galactic Center, the spatial distribution of HVS ejections should be isotropic. If HVS progenitors come from a stellar disk, the spatial distribution of HVSs should lie on a plane. In the binary SMBH ejection scenario, the highest-velocity ejections occur from the SMBH orbital plane. An eccentric binary SMBH may produce a jet of HVSs. As the binary SMBH hardens, its orbital plane is perturbed on rapid timescales, and HVS ejections should become more energetic and isotropic. Thus, the signature of the binary SMBH in-spiral scenario may be a ring, jet, or shell of HVSs with similar flight times. Even if HVSs are ejected isotropically, a non-spherical Galactic potential can modify the observed distribution of HVSs. The predicted spatial distribution depends on the axis ratio and the rotation direction of the potential. A clump of unbound stars that share a common flight time may be explained as the tidal debris of a dwarf galaxy that experienced a very close pericenter passage to the Milky Way. Massive (> M ) dwarf galaxies are the most likely progenitors of such unbound stars. A supersonic gas cloud, launched from an active galactic nucleus (AGN), can collapse to form HVSs. This scenario implies that HVSs should be found in discrete clumps with common velocities. Disk runaways can be launched in any direction, but the fastest runaways are those ejected in the direction of Galactic rotation. Simulations predict that 90% of hyper-runaways are to be found at Galactic latitudes b < 25 ; thus, the signature of these objects is a flattened spatial distribution of unbound stars with all possible flight times, 30

34 1.8 Constraints on the Origin of HVSs reflecting an origin from the rotating disk. All these different ejection mechanisms produce different spatial distributions of HVSs that are illustrated in Figure 1.6. The HVS Survey provides a clean test Figure 1.6: Spatial distribution of different ejection mechanisms. (Brown, W. R., 2015)[6]. See Section for details. of spatial distribution for a uniform set of unbound stars, complete over 29% of the sky. Figure 1.7 shows the distribution of HVS Survey stars. Unbound stars in the HVS Survey are distributed equally across Galactic latitude. There is no statistical difference between the latitude distribution of the unbound stars and the underlying survey stars. A uniform latitude distribution is inconsistent with a disk runaway origin, but consistent with a SMBH origin. However, the unbound stars in the HVS Survey are clumped in Galactic longitude. Specifically, 11 of the 21 unbound stars are located in 31

35 1 Introduction to the Physics of Hypervelocity Stars a region (5% of survey area) centered around (RA, Dec) = (167.5, 3.0 ) in the direction of the constellation Leo. Unfortunately, the flight time uncertainties are too large to determine whether the clump of unbound stars share a common flight time. At present, there is no good explanation for the anisotropic distribution of unbound late B-type stars in the halo. The tidal debris scenario is one possibility; however, it requires a gas-rich star-forming dwarf galaxy to account for the short-lived B stars, and no other tidal debris is observed. A full understanding of the spatial distribution requires a Southern Hemisphere survey. Figure 1.7: Polar projection, in Galactic coordinates, showing the spatial distribution of HVS Survey stars: unbound HVSs are marked by magenta stars (Brown, W. R., 2015) [6]. 32

36 1.9 Summary of Observations Proper motion Proper motions, combined with distance and radial velocity, provide the full space trajectory of a star. Interestingly, a Galactic Center origin is often well separated in proper motion from a disk origin. This separation occurs because disk ejections are limited to those locations where the escape velocity from the Galaxy is less than the escape velocity from the surface of the star, 600 km/s. Thus, for the fastest known HVSs, possible disk ejection locations are limited to the outer disk. Unfortunately, known HVSs are distant and on radial trajectories. Their expected proper motions are too small to be measured with ground-based surveys ( 1 mas/year). Proper motions with <0.1 mas/year uncertainties, like those achievable by the Gaia mission, can provide a clean separation between Galactic disk and Galactic Center origins. Figure 1.8 illustrates the case of HVS5. The proper motion of this star (v rf = 650 ± 4.3 km/s, R = 50 ± 5 kpc, m = 3.6 M ) is not yet known. The Figure shows how different trajectories map into proper motion space. Green and blue ellipses are the locus of HVS5 trajectories that pass within 8 and 20 kpc, respectively, of the Galactic Center (GC). The trajectory that passes through the GC is marked with a plus symbol (+). The locus of HVS5 trajectories with disk plane ejection velocities of 600 km/s (i.e. the escape velocity from the surface of the star) is drawn by the red contour. Physically allowed disk ejections are thus limited to the outer disk. The black circle is the size of the predicted Gaia proper motion measurement error, which will cleanly discriminate between outer disk and GC origins. 1.9 Summary of Observations HVSs are stars ejected at unbound speeds by the Milky Way s SMBH. The unbound stars that are observed in the halo support the SMBH ejection origin. The supporting observations include the velocities 33

37 1 Introduction to the Physics of Hypervelocity Stars Figure 1.8: Illustration of HVS5 possible trajectories in proper motion space (Brown, W. R., 2015)[6]. of the unbound stars, velocities that in certain cases exceed the escape velocity from the surface of the stars. The velocity distribution of both unbound and bound stars in the HVS Survey is consistent with predicted HVS ejection velocity distributions. The unbound B stars are, on the basis of their rotation, main-sequence stars and must originate from a region of ongoing star formation like the Galactic Center. The observed number of unbound B stars is consistent with theoretically predicted HVS ejection rates, but inconsistent with hyper-runaway ejection rates. The distribution in Galactic latitude of unbound B stars is consistent with Galactic Center ejections, but inconsistent with disk ejections. 34

38 1.9 Summary of Observations The four brightest HVSs in the HVS Survey were ejected 100 Myr after they were formed, which is a timescale consistent with a Galactic Center origin, but inconsistent with scenarios that require a prompt ejection as in the runaway or AGN scenarios. Finally, the nature, number, and orbital distribution of S-stars orbiting the SMBH in the central arc-second of the Galaxy are consistent with their being the former companions of HVSs. Other observations, such as the distribution of v e sin(i), flight time, metallicity, and proper motion of unbound B stars, currently provide no significant constraint on origin, but can be further improved. Taken together, the current set of observations is consistent with a SMBH origin for unbound HVSs. 35

39 36

40 Chapter 2 Objectives: Hypervelocity Stars as a Probe of the Milky Way Mass Distribution The context of this Thesis is the investigation of the Milky Way mass distribution through the kinematical properties of hypervelocity stars. The understanding of the Milky Way gravitational potential is crucial in order to either investigate the existence and the shape of dark matter halo, or to test modified gravity theories. The dark matter halo interacts with the other components of the Galaxy only through the gravitational force: therefore, stellar dynamics is one of the few possible tools to study this mysterious yet crucial component. Hypervelocity stars are perfect tracers of the Milky Way potential well because they start their journey at the Galactic Center and, thanks to their unbound velocities, they reach the outer halo of the Milky Way. The study of the time evolution of the positions and velocities of HVSs is crucial to constrain the features of the Galactic potential. 37

41 2 Objectives: Hypervelocity Stars as a Probe of the Milky Way Mass Distribution The main goal of my Thesis is the implementation of a numerical code that integrates the equations of motion and computes the trajectories of a sample of hypervelocity stars from the time of ejection up to a given time of observation. I will then study the distribution of radial velocities in a well-defined survey volume. My simulations are performed under the assumption that the HVSs are produced through the mechanism proposed by Hills in 1998, according to which a HVS is generated from an encounter of a binary star system with the SMBH located at the Center of the Galaxy (see Section 1.2 for details). I will use this code to investigate how the Milk Way mass distribution influences the spectrum of radial velocities, by considering two different models for the Galactic potential: a spherically symmetric potential (see Section 3.4) and an axisymmetric potential (see Section 3.5). Finally, I will also compare the two potential models, in order to test whether they are equivalent in describing the MW mass distribution. My Thesis work is a starting point of a long-term project, whose final objective will be the building of HVSs mock catalogs including star positions, radial velocities, and proper motion, for different ejection mechanisms (the Hills mechanism and other mechanisms) and in different Galactic potentials (including more complex, non sphericallysymmetric models for the dark matter halo potential). These mock catalogs will be used both for a comparison with the HVS data coming from the Gaia mission and for forecasts for future astrometric missions (e.g., Theia 1 ). 1 In the context of the ESA M5 (medium mission) call, a new satellite mission, Theia, has been proposed. Theia will be based on relative astrometry and extreme precision to study the motion of very faint objects in the Universe. Theia is primarily designed to study the local dark matter properties, the existence of Earth-like exoplanets in our nearest star systems and the physics of compact objects. Furthermore, about 15% of the mission time was dedicated to an open observatory for the wider community to propose complementary science cases. With its unique metrology system and point and stare strategy, Theia s precision is expected to reach the sub micro-arcsecond level. This is about

42 times better than ESA/Gaia s accuracy for the brightest objects and represents a factor improvement for the faintest stars [22]. 39

43 40

44 Chapter 3 Numerical Simulations of the Journey of Hypervelocity Stars In order to investigate the effects of different models of the MW mass distribution on the time evolution of the distribution of the radial velocity, I implement a numerical code that, for any gravitational potential, simulates the time evolution of the position, the velocity, and the acceleration of a sample of stars whose initial position, velocity, and acceleration are known. The code is implemented in C++ programming language. 3.1 The Integration Method The time evolution of a dynamical system can be studied by solving the classical equation of motion, Newton s second law, F = d P /dt. Here F is the force acting on the classical particle and P is the momentum of the particle, P = m v (with m and v the mass and the velocity of the particle, respectively). The equation of motion is a second order differential equation for the position of the particle as a function of time. 41

45 3 Numerical Simulations of the Journey of Hypervelocity Stars In the case of the gravitational potential, the force is conservative and can be written as the gradient of a scalar function, the potential energy, U( x). The force is then only a function of the coordinate of the particle, F ( x), and is not dependent on the velocity: a( x(t)) = d2 x(t) dt 2 = F ( x(t)) m = (U( x(t))) m. (3.1) Therefore, the problem of calculating the time evolution of such a system reduces to the problem of solving a second order differential equation. I solve the equation of motion (in a Galactocentric reference system and in Cartesian coordinates) with the Velocity Verlet algorithm [23] Derivation of the Verlet algorithm using Taylor s expansions Given a time t and a subsequent time t + h, the position of a particle x(t + h) can be written as a Taylor expansion about x(t): x(t + h) = x(t) + h x(t) + h2 2 x(t) + h3... x + O(h 4 ). (3.2) 6 Expanding x(t h) gives a similar expression, except for the odd terms that have a minus sign: x(t h) = x(t) h x(t) + h2 2 x(t) h x + O(h 4 ). (3.3) When the two expressions are added, the odd terms cancel out: Therefore, x(t + h) + x(t h) = 2 x(t) + h 2 x(t) + O(h 4 ). (3.4) x(t + h) = 2 x(t) x(t h) + h 2 x(t) + O(h 4 ). (3.5) 42

46 3.1 The Integration Method This expression is general, and holds for any function x(t) that is differentiable enough times. The equation of motion for classical dynamics enables us to know the expression for a( x(t)). The Verlet algorithm is obtained by neglecting the fourth and higher order terms. It turns out that this algorithm is stable, i.e. the error made at any given step tends to decay rather than magnify later on Derivation of the Velocity Verlet algorithm In classical mechanics it is possible to know simultaneously both the position and velocity of particles. A classical system is often represented as a point in phase space, a space of all the coordinates and momenta of particles in the system. Knowing the position and velocity (or momentum) at one point in time completely specifies the state of the system and, through the equation of motion, the future (and past) motion of the particles, as long as the forces acting on the particles are known. The velocity, however, does not appear explicitly in the Verlet algorithm. If it is needed, for example, to calculate the kinetic energy, a central finite difference estimate can be used: x k = ( x k+1 x k 1 ) 2h ; (3.6) here k is an index that runs on all the time steps. The Velocity Verlet algorithm, on the other hand, explicitly includes the velocity at each step and is self-starting from the position and velocity at the initial time. It is mathematically identical to the original Verlet algorithm in the sense that it generates the same trajectory in the absence of roundoff errors in the computer. Any n th order differential equation can be reduced to a set of n first order differential equations. In particular, the classical equation of motion x(t) = F ( x(t))/m is second order and can be reduced to two 43

47 3 Numerical Simulations of the Journey of Hypervelocity Stars first order equations. Let v(t) = x(t) and the two equations are: x(t) = v(t) (3.7) v(t) = a( x(t)). (3.8) Using a Taylor expansions and starting with x(t + h), one gets: x(t + h) = x(t) + h x(t) + h2 2 x(t) + O(h 3 ). (3.9) Using v(t) and v(t) gives: x(t + h) = x(t) + h v(t) + h2 2 a( x(t)) + O(h3 ). (3.10) Then, expanding the second function, v(t + h), one gets: v(t + h) = v(t) + h v(t) + h2 2 v(t) + O(h 3 ). (3.11) One can use a to eliminate v, but needs to find an expression for v in terms of known quantities. This can be done by expanding v(t + h): v(t + h) = v(t) + h v(t) + O(h 2 ). (3.12) It is enough to go up to order h 2 here, because one only needs an approximation that is good to order h 3 to the quantity h 2 v(t)/2. Multiplying by h/2 and rearranging, gives: h 2 2 v(t) = h 2 ( v(t + h) v(t)) + O(h 3 ) ; (3.13) thus, the expression for v(t + h) becomes: v(t + h) = v(t) + h v(t) + h 2 ( v(t + h) v(t) + O(h 3 ). (3.14) 44

48 3.2 Gravitational Potential of Point Sources Using the equation of motion, this can finally be rewritten as: v(t + h) = v(t) + h 2 ( v(t + h) + v(t) + O(h 3 ). (3.15) Schematically, the velocity Verlet algorithm is as follows. Given x k and v k and an expression for a( x), the algorithm: 1. calculates x k+1 = x k + h v k + h 2 a( x k); 2. evaluates a( x k+1 ) from the interaction potential; 3. calculates v k+1 = v k + h 2 ( a( x k) + a( x k+1 )). When all the quantities for the new step, k+1, have been found, the calculations 1-3 are repeated for the step k+2, and so on. 3.2 Gravitational Potential of Point Sources As a first step, I implement a numerical code that is able to simulate the trajectory of a single star in the potential well generated by a point-like mass distribution, as the one that can be associated to the SMBH located at the Galactic Center (assuming to be sufficiently far from the SMBH, where the Newtonian approximation holds). The spherical potential generated by a SMBH of M BH = M takes the form: Φ(r GC ) = G M BH r GC, (3.16) with G the gravitational constant and r GC the distance from the SMBH (and from the Galactic Center). Therefore the acceleration takes the form: a(r GC ) = Φ(r GC) m = G M BH r 2 GC. (3.17) Equation (3.17) is the equation of motion that enters the code. To integrate this equation, I use the Velocity Verlet algorithm (described 45

49 3 Numerical Simulations of the Journey of Hypervelocity Stars in Section 3.1) and I test the reliability of the code by reproducing some known cases (elliptical and circular, close orbit, and open orbit). In all the simulations the star begins its journey at x = 1 pc, y = 0 pc, and z = 0 pc (thus at r GC = 1 pc); its initial velocity, v = v y = v ê y, lies on the y axis (the values are reported in Table 3.1 in units of GMBH, which is the initial velocity in the case of circular orbit). r GC v y [ GM BH r GC ] Circular orbit 1 Elliptical orbit 1.3 Open orbit 1.5 Table 3.1: Initial values of the star s velocity ( v y ) in the simulation of elliptical, circular, and open orbit. For each time step of the integration, the code computes the position, the velocity, the acceleration, the Hamiltonian, and the magnitude of the angular momentum of the star. All the physical quantities that enter this and the following simulations are expressed in appropriate units: spatial scales in parsec (pc), masses in solar masses (M ), and time intervals in mega-years (Myr). I use Cartesian coordinates in a Galactocentric reference frame. 3.3 Code Reliability In order to verify the code reliability, I check if the code is able to conserve the Hamiltonian per unit mass (H = 1 2 v2 + Φ). The code calculates the relative difference between the Hamiltonian at a time t and the Hamiltonian at the initial time, t 0 : H H 0 = H(t) H 0 H 0 (3.18) 46

50 3.3 Code Reliability Y [pc] X [pc] Figure 3.1: Circular (orange, solid line), elliptical (green, dashed line), and open (blue, dotted line) orbits of a test particle (a star), moving in the potential well generated by a point-like mass (the SMBH), indicated by the black point, located in (0,0). See Section 3.2 for details. with H 0 = H(t = 0) = 1 2 v2 (t = 0) + Φ(r GC (t = 0)). In all the case studies, the simulation stops at t final = 0.5 Myr. 47

51 3 Numerical Simulations of the Journey of Hypervelocity Stars For the open orbit, in which the star s motion is outward from the SMBH, the time step is initialized with dt = Myr and then it grows exponentially. On the other hand, for the circular and for the elliptical, close orbit, the time step is constant (dt = 10 7 Myr). The code conserves the Hamiltonian with an accuracy H H 0 of the order of 10 10, 10 13, and 10 10, respectively, for the open (Figure 3.2), the circular (Figure 3.3), and the elliptical (Figure 3.4) orbit. The two peaks in H H 0 that can be seen in the case of the elliptical, close orbit (see Figure 3.4) correspond to the time when the star closely approaches the SMBH: its velocity changes very rapidly and the time step, although very small, becomes less appropriate to perform the integration efficiently. So the H H 0 drops and then sets around ; this value remains almost unvaried until the star closely approaches again the SMBH, at the beginning of the second orbit. 2.5x10-10 open orbit 2x x10-10 ΔH/H 0 1x x t [Myr] Figure 3.2: Open orbit: relative difference of the Hamiltonian as a function of time. 48

52 3.3 Code Reliability 1.5x10-13 circular orbit 1x x10-14 ΔH/H 0 0-5x x x t [Myr] Figure 3.3: Closed, circular orbit: relative difference of the Hamiltonian as a function of time. 49

53 3 Numerical Simulations of the Journey of Hypervelocity Stars ΔH/H 0 2x10-11 elliptical orbit 0-2x x x x x x x t [Myr] 1 elliptical orbit 0-1 X [kpc] t [Myr] Figure 3.4: Closed, elliptical orbit. Top panel: relative difference of the Hamiltonian as a function of time. Bottom panel: position along the x-axis as a function of time 50

54 3.4 Gravitational Potential of Extended, Spherical Sources 3.4 Gravitational Potential of Extended, Spherical Sources Once the reliability of my code is verified in the simple case of a point source, I improve the code itself by implementing the case of a continuous mass distribution. This mass distribution is more appropriate to study the motion of a HVS through the entire Galaxy. Following Bromley et al. (2006) [5] and references therein, I first consider a simple parametrization for the density profile of the Milky Way: ρ(r GC ) = ρ ( r GC a c ) 2, (3.19) where r GC is the distance to the Galactic Center, a c = 8 pc is a core radius (a length scale for the model), and ρ 0 = M pc 3 is the central density. This density profile is displayed in Figure 3.5. The above choices of the model s free parameters give a mass within 10 pc of the Galactic Center of M, as inferred from stellar kinematics in the region of Sgr A*, and a circular rotation speed of 220 km/s in the solar neighborhood. This density profile is a simplified profile with spherical symmetry and only three free parameters. It does not account separately for the contributions of the different components of the MW mass distribution, as the bulge, the disk, and the dark matter halo, but accounts for all of these components with a single density profile, designed to fulfill a number of observational constraints (see above). Even though it may not be the ideal model of the MW mass distribution to study the HVS trajectories, it is a good model to study the evolution of the velocity distribution of HVSs at distances 10 kpc from the Galactic Center, where the dark matter gravitational effect is dominant. Moreover, the use of this toy model enables me to test my numerical code by comparing my results with those by Bromley et al. (2006). The density profile of Equation (3.19) yields a potential: 51

55 3 Numerical Simulations of the Journey of Hypervelocity Stars ρ(r GC ) [M /pc 3 ] r GC [pc] Figure 3.5: Density profile of the Milky Way as a function of the distance to the Galactic center, according to Equation (3.19). Φ(r GC ) = 2πGρ 0 a 2 c [ 2 a c r GC arctan ( ) rgc a c + ln ( 1 + r2 GC a 2 c )] + C, (3.20) with C an additional constant. Figure 3.6 displays this potential as a function of the distance to the Galactic Center, r GC. Because the density profile of Equation (3.19) is spherically symmetric, this gravitational potential is also spherically symmetric. From the density profile, I obtain the potential Φ, by integrating the Poisson equation: 2 Φ(r GC ) = 4πG ρ(r GC ), (3.21) where the Laplacian of the potential is: 2 Φ(r GC ) = 1 r 2 GC r GC ( Φ rgc 2 r GC ). (3.22) 52

56 3.4 Gravitational Potential of Extended, Spherical Sources Φ(r GC ) [pc²/myr²] r GC [pc] Figure 3.6: Gravitational potential of the Milky Way generated by a spherically symmetric distribution of mass as a function of the distance to the Galactic Center (Equation (3.20), with C = 0). Equation (3.21) yields: r GC ( Φ rgc 2 r GC ) = 4πG r 2 GC ρ(r GC ) ; (3.23) Because Φ depends only on r GC, integrating over dr GC ( ) rgc d rgc 2 Φ rgc = G 4π rgc 2 ρ(r GC ) dr GC 0 0, (3.24) implying r 2 GC and r GC Φ = G M(r GC ) = 4πG a 2 r cρ 0 GC Φ r GC = G M(r GC) r 2 GC 53 [ r GC a c arctan ( )] rgc a c (3.25). (3.26)

57 3 Numerical Simulations of the Journey of Hypervelocity Stars Integrating once again over dr GC yields the potential Φ: G M(rGC ) Φ(r GC ) = dr GC + C (3.27) r 2 GC that takes the form: Φ(r GC ) = 2πGρ 0 a 2 c [ 2 a c r GC arctan ( ) rgc a c + ln ( 1 + r2 GC a 2 c )] + C. (3.28) The acceleration of a particle in this potential well takes the form: a(r GC ) = Φ(r GC ) = 4πGρ 0a 2 c r GC [ 1 a c r GC arctan ( )] rgc a c. (3.29) In my simulation, I use Cartesian coordinates; thus, I decompose the acceleration along the three axes. Figure 3.7 shows the gradient of the potential as a function of the distance to the Galactic Center, r GC. Integrating equation (3.21) enables to build the trajectory of a star in the potential Φ. In particular, following Bromley et al. (2006), I implement the numerical integration of the equation of motion (Equation (3.29)) to study the trajectories and the velocity distribution of an ensemble of N = 10 5 stars injected into the Galactic potential by an encounter with the SMBH (Hills mechanism). To this aim, it is crucial to properly set the initial conditions, i.e. the positions and velocities of the stars injected into the Galaxy through the Hills mechanism. The setting of the initial conditions will be explained in the following Sections Initial distribution of velocities of ejected stars With my code, I simulate the evolution of positions and velocities of a sample of N = 10 5 stars. 54

58 3.4 Gravitational Potential of Extended, Spherical Sources - a(r GC )= Φ(r GC ) [pc/myr²] r GC [pc] Figure 3.7: Gradient of the potential as a function of the distance to the Galactic center, r GC (the conversion factor from pc/myr 2 to cm/s 2 is ). At each star, I associate an initial velocity, i.e. a velocity of ejection from the Galactic Center, consistent with the value expected from the Hills mechanism: ( abin v ej = 1, AU with ) ( ) 1/2 1/3 ( mp + m s M BH 2M M ) 1/6 f R km s, (3.30) f r = ( ( ( (3.31) +( D)D)D)D)D, where the Hills parameter D (previously defined in Section 1.2) is: D = ( Rmin ) ( 10 6 ) 1/3 (m p + m s ) ; (3.32) a bin 2M BH 55

59 3 Numerical Simulations of the Journey of Hypervelocity Stars here, a bin is the binary s semi-major axis, R min is the closest approach distance between the binary and the SMBH, m p and m s are the masses of the primary and the secondary stars of the binary system, respectively, and M BH is the mass of the SMBH. D is related to the probability of ejection through the relation P ej = 1 D/175. During the ejection process of the HVS, the primary and secondary stars of the binary system have an approximately equal chance of ejection. The key factors in the ejection are the orbital phase and the orientation of the orbital plane when the binary comes under the grip of the black hole s tidal field. If a binary is disrupted, then the star on the orbit closest to the black hole tends to be captured; its partner is ejected. For my simulation, I choose an equal-mass binary star system, with masses of the members equal to m p = m s = 4 M. This assumption simplifies the problem, because it becomes irrelevant to understand what star is ejected by the SMBH. The case of a 4M 4M binary is among the cases studied by Bromley et al. (2006). Furthermore, it is a useful choice for possible future comparisons with data: indeed, the majority of observed HVSs are B-type stars, and 4M is a typical mass value for this type of stars. This is due to the fact that B-type stars are among the most luminous stars (only O-type stars have higher luminosities), and this leads to a larger chance of detection; furthermore, B-type star lifetime is longer than O-type star lifetime: this enables B-type HVSs to reach the survey volume and be detected before their death. In my simulation, I use the value of the SMBH mass adopted by Bromley et al. (2006), i.e. M BH = M. The ejection speed, v ej, in Equations (3.30) represents an average velocity; it is the theoretical speed at infinite distance from the SMBH in the absence of other gravitational sources. This speed will be referred to as v avg in the following. Repeated trials for a wide range of binary configurations performed by Bromley et al. (2006) indicate 56

60 3.4 Gravitational Potential of Extended, Spherical Sources that the Gaussian model with a σ v = 0.2 v avg generally gives an adequate description of the ejection speed distribution. The shape of the velocity distribution reflects the detailed dependence of an ejection event on the orbital phase and orientation of the binary orbit relative to the SMBH. The extended tails of the distribution correspond to infrequent encounters in which the binary angular momentum is either strongly aligned or counter-aligned with its center-of-mass angular momentum about the SMBH. Otherwise, encounters yield velocities near the mean, causing a peak in the distribution near v avg. Figure 3.8 shows the distribution of ejection speeds from a binary with a 4 M primary star, just after encountering a SMBH with M BH = M. The original binary has a semi-major axis of 0.1 AU, and its center of mass is targeted to reach a minimum distance R min of 5 AU from the SMBH. In my simulation, for each star of the sample, I evaluate v ej according to Equation (3.30) and then randomly sample a true ejection velocity, v true, from a Gaussian distribution whose mean is v avg = v ej. For a sample of N stars, the distribution of ejection velocities, v ej, clearly depends on the distributions of the binary semi-major axis, a bin, and on the minimum distance of the binary from the SMBH, R min. Surveys of large samples of binary systems with solar-type primary stars suggest that the probability density function for binary semimajor axes, a bin, is roughly: p(a bin ) da bin da bin a bin (3.33) for a bin AU. Specifically, the chosen range of binary semimajor axes is 0.05 AU a bin 4 AU; the lower limit is suggested by the physical radius of the 4M star (about half this length); the upper limit arises because I am interested in stars with sufficient ejection speeds to populate the Galactic halo beyond 10 kpc. The probability density function of closest approach distances, R min, 57

61 3 Numerical Simulations of the Journey of Hypervelocity Stars Figure 3.8: Distribution of ejection speeds from a binary star system with a 4 M primary star, just after encountering a SMBH with a mass of M. The original binary has a semi-major axis of 0.1 AU, and its center of mass is targeted to reach a minimum distance of 5 AU from the SMBH (Bromley et al. (2006) [5]). to Sgr A* varies linearly with R min, p(r min ) dr min R min dr min (3.34) as a result of gravitational focusing (Hills 1988 [1]). The gravitational focusing is the enhancement in the likelihood that two particles will collide, due to their mutual gravitational attraction. Without gravity, 58

62 3.4 Gravitational Potential of Extended, Spherical Sources the probability of a collision scales with the cross-sectional area of the two particles, but with gravity, some particles that would have missed each other can still collide. In other words, the particles have effective cross sections larger than their physical cross sections. Thus, the cross-section for two particles (the center of mass of the binary star system and the SMBH), having a relative velocity at infinity of v, to pass within a distance R min is given by: ( ) σ = πrmin v2 (3.35) v 2 where v is the relative velocity of the binary center of mass and the SMBH at closest approach distance (R min ) in a parabolic encounter: v 2 = 2G m b M BH R min, (3.36) where m b is the mass of the binary. In Equation (3.35) the second term arises from the attractive gravitational force, and is referred to as gravitational focusing. In the case of an encounter with a SMBH, the gravitational focusing is important (v v ). Thus, (1+v 2 /v 2 ) v 2 /v 2 and the the cross-section for the center of mass of the binary star system to pass within a distance R min from the SMBH becomes proportional to the distance itself: σ R min. (3.37) Once the a bin value is set, the maximum value of R min that could results in a nonzero ejection probability can be inferred from Equation (3.32), imposing P ej > 0. Since P ej > 0 when D < 175, one gets: R min < R max min = 175 a bin ( 10 6 ) 1/3 (m p + m s ). (3.38) 2M BH Thus, for each star, once the value of a bin is generated according to the probability distribution, the maximum value of R min will be 59

63 3 Numerical Simulations of the Journey of Hypervelocity Stars inferred; for the minimum value of R min, I choose the same value adopted by Bromley et al. (2006), i.e. 1 AU. Hence, every time I sample a value of a bin, the range of R min changes accordingly. Once the a bin and R min values are generated, the code calculates D and v avg. From the value of D, the code calculates the probability of ejection, P ej = 1 D/175. Then using a uniform random distribution, the code generates a number between 0 and 1. If this number is less than the value obtained for the probability of ejection, the star will be ejected with an initial velocity randomly sampled from the Gaussian distribution around v avg (Normaldev structure in C++); otherwise the star will be rejected. The structures and the transformation method To generate a random number from the Gaussian probability distribution, I use the structure Normaldev taken from the Numerical Recipes [24]. To randomly sample the semi-major axis and closest approach distance probability distribution, I create two new structures (Abindev, Rmindev), using the transformation method. In a random deviate with a uniform probability distribution, the probability of generating a number between x and x + dx is p(x)dx, and is given by: dx 0 < x < 1 p(x)dx = (3.39) 0 otherwise. The function p(x) is normalized such that: p(x)dx = 1. (3.40) By using the transformation method, and starting from a random deviate with a uniform probability distribution, it is possible to generate some arbitrary distribution of y s, one with p(y) = f(y), for 60

64 3.4 Gravitational Potential of Extended, Spherical Sources some positive function f whose integral is 1. The probability distribution of y is determined by the fundamental transformation law of probabilities: p(y)dy = p(x)dx (3.41) Thus, it is necessary to solve the differential equation: dx dy = f(y) (3.42) The solution is x = F (y), with F (y) indefinite integral of f(y). The desired transformation which takes a uniform deviate into one distributed as f(y) is: y(x) = F 1 (x) (3.43) where F 1 is the inverse function of F. This equation has a geometric interpretation. F (y) is the area under the probability curve to the left of y. Thus, equation (3.43) prescribes to choose a uniform random x, then to find the value of y that has that fraction x of probability area to its left, and finally to return the y value. This geometric interpretation is represented in Figure 3.9. In my case, neither p(a bin ) nor p(r min ), integrated between and +, are equal to 1. But I have an interval in a bin and R min and I require that: a max bin p(a bin )da bin = 1 (3.44) and that: a min bin R max min R min min p(r min )dr min = 1. (3.45) Therefore, the only difference between my case and the general case is that in the general case the x is uniformly, randomly chosen between 0 and 1, whereas in my case I have to choose x between F (y min ) and F (y max ). Thus: 61

65 3 Numerical Simulations of the Journey of Hypervelocity Stars Figure 3.9: Geometric representation of the transformation method [24]. 1. For the semi-major axis probability distribution I get: f(y) = 1/y, x = F (y) = ln(y), y(x) = e x, and x is uniformly, randomly drawn between ln(y min ) and ln(y max ). 2. For the minimum approach distance probability distribution I get: f(y) = y, x = F (y) = y 2 /2, y(x) = (2x) 1 2, and x is uniformly, randomly drawn between y 2 min/2 and y 2 max/2. Before using these structures in the code, I check their reliability. Figures 3.10, 3.11, and 3.12 show the distributions of the random numbers generated with Normaldev, Abindev and Rmindev, respectively. To recap The code generates a sample of N = 10 5 stars. For each of these stars, the code: 1. creates a value of semi-major axis a bin, by means of Abindev; 62