M11, NGC 6705: Total Mass About 10000 M(Sun), 200 Myr Orion Nebula, Distance about 450 pc, Total Mass about 5000 M(Sun), Diameter about 3 pc Cluster formation • Observations versus Models • Important parameters 1. Time scale 2. Total mass 3. Initial Mass Function (IMF) 4. Velocity distribution 5. Binary fraction 6. Diameter 7. Density distribution Heuristic Approach • We know of 14 Open Clusters which are younger than 10 Myrs within 1000 pc around the Sun (Source: WEBDA) • There are also five star forming regions • Open Clusters still have to form within the solar vicinity • Total masses: up to 40 000 M(Sun) • Stable for some Gyrs • Evolutionary theory has to explain these facts Several spurious entries like the “ASCC clusters” Distribution of young open clusters and star forming regions from Alfaro et al., 2009, Ap&SS, 324, 141 Sun at [0,0] Distribution of star forming regions from Preibisch & Mamajek, 2008, Handbook of Star Forming Regions, Volume II Massive stars hotter than B0 (filled circles) and B0 to B2 (open circles) This means they are formed in the last few Myrs Star formation in the solar neighborhood is still ongoing Giant Molecular Clouds • Star Clusters can only form within „Giant Molecular Clouds“ (GMC) with a high enough initial mass • The stellar formation rate in the solar neighborhood is very low but not zero • But still there have to exist several GMCs to form Star Clusters • Is the formation process the same for all observed Galaxy types? Giant Molecular Clouds Stark & Lee, 2006, ApJ, 641, L113 Recent investigation of the 13CO Gas within 2000 pc around the Sun The number of young OCLs can be very well explained Formation rate of 0.45 OCLs per kpc-2 Myr-1 in the Galactic disk within 2 kpc around the Sun Battinelli & Capuzzo-Dolcetta, 1991, MNRAS, 249, 76 Star Clusters Single + Binaries NGC 6611 (M16) d = 1750 pc t = 8 Myr Star formation „live“ Initial Mass Function • The „Initial Mass Function“ (IMF) describes the mass distribution for a population of stars when they are formed together • Relevant astrophysics: 1. Size, total mass and metallicity of the initial GMC 2. Fragmentation of the GMC 3. Conservation of the angular momentum 4. Local and global magnetic fields 5. Accretion in the Pre-Main Sequence phase • The only observational parameter for the test of stellar formation and evolution models • We observe a luminosity function which has to be transformed to the IMF Initial Mass Function • Several most important questions are still not solved 1. Is the IMF homogeneous within the Milky Way? 2. Is the IMF constant throughout time? 3. What is the influence of the local and global magnetic field on the IMF? 4. What is the influence of the local and global metallicity on the IMF? Initial Mass Function The IMF q(m), often called „Present-Day Mass Function“ (PDMF), is defined as: dN = q(m) dm dN is the number of all stars per cubic parsec on the main sequence with a mass between M and (M + dm). But we observe not the masses of stars but their magnitudes and colors (relative and absolute) or luminosities. So we have to define the luminosity function and transform it into the IMF. Evolved stars In each row (MV + dM) there is a mixture of main sequence and evolved objects. For the IMF, we need the main sequence only. Evolved stars Luminosity function The luminosity function Y(MV), is defined as: dN = -Y(MV) dMV dN is the number of all stars per cubic parsec on the main sequence with an absolute magnitude between MV and (MV + dMV). The transformation to the IMF is given as: q(m) = -Y(MV)[dm(MV)/dMV]-1 The second term is the derivation of the Mass-Luminosity function m(MV). It is also depending on the age (t), metallicity (Z) and rotation (vrot) m(MV) = m(MV, Z, t, vrot) With which masses are these giants born? Correction of the observations We have to correct the complete observations for the evolved objects. There are three possibilities: 1. Take a statistical sample with a well known luminosity function (star clusters) 2. Take a statistical sample with well known photometric magnitudes and distances 3. Take isochrones = theoretical star evolution = models based on observations = circular argument All these methods are not self consistent and always introduce an unknown error to the analysis Salpeter, 1955, ApJ, 121, 161 Results of classical spectral classification, only 10% of stars with MV = -4.5 mag are on the main sequence! These values are depending on the chosen sample for the spectral classification and which classification scheme is applied. The errors are rather large. All observations have to be normalized to one “standard system” which means essentially to one “time scale”. The observations show, that this heuristic law describes them very well q(m) ≈ m-G Salpeter law (1955) Star cluster are one of the most important observational test for the IMF because they, normally, have well defined ages, distances and metallicities. However, the errors are still quiet large. But there is still no homogeneous IMF determination for open clusters taking into account the available data. Bastian et al., 2010, Annual Review of Astronomy and Astrophysics, 48, 339 Kroupa (2002, Science, 295, 82) most often used nowadays The observations show that the IMF cannot be described with a simple law for all masses TYCHO2 data Sanner & Geffert, 2001, A&A, 370, 87 Widely different G values Saurabh et al., 2008, AJ, 135, 1934 Typical values and errors No statistical analysis of star clusters with Gaia data is available, yet Magnetic field – star formation • Price & Bate, 2009, MNRAS, 398, 33 • Effects of magnetic pressure on fragmentation • Stronger magnetic fields seems to favor single star formation (see next page) Increasing magnetic field strength (B) Increasingmagneticfieldstrength Star formation Gravitation „wins“ Magnetic field, Shock wave Protostar FREE GAS NO FREE GAS Star formation • The detection of free gas in a Star Cluster is an excellent indicator for the time scale of continuous stellar formation Hartmann et al., 2001, ApJ, 562, 852 Star formation lasts 3 to 4 Myrs and is continuous This is also the “intrinsic” error of an age determination Numerical simulation of star formation in Giant Molecular Clouds • Hypothesis: the formation of all members of a star cluster is continuous for 3 to 4 Myrs within one GMCs • Is this a realistic approach? • Is it possible to simulation the formation of star clusters and compare the results with observational data within the solar vicinity? Numerical simulation of star formation in Giant Molecular Clouds • Detailed paper by Bate & Bonnell, 2005, MNRAS, 356, 1201 • Basis: Orion Nebula and Taurus star forming region • “Complete” astrophysical numerical simulation including Shock Waves, dynamical parameters and 3D-Hydrodynamics, Jeans Mass < 1 M(Sun) • The numerical simulations are astonishing close to the observations Numerical simulation of star formation in Giant Molecular Clouds Input parameter: 1. Mass (GMC) = 50 M(Sun), limited by CPU time 2. Diameter = 0.375 pc, limited by CPU time 3. Time for the gravitational collapse: 19000 years 4. Random turbulence field with a 3D Gaussian distribution „Stars“: Mass > 0.084 M(Sun) Brown Dwarfs: Mass < 0.084 M(Sun), no Hydrogen burning More low mass stars formed due to the IMF For star clusters it is essential to know the internal velocity distribution because of their evolution (see later) Continuous star formation in time The formation of Binary systems Binaries are connected with a line The rms velocity dispersion of the simulations is 4.3 km s-1 Such observational data for d > 500 pc are still not available => Gaia DR3 will (hopefully) provide it Situation – Gaia DR2 So can we use this for studying the internal velocity distribution (rms = 4.3 km s-1 )? Situation – Gaia DR2 Huge spread and only a few members per cluster Situation – Gaia DR2 Not useable Formation of Globular Clusters • Globular Clusters also formed from one GMC • But how are GCLs formed in Galaxies? Taniguchi et al., 1999, ApJ, 526, L13 Formation of Globular Clusters My interpretation: the results are inconclusive Evolution of Star Clusters • Star Clusters form with the following characteristics 1. Total Mass: IMF 2. Metallicity 3. Kinematics of the Cluster center: location within the Galaxy 4. Internal velocity dispersion • How does a Star Cluster evolve with these starting parameters? • Each member (= star) evolve “as an individual”, some important topics 1. Binary Evolution 2. Mass Loss (hot stars) 3. AGB Evolution 4. Planetary Nebula (cool stars) 5. Supernovae explosions • In Star Clusters, collisions are very uncommon (see later), almost no new multiple (binary) systems form during the later evolution • Star Clusters, normally, follow Galactic Rotation Each member (= star) evolve “as an individual”, so we have to take into account all these stages these stages And we should find White Dwarfs or Planetary Nebulae in Star Clusters Planetary Nebulae Majaess et al., 2007, PASP, 119, 1349 PNs exist in Open Clusters From Chandra, Diameter 4 pc Important topic of how SN explosions affect the cluster evolution Shockwaves Mass flow Statistically, SN explosions are rather common SN Remnants • Catalogue of Galactic SNRs: http://www.mrao.cam.ac.uk/surveys/snrs/ • 294 entries • Complete list of papers for Open Clusters 1. Pauls, 1977, A&A, 59, L13: NGC 559? 2. Kumar, 1978, ApJ, 219, L13: Tr 18 and 21? 3. Peterson et al., 1988, MNRAS, 235, 1439: Lynga 1, Pismis 20, Stock 14, and Trumpler 21, none conclusive • No new papers available • They exist but are extremely rare Muno et al., 2006, ApJ, 636, L41: Westerlund 1 d = 5200 pc, log t < 6.4 Pulsar, V fainter than 25th mag • White Dwarfs were detected in Open Clusters • The number is compatible with a common stellar evolution scenario, but the membership determination is very difficult • The absolute magnitude of WDs is about 10 magnitudes fainter than the corresponding Main Sequence von Hippel, 1998, AJ, 115, 1536 Single Multiple In total, 41 WDs until 1998 found, no firm improvement after that Why do Star Clusters dissipate? Virial Theorem: Kinetic Energy: Potential Energy: yielding: kinE2 centerclusterthetorelative memberstheofmean 2 22 vv vMvmnE ikin   2 2 2 2 2 2 1 R MG v R MG     Escape Velocity: Collisions: Density r and cross section s: 22 4 vv  v tcoll   sr 1 vRN R tR R N coll   2 * 3 2 *3 4 4  sr Example of a typical Open Cluster: 5pcR,5.2R,kms10,1000 * -1  SunRvN tcoll = 1025s => Collisions play no role Even in the most inner core parts, collisions are highly improper, but could occur Conclusions: 1.Binary and Multiple systems are not results of collisions in later stages but form already at the very beginning 2.Members do, in general, not escape due to collisions (swing-by effect), but their peculiar velocity component is part of the cluster formation or due to a SN explosion Crossing Time: Members can escape from a Star Cluster on a reasonable time scale Reasons: Velocity dispersion caused by the cluster formation and SN events (“angular momentum kick”) v R tcross   yrtv cross 8-1 109.45pcRandkms10  Tidal Forces due to Differential Galactic Rotation Total Mass of the Milky Way: MG = 2x1011 M(Sun) Gravitational acceleration of the complete star cluster gG and the individual member g* : The difference of these two values, is the force, of which “the Milky Way” tries to pull away a star from the cluster  2 rR MG g GC G   2 GC G G R MG g   On the other side we have the gravitational force of the open cluster. The stability radius rS is defined as: For 1000 M(Sun) => Diameter 20 pc 3/1 23 2 2           G OC GCS s OC GC SG M M Rr r MG R rMG GC GC G G Rr R rMG g    for 2 3,* ]pc,[Minkpc8for 1000 9.10 Sun 1/3        GC OC S R M r Tidal Forces due to Differential Galactic Rotation • Star Clusters lose members on their way following the Galactic rotation due to differential rotation and internal velocity distribution as well as “SN kicks” • These stars become “Galactic field stars” • For some Star Clusters, also tidal tails were found. These are former members still very weakly bound to the Star Cluster following its path Tidal Trails - Coma Berenices Tang et al., 2019, ApJ, 877, 12 Open Cluster, age about 400 Myr Tidal Trails – Palomar 5 Bonaca et al., 2020, ApJ, 889, 70 Globular Cluster, age about 11 Gyr Summary • Star Cluster dissipate because of 1. Differential Galactic Rotation 2. Internal Velocity Dispersion 3. Collisions in the first few Myrs 4. SN Explosions and corresponding Shock Waves 5. ( Collisions with “Field Stars”) • Explains the existence of Globular Clusters • Valid for all Spiral Galaxies