Constraints on the Dynamical Evolution of the Galaxy Group M81 3

arXiv:1701.01441v1 [astro-ph.GA] 5 Jan 2017 o.Nt .Ato.Soc. Astron. R. Not. Mon. aur 2017 January 9 Republic Czech o srnm Af) u e ¨gl7,D511Bn,Ger- Bonn, D-53121 Hügel 71, dem Auf many (AIfA), Astronomy for ‡ Germany Bonn, D-53121 † ⋆ mat- dark particle of presence the to due obser at about the attempt comes that deficit hypothesize One to galaxies. is of extrapolation curves this t rotation solving as the such ex- of observations, the the shape regime flat for account curvature to weak to (Koyama fails very regimes trapolation magnitude, the curvature in of weak However, very orders 2016). and by strong system. very extrapolated, Solar the commonly the within is derived It Ein- empirically as by been reformulation matter has through its stein distortion and space-time Newton geometrical by a gravitation of law The INTRODUCTION 1 .Oehm W. galaxy the of M81 evolution group dynamical the on Constraints 3 2 1 [email protected],Psa drs:Aglne Ins Argelander address: Postal H¨[email protected], dem Auf address: Postal [email protected], [email protected] hre nvriyi rge aut fMteaisadPh and Mathematics of Faculty Prague, in University Charles emot-ntttfu talnudKrpyi (HISKP), Kernphysik und für Strahlen Helmholtz-Institut csf G letNslrSr 0 -63 alrh,Ge Karlsruhe, D-76131 10, Albert-Nestler-Str. AG, scdsoft 1 ⋆ .Thies I. , 000 0–0 00,acpe)Pitd9Jnay21 M L (MN 2017 January 9 Printed accepted) (0000, 000–000 , 2 bevdpstoso h he aaistesltosfudcompr words: found Key solutions the galaxies motions. three proper genetic the calcu their co the of (N-body are and systems positions reached self-consistent observed conclusions indepen method live The of and Carlo environment. simulations separate Monte resolution system two chain SAP of the Markov employment rece using a the the namely within by from methods, other derived arriving each galaxy– are encounter are multiple simultaneously results which to dyna allow only, happen account which galaxies and into constellations unbound taking comprise initial T statistically encounters non-merging profiles. examined Navarro-Frenk-White living, compr are on 30 Long a galaxies NGC based cover and these model M82 to of M81, three-body order members dyn a core In possible three as inner for model. the studied standard conditions, here initial dark-matter is the group near interact This within the galaxy–galaxy occurred. significant throughout having typ past distribution by merger for matter gas evidence baryonic shows observed visible par galaxies the The of of model magnitude. size standard the of the and beyond mass embed particles the are of extending galaxies made cosmology, are of which model halos standard the to According ABSTRACT uainlmethods putational † n .Kroupa P. and aatcdnmc akmte tnadmdlo omlg co - cosmology of model standard - matter dark - dynamics galactic titute ved 71, usle 41,D515Bn,Germany Bonn, D-53115 14-16, Nussallee he rmany 2 , 3 sc,Atooia nttt,VHoleˇsoviˇckách V Institute, CZ-1 Astronomical 2, ysics, ‡ 04adRcte ta.2011). al. et Richtler and 2014 h yaia vdne(..Smrv´ azgr2005, Danziger Samurović Samurović & & (e.g. 2006, Samurović evidence of conf dynamical lack anisotropies the the velocity of because because and ellipti- constrained tracers around observable less much gravitation potentials are galaxies non-Newtonian the cal 2012), or McGaugh halos & either (Famaey his- matter imply unambiguously long large dark galaxies to a particle disk curves has in rotation observed observationally flat radii approximately inferred the be While tory. can halos the ter of particles known physics. of particle variety of the model with with standard weakly and most at other and each particles gravitationally cold- new interact plus the only (LCDM) dark-energy must cosmology standard of the model In dark-matter model physics. standard the particle of of extension an proposing therewith ter, n motn etbeipiaino h atcedark particle the of implication testable important One h usinwehrteeitneo akmat- dark of existence the whether question The A T E tl l v2.2) file style X ikv´ 08 Samurović 2008, Cirković ´ e ndr matter dark in ded osbtwtota without but ions ain) ie the Given lations). s rdcin for predictions ise orbits possible he il hsc,thus physics, ticle yM1gopof group M81 by t50Mr Our Myr. 500 nt fimdb high- by nfirmed clytoorders two ically mclsolutions amical hniestof set ehensive etstatistical dent ia friction. mical a distance far a 7aetreated are 77 00 rh 8, Praha 00 80 algorithm galaxy use m- 2 W. Oehm, I. Thies and P. Kroupa matter halos is that they imply significant dynamical friction the putative dark matter halos. For this purpose the three- when a galaxy encounters another galaxy (Kroupa 2015). particle integration is combined in a novel approach with Indeed, “Interacting galaxies are well-understood in terms the genetic algorithm and the Markov chain Monte Carlo of the effects of gravity on stars and dark matter” (Barnes method to yield robust estimates of the allowed pre-infall 1998), and interacting major galaxies merge within about 1- configurations. 1.5 Gyr after infall (Privon et al. 2013). This is the sole ori- Thus, as a first step, we perform numerical simulations gin of the hierarchical, merger-driven formation of structure of the possible orbits of the three core members M81, M82 in the LCDM model, which is popularly assumed to be the and NGC 3077, treated as a three-body problem with rigid main physical process driving the formation and evolution Navarro-Frenk-White profiles (NFW, Navarro et al. 1995) of the galaxy population observed at the present cosmolog- for the DM halos. Based on the knowledge of today’s plane ical epoch. Thus, if observed galactic systems show either of sky (POS) coordinates, the distances and the radial veloc- evidence for dynamical friction or absence of this important ities, as well as the knowledge about the North Tidal Bridge process then this would test the presence of dark matter (NTB) and South Tidal Bridge (STB) obtained by radio halos (Kroupa 2015). astronomical observations, two different methods (Markov Located about 3.6 Mpc from the Local Group of galax- chain Monte Carlo (MCMC) and genetic algorithm(GA)) ies, the M81 group of galaxies is the closest similar con- are utilized for the delivery of statistical populations for the stellation of galaxies. It therefore offers opportunities for unknown, or just roughly known physical entities under the the investigation of the dynamical behavior of such loose constraint that known conditions be fulfilled. For a com- but typical groups of galaxies, based on thorough available prehensive statistical evaluation, the simplified three-body observational data. Thus, if information exists which con- model serves as a first basis since numerical N-body simula- strains the past encounters between group members then tions wouldn’t allow a comprehensive scan of initial parame- the process of dynamical friction can be tested for. For ex- ters within an appropriate project time. By then comparing ample, the Local Group of galaxies is constrained by the individual solutions with high-resolution simulations of live two major galaxies not allowed to have merged, such that self-consistent systems the results of these estimates are ver- Andromeda and the Milky Way never had a close encounter ified. in the past if the LCDM is valid (Zhao et al. 2013). In Section 2 we briefly discuss the impact of dark mat- The M81 group of galaxies is comprised of a ter (DM) halos on the orbits of bodies upon intruding their major Milky-Way type galaxy (M81, baryonic mass interior and present the model for the Coulomb logarithm 10 MM81 3 10 M⊙) and four companion galaxies, two examined in this publication. Section 3 presents facts about ≈ · of which are themselves dwarf but nevertheless substan- the M81 group important for our investigations, and our 10 tial star-forming disk galaxies (M82, MM82 1 10 M⊙ approach how to achieve meaningful statistical results con- 9 ≈ · and NGC 3077, MN3077 2 10 M⊙). The interesting ob- cerning their dynamical behaviour. Sections 4 and 5 deal ≈ · servational finding is that these galaxies are enshrouded with the details of our application of the mentioned sta- by HI-emitting gas spanning the dimensions of the inner tistical methods, MCMC and GA, to the inner M81 group group which is about 50-100 kpc. This requires the galaxy and the results obtained there, followed by a general discus- pairs M81/M82 and M81/NGC 3077 to have interacted sion comparing those methods (Section 6). In Section 7 we closely at least once, in order to strip the outer gaseous present the results of individual N-body calculations com- disks and rearrange the matter in the long tidal features pared to the orbits of our three-body calculations, randomly (Cottrell 1977, Gottesman & Weliachev 1977, van der Hulst extracted from the statistical population. Finally, in Sec- 1979, Yun et al. 1993 and Yun et al. 1994). This constrains tion 8 we discuss the results obtained. For easy reading, the distribution of dark matter in the group by the pro- Appendix B provides a list of acronyms used in this publi- cess of dynamical friction, because the time span between cation. the encounters and the merging due to dynamical friction is constrained to be at least about a crossing time, i.e. about 0.5–1 Gyr (for a typical velocity of 200 km/s and over a spa- 2 DARK MATTER HALOS AND DYNAMICAL tial range of 50–100 kpc). FRICTION The M81 group must have assembled through the infall Exploring the dynamics of bodies travelling along paths in of the individual galaxies according to the merger tree which the interior of DM halos implies that the effects of dynam- is a necessary part of the hierarchical structure formation in ical friction have to be taken into account in an appropri- the LCDM model. The currently observed configuration of ate manner (Chandrasekhar 1942). For isotropic distribu- the M81 group constrains the pre-infall initial conditions tion functions the decelaration of an intruding body due since for example tightly bound initial conditions are most to dynamical friction is described by Chandrasekhar’s for- likely ruled out given that the M81 group has not merged mula (Chandrasekhar 1943), which reads for a Maxwellian yet to an early-type galaxy. Full-scale simulations of live self- velocity distribution with dispersion σ (for details see gravitating disk galaxies, each with their inter-stellar media Binney & Tremaine (2008), chap. 8.1): are too prohibitive, given that only four dimensional ob- 2 servational information is available in six-dimensional phase d~vM 4πG Mρ 2X 2 −X ~v space, to allow a wide scan of parameters to seek those pre- = 3 lnΛ erf(X) e M , (1) dt − vM − √π infall configurations which lead to the presently observed group structure. Therefore, as a first step simplified but very with X = vM /(√2σ). The intruder of mass M and relative rapid simulations are performed here taking into account velocity ~vM is decelareted by d~vM /dt in the background the essential physical process, namely dynamical friction on density ρ of the DM halo. Constraints on the dynamical evolution of the galaxy group M81 3

Simulating galaxy-galaxy encounters Petsch & Theis 3.2 Previous Results (2008) showed that a modified model for the Coulomb log- Thomasson & Donner (1993) simulated the tidal interaction arithm lnΛ, originally proposed by Jiang et al. (2008), de- between M81 and NGC 3077, reproducing the STB and the scribes the effects of dynamical friction in a realistic manner. spiral structure of M81. The orbit generating the best fit This mass- and distance-dependent model reads: shows a pericentre passage with a distance of 22 kpc about M (r) 400 Myr ago (i.e. at time 400 Myr). lnΛ = ln 1+ halo , (2) − M Yun (1999) additionally considered the tidal interaction between M81 and M82, too, reproducing the where Mhalo(r) is the mass of the host dark matter halo NTB as well as the STB. The pericentre passages oc- within radial distance r. cur at -220 Myr with a separation of 25 kpc and at Chandrasekhar’s formula only gives an estimate at -280 Myr with a separation of 16 kpc for M82 and hand. However, high-resolution simulations of live self- NGC 3077, respectively. (A GIF-movie is still available at consistent systems presented in Section 7 confirm our ap- http://www.aoc.nrao.edu/ myun/movie.gif). ∼ proach of employing this semi-analytical formula in our Full N-body simulations by Thomson et al. (1999) for three-body calculations. the three core members did not yield satisfactory results because the galaxies merge. Indeed, due to the effect of dynamical friction within the DM halos, the solutions found by Yun (1999) and Thomson et al. (1999) would result in merged 3 THE INNER GROUP galaxies2. Obviously, this begs the question how likely it is As already indicated, the three core members M81, M82 and to catch the M81 system, which is also the next and nearest NGC 3077 are treated as a three-body system, based on rigid Local Group equivalent, just in this special evolution phase NFW-profiles for the DM halos. In our notation the indices after the close encounters between its inner three members 1, 2 and 3 represent M81, M82 and NGC 3077, respectively. and before any one of them has merged. Further investiga- M81 appears to be the most massive object accompanied by tions of the behaviour of the inner M81 group have not been M82 and NGC 3077. published since Yun (1999) and Thomson et al. (1999).

3.1 Observational Data 3.3 DM halo masses The observational data for the plane-of-sky (POS) co- Following Behroozi et al. (2013) we derived the baryonic and ordinates and the radial velocities are taken from the DM Halo masses from the bolometric luminosities, as ex- NASA/IPAC Extragalactic Database (NED)1, query sub- tracted fom NED (query submitted on 3 Nov 2015). The mitted on 8 Feb 2014, and are presented in Table 1. The results are shown in Table 1. distances, however, are only roughly known: even for M81 The masses of the DM halos listed in Table 1 may an uncertainty of about 10% due to the variation of recent appear very high compared to the baryonic masses. How- results published over the last decade can be extracted from ever, following fig. 7 and 8 in Behroozi et al. (2013) DM-to- NED. In our context we are interested in the relative dis- baryonic mass ratios of the order of 100:1 are expected for tances of the three core members related to the M81 frame the baryonic masses listed in Table 1. The lowest ratio is 12 at present, rather than absolute distances. Therefore we take about 40:1 at a DM halo mass of 10 M⊙, as can be seen the average value for M81 as granted and cater for the men- for M81. tioned uncertainty by means of appropriate ranges for the This is in line with Binney & Tremaine (2008) (p. 18, distances of the companions M82 and NGC 3077, as shown table 1.2), who quote for the M81 comparable Milky Way 10 in Table 1. As a final remark regarding the phase space, the a baryonic mass of 5 10 M⊙, and a DM halo mass of 12 · POS velocity components are not known at all. 2 10 M⊙. · Apart from optical observations, radio astronomical data are at our disposal. M81, M82 and NGC 3077 are known to be embedded in a large cloud of atomic 3.4 The Model hydrogen as observations of the 21cm HI emission line The DM halo of either galaxy is treated as a rigid halo with have shown (Appleton et al. 1981). Cottrell (1977) and a density profile according to Navarro et al. (1995) (NFW- Gottesman & Weliachev (1977) identified the NTB connect- profile), truncated at the the radius R200: ing M81 and M82, as van der Hulst (1979) did for the STB between M81 and NGC 3077. The results of those obser- ρ0 ρ(r)= 2 , (3) vations were confirmed by Yun et al. (1993) and Yun et al. r/Rs (1 + r/Rs) (1994) by investigating the inner M81 group with the Very with Rs = R200/c, R200 denoting the radius yielding an Large Array Telescope. Since then, the fact that both com- average density of the halo of 200 times the cosmological panions must have experienced close encounters with M81 critical density in the recent cosmological past has been commonly agreed to be an established feature of the inner group. A review of the entire situation is presented by Yun (1999). 2 ”...this simulation resulted in the mergers of the companions onto M81, probably because the dynamical dissipation associ- ated with the truncated isothermal model halo used was too effi- 1 http://ned.ipac.caltech.edu/ cient.”(Yun 1999, p. 88). 4 W. Oehm, I. Thies and P. Kroupa

Object RA DEC Heliocentric Distance Lv Baryonic Mass DM Halo Mass (EquJ2000) (EquJ2000) Velocity (visual) (M⊙) (M⊙)

M81 09h55m33.2s +69d03m55s -34 km/s 3.63 Mpc 2.04 · 1010 3.06 · 1010 1.17 · 1012 M82 09h55m52.7s +69d40m46s 203 km/s (3.53 ± 0.6) Mpc 8.70 · 109 1.305 · 1010 5.54 · 1011 NGC 3077 10h03m19.1s +68d44m02s 14 km/s (3.83 ± 0.6) Mpc 1.95 · 109 2.925 · 109 2.43 · 1011

Table 1. Observational and derived data for the inner M81 group members.

3 Object R200(kpc) ρ0 (M⊙/pc ) c P1 P2 P3 P4 P5 P6

−3 M81 210.4 7.210 · 10 10.29 v2x v2y z2 v3x v3y z3 M82 164.0 8.810 · 10−3 11.17 −2 NGC 3077 124.6 1.100 · 10 12.22 Table 3. Open parameters for the inner M81 group.

Table 2. NFW proﬁle parameters for the inner M81 group members. equations thereafter are presented in Appendix C. Figure 2 shows the forces between the galaxies due to gravity and dynamical friction, dependent on the relative velocities. 2 1.8 1.6 1.4 3.5 Approach 1.2

Λ 1 At first, calculating three-body orbits backwards up to ln 0.8 7 Gyr, statistical populations for the open parameters − 0.6 (Table 3) are generated by means of the methods of Sec- 0.4 tions 4 (MCMC) and 5 (GA). Following the results of the 0.2 0 previous publications mentioned in Section 3.2 we added, 0 50 100 150 200 additionally to the known initial conditions at present, the r [kpc] rather general condition (later referred to as COND) that both companions M82 and NGC 3077 encountered M81 Figure 1. The Coulomb logarithm lnΛ for M82 (lower curve) within the recent 500 Myr at a pericentre distance below and NGC 3077 (upper curve) penetrating the DM halo of M81, 30 kpc. shown as a function of the distance from the DM halo centre of M81. Each three-body orbit of those populations is fully de- termined by all the known and open parameters provided by either MCMC or GA. Starting at time 7 Gyr and calculat- 3H2 − ρcrit = , (4) ing the corresponding three-body orbits forward in time up 8πG to +7 Gyr, the behaviour of the inner group is investigated and the concentration parameter c with respect to the question of possibly ocurring mergers in the future. Mvir log10 c = 1.02 0.109 log10 12 (5) − 10 M⊙ (see Macciòet al. (2007)). The parameters for the three 4 STATISTICAL METHODS I: MARKOV NFW-profiles are presented in Table 2. CHAIN MONTE CARLO As explained in Section 2 in case of overlapping DM halos, not only the gravitational force derived from the NFW We follow a methodology proposed by Goodman & Weare profiles but also the effect of dynamical dissipation has to be (2010) employing an affine-invariant ensemble sampler for taken into account. As an example, the Coulomb logarithm the Markov chain Monte Carlo method (MCMC). A solid (Eq. 2) to be incorporated in Eq. 1 is displayed in Figure 1 instruction ”how to go about implementing this algorithm” for either companion M82 and NGC 3077 penetrating M81’s can be found in Foreman-Mackey et al. (2013), who also pro- DM halo. vide an online link to a Python implementation. However, In the M81 reference frame, at present, with x1= y1= we decided to realize this algorithm on our own utilizing the ABAP development workbench (see Appendix A). The ba- z1 = 0 and v1x = v1y = v1z = 0, our three-body problem is described by the POS coordinates and radial veloc- sics of applying this formalism to our situation are outlined ities of the two companions M82 and NGC 3077 relative in Appendix D. to M81, deducted from Table 1, their unknown POS velocity components, and their roughly known relative distances 4.1 Definition of the Posterior Probability also derived from Table 1. Table 3 shows the full set of open Density parameters. The details regarding the numerical solution for the cal- First of all, concerning the open parameters, we need to culation of the forces between the galaxies, as well as the account for the distance ranges specified in Table 1 as well as details concerning the numerical solution of the dynamical the fact that realistic velocities should be considered. This is Constraints on the dynamical evolution of the galaxy group M81 5

Gravitational force between M81 and M82 Gravitational force between M81 and NGC 3077 Gravitational force between M82 and NGC 3077 2e+011 1.2e+011 8e+010 1.8e+011 1.6e+011 1e+011 7e+010 1.4e+011 6e+010 8e+010 1.2e+011 5e+010 1e+011 6e+010 4e+010 8e+010 3e+010 6e+010 4e+010 4e+010 2e+010 2e+010 2e+010 1e+010 0 0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 Distance [kpc] Distance [kpc] Distance [kpc]

Dynamical Friction of M82 in M81 Dynamical Friction of NGC 3077 in M81 Dynamical Friction of NGC 3077 in M82 2e+011 1.2e+011 8e+010 1.8e+011 1.6e+011 1e+011 7e+010 1.4e+011 6e+010 8e+010 1.2e+011 5e+010 1e+011 6e+010 4e+010 8e+010 3e+010 6e+010 4e+010 4e+010 2e+010 2e+010 2e+010 1e+010 0 0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 Distance [kpc] Distance [kpc] Distance [kpc]

Dynamical Friction of M81 in M82 Dynamical Friction of M81 in NGC 3077 Dynamical Friction of M82 in NGC 3077 2e+011 1.2e+011 8e+010 1.8e+011 1.6e+011 1e+011 7e+010 1.4e+011 6e+010 8e+010 1.2e+011 5e+010 1e+011 6e+010 4e+010 8e+010 3e+010 6e+010 4e+010 4e+010 2e+010 2e+010 2e+010 1e+010 0 0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 Distance [kpc] Distance [kpc] Distance [kpc]

Forces between M81 and M82 at a distance of 10 kpc Forces between M81 and NGC 3077 at a distance of 10 kpc Forces between M82 and NGC 3077 at a distance of 10 kpc 2e+011 1.2e+011 8e+010 1.8e+011 1.6e+011 1e+011 7e+010 1.4e+011 6e+010 8e+010 1.2e+011 5e+010 1e+011 6e+010 4e+010 8e+010 3e+010 6e+010 4e+010 4e+010 2e+010 2e+010 2e+010 1e+010 0 0 0 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Relative velocity [pc/Myr] Relative velocity [pc/Myr] Relative velocity [pc/Myr]

Figure 2. The variety of forces between the galaxies presented in galactic units (M⊙, pc, Myr). First row: The gravitational forces between M81 and M82 (left), M81 and NGC 3077 (centre) and M82 and NGC 3077 (right). The distances between the centres of the galaxies are extended by 50 kpc beyond the separation distance of the DM halos, thus visualizing the continous transition to Newton’s law of gravitation for point masses upon separation. Second row: Forces, caused by dynamical friction, on M82 moving in the DM halo of M81 (left), NGC 3077 in M81 (centre) and NGC 3077 in M82 (right), for relative velocities of 100 pc/Myr (top curves), 300 pc/Myr (middle curves) and 500 pc/Myr (bottom curves). Third row: Same as second row, but vice versa. Fourth row: Forces due to gravitation (constant values) and dynamical friction at a separation distance of 10 kpc between the centres of the DM halos, shown in dependence on the relative velocity. Left: M82 in M81 (top curve) and M81 in M82 (bottom curve). Centre: NGC 3077 in M81 (top curve) and M81 in NGC 3077 (bottom curve). Right: NGC 3077 in M82 (top curve) and M82 in NGC 3077 (bottom curve).

ensured by an appropriate deﬁnition of the prior distribution with the appropriate values for zmin and zmax for either p(X~ ). companion derived from Table 1, and a value of 100 kpc Regarding the distance ranges for M82 and NGC 3077 for z0. To account for realistic velocities vi = ~vi (about | | we deﬁne (i = 2, 3) 6 500 km/s) in the centre-of-mass frame we implemented (i = 2, 3) 2 (zi zmin) exp − 2 , zi zmax , 2  2 ~ (vi vmax)  − 2 z0 pv(X)i (7) · ∝  exp − 2 , vi > vmax , 1, otherwise ,  − 2 v0  ·    6 W. Oehm, I. Thies and P. Kroupa

with vmax = 400 pc/Myr and v0 = 100 pc/Myr. All together MCMC: Autocorrelation functions our prior distribution is given by 1

3 0.8 p(X~ ) pz(X~ )i pv(X~ )i . (8) ∝ · i=2 Y 0.6 Exploiting the minimal distances d12 and d13 within j the recent 500 Myr between M81/M82 and M81/NGC 3077, ρ 0.4 respectively, the condition COND speciﬁed in Section 3.5 is 0.2 incorporated by the following contribution to the likelihood function (i = 2, 3): 0

1, d1i 6 dper , 2 -0.2 PC (X~ D)i (d1i dper) (9) 0 10000 20000 30000 40000 50000 60000 70000 80000 | ∝  exp − 2 , d1i >dper , t − 2 d0  · Figure 3. Autocorrelation functions ρ (t) (j = 1,...,6), calculated with dper = 25 kpc and d0 = 5 kpc. Denoting the initial j velocities at 7 Gyr (prior to the encounters) in the centre- for 300 000 ensembles of 100 walkers and displayed for the first − 80 000 ensembles. of-mass frame by ui = ~ui , we generate realistic values for | | those by mostly within the range of ( 0.1, 0.1). A detailed discussion 1, ui 6 vmax , − 2 of this behaviour is left for Section 6. Pu(X~ D)i (ui vmax) (10) | ∝  exp − 2 , ui > vmax , As explained in Section 3.5 the ensemble chains serve as − 2 v0  · input for three-body integration runs into the future. Tak- ing each of the ensembles t = 75 000, 100 000, ..., 300 000 taking the same values for vmax and v0 as in Eq. 7. Assem- bling both aspects we arrive at the likelihood function as a starting point for continuing the MCMC procedure, we extracted the first 1000 occurences of walkers fulfilling con- 3 dition COND in each case. This produces ten independent P (X~ D) PC (X~ D)i Pu(X~ D)i . (11) | ∝ | · | follow-up sets of walkers. As soon as one of the companions i=2 Y is seperated less than 15 kpc (the baryonic radius of M81’s disc) from M81 and does not leave this spatial extension 4.2 The First Ensemble thereafter anymore, we regard this time as the time of the merger process. Generating random numbers n 0, ..., 1000 separately for In accordance with the statements concerning the be- ∈ { } each walker and open parameter, the first ensembles are cre- haviour of the autocorrelation functions we present the ated according to (i 1, ..., 6 ) merger rates within the forthcoming 7 Gyr in Figure 4. Ap- ∈ { } parently no full convergence is achieved, and the results ap- Pi =(Pi)min + n [(Pi)max (Pi)min] /1000 . · − pear ”to walk around”. As an attempt to achieve meaningful The minimum and maximum values for the distances are results we calculated the averages of the ten follow-up sets derived from Table 1, and for the POS velocity components and show them in Figure 5 and Table 4. chosen to be 500 pc/Myr. Instead of applying clustering or alike methods in order ± to achieve full convergence, we decided to employ another statistical method, the genetic algorithm, which overcomes 4.3 Results difficulties like the problem of local maxima and the above mentioned one. As we will see, the results obtained there The comparison of the autocorrelation functions (Eq. D5) are in good agreement with those obtained by the MCMC for ensembles of either 100 or 1000 walkers (NL = method. 100, 1000), creating a chain of 50 000 ensembles in either case, did not show a significant advantage for the choice of the larger ensemble size over the smaller one. Therefore the evaluations were performed with NL = 100 creating a 5 STATISTICAL METHODS II: GENETIC chain of 300 000 ensembles. Figure 3 shows the behaviour of ALGORITHM the corresponding autocorrelation functions. The acceptance General aspects of the genetic algorithm (GA) are explained rates for the succeeding walkers, generated by the stretch in detail in Charbonneau (1995). As pointed out there, the moves, appeared to be 0.3, in accordance with the fact ≈ major advantage of GA is perceived to be its capability of that a range of (0.2, 0.5) is considered to be a good value in avoiding local maxima in the process of searching the global that regard. maximum of a given function (fitness function). A precise Examining the autocorrelation functions, one realizes description of the algorithm, especially instructions for the that a fairly good convergence towards the requested value implementation, can also be found in Theis & Kohle (2001). of 0 is already achieved after a couple of thousands of ensem- In our case each open parameter from Table 3 is mapped bles, except for parameter 6 which represents the distance to a 4-digit string (”gene”) [abcd] (i 1, ..., 6 ): of NGC 3077, where it takes about 60 000 ensembles. How- i ∈ { } ever, afterwards the autocorrelation functions keep varying, Pi =(Pi)min + [abcd] [(Pi)max (Pi)min] /10 000 . (12) i · − Constraints on the dynamical evolution of the galaxy group M81 7

75000 ensembles 100000 ensembles GA GA

14 14 14 1000 12 12 12 800 10 10 10 8 8 8 600 6 6 6 400 4 4 4 200 2 2 2 Cumulated number of mergers Number of mergers per 10 Myr 0 Number of mergers per 10 Myr 0 Number of mergers per 10 Myr 0 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 Myr Myr Myr Myr

125000 ensembles 150000 ensembles Figure 6. GA: Merger rates and cumulated number of mergers 14 14 12 12 displayed for the forthcoming 7 Gyr, based on a population of 10 10 1000 solutions. 8 8 6 6 4 4 2 2 dures of cross-over, mutation and ranking within a popula- Number of mergers per 10 Myr 0 Number of mergers per 10 Myr 0 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 tion of Npop genotypes. The first generation is established Myr Myr by randomly creating Npop 6 4-digit strings. 175000 ensembles 200000 ensembles × 14 14 12 12 10 10 5.1 Definition of the Fitness Function 8 8 6 6 4 4 In contrast to Section 4, where the stretch moves may take 2 2 walkers outside the ranges specified in Table 1, we need not Number of mergers per 10 Myr 0 Number of mergers per 10 Myr 0 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 take care of those ranges explicitly because Eq. 12 a priori Myr Myr guarantees that the distances are confined to their intervals. 225000 ensembles 250000 ensembles Otherwise we follow the goal to establish GA evaluations 14 14 12 12 compatible to what has been performed by means of MCMC. 10 10 Dropping Equation 6 we therefore arrive at the following 8 8 6 6 definition for the fitness function 4 4 2 2 3 Number of mergers per 10 Myr 0 Number of mergers per 10 Myr 0 ~ ~ ~ ~ 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 (X D)= pv(X)i PC (X D)i Pu(X D)i , (13) Myr Myr F | · | · | i=2 275000 ensembles 300000 ensembles Y

14 14 where the individual components are taken from Equa- 12 12 tions 7, 9 and 10. Like in Section 4, X~ denotes the parameter 10 10 8 8 vector while further entities incorporated are indicated by 6 6 D. 4 4 2 2

Number of mergers per 10 Myr 0 Number of mergers per 10 Myr 0 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 Myr Myr 5.2 Results Figure 4. MCMC: Merger rates based on the begin times as of We generated a set of 1000 solutions fulﬁlling condition the present displayed for the next 7 Gyr. (For further details refer COND. Upon detection of a ﬁrst solution the GA procedure to the text.) is being repeated by randomly creating a new population as a starting point, as explained above, until 1000 appropriate

MCMC MCMC three-body orbits are collected.

14 1000 The fitness function corresponds to the posterior proba- 12 800 10 bility density of Section 4. In Figure 6 the merger events oc- 8 600 curring within the next 7 Gyr are presented for that model. 6 400 4 The comparison of Figures 5 and 6 confirms the statement 200 2 that both statistical methodes applied, namely MCMC and Cumulated number of mergers Number of mergers per 10 Myr 0 0 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 GA, yield similar results with respect to the merger rates, Myr Myr and key numbers are presented in Table 4. Figure 5. MCMC: Merger rates and cumulated number of merg- It is worthwhile to investigate the pre-infall phase space ers displayed for the forthcoming 7 Gyr. In total 1000 walkers at -7 Gyr. For this purpose we continued the GA-runs until were considered by randomly extracting 100 walkers out of each 1000 merging solutions and 1000 solutions not merging were of the ten follow-up sets. collected. Figure 7 shows the corresponding 3D-plots which were rotated in a way that structures are efficiently visu- alized. Roughly evaluating Figure 8 for the spatial coordi- Here the minimum and maximum values for the parameters nates and Figure 9 for the velocity components of M82 and Pi are choosen to be the same as in Section 4.2. All genes NGC 3077 one might draw the conclusion that the phase together define the genotypes as 6 4-digit strings × space volume for the population of the merging solutions fairly exceeds the phase space volume for the population of [abcd]1 ... [abcd]6 , the non-merging solutions. However, for the population of which, when creating a new generation, undergo the proce- the merging solutions a higher correlation in phase space 8 W. Oehm, I. Thies and P. Kroupa

Positions of M82 at -7 Gyr (merger) Positions of N3077 at -7 Gyr (merger)

500 z -500 1000 -1500 z 0 3000

-2000 -1000 1000 y -1000 0 x 0 -1000 1000 -2000 -1000 -3000 y -3000 -1000 1000 2000 x

Positions of M82 at -7 Gyr (non-merger) Positions of N3077 at -7 Gyr (non-merger)

500 1000 z -500 z 0 -1500 0 3000 -1000 -2000 -1000 1000 y -1000 -2000 0 y -3000 -1000 -1000 x x 1000 1000 -3000 2000

Velocities of M82 at -7 Gyr (merger) Velocities of N3077 at -7 Gyr (merger) z v 0

-200 200 200 z 0 v 0 -300 -200 v -200 400 -400 -200 y -100 200 0 -400 vx 0 100 v -200 y -200 0 200 400 200 vx

Velocities of M82 at -7 Gyr (non-merger) Velocities of N3077 at -7 Gyr (merger) z v 0 -200 200 200 z 0 -300 v -200 0 -200 400 v -400 -200 y -100 200 0 vx 0 -400 100 v -200 y -200 0 200 400 200 vx

Figure 7. GA: Distribution of the pre-infall phase space at -7 Gyr for M82 and NGC 3077 in today’s M81 reference frame. A population of 1000 solutions merging within the forthcoming 7 Gyr (i.e. within 14 Gyr since -7 Gyr) is displayed in the ﬁrst and the third row for the coordinates (kpc) and the velocities (pc/Myr), respectively. Accordingly the second and fourth row refer to a population of 1000 solutions not merging. Constraints on the dynamical evolution of the galaxy group M81 9

MCMC GA

solutions not merging within next 7 Gyr 118 278

solutions not merging within next 7 Gyr and: neither M82 nor N3077 bound to M81 7 Gyr ago 117 276

solutions not merging within next 7 Gyr and: one companion bound to M81 7 Gyr ago 1 2

solutions for: M82 and N3077 bound to M81 7 Gyr ago 66 70

longest lifetime from today for: M82 and N3077 bound to M81 7 Gyr ago 2.7 Gyr 2.8 Gyr

average lifetime from today for: M82 and N3077 bound to M81 7 Gyr ago 1.7 Gyr 1.3 Gyr

Table 4. Key numbers for both statistical methods MCMC and GA, based on populations of 1000 solutions in either case. Actually, the three solutions not merging within the next 7 Gyr where one companion is bound to M81 merge after 7.3 Gyr (MCMC), and 7.8 and 8.2 Gyr (GA).

GA (extended fitness function) GA (extended fitness function) 5.4 Proper motions

14 1000 12 800 The proper motions of the galaxies are extracted from the 10 8 600 available solutions. Since the center-of-mass motion of the 6 400 M81 group is a free parameter in these calculations the 4 200 2 proper motions of M82 and NGC 3077 in the solutions for Cumulated number of mergers Number of mergers per 10 Myr 0 0 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 GA are calculated with respect to M81 at t = 0 (i.e. present Myr Myr day) in µas yr−1 (1 µas yr−1 corresponds to 17.2 km s−1 based on the distance of M81 being 3.62 Mpc). The results Figure 10. Same as Figure 6, but fitness function FE (Eq. 15). are shown in Fig. 11. The velocities are clearly limited to certain directions, at least for higher velocities. In the low- velocity end the direction angles are rather randomly dis- is evident from Figure 7 (especially for NGC 3077). There- tributed. The distribution of solutions in Fig. 11 allows to fore a solid statement regarding the ratio of the phase space place additional constraints on the models with dark matter, volume cannot be readily extracted. once the proper motion of M82 and/or of NGC3077 have been measured. In addition, the solutions computed with RAMSES (Sec. 7) are included as filled squares (M82) and 5.3 Extended Fitness Function filled circles (NGC 3077) while the corresponding GA solutions are depicted with open symbols, mutually connected In order to facilitate that the GA evaluations preferably gen- by lines. The dynamically self-consistent RAMSES solutions erate solutions where at least one of the companions M82 are in good agreement with the GA solutions. and NGC 3077 is bound to M81, we enhance the fitness function by a function depending on the three-body energy E in the centre-of-mass system before the encounter

1,E 6 Emin/2 , 6 STATISTICAL METHODS III: DISCUSSION 2 OF MCMC VS. GA fE = (E Emin/2) (14)  exp − 2 , E>Emin/2 . − 2 (Emin/2) As presented in Section 4.3 apparently no full convergence  · is achieved in our approach of applying the MCMC method Here Emin is the minimum three-body energy possible, de-  to the inner M81 group. The autocorrelation functions keep pending on the particular distances of M82 and NGC 3077 varying, though at low values (Figure 3), and the merger resulting from the genotypes. All together the extended ﬁt- rates appear ”to walk around” (Figure 4). ness function reads We expect that this phenomenon is caused by our choice

E (X~ D)= fE (X~ D) . (15) of the various contributions to the posterior probability den- F | ·F | sity. To be precise, all of them consist partly of a region with Thus avoiding the unlikely constellations that both com- a constant value 1, simply because we do not really know, panions are arriving from far distances fulfilling condition and any procedure defining preferred values there would COND by collecting solutions with a three-body energy mean a non-justified interference. As a result, for each open E 6 0 in the centre-of-mass system at 7 Gyr, the re- parameter the individual walkers have no direction so long − sulting merger rates for this fitness function are presented they find themselves in regions of constant value, and unless in Figure 10. After 12.6 Gyr, i.e. 5.6 Gyr as from today, all they are taken out from there again by a stretch move. The solutions will have merged. missing directions ”where to move to” in regions of constant 10 W. Oehm, I. Thies and P. Kroupa

Pre-infall coordinates of M82 at -7 Gyr (merger) Pre-infall coordinates of M82 at -7 Gyr (merger) Pre-infall coordinates of M82 at -7 Gyr (merger) 30 16 16

14 14 25 12 12 20 10 10

15 8 8

6 6 10 4 4

No. of occurrences per 10 kpc 5 No. of occurrences per 10 kpc No. of occurrences per 10 kpc 2 2

0 0 0 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 500 1000 -1500 -1000 -500 0 500 x (kpc) y (kpc) z (kpc)

Pre-infall coordinates of M82 at -7 Gyr (non-merger) Pre-infall coordinates of M82 at -7 Gyr (non-merger) Pre-infall coordinates of M82 at -7 Gyr (non-merger) 25 14 25

12 20 20 10

15 15 8

6 10 10

4 5 5 No. of occurrences per 10 kpc No. of occurrences per 10 kpc 2 No. of occurrences per 10 kpc

0 0 0 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 500 1000 -1500 -1000 -500 0 500 x (kpc) y (kpc) z (kpc)

Pre-infall coordinates of N3077 at -7 Gyr (merger) Pre-infall coordinates of N3077 at -7 Gyr (merger) Pre-infall coordinates of N3077 at -7 Gyr (merger) 16 14 16

14 12 14

12 12 10 10 10 8 8 8 6 6 6 4 4 4

No. of occurrences per 10 kpc 2 No. of occurrences per 10 kpc 2 No. of occurrences per 10 kpc 2

0 0 0 -3000 -2500 -2000 -1500 -1000 -500 0 500 -500 0 500 1000 1500 2000 2500 3000 -1000 -800 -600 -400 -200 0 200 400 600 800 x (kpc) y (kpc) z (kpc)

Pre-infall coordinates of N3077 at -7 Gyr (non-merger) Pre-infall coordinates of N3077 at -7 Gyr (non-merger) Pre-infall coordinates of N3077 at -7 Gyr (non-merger) 14 16 14

12 14 12

12 10 10 10 8 8 8 6 6 6 4 4 4

No. of occurrences per 10 kpc 2 No. of occurrences per 10 kpc 2 No. of occurrences per 10 kpc 2

0 0 0 -3000 -2500 -2000 -1500 -1000 -500 0 500 -500 0 500 1000 1500 2000 2500 3000 -1000 -800 -600 -400 -200 0 200 400 600 800 x (kpc) y (kpc) z (kpc)

Figure 8. GA: Distribution of the pre-infall coordinates at -7 Gyr for M82 and NGC 3077 in today’s M81 reference frame. A population of 1000 solutions merging within the forthcoming 7 Gyr (i.e. within 14 Gyr since -7 Gyr) is displayed in the first and the third row for M82 and NGC 3077, respectively. Accordingly the second and fourth row refer to a population of 1000 solutions not merging. value probably prevents, in our regard, the MCMC formal- erate solutions at all while establishing our concepts we de- ism from full convergence. cided to implement the Markov chain Monte Carlo method, As already indicated at the end of Section 4.3 we sacri- too. And in fact we were able to generate solutions by em- ficed on-top methodoligies to MCMC in order to overcome ployment of the MCMC method quickly. In our experience, this problem. The transfer to the genetic algorithm, im- MCMC is the stronger search engine in comparison to GA plemented as an additional independent statistical method, and therefore helped signifficantly to ”get feet on ground” turned out to be extremely valuable. For, this formalism during the early phase of our project. apparently delivers stable statistical results under circum- stances mentioned above. To establish this assesment we repeated the GA-runs for a second time and present the results in Table 5. Only slight deviations of the merger rates in comparison to the first evaluation are produced by the second one. To summarize our assessment: Both statistical methods However as a matter of fact, at the beginning of our im- turned out to be very valuable in our regard. MCMC helped plementation we started with the genetic algorithm. Since, to establish the concepts, and at the end GA delivered more by means of this algorithm, it appeared to be diffcult to gen- stable results. Constraints on the dynamical evolution of the galaxy group M81 11

Pre-infall velocities of M82 at -7 Gyr (merger) Pre-infall velocities of M82 at -7 Gyr (merger) Pre-infall velocities of M82 at -7 Gyr (merger) 140 70 60

120 60 50

100 50 40 80 40 30 60 30 20 40 20

20 10 10 No. of occurences per 10 pc / Myr No. of occurences per 10 pc / Myr No. of occurences per 10 pc / Myr

0 0 0 -300 -200 -100 0 100 200 300 -400 -300 -200 -100 0 100 200 300 400 -300 -200 -100 0 100 200

vx (pc / Myr) vy (pc / Myr) vz (pc / Myr)

Pre-infall velocities of M82 at -7 Gyr (non-merger) Pre-infall velocities of M82 at -7 Gyr (non-merger) Pre-infall velocities of M82 at -7 Gyr (non-merger) 140 40 90 80 120 35 30 70 100 60 25 80 50 20 60 40 15 30 40 10 20 20 5

No. of occurences per 10 pc / Myr No. of occurences per 10 pc / Myr No. of occurences per 10 pc / Myr 10 0 0 0 -300 -200 -100 0 100 200 300 -400 -300 -200 -100 0 100 200 300 400 -300 -200 -100 0 100 200

vx (pc / Myr) vy (pc / Myr) vz (pc / Myr)

Pre-infall velocities of N3077 at -7 Gyr (merger) Pre-infall velocities of N3077 at -7 Gyr (merger) Pre-infall velocities of N3077 at -7 Gyr (merger) 60 60 70

50 50 60

50 40 40 40 30 30 30 20 20 20

10 10 10 No. of occurences per 10 pc / Myr No. of occurences per 10 pc / Myr No. of occurences per 10 pc / Myr

0 0 0 -300 -200 -100 0 100 200 300 400 -500 -400 -300 -200 -100 0 100 200 300 400 500 -400 -300 -200 -100 0 100 200 300

vx (pc / Myr) vy (pc / Myr) vz (pc / Myr)

Pre-infall velocities of N3077 at -7 Gyr (non-merger) Pre-infall velocities of N3077 at -7 Gyr (non-merger) Pre-infall velocities of N3077 at -7 Gyr (non-merger) 50 70 70

60 60 40 50 50

30 40 40

30 30 20

20 20 10 10 10 No. of occurences per 10 pc / Myr No. of occurences per 10 pc / Myr No. of occurences per 10 pc / Myr

0 0 0 -300 -200 -100 0 100 200 300 400 -500 -400 -300 -200 -100 0 100 200 300 400 500 -400 -300 -200 -100 0 100 200 300

vx (pc / Myr) vy (pc / Myr) vz (pc / Myr)

Figure 9. GA: Same as Figure 8, but for the velocity components.

Table 5. GA: Percentages of mergers for selected periods of time from the present until maximally 7 Gyr. The original GA-evaluations are denoted by #1, and the repitition runs by #2.

Fitness function GA-run 0-1 Gyr 0-2 Gyr 0-3 Gyr 0-4 Gyr 0-5 Gyr 0-6 Gyr 0-7 Gyr

Eq. 13 #1 15.1% 42.8% 56.6% 63.3% 67.5% 69.8% 72.2% #2 13.9% 40.4% 54.0% 61.1% 67.2% 70.9% 73.2%

Eq. 15 #1 22.3% 62.5% 88.8% 96.8% 99.3% 100% 100% #2 23.2% 60.9% 86.1% 96.8% 99.2% 99.9% 100%

7 NUMERICAL SIMULATIONS WITH up as NFW proﬁles (Navarro, Frenk & White 1996) as in RAMSES the semi-analytic model described in Section 3.4. The virial masses are the same as in the semi-analytic model (see Ta- To validate the semi-analytic approach described in the bles 1 and 2). The total masses are taken as the DM halo previous sections, we computed numerical models with the masses, i.e. neglecting the baryonic content. The initial po- well-tested adaptive mesh reﬁnement (AMR) and particle- sitions and velocities are listed in in Table 6. In total 1.967 mesh code RAMSES (Teyssier 2002). The halos are set million particles are used for the models shown here, with 12 W. Oehm, I. Thies and P. Kroupa

250 30 200 20 150

10 100 -1 50 as yr

µ 0 / deg /

α 0 DEC v -10 -50

-100 -20 -150 -30 -200 30 20 10 0 -10 -20 -30 0 100 200 300 400 500 v / µas yr-1 -1 RA vabs / km

Figure 11. Proper motions of M82 (crosses) and NGC 3077 (open circles) of the GA solutions, all with respect to M81. Left panel: Proper motions in RA and DEC in µas yr−1. Note that the maximum velocity of the solutions in each x,y component is ±500 km s−1, corresponding to about ±30 µas yr−1. The solutions computed with RAMSES are shown as ﬁlled squares (M82) and ﬁlled circles (NGC 3077) while the corresponding GA solutions are depicted with open symbols, mutually connected by lines. Right panel: The −1 distribution of the absolute velocities, vabs in kms , and the position angles, α, in degrees, measured counterclockwise from the North Celestial Pole.

x y z vx vy vz

Model 1728-2 M81 153.33 -116.46 499.62 -19.04 16.67 -62.78 M82 268.33 -385.66 -1143. -38.02 46.11 146.66 NGC3077 -1350. 1439.94 200.22 178.35 -185.38 -32.09

Model 1728-6 M81 13.04 409.78 475.99 -3.82 -49.76 -61.98 M82 -166.98 -1100.0 -1009.9 16.39 139.63 132.02 NGC3077 317.92 534.88 10.61 -18.97 -78.76 -2.57

Model 1728-16 M81 265.72 597.69 256.07 -27.57 -81.71 -31.57 M82 -245.68 -1731.1 -567.83 29.68 230.47 72.63 NGC3077 -719.28 1068.89 61.64 65.10 -132.00 -13.59

Model 1729-633 M81 -18.84 205.98 283.40 -2.07 -12.41 -20.41 M82 68.10 -467.72 -530.50 -13.74 31.52 38.45 NGC3077 -64.56 74.58 -155.10 41.29 -12.12 10.59

12 11 Table 6. Initial positions, and velocities of the halos hosting the galaxies M81 (virial mass = 1.17 · 10 M⊙), M82 (5.54 · 10 M⊙), and 11 6 NGC3077 (2.43 · 10 M⊙). The positions are in kpc, the velocities in km/s. Each halo is modelled by particles of 1 · 10 M⊙, i.e., the M81 halo contains 1.17 million particles, M82 554,000 particles and NGC3077 243,000 particles.

6 each particle representing 10 M⊙. Each simulation takes (1985). The density centre method rather than the centre of between about 10 and 30 hours with 32 parallel threads mass has been chosen to determine the positions and mu- used. Additional runs with 196,700 and 19,670 particles have tual distances of the halos since the in-falling halos lose large been performed to conﬁrm the models to be essentially in- amounts of mass upon encounter (see Fig. 12). The resulting dependent of the particle number, which is indeed the case. steady shift of the centre of mass would render its position The halos are allowed to settle for several Gyr before being useless for the purpose of this study. In addition, the density included in the initial conditions for the simulations. The centres are the best representations of the positions of the position of each halo is represented by the density centre embedded galaxies which can be observed today. calculated via the method described by Casertano & Hut The models studied in this section correspond to the 3- Constraints on the dynamical evolution of the galaxy group M81 13

1000 -7000 Myr -3000 Myr -1000 Myr

500

-500

1000

y kpc 0 Myr 2000 Myr 7000 Myr

500

-500

-1000 -1000 -500 0 500 1000 -500 0 500 1000 -500 0 500 1000 x kpc

Figure 12. Snapshots of the merging halos of NGC 3077 (red) and M82 (green) with M81 (blue). The initial conditions correspond to model 1729-633. It can be seen that the density centres merge within several Gyr after their first encounter with M81. Both halos lose substantial amounts of mass, partially ejected as tidal streams (NGC 3077), partially as a wide-angle spray cloud (M82). Also shell-like DM structures form (lower right panel). body genetic algorithm solutions No. 1728-2, 1728-6, 1728- large mass loss along two dense tidal streams which causes 16, and 1729-633 for fitness function Eq. 13, and to the so- a larger energy loss during the encounter. This seems to lutions No. 1750-314, 1750-224, 1750-423, and 1750-445 for contradict Boylan-Kolchin et al. (2008) who predict longer the extended fitness function Eq. 15. They are shown on the merger times especially for large satellite-to-host mass ra- left side of Figure 13 and 14, respectively, and were chosen tios compared to the results of semi-analytic calculations randomly from the full set of GA solutions. using Chandrasekhar’s formula. In the Coulomb logarithm model used in the previous sections, on the other hand, the merger timescale for the semi-analytic calculations are 7.1 Results slightly longer than in the RAMSES simulation rendering Figure 12 shows snapshots of the encounter leading to the that Coulomb logarithm more conservative. The halos merge merger of all halos for model 1729-633. As can be seen the as fast as in GA and even faster in the right panel, thus val- dynamical friction causes the in-falling halos of NGC 3077 idating the GA solutions. There are some additional effects (red) and M82 (green) to decelerate quickly and merge to be noted: with the halo of M81 (blue) after a few oscillations. Fig- The in-falling halos of M82 and NGC 3077 suffer ure 13 and Figure 14 show the mutual distances for the • galaxies. The left-hand frames depict the results from the substantial mass loss during the merging process. The 3-body GA calculations while the right columns contain ejected DM forms tidal streams in almost opposite direc- the results of the RAMSES N-body simulations. The solid tions (NGC 3077, see Fig. 12) as well as a wide spray (M82), line refers to the distances of M82 to M81, the dotted line depending on the encounter parameters. The density centres of M82 and NGC 3077 first oscil- those of NGC3077 to M81, and the distances between the • minor components M82 and NGC3077 are shown by the late around M81 and then merge shortly after the present dashed line. All mutual distances follow a time evolution (marked by time = 0). very similar to the semi-analytic solution. The RAMSES All halos merge faster or lose more kinetic energy in the • N-body computations lead to slightly to substantially lower full N-body computations than in the semi-analytic model post-encounter velocities compared to the semi-analytical (right panels of Fig. 13). This emphasises the importance of 3-body calculations. The reason for this is probably the dynamical friction. 14 W. Oehm, I. Thies and P. Kroupa