Linear stability of rotating spherical halos

Simon Rozier

J.-B. Fouvry, P. G. Breen, A. L. Varri, C. Pichon, D. C. Heggie

News from the Dark - LUPM - 20-22/03/2019

Simon Rozier 1 20.05.2019 5594 S. Kamann et al. 5598 S. Kamann et al. Downloaded from https://academic.oup.com/mnras/article-abstract/473/4/5591/4562634 by UPMC user on 11 March 2019 Downloaded from https://academic.oup.com/mnras/article-abstract/473/4/5591/4562634 by UPMC user on 11 March 2019

Spheriodal stellar systems are rotating

Globular Clusters Elliptical & SpheroidalThe SAURON Galaxies project – IX 403 Nuclear Star Clusters

1000 NGC 4244 SETH ET AL. Vol. 687 (nearby, spiral, edge on) Downloaded from https://academic.oup.com/mnras/article-abstract/379/2/401/1022477 by UPMC user on 11 March 2019

8 Sollima et al.

Table 3. Detection of rotation in GCs from literature results: checkmarks, crosses and tildes mark positive, negative and 2σ detections, respectively. Checkmarks within parenthesis mark those GCswithuncertaindetectionsfoundinthiswork.References:Lane et al. (2010, L10), Bellazzini et al. (2012,B12),Fabricius et al. (2014,F14),Lardo et al. (2015,L15),Kimmig et al. (2015,K15),Kamann et al. (2018, K18), Ferraro et al. (2018,F18),Gaia CollaborationKamann et al. (2018b ,G18),et al.Bianchini 2017 et al. (2018,B18),Vasiliev (2018,K18).OnlyGCs Figure 1. GRI mosaics of all observed clusters, created from the reducedFigure MUSE 5. Results data of and the kinematic ordered analysis by increasing forwithat three least a clusters 2σ detection NGC from have been number. our listed. sample, Note NGC 104 that (top), for NGC NGC 6441 104 (middle) (47 and NGC 7089 (bottom). The left-hand Tuc) and NGC 5139 (ω Centauri), two and three pointings, respectively, withpanels larger show the distances radial rotation to and the dispersion cluster profiles, centres respectively.Name are not L10 The shown. B12 dashed F14 L15In and each K15 dotted K18 mosaic, vertical F18 G18 lines B18 north indicate V18 isthis the work up core and half-light radii of each cluster; NGC 104 !! !! !!! ! 10 pc all values were taken from Harris (1996). The central panelNGC shows 288 the X position X angle of X the rotation! curveXX and its uncertainty X for each radial bin. A blue dashed and east is left. NGC 362 X ∼ ! XX X line is used to indicate the cluster’s photometric semi-majorNGC axis 1261 angle as determined in Section∼ 6.1, with the blue shadedX area indicating the uncertainty and the length of the line scaling with cluster ellipticity. The right-handNGC 1851 panels!! show Voronoi-binned ! maps of∼ theXX mean velocity and the velocity dispersion across NGC 1904 X ! X(!) 4042 A. Feldmeier-Krause et al. NGC 2808 !!X ! XX ! Seth et al. 2008 the footprint covered by the MUSE data. The dashed circlesNGC indicate 3201 again!!! the core radii of the clusters.∼ SimilarX plots∼ for the remaining clusters of oursample are presented in Appendix A. NGC 4372 X ! ∼∼ NGC 4590 X ! XXXX 4DATAANALYSIS The spectra were extractedNGC from 5024 the X Xcubes! inX a multi-step process.∼ Milky Way NGC 5139 !!!!!! NGC 5272 ! X !!!∼ (!) First, the subset of sources fromNGC 5286 the input catalogue that is∼ resolvedXX In large parts, the analysis of the data cubes was done in a similar NGC 5466 ! X in our sample rotate. If we perform a visual classificationNGC 5824 of our Mayor 1997; van de Ven et al.(!) 2006; Sollima et al. 2009, see dis- sample intoat non-rotating the lower or spatial rotating clusters,resolution we findNGC of 5904 that the about MUSE!! datacussion ! is ! below). identified. !!! Beyond Using the half-light ! radii, our data lack the radial way to our pilot study on NGC 6397. Details on the individual steps NGC 5927 ! ∼ XX X NGC 5986 XX ∼ 60 per centthose, (13/22) of in the a second clusters show step, obvious a common rotation,NGC 6093 while PSF and!!coverage a coordinate to investigate∼ transfor-X( any further!) trends. In this respect, it is inter- of the spectrum extraction and the spectral analysis can be found NGC 6121 !! ∼∼ ∼XX the remaining ones appear consistent with no rotation.NGC 6171 Further in-∼ esting to note! thatXX some of the X clusters that we visually classified mation from the input catalogueNGC 6205 are fitted! to the MUSE data.∼ X The! in Papers I and II. In the following, we restrict ourselves to a briefspection of the radial profiles of the rotating clustersNGC reveals 6218 Xa pro- X ! as non-rotatingXXX( (NGC 3201, NGC!) 6121 and NGC 6254) have large NGC 6254 X !!∼∼XX nounced similaritywavelength in that the dependences rotation signal increases of thoseNGC with 6266 quantitiesdistance arecore modelled radii! so that as our∼ smooth data!! coverage is basically restricted to the summary of both steps and put emphasis on new aspects that have NGC 6273 !! ! to the clusterfunctions centre. It tends of to the disappear wavelength inside the afterwards.coreNGC 6293 radius and Finally,areas inside this! information the core radii.X Hence, is we cannot exclude the possibility NGC 6341 ! ∼ XX (!) been introduced since the publication of these papers. steadily increases between the core and the half-lightNGC radius. 6388 This!!that the clusters rotateXX at larger X radii. This, and the fact that projec- used to extract all spectra. TheNGC 6397 number X of spectra that∼ couldX be! behaviour is in general agreement with the evolutionaryNGC 6402 globular tion X effects may also∼ limitX the∼ amount of visible rotation in some NGC 6441 ! X ∼ XX cluster modelsextracted of Fiestas et in al. this (2006 way) or the varied equilibriumNGC with 6496 models the of seeingclusters and [the the∼ inclination densities of most ofX clusters is not known, but see van NGC 6522 ∼ XX

Lagoute & Longaretti (1996) or Varri & Bertin (NGC2012 6539). It was de Ven et al. (2006)forNGC5139orBellinietal.(∼ X ∼ 2017)forNGC Downloaded from https://academic.oup.com/mnras/article-abstract/466/4/4040/2769504 by Institut d'astrophysique de Paris user on 11 March 2019 the clusters and was typicallyNGC 6541 between 1000 and! 5000∼ starsX per! also found in detailed studies of individual clustersNGC like 6553 NGC 104 104], leads us to concludeX that! the fraction of rotating clusters in NGC 6626 ! XX ! 4.1 Spectrum extraction or NGC 5139pointing. (e.g. Meylan & Mayor 1986; Merritt,NGC 6656 Meylan!! & ourX sample!!!!! is probably significantly higher than 60 per cent. NGC 6681 ∼ XX In contrast to most spectroscopicNGC 6715 surveys,! XXX we do not perform NGC 6723 ∼ XX Fig. 2.—Top: Color indicates the measured radial velocity for the Voronoi Fig. 3.—Top:200 ; 200 image of the F606W F814W color map from the HST NGC 6752 X X X X ∼ !!∼ (!) À The extraction of the individual stellar spectra from the data cubes NGC 6779 ! ∼ XX binned data using templates observed with NIFS. Gray areas indicate spectra with ACS data described in Paper I. The image has been rotated to match the orienta- MNRAS 473,any5591–5616 pre-selection (2018) of the observedNGC 6809 stars,∼∼ but instead ∼ aim!! to obtain∼ (! a) 1 1 S/N < 10 and/or errors >15 km sÀ . Rotation of 30 km sÀ is clearly visible tion of the NIFS data. The contours indicate the F814W surface brightness and was performed with our dedicated software presented in Kamann NGC 6838 ∼ XXX∼ $ spectrum of every star in theNGC field 6934 of view.! Consequently,X spectra along the major axis. Contours show the K-band image at 15.5, 14.5, 13.5, and were chosen to roughly match the K-band contours in Fig. 2 and in the bottom NGC 7078 !!!!!!!! 12.5 mag arcsec 2.Theblackbarindicates10pc(0:47 ). Bottom:Velocitydis- panel. The black bar indicates 10 pc (0:47 ). Bottom:2 ; 2 image of CO line et al. (2013). It determines the positions of the sources as well as NGC 7089 !! !! !1 À 00 00 00 00 ! 1 are extracted over a wide rangeNGC 7099 of X signal-to-noise X X ∼ ratiosXX (S/N), () persion measurements with S/N > 15 and errors < 10 km sÀ . The dispersion strength, with larger values indicating deeper CO lines. The contours show K-band Terzan 5 ! the shape of the point spread function (PSF) as a function of wave- drops away fromFeldmeier-Krause the center, indicating a relatively et al. cold 2017 disk population. contours as in Fig. 2. including many spectra for which the S/N is too lowFigure 1.toSAURON perform stellar velocity fields for our 48 E and S0 galaxies (see Paper III), the global outer photometric axis being horizontal. Colour cuts were ”uncertain”in our work (NGC 5272, NGC6093,Sollima NGC 6341, et tional. could 2019 be detected by Vasiliev (2018)inthe6rotating Emsellem et al. 2007 length and uses this information to optimally extract the spectrum tuned for each individual◦ galaxy as to properly emphasize the observed velocity structures. A representative isophote is overplotted in each thumbnail as any meaningful analysis.NGC 6752, NGCIn 6809), a first three show cut, a 2σ significant we rotationexcludeGCs with all inclination spectra angles i> with65 (NGC 2808, NGC 6205, (NGC 4372, NGC 5986, NGC 6341), one is found to be non- NGC 6397,a black NGC solid 6541, line, NGC and 6553, the NGC centre 6626), is markedwhile Ter with 5 a cross. Galaxies are ordered by increasing value of λRe (from left to right-hand side, top to bottom; see et al. 2005). Nonetheless, we are able to make an accurate pect to be able to detect any non-Gaussian signatures (Cappellari of each resolved star. For this method to work, an input catalogue rotating (NGC 6388) and one is not included in our sample is not includedSection in 3). his Slow sample. rotators are galaxies on the first two rows. NGC numbers and Hubble types are provided in the lower-right and upper-right corners of each &Emsellem2004). S/N < 5 from any further(IC 1276). As analyses. for the Bianchini et al. This(2018)work,norota- left us withpanel, about respectively. 813 Tick 000 marks correspond to 10 arcsec. measurement of the central dispersion of the cluster,  28 Among works based on line-of-sight velocities, 1 ¼ Æ of sources is needed that provides astrometry and broad-band mag- 2kmsÀ . Convolution of the templates with this dispersion sub- spectra of about 273 000Simon stars. Rozier The cumulative histograms of theMNRAS 000, 1–19 (2018) !2 2 20.05.2019 3.3. Stellar Populations and Connection to Kinematics nitudes across the MUSE field of view. Where possible, we used stantially (by a factor of 2) improves the of their fit to the cen- remaining spectra and stars are shown in Fig. 2. Note that the shape tral spectrum of the cluster, giving a reduced 2 of 0.9 and 1.1 From the optical data described in Paper I, the NGC 4244 NSC HST data from the Advanced Camera for Surveys (ACS) of Galactic using the GNIRS and NIFS templates, respectively. The bottom has two distinct stellar populations—a bluer population domi- of the histograms, a steep rise at low S/N that gets shallowerC 2007 The Authors. when Journal compilation C 2007 RAS, MNRAS 379, 401–417 ⃝ ⃝ panel of Figure 2 shows a map after clipping data with S/N < 15 nates in the midplane with a redder population dominating above globular clusters (Sarajedini et al. 2007;Andersonetal.2008)as 1 moving to higher S/N, is a direct consequence of the luminosity or errors in the dispersion >10 km sÀ .Thevelocitydispersion and below the plane. This is clearly seen in the HST-based color input. However, some of our clusters (NGC 1904, NGC 6266, NGC drops along the major axis to values of 10Y20 km s 1 at precisely map shown in the top panel of Figure 3. We would expect the function of the cluster stars. The number of stars per magnitude bin À 6293, NGC 6522) were not included in the survey. In addition, our the area where rotation is seen most strongly indicating that we NIFS data to resolve these different populations despite the 2 times increases when moving to fainter magnitudes. are indeed seeing a relatively cold rotating disk. This dispersion lower resolution. If we make a map of the CO line strength (Fig. 3, outer pointings in NGC 104 and NGC 5139 are located outside of is close to the limit of what we can reliably recover from our rela- bottom), the bluer areas of the HST image clearly have stronger the footprint of their ACS observations. In those cases, we obtained tively low resolution observations, thus these values may repre- CO absorption than the redder areas above the midplane. The CO sent upper limits. It is unclear whether the increased dispersion line strength is determined by comparing the average depth of archival HST images and analysed them with the DOLPHOT software we see in the center is a result of unresolved rotation or a gen- the line at rest wavelengths of 22957 52 8 to the continuum Æ 12 package (Dolphin 2000). An overview of the additional HST data 1 As in Paper I, the S/N we provide is the average value for each extracted uinely hot component. measured just to the blue of the CO band head [ CO (2, 0); The use of the templates from NIFS and GNIRS give very sim- Kleinmann & Hall 1986]. We also measured these values for the that were used can be found in Table 2. spectrum determined with the method of Stoehr et al. (2008). ilar results for both the velocities and dispersions. The GNIRS GNIRS template stars for comparison. The CO line strength val- templates selected for fitting the spectra are primarily at spectral ues for the NSC range from 0.10 above and below the major $ types close to those (KYM giants) represented by the three NIFS axis typical of late G-type giant stars (TeA 4800 K), to 0.25 $ $ spectra. For almost all the spectra, our measured non-Gaussian along the major axis, typical of K3YK6 giants (TeA 4100 K). MNRAS 473, 5591–5616 (2018) h3andh4 components of the LOSVD were consistent with zero. We caution that the CO depth depends on the gravity$ of the stars FigureHowever, 1. Kinematic due to our low data S/N and (top resolution, row) and we would respective not ex- uncertaintiesas well as their effective (bottom temperature. row). We The note that columns while Mg, Ca, denote velocity V,velocitydispersionσ,Gauss–Hermitemoments h3 and h4. White pixels are due to excluded bright stars.

Schodel¨ et al. (2014). We used the extinction and emission cor- Table 1. The MGE fit parameters for the 4.5 µm Spitzer/IRAC dust ex- rected 4.5 µm image to measure the light distribution. The image tinction and PAH emission corrected image in combination with the NACO 1 H-band mosaic scaled to Spitzer flux. I is the peak surface brightness was smoothed to a scale of 5 arcsec pixel− ,andextendsover scaled 270 pc 200 pc. We excluded a central circle with r 0.6 pc used in the dynamical modelling, σ MGE is the standard deviation and qMGE (∼ 15.4 arcsec)× to avoid contribution from ionized gas emission= and is the axial ratio of the Gaussian components. Iunscaled is the peak surface brightness before scaling to Fritz et al. (2016). young∼ stars. In addition, we masked out the young Quintuplet star 1 cluster (Figer, McLean & Morris 1999), and the dark 20-km s− Iscaled σ MGE qMGE Iunscaled cloud M-0.13-0.08 (Garc´ıa-Mar´ın et al. 2011). 4 2 4 2 (10 L ,4.5 µm pc− )(arcsec) (10L ,4.5 µm pc− ) We used the MGE_FIT_SECTORS package (Cappellari 2002) to derive ⊙ ⊙ the surface brightness distribution. The Multi-Gaussian-Expansion 0.86 1.7 0.30 312 (MGE) model (Emsellem, Monnet & Bacon 1994)hastheadvan- 32.4 10.4 0.34 164 tage that it can be deprojected analytically. We measured the pho- 89.8 15.0 0.82 257 tometry of the two images along the major axis and the minor axis. 18.5 52.1 0.95 30.0 17.0 98 0.36 29.3 We assumed that the cluster’s major axis is aligned along the Galac- 7.1 154 0.95 7.4 tic plane, as found by Schodel¨ et al. (2014), and constant. The centre 4.8 637 0.36 4.9 is the position of Sgr A*, which is the radio source associated with 3.2 2020 0.30 3.2 the Galactic Centre supermassive black hole. We fitted a scale fac- 1.3 4590 0.81 1.3 tor to match the photometry of the two images in the region where they overlap (16–27.8 arcsec). Then we measured the photometry on each image along 12 angular sectors, and converted the NACO photometry to the Spitzer flux. Assuming fourfold symmetry, the and M153 filters) and public VISTA Variables in the Via Lactea measurements of four quadrants are averaged on elliptical annuli Survey images (H and KS bands; Saito et al. 2012). We lowered the with constant ellipticity. Using the photometric measurements of intensities of the central Gaussians by scaling our averaged profile the two images, we fitted a set of two-dimensional Gaussian func- to the one-dimensional flux density profile of Fritz et al. (2016). tions, taking the point spread function (PSF) of the NACO image As a result the amplitudes of the inner Gaussians become smaller, into account. but the outer Gaussians (σ MGE > 100 arcsec 4pc)arenearly A comparison with the surface brightness profile of Fritz et al. unchanged. We list the components of the MGE∼ in Table 1 and (2016, their fig. 2) showed that our profile is steeper in the central plot the surface brightness profile in Fig. 2 (upper panel). We also 30 arcsec. A possible reason is the small overlap region of the show the projected axial ratio q as a function of radius in the ∼ proj Spitzer and NACO images, and that the Spitzer flux could be too lower panel of Fig. 2.Outtothecentral1pc,qproj is increasing high at the centre. Maybe the PAH emission correction of the Spitzer from 0.4 to 0.7. Schodel¨ et al. (2014)foundameanaxialratio image was too low. The mid-infrared dust emission is significant of 0.71 0.02 for the nuclear star cluster. This is in agreement ± out to almost 1 arcmin. Fritz et al. (2016)usedNACOH-and with our maximum value of qproj.However,qproj decreases at larger K -band images in the central r 20 arcsec. Out to 1000 arcsec radii, as the contribution from the nuclear stellar disc becomes more S = ( 39 pc) they used Hubble Space Telescope WFC3 data (M127 important and the light distribution therefore flatter. ∼

MNRAS 466, 4040–4052 (2017) DM halos are rotating

J 0 = p2MVR

20 λ’0=0.035 σ=0.57 λ’0=0.0347 σ=0.629

15 ’) λ

10P(

5

0 0.00 0.05 0.10 0.15 λ’ Bullock et al. 2001 Aubert et al. 2004

Simon Rozier !3 20.05.2019 Rotation in DM halos

• Teory: Rotation emerging from tidal torques at the time of structure formation. Peebles 1969 Accretion of satellites (assembly). Vitvitska et al. 2002

• Simulations: Tidal torquing and assembly do not seem to be independent. Lopez et al. 2019

• Observations: Open question. Imprint of the rotating halo on the baryon dynamics?

• Problem: Rotation has an infuence on the dynamical evolution of the halo, and its shape. Can transform a spherical halo into a triaxial one, e.g through instabilities.

Simon Rozier !4 20.05.2019 Linear Response Teory: the response matrix How does a stellar system respond to an exterior perturbation?

@f @f @ @f + v =0 ⟶ Angle-Action variables Galaxies are self-gravitating @t · @x @x · @v (J, ✓) Self-gravitating amplification Galaxies are self-gravitating • Self-gravitating amplification Collective e↵ects• Collective e↵ects Boltzmann det[I M(!)] = 0 ext SecularLinear response Evolution F ======Boltzmann = = Mode? (or linearext instability) ) Secular Evolution F ======Complex = = = = = frequency ======! = !0 + i⌘ (or linear instability) ) dv !0 = precession rate ⌘ c Z dv = growth rate ⌘ < 0 stable; ⌘ > 0 unstable Z ! ! self ⇢self Poisson self ⇢self Poisson (p) Matrix method - (Kalnajs (1976)) (p) • = Representative basis ( (p), ⇢(pMatrix)) method - (Kalnajs (1976)) • to) solve Poisson=4 once⇡G for⇢ all.⟶ Projection= Representative basis ( (p), ⇢(p)) to) solve Poisson once for all. (p) =4⇡G⇢(p) , (p) (p) (p) (q) q =4⇡G⇢ , 8 dx ⇤(x) ⇢ (x)= p . < (p) (q) q Z Kalnajs 1976 8 dx ⇤(x) ⇢ (x)= p . Basis < (p) (q) q : Zdx ⇤(x)⇢ (x)= 10 / 47 p : 10 / 47 Saturday, 10June,Simon 17 Rozier Z !5 20.05.2019

Saturday, 10June, 17 Key points of the method

Angle-action variables: Guarantee a simple (proportional) relation between the response and the perturbation

∂f ψ˜ 1 (J, ω) f˜1 (J, ω) = m ⋅ 0 m m ∂J m ⋅ Ω − ω

Projection onto Bi-Orthogonal Basis: Transforms the differential equations into a vectorial problem ψ˜ resp(ω) = M(̂ ω)( I ̂ − M(̂ ω))−1 ⋅ ψ˜ ext(ω)

∂f0 m ⋅ ∂J Response matrix: M ̂ (ω) = − (2π)3 dJ ψ(p)*(J)ψ(q)(J) pq ∑ ∫ m m m m ⋅ Ω − ω

̂ ̂ resp iωmt Unstable mode: det[ I − M(ωm)] = 0, Im(ωm) > 0 ψ (t) ∝ e

Simon Rozier !6 20.05.2019 computation

Simon Rozier !7 20.05.2019 computation

Sum over resonance vectors

Simon Rozier !8 20.05.2019 computation

Sum over resonance vectors Integral over action space

Simon Rozier !9 20.05.2019 computation

Sum over resonance vectors Integral over action space Gradient of the distribution function

Simon Rozier !10 20.05.2019 computation

Sum over resonance vectors Integral over action space Gradient of the distribution function Resonant denominator at the intrinsic frequencies

Simon Rozier !11 20.05.2019 computation

Sum over resonance vectors Integral over action space Gradient of the distribution function Resonant denominator at the intrinsic frequencies Potential basis functions

Simon Rozier !12 20.05.2019 computation Assumptions and Technical challenges

Computing the frequencies: Requires to sample the integral in peri- and apo-centre space.

Computing the integral: Computation on a static grid. 4 parameters allow some fexibility in the choice of the grid in peri- and apo-centres. Te regions where the denominator resonates have to be particularly well sampled.

Simon Rozier !13 20.05.2019 computation Assumptions and Technical challenges

Computing the Bi-Orthogonal Basis: Basis functions defned through nested integrals. Runge-Kuta scheme implemented, with a signifcant gain in performance as compared to previous response matrix codes.

Simon Rozier !14 20.05.2019 computation Assumptions and Technical challenges

Truncation of the sum: Infnite sum transformed into fnite by assuming that only the frst several resonance vectors contribute. Motivated by the literature (Lynden-Bell, Polyachenko): signifcant

resonances should have |n1 |, |n2 |, |n3 | ≤ 2 1 additional parameter

Simon Rozier !15 20.05.2019 computation Assumptions and Technical challenges

Adding rotation to the theory:

No rotation: f(Jr, L) With rotation: f(Jr, L, Lz)

System: Spherical cluster (ψ 0 ( r ) , Plummer potential), steady state with tunable anisotropy ( f q ( E , L ) ) and rotation ( f q , α ( E , L , L z ) ): Dejonghe 1987 Lynden-Bell 1962 α rotation parameter : no rotation fq,α(E, L, Lz) = fq(E, L)(1 + α Sign(Lz /L)) α = 0 α = 1 : maximal rotation

Simon Rozier !16 20.05.2019 Response matrix for rotating systems α rotation parameter ̂ 0̂ 1̂ : no rotation M pq(ω) = M pq(ω) + α M pq(ω) α = 0 α = 1 : maximal rotation p and q described by their harmonic numbers mp, ℓp, np, mq, ℓq, nq

0 p q ℓq Good properties of M : independent of m , m ; proportional to δℓp M1 does not have these properties anymore: the matrix is much larger 0 1 M pq M pq ℓq=2 ℓq=3 ℓq=4 ℓq=2 ℓq=3 ℓq=4 2 2 = = p p ℓ ℓ 3 3 = = p p ℓ ℓ 4 4 = = p p ℓ ℓ

Simon Rozier !17 20.05.2019 computation Assumptions and Technical challenges

Truncation of the matrix: Infnite matrix truncated in m p , ℓ p , n p , m q , ℓ q , n q . • m p = m q = 2 : searching for bars and two-armed spirals • Cut-off in ℓ p , ℓ q : assuming the frst resonances only mater + the background and the instabilities are well-described by low-order ℓ terms. • Cut-off in n p , n q : assuming the frst radial basis functions are sufficient to describe both the background and the instabilities. 2 more parameters + 1 parameter controlling the characteristic radius of the basis.

Summary: 8 parameters controlling 1 computation of M 0 ̂ ( ω ) and M 1̂ (ω) Convergence study needed on each of these parameters.

Simon Rozier !18 20.05.2019 Identifying unstable modes in rotating systems Nyquist diagrams: Infuence of rotation 0.06 10 changing changing 0.04

8 η ω 0.040 0 0.045 0.02 0.049 6 0.054 -0.0022 0. ●● 0 4 0.0022 -0.02 2

0.0055 0.0060 -0.04 0.0067 0.0074 0 True diagram ●●●● Not rotating -0.06 -6 -4 -2 0 2 4 -0.03 -0.02 -0.01 0. 0.01 0.02 0.03 0.06

1.0 0.04

0.02

0.5 0.0024 0.542 0.0033 0.544 0.547 0. ●● 0.0045 0.550 0.553 0.0060 0.0 ●●●0.555 ● 0.0081 0.054 -0.02 0.049 0.045 0.040

0.0074 0.0082 -0.5 -0.04 0.0090 0.010 Logarithmic rescaling Rotating -1.0 -0.06 -1.0 -0.5 0.0 0.5 1.0 !19 -0.03 -0.02 -0.01 0. 0.01 0.02 0.03 Results: instability mapped in anisotropy-rotation space η(�,q) = growth rate

Linear Response Teory N-body simulations

q: anisotropy; �: rotation

Simon Rozier !20 20.05.2019 Results: instability mapped in anisotropy-rotation space

Tangentially anisotropic Fast rotation

Radially anisotropic (ROI) Linear Response Teory N-body simulations

q: anisotropy; �: rotation

Simon Rozier !21 20.05.2019 Instability in tangentially- biased systems

η

Simon Rozier !22 20.05.2019 Radial Orbit Instability: 2 regimes

Slow ROI

η

Fast ROI

Simon Rozier !23 20.05.2019 ROI in 2 sketches

Simon Rozier !24 20.05.2019 ROI in 2 sketches

Simon Rozier !25 20.05.2019 ROI in 2 sketches

Competition between atracting torque (destabilizing) and kinetic pressure in tumbling rates (stabilizing)

Simon Rozier !26 20.05.2019 ROI in rotating spheres: qualitative argument

PDF(ΩILR) 12 q= 0 q= 1 q= 2 10 α= 0 α= 0.5 8 α= 1 6

4

2

ΩILR -0.1 -0.05 0 0.05 0.1

When q or � increases, the dispersion in tumbling rates decreases

Simon Rozier !27 20.05.2019 Conclusion - Prospects

• Stability analysis of Plummer spheres with various rotation & anisotropy. Checked against N-body. ⟶ Generalized the radial orbit instability to rotating systems. ⟶ Discovery of a new regime of instability in tangentially anisotropic fast rotators.

• Next step: Explain the mechanisms behind the instabilities (resonances?) Switching off some resonance vectors gives good results in the case of the ROI.

Simon Rozier !28 20.05.2019 Tanks for your atention

Simon Rozier !29 20.05.2019 Backup

!30 0.5 PDF(ΩILR) ϕ v 0.0 12

-0.5 q= 0 α= 0 α= 0 α= 0 q= 1 q=-6 q= 0 q= 2 q= 2 10 α= 0 0.5 α= 0.5 8

ϕ α= 1

v 0.0

-0.5 6 α= 0.5 α= 0.5 α= 0.5 q=-6 q= 0 q= 2 4 0.5 ϕ v 0.0 2 -0.5 α= 1 α= 1 α= 1 q=-6 q= 0 q= 2 ΩILR -0.5 0.0 0.5 -0.5 0.0 0.5 -0.5 0.0 0.5 -0.1 -0.05 0 0.05 0.1

vr vr vr

Simon Rozier !31 20.05.2019 Galaxies are self-gravitating Self-gravitating amplification • Collective e↵ects

ext Boltzmann Secular Evolution F ======Te matrix formalism(or linear instability) ) dv Z Projection of the potential and density onto a bi-orthogonal self basis⇢self solving the Poisson equation Poisson

(p) Matrix method - (Kalnajs (1976)) • =4⇡G⇢ ⟶ Projection= Representative basis ( (p), ⇢(p)) to) solve Poisson once for all. (p) =4⇡G⇢(p) , (p) (q) q 8 dx ⇤(x) ⇢ (x)= p . Kalnajs 1976 Basis < (p) (q) q Zdx ⇤(x)⇢ (x)= p Z: 10 / 47

Saturday, 10June, 17 Self-induced perturbation External perturbation Total perturbation s1 (p) e (p) 1 (p) c = a + b ψ (x, t) = ∑ ap(t)ψ (x) ψ (x, t) = ∑ bp(t)ψ (x) ψ (x, t) = ∑ cp(t)ψ (x) p p p p p p

a (t) = − dx ρs1(x, t) ψ(p)*(x) = − dx ρs1(J, t)eim⋅θ ψ(p)*(x) = − dxdv f1 (J, t)eim⋅θ ψ(p)*(x) p ∫ ∑ ∫ m ∑ ∫ m m m Bi-orthogonal basis im⋅θ ρ = dv f X(θ, J, t) = ∑ Xm(J, t)e ∫ m

a (t) = − dJdθ f1 (J, t)eim⋅θ ψ(p)*(x) = − (2π)3 dJ f1 (J, t) ψ(p)*(J) p ∑ ∫ m ∑ ∫ m m m m Change of canonical variables im⋅θ X(θ, J, t) = ∑ Xm(J, t)e m

Simon Rozier !32 20.05.2019 Te response matrix

∂f ψ˜ 1 (J, ω) 3 ˜1 (p)* 3 0 m (p)* Laplace transform a˜p(ω) = − (2π) dJ f m(J, ω) ψm (J) = − (2π) dJ m ⋅ ψm (J) ∞ ∑ ∫ ∑ ∫ ∂J m ⋅ Ω − ω iωt m m 1 X˜(ω) = dt X(t) e ∂f0 ψ˜ m(J, ω) ∫ Linearized Vlasov: f˜1 (J, ω) = m ⋅ −∞ m ∂J m ⋅ Ω − ω ∂f 1 a˜ (ω) = − (2π)3 c˜ (ω) dJ m ⋅ 0 ψ(p)*(J)ψ(q)(J) p ∑ q ∑ ∫ m m q m ∂J m ⋅ Ω − ω 1 (q) ψ˜ m(J, ω) = ∑ c˜q(ω)ψm (J) q

Matrix equation a˜(ω) = M(̂ ω) ⋅ c˜(ω) or a˜(ω) = M(̂ ω)( I ̂ − M(̂ ω))−1 ⋅ b˜ (ω)

∂f0 m ⋅ ∂J Response matrix M ̂ (ω) = − (2π)3 dJ ψ(p)*(J)ψ(q)(J) pq ∑ ∫ m m m m ⋅ Ω − ω

̂ ̂ iωmt Condition for an unstable mode: det[ I − M(ωm)] = 0, Im(ωm) > 0 a(t) ∝ e

Simon Rozier !33 20.05.2019 Linearizing the Vlasov equation

System:

Spherical cluster ( ψ 0 ( r ) , Plummer potential), steady state with tunable anisotropy ( f q ( E , L ) ) Dejonghe 1987 and rotation (f q , α ( E , L , L z )): Lynden-Bell 1962 fq,α(E, L, Lz) = fq(E, L)(1 + α Sign(Lz /L)) Angle-action variables: ∂H Te actions J are constants of the motion, so the angles do not appear in the Hamiltonian. q· = ∂p Spherically-symmetric system ⟶ 3 actions available Binney & Tremaine 2008 Ten the corresponding angles are straightforward (linear in time).

1 ra Usual choice: J = J = v dr J = J = L = r v J = J = L ⋅ e 1 r ∫ r 2 ϕ T 3 z z π rp

1 1 1 ∂f ∂f ∂f ∂f ∂f0 ∂ψ Vlasov: + ⋅ Ω(J) = 0 ∂H Linearized Vlasov: + ⋅ Ω(J) − = 0 Ω(J) = ∂t ∂θ ∂J ∂t ∂θ ∂J ∂θ

Fourier transform in angles: 2�-periodicity implies all functions can be writen im⋅θ X(θ, J, t) = ∑ Xm(J, t)e ∞ m Laplace transform in time: X˜(ω) = dt X(t) eiωt Complex frequency ! = ! + i⌘ ∫−∞ 0 !0 = precession rate ⌘ = growth rate 1 ⌘ < 0 stable; ⌘ > 0 unstable ∂f ψ˜ (J, ω) ! ! f˜1 (J, ω) = m ⋅ 0 m m ∂J m ⋅ Ω − ω

Simon Rozier !34 20.05.2019