The central kinematics of NGC 1399 measured with 14pc resolution

a r X i v :a s t r o -p h /0510278v 1 10 O c t 2005

Mon.Not.R.Astron.Soc.000,1–18(2005)Printed 5February 2008

(MN L A T E X style ?le v2.2)

The central kinematics of NGC 1399measured with 14pc

resolution

R.C.W.Houghton 1,J.Magorrian 2,M.Sarzi 1,N.Thatte 1,R.L.Davies 1,D.Krajnovi′c 1

1

University of Oxford,Denys Wilkinson Building,Keble Road,Oxford,OX13RH

2University

of Oxford,Rudolf Peierls Centre for Theoretical Physics,1Keble Road,Oxford,OX13RH

29th September 2005

ABSTRACT

We present near infra-red (NIR)adaptive optics assisted spectroscopic observa-tions of the CO (?μ=2)absorption bands towards the centre of the giant elliptical galaxy NGC 1399.The observations were made with NAOS-CONICA (ESO VLT)and have a FWHM resolution of 0.′′15(14pc).Kinematic analysis of the observations re-veals a decoupled core and strongly non-Gaussian line-of-sight velocity pro?les (VPs)in the central 0.2arcsec (19pc).NIR imaging also indicates an asymmetric elongation of the central isophotes in the same region.

We use spherical orbit-superposition models to interpret the kinematics,using a set of orthogonal “eigenVPs”that allow us to ?t models directly to spectra.The models

require a central black hole of mass 1.2+0.5?0.6×109

M ⊙,with a strongly tangentially biased orbit distribution in the inner 40pc.

Key words:instrumentation:adaptive optics,Galaxies:kinematics and dynamics,galaxies:individual:NGC 1399

1INTRODUCTION

Super-massive black holes (SMBHs)are thought to be the only viable candidates for the massive dark object (MDO)observed at the centres of many nearby galaxies.Indeed,re-cent near infra-red (NIR)observations of the centre of the Milky Way have resolved individual stars orbiting in close proximity to the central MDO (which coincides with the ra-dio source,Sgr A ?)and these data rule out all other plausible explanations for a MDO,other than a SMBH (Sch¨o del et al.2003).

Signi?cantly,a relationship between the mass of the SMBH and the bulge luminosity of the host galaxy was dis-covered (Kormendy &Richstone 1995)and subsequently,a tighter correlation between the SMBH mass,M ?,and the velocity dispersion of the bulge,σ(the M ?–σrelation)was measured (Ferrarese &Merrit 2000;Gebhardt 2000)of the form

log M ?/M ⊙

=α+βlog(σ/σ0).

(1)

with σ0=200km s ?1.Tremaine et al.(2002,hereafter T02)?nd α=8.13and β=4.02and Ferrarese &Ford (2005,hereafter FF05)?nd α=8.22and β=4.86.Similar re-lations with low scatter have also been found between M ?and the infrared luminosity L IR (Marconi &Hunt 2003)and between M ?and bulge mass (H¨a ring &Rix 2004).

It is believed that the mass accretion history of a

SMBH is linked to the formation and evolution of its host (Haehnelt &Kau?mann 2000;de Zeeuw 2003)and so such a precise relation,connecting quantities on vastly di?erent scales,provides an important constraint on models of galaxy assembly.It would be particularly remarkable if it holds true for galaxies of di?erent morphological types,which most likely underwent very di?erent formation and evolu-tion histories.For example,Faber et al.(1997)suggest that power-law ellipticals and spiral bulges formed dissipatively whereas core-like ellipticals formed from mergers,yet both appear to follow the same relation.In practice,the rela-tion can also be used to measure the mass of a host galaxy black hole (BH)where a dynamical estimate is not possible (Aller &Richstone 2002;Yu &Tremaine 2002).1.1

Contention

There has been considerable debate over the values of the pa-rameters αand β(Ferrarese &Merrit 2000;Gebhardt 2000;Tremaine et al.2002;Ferrarese &Ford 2005).The slope βis crucial for comparison with theoretical models that attempt to explain the M ?–σrelation,but currently,the sample of galaxies used by T02(Fig.1)and FF05is somewhat limited and biased.Of the 31galaxies in the sample of T02,18are elliptical,9are lenticular and 4are spiral (see Fig.1).Al-though there are roughly equal numbers of power-law and core ellipticals,there remain few high dispersion and low

2R.C.W.Houghton,J.Magorrian,M.Sarzi,N.Thatte,R.L.Davies,D.Krajnovi′c dispersion galaxies where any deviations from the canonical

slope will be most obvious.Such bias is not surprising con-

sidering that,until recently,only one telescope could per-

form dynamical mass estimates with the required spatial

resolution:the Hubble Space Telescope(HST).

In order to reliably estimate the mass of the black hole,

it is necessary to resolve kinematics in the region of space

where the black hole potential dominates over the potential

of the stars,a point stressed by FF05.The radius of this

sphere of in?uence(SoI)is of order

GM?

r～

3

the case of a spherical https://www.wendangku.net/doc/4716929404.html,ing the Jeans equa-

tion,the mass enclosed within radius r can be written as (Kormendy &Richstone 1995)M (r )

=

v 2r

G

?

d ln j

d ln r

?

1?

σ2θ

σ2r

,

(3)

where v is the rotation velocity,σr ,σθ,σφare the radial and azimuthal components of the velocity dispersion and j is the deprojected luminosity density.The ?rst two terms in square brackets can be estimated almost directly from obser-vations.Both are positive for the vast majority of galaxies.The last two terms,however,depend on the the unknown anisotropy and can take either sign.Their e?ect on M (r )is minimised for galaxies with steep j (r )pro?les,steep ve-locity dispersion pro?les σr and rapid rotation v =0,all of which tend to be satis?ed in power-law galaxies.Core galaxies like NGC 1399,however,tend to be non-rotating with shallow density and dispersion pro?les.For such galax-ies it is particularly important to constrain the anisotropy by modelling at the detailed shape of the galaxy’s line-of-sight velocity pro?les (VPs),for which high signal-to-noise spectra are essential Gerhard (1993).

Using the NAOS AO system coupled with the CON-ICA NIR imager /spectrograph at the European Southern Observatory’s Very Large Telescope (ESO VLT)we have resolved the SoI of NGC 1399and measured its stellar kine-matics using the CO absorption bands at 2.3μm and the CaI absorption feature at 2.26μm.The SNR of the spectra range from ～90to ～20over the region used to extract kinemat-ics,with a SNR of ～70at the CO bandhead (2.3μm).We use these kinematics to construct a spherical orbit super-position model for the galaxy to estimate the mass of the central MDO.Throughout this paper we assume a distance of 19.9Mpc to NGC 1399(Tonry et al.2001);the reader is reminded that BH mass scales linearly with assumed dis-tance.

The structure of this paper is as follows.The data re-duction techniques are discussed in Section 2;the kinematic analysis is discussed in Section 3;the imaging and kinemat-ics are presented in Section 4and the discussion of their implications is contained in Section 5.The dynamical mod-elling is described and discussed in Section 6.Finally,Section 7concludes.

2DATA AND REDUCTION 2.1

Observations

AO assisted K band images (Ks ?lter)and K band long-slit spectra (SK ?lter)of the nuclear region of NGC 1399were obtained with NAOS-CONICA (Rousset et al.1998;Lenzen et al.1998)on the nights of 30/11/03and 01/12/03.The spectral range extended from 1.79μm–2.45μm,with a scale of 0.972nm per pixel,although atmospheric transmis-sion limits high SNR data to 1.95μm–2.45μm.The spatial scale of the spectroscopy was 0.′′0543per pixel;the scale of

the imaging was 0.′′

027per pixel.The slit width was 0.′′172(17pc)corresponding to an instrumental resolution (λ/?λ)

Figure 1.The sample of T02and their best ?t correlation of the form of (1)with α=8.13and β=4.02.Left:the symbols indicate the technique used to derive the black hole mass.Right:the symbols represent the di?erent morphological types.

of 880at 2.3μm (as measured from the width of the arc lines)and an instrumental broadening of σinst =145km s ?1.The conditions while observing were excellent with the seeing varying between 0.′′4and 0.′′6.

A total of 27000seconds (7.5hours)of useful on-source spectroscopic integration was achieved with individual ex-posures lasting 300seconds each.Airmass ranged from 1.0to 1.4,although 85%of exposures were made with airmass <1.2.The standard ABBA technique of nodding back and forth along the slit removed the need for separate sky expo-sures and Fowler readout mode was used to minimise read-out noise.AO assisted images of NGC 1399were also taken for a total of 80seconds with the Ks ?lter together with an equivalent number of sky exposures.

In order to remove the complex atmospheric transmis-sion curve (referred to as telluric absorption)from the object spectra,it was necessary to observe several telluric standard stars (see Sec.2.3)which have almost intrinsically feature-less spectra in the NIR (e.g.very hot O or B stars).As the strength of telluric absorption depends on the airmass,on the ?rst night telluric standards were observed at air-masses of 1.02(HD25631)and 1.40(HD25631),and on the second night telluric standards were observed at airmasses of 1.12(HD480)and 1.25(HD41814).The di?erence in air-mass between telluric and galaxy exposures on each night was never more than 0.2airmasses.Two kinematic template stars were also observed at low airmass:HD11931(K4III)and HD25840(M0III).

All stars (telluric standards and kinematic templates)were observed with an identical spectrograph con?guration to the galaxy observations.However,for the stellar observa-tions alone,it was necessary to reduce the higher order gain of AO correction to ensure that the FWHM of the point spread function (PSF)was larger than the width of the slit to match the spectral resolution of the galaxy and stellar observations.With full AO correction,the FWHM of the PSF was ～0.′′1which would have been signi?cantly smaller

than the 0.′′

172wide slit.

The position angle (PA)of the slit was 5.06?so as to include the galaxy centre and AO reference star in the slit.This allowed us to monitor and asses the AO correction.We assume that the SMBH lies at the most luminous region of the galaxy (also assumed to be at the centre of the galaxy)so it is important to position the slit to sample this region or

4R.C.W.Houghton,J.Magorrian,M.Sarzi,N.Thatte,R.L.Davies,D.Krajnovi′c to be able to quantify any o?set from it.Prior to acquiring

the galaxy,an image of the slit on the detector was made

by removing the grism and illuminating the slit with the

?at?eld lamp.The slit image was then used as a bias for

the subsequent acquisition images to precisely align the slit.

Special care was taken with the?rst acquisition of the?rst

night to ensure the PA of the slit intersected the AO refer-

ence star and the brightest part of the galaxy.The PA was

subsequently held?xed for all observations and only shifts

perpendicular to the slit length(i.e.along the slit’s minor

axis)were made to maintain the slit position on the star and

the galaxy centre.On the second night,the same PA was

veri?ed to hold the star and the galaxy centre in the slit and

then shifts were made along the slit’s minor axis as before.

2.2Reduction Techniques

The data reduction was completed with the aid of the IRAF1

and ECLIPSE2packages as well as custom IDL3scripts,

incorporating use of the IDL Astronomy User’s Library

(Landsman1993)4.

The spectroscopic data were initially reduced follow-

ing the standard ABBA technique for NIR data reduction5

which has many advantages:the timescale on which the

background subtraction is achieved is as short as possible,

helping to correct for the variability of the NIR sky;any

residual sky in a single A-B frame cancels with the residual

in the B-A frame,assuming a uniform sky?eld;the pixel-

to-pixel subtraction is very well suited to removing system-

atic errors.However,such a technique does not optimise the

random noise(sky,thermal,dark,readout)as a background

exposure of the same duration as the object exposure is sub-

tracted from each pixel.

In an e?ort to increase the SNR,the background level

of each pixel was interpolated as a function of time from

all the data frames.Pixels with signi?cant source(galaxy)

counts were excluded when?tting a3rd order polynomial as

a function of time to each pixel position.In order to gauge

the change in the SNR from interpolation,the random noise

of the sky dominated region between galaxy and the AO ref-

erence star was measured as a function of wavelength and

compared to the noise of the frames without background

interpolation:the interpolated background showed a signif-

√

icant decrease in random noise(～

5

A′(λ)=1.0?F 1.0?A(λ) (4) where F is a free parameter,to account for small variations in airmass between the object and telluric standard star ob-servations.Note that this linear correction for airmass is only an approximation to a more complicated response and is only e?ective for very small di?erences in airmass.The ini-tial normalisation was then removed by multiplying back by the original linear?t.Every telluric spectrum also required a small shift in wavelength(around a few tenths of a pixel)to compensate for di?erences in wavelength zero-point between the object and telluric spectra.The optimisation of airmass and wavelength was initially automated by minimising resid-uals around the strong telluric features at(2.0-2.1)μm in the telluric divided object spectrum.Fine adjustments were then made to further optimise the removal of prominent tel-luric absorption features at the region of interest(the CO band-heads after2.3μm).Note that in practice,while one can correct individual stellar exposures(due to the high SNR in each frame),galaxy exposures must be coadded in groups of similar airmass to increase the SNR before an accurate telluric correction can be determined.

Telluric correction is not without complication though. The telluric correction is optimised for the removal of sharp, prominent,high frequency absorption features.Accordingly, there is likely to be minimal residuals from such in the cor-rected spectra.However,variation in the continuum normal-isation of a galaxy spectrum is known to introduce system-atic e?ects into the derived kinematics(van der Marel et al. 1994).Due to the blind division of a blackbody spectrum and the lack of continuum reference in the telluric spectra after 2.3μm,error in telluric correction will most likely manifest itself as error in the continuum normalisation of the telluric spectrum after the application of(4).This would propagate into an error in the continuum level of the object spectrum (galaxy or kinematic template).Such an error may not be uniform over the length of the spectrum:there is likely to be a higher chance of error where the normalisation was com-pletely extrapolated,after2.3μm.To help compensate for such systematic e?ects,we include a polynomial continuum correction when extracting kinematics(see Sec.3),but the constraint for the correction is the minimisation ofχ2s(7), which does not guarantee to choose a solution free of sys-tematics.Furthermore,this continuum correction is additive whereas any real error would be divided into the object spec-trum.

Di?erent functions were used to normalise the galaxy and kinematic template spectra after telluric correction.The galaxy continuum appeared linear with wavelength and so at every position along the slit,a linear?t to the continuum shortward of the CO bands was su?cient.The continuum of each kinematic template shortward of the CO bandhead was clearly non-linear but was well?tted(and removed)by a blackbody spectrum.As the continuum after2.3μm must be extrapolated,it was necessary to?t slowly varying functions with relatively little freedom.Note that with all continuum ?ts,care was taken not to include obvious absorption or emission features,or areas of signi?cant telluric absorption.2.4Kinematic Templates

The choice of kinematic template(s)is important to accu-rately extract kinematics from the galaxy spectra.The sys-tematic errors introduced into kinematics by use of a poor template are well studied(van der Marel et al.1994),al-though it is di?cult to numerically quantify such e?ects. At best,we can say that di?erent templates appear to in-troduce systematic o?sets into the VP parameters.Ideally, a large library of spectral types should be available so that an optimal mix can be found,but in this case,due to time constraints,only two templates were observed with the same instrumental setup as the NGC1399observations.However, it is possible to check if the templates are well matched to the luminosity weighted population of the galaxy.

The CO(2–0)band head at2.2935μm is an indica-tor of stellar type:a linear relation has been found be-tween the equivalent width(EW)of the?rst CO feature, W CO(2?0)and the stellar type(Kleinmann&Hall1986; Origlia,Moorwood and Oliva1993).Although this relation is based on observations of individual stars,it can be ex-tended to galaxy populations by applying a correction based on the velocity dispersion of the system(Oliva et al.1995; Thatte,Tecza and Genzel2000).The relationship di?ers be-tween giants and supergiants but this should not be a prob-lem for the old,giant dominated,population of an elliptical galaxy such as NGC1399.To estimate the galaxy EW,the (rest frame)wavelength range over which the CO(2-0)EW is de?ned must be shifted to the velocity frame of the galaxy and the EW measurement must be corrected for the galaxy’s velocity dispersion.The kinematic properties of the galaxy are not known a priori and depend on the template used to extract them.However,it is possible to account for this and calculate reasonable limits on the EW of the galaxy.

Such analysis was performed for NGC1399.The vari-ation of EW with velocity dispersion was simulated using the kinematic templates and the necessary quadratic cor-rection(inσ)found;this was then applied to the galaxy EW measurements.Furthermore,all measurements were scaled by a constant factor(the correction of Oliva et al. (1995))to correct for the instrumental resolution of CON-ICA,allowing direct comparison with the relation of Origlia,Moorwood and Oliva(1993).The results are pre-sented in Fig.4and discussed in Sec.4.

2.5AO correction and PSFs

The quality of the AO correction can be estimated from the reference star,which was observed in the slit simultaneously with the galaxy.The1D pro?le of the star(calculated from summing the?ux over the same wavelength range that the kinematics are extracted from)is well?t by a double Gaus-sian as shown in Fig.2.

However,the correction and PSF vary further away from the reference star.Although the exact‘o?-source’PSF at the centre of the galaxy is unknown,it can be estimated. Using the NAOS preparation software v1.746,one is able to simulate how the PSF varies with seeing,airmass and dis-tance from the reference star.However,the simulated PSFs 6https://www.wendangku.net/doc/4716929404.html,/observing/etc/naosps/doc/NAOS-PS-tool.html

6R.C.W.Houghton,J.Magorrian,M.Sarzi,N.Thatte,R.L.Davies,D.Krajnovi′c

Parameter

On-source

O?-source

2πσx 1σy 1

exp

?1

σx 1

2

+

y 2

2πσx 2σy 2

exp

?

1

σx 2

2

+

y

2

?

G i

2

,(7)

where ?G i is the measurement error in the i th galaxy spec-trum pixel,N p is the number of pixels being ?tted and the

third term corrects for any continuum that escapes our ini-tial continuum division.The parameter k ?1accounts for di?erences in normalisation between G and T :we want our

7

VPs to be normalised with

L (v )d v =1.We further as-sume that T (λ)is well approximated by a weighted aver-age of known stellar templates.As we only have two stellar templates available for kinematic extraction,we de?ne our optimal template to be T (λ)=fT 1(λ)+(1?f )T 2(λ)

(8)

where f de?nes the relative fraction of each of the two avail-able templates (T 1,T 2).

Unlike Fourier

methods (Sargent et

al.1978;

Richstone

&

Sargent

1972;Franx,

Illingworth

&Heckman

1989;Bender 1990;van der Marel &Franx 1993),pixel ?tting does not require any assumptions about window functions and allows us to propagate the pixel-to-pixel error estimates in our measured spectra directly into uncertainties in our LOSVDs.Note that the NaI doublet at 2.21μm does not appear to be well ?t by our templates (the line strength in the galaxy spectra is much higher than that in our templates).Thus the ?tting range for extraction of kinematics is from 2.249μm to 2.438μm (rest frame)to include the CaI feature and the CO bands.We use two di?erent parametrisations for L (v ),each of which is discussed separately below.3.1

Gauss–Hermite parametrisation of the VP

VPs are expected to be reasonably close to Gaussian.A convenient way of parametrising a VP L (v )is by us-ing a truncated Gauss–Hermite expansion (Gerhard 1993;van der Marel &Franx 1993),L GH (v )=

γ2πσ

exp

?1

σ

2 N

i =0

h i H i

v ?V

2π

∞

?∞

exp(?x 2)H i (x )H j (x )d x =

1π

δij ,(10)

from which it is easy to see that choosing h i =

12γ

∞

?∞

exp

?

1

σ

2

L (v )H i

v ?V

?G i

2

,(13)

where L (v )is the galaxy’s real underlying VP and n i ≡

G i ?(T ?L )i is the noise in the i th pixel.Now if we ?x (γ,V,σ)and minimise (13)with respect to the {h i },then:(i)in the absence of noise,the coe?cients h i are still the modi?ed moments (11)of L (v );

(ii)the h i are not independent since the Hessian ?2χ2

s /?h i ?h j is no longer diagonal;

(iii)since the h i are not independent,in the presence of noise there is a di?erent set of h i for each choice of N ;

(iv)if we choose (γ,V,σ)to be the parameters of the best-?t Gaussian to (7),then the minimum χ2s will not occur at precisely (h 0,h 1,h 2)=(1,0,0).

Our procedure for ?tting Gauss-Hermite coe?cients is motivated by these points and by our desire to have a set of parameters that depend linearly on L (v ).The procedure is as follows:

(i)Choose (γ,V,σ)to be the parameters of the best-?t Gaussian to the VP:?nd (γ,V,σ),template fraction f and continuum parameters c l that minimise (7)with k =h 0=1and h 1=h 2= 0

8R.C.W.Houghton,J.Magorrian,M.Sarzi,N.Thatte,R.L.Davies,D.Krajnovi′c (ii)Holding(γ,V,σ)?xed at their best-?t values,we?nd

the h i,c l and f that minimise(7).Having found these h i,

the normalisation is k=γ(h0+h2/

√3/8+···).

(iii)Finally,we divideγand the h i by k.

We use a standard Levenberg-Marquardt routine to

carry out the minimisations in the?rst two steps.Our best-

?t parameters are(h0,...,h N),along with their covariances

and the choice of(γ,V,σ).Note that there are no errors as-

sociated with(γ,V,σ)in our version of the Gauss–Hermite

parametrisation:they merely re?ect the Gaussian around

which we have chosen to expand L(v).Choosing the best-?t

Gaussian here lets us make a straightforward comparison of

our kinematics with earlier work.In practice we?nd that

our method yields(h0,h1,h2)that di?er from(1,0,0)by

around(0.1,0.04,0.04),but with very strongly coupled er-

rors among all the even h i.We describe how we deal with

these covariances in section6.3below.

We have tried using standard simultaneous N+1pa-

rameter?ts to(γ,V,σ,h3,...,h N)(van der Marel&Franx

1993),but our VPs are so strongly non-Gaussian at the cen-

tre of the galaxy that the usual linear approximations among

the errors in these parameters(van der Marel&Franx

1993)break down and we?nd multiple minima inχ2S(7).

For spectra at(-0.′′08,0.′′02)this process can yield h4>0.6

and alarmingly low values ofσ(～250km s?1)which de-

scribe a triple peaked VP.In fact,for such values of h4it

can be shown that even the idealisedχ20(12)has multiple

minima inσand h4.

3.2VP histograms

One might expect that the putative BH in NGC1399

would cause high-velocity wings in the central VPs,which

might not be captured well by the low-order Gauss–

Hermite parametrisation above.Therefore we also?t“non-

parametric”VPs,in which we choose N regularly spaced

velocities v1

histogram

L H(v)=

n v

i=1L i S i(v),(14)

where the step function S i(v)=1if v i

Given parameters L1,...,L n

v

it is straightforward to calculate the convolution(6)of this L H(v)with a stellar tem-plate.For any given galaxy spectrum G there will be many sets of parameters that produce good?ts to the spectrum, but most of them will be unrealistically jagged.Therefore, instead of minimising theχ2s given by eq.(7)directly,we minimise the penalisedχ2p=χ2s+P[L i],where the penalty function

P[L i]=α i(L i+1?2L i+L i?1)2(15)

uses the mean-square second derivative of L(v)as a measure of the jaggedness of the solution.

Our procedure for?tting L i is simple:

(i)Find the best-?t smooth L i,continuum parameters c l and template fraction f by minimising the penalisedχ2p=χ2s+P[L i]with k=1.(ii)Set the normalisation factor to k?1= i(v i+1?v i)L i and rescale the L i.

This makes no attempt to impose the obvious non-negativity constraint on the L i.While the resulting VP his-tograms are?ne for“by-eye”comparisons of one VP against another,they are not suitable for direct comparison against models.So,in§6.3we describe a variation on this?tting pro-cedure that takes account of the correlations among the L i and removes the bias introduced by the penalty function.

Based on an average separation of300?A between the CO bands in the?tting range(12CO(2?0),12CO(3?1), 12CO(4?2),12CO(5?3),12CO(6?4)),the maximum relative velocity we can reasonably hope to measure is ～1900km s?1.The systematic velocity of NGC1399is ～1500km s?1so we divide each LOSVD into n=50eq-uispaced velocity points between v1=?1000km s?1and v n=4000km s?1.We chooseα=4×107,which is the min-imum value required to give smooth non-parametric VPs consistent with our outer Gauss-Hermite VPs.

4RESULTS:IMAGING AND KINEMATICS

An80second,AO corrected,Ks exposure of NGC1399is shown in Fig.3.It has the same PA as the long-slit ob-servations and a slit image(0.′′172wide and centred on the AO reference star)has been overlaid.If the reference star is perfectly centred in the slit,there is a maximum posi-tion error of0.3pixels(1.6pc)on the brightest region of the galaxy from error in the PA.However,the acquisition images(incorporating?at?elding etc.and using the slit im-ages)indicate that the centring of the star along the minor axis of the slit was accurate to only a pixel(5.2pc);the slit was approximately3pixels wide(0.′′172or17pc).Hence,the dominant error in aligning the slit on the galaxy centre is from shifts along the minor axis,not from error in the PA. The former is expected to be random about the centre of the slit;the later would give a systematic di?erence for all observations.The?nal position accuracy is therefore good: the slit was aligned on the brightest point of the galaxy to an accuracy of around±0.′′06(±6pc),with no signi?cant systematic o?set.

Two globular clusters are also seen in the image data close to the centre of the galaxy.The nearest is1.′′15to the east of the galaxy centre.A Gaussian?t to this globular cluster yields an estimate on the FWHM of the image PSF to be0.′′078(7.5pc).This is equal to the on-source FWHM predicted by the NAOS preparation software in Sec.2.5, suggesting that the single Gaussian was probably?t to the di?raction limited component of the globular cluster pro-?le and ignored the seeing limited halo,although the atmo-spheric conditions were excellent around the time of obser-vation,with the seeing occasionally dropping below0.′′4.

The parametrised kinematics of the central few arcsec-onds of NGC1399are shown in Fig.4.Where possible,pre-vious data is over plotted(Graham et al.1998;Longo et al. 1994).The?ux is calculated by summing over all available wavelengths at each point along the slit.The parameters (γ,v,σ)are chosen to be the best-?t Gaussian parameters and the Gauss-Hermite coe?cients h n are derived using these best-?t Gaussian parameters.The velocity v is given

9

Figure 3.Top :A Ks band image of NGC 1399showing the slit alignment (white)on the galaxy and the AO reference star (saturated on this colormap);the PA of the slit is 5.06?so that the reference star is approximately due north of the galaxy nucleus;the nucleus of the galaxy and the star are separated by 17.′′5.Bottom :the same as the above but magni?ed and centred on the nucleus of NGC 1399with isophote ellipses over plotted in blue.Note the elongation of the nucleus to the south-east.Each pixel is 27mas wide (the pixel size of the spectroscopic data was twice this at 54mas).The angular scale is given in arcseconds.

relative to the mean heliocentric velocity of 1467±4km s ?1.Note the error quoted for this velocity is an estimate of the random error only.

Fig.5compares the non-parametric and parametric VPs for the central arcsecond of the galaxy.Error bars are given for the non-parametric VPs.The galaxy spectra and best-?t broadened template (constructed from the Gauss-Hermite VP)for the central arcsecond are also shown.

4.1Highlights

Many interesting features are present in the above ?gures.The parametrised kinematics show a decoupled velocity structure across the galaxy centre,a double peaked veloc-ity dispersion across the centre separated by 0.′′2or 19pc (with the dip in σhalf a pixel o?the galaxy centre)and signi?cant variations in the h i with radius.The imaging shows o?set asymmetric isophotes at the galaxy centre and the non-parametric VPs (while generally in good agreement with the parameteric VPs)show strong high velocity wings

at r =?0.′′08and lopsided velocity structure at r =0.′′14,which are not detected with the parametrised ?t.

4.2Further Details

The kinematic features described above persisted when the spectra were reduced using the standard sky subtraction technique for ABBA sequence observations (rather than us-ing an interpolated sky),albeit with increased noise.Angu-lar distances in Figs.4and 5have been shifted such that the brightest region of the galaxy lies at r =0′′.This position was determined by scaling to the (outer)centroid of a Nuker pro?le (Lauer et al.1995)?tted to the ?ux of Fig.4.The photometry is not symmetric at the centre which could bias the calculation,although when the inner 0.′′5of the light pro-?le were omitted the change in the centroid was less than 0.1spectroscopic pixels (5mas or 0.5pc).The centroids of the el-lipses ?tted to Fig.3are not constant and ?uctuations of up to 0.5image pixels (13.5mas or 1.3pc)from the mean are present.The scale on Fig.3is aligned to the mean x-axis centroid of the ellipses.Registration between the angular

10

R.C.W.Houghton,J.Magorrian,M.Sarzi,N.Thatte,R.L.Davies,D.Krajnovi′c

F l u

x

γ

v (k m /s

)

σ(k m /s

)γh

0γh

1

γh

2γh

3γh

4W C O (2?0

)T e m p l F r a c

.

Radius from Nucleus (arcsec)

Figure 4.The nuclear kinematics of NGC 1399:The top plot indicates the ?ux received at each position on the slit,the next 8plots show the variations in the parameters of (9)and the lower two show the template and galaxy EWs and the optimal kine-matic template composition as a function of radius from the nu-cleus.Where applicable,previous data is shown as orange trian-gles (Longo et al.1994)and blue squares (Graham et al.1998).In the EW plot,the kinematic template EWs are shown in orange (M0III)and blue (K4III);the width represents a 20km s ?1error in v and σinst .The EW of NGC 1399is in shown in black (red shows the combined e?ect of 20km s ?1errors in v and σ).The PA of the slit was 5.06?so that the positive radius is northward.

See seperate ?gure

Figure 5.The spectra and VPs of the central arcsecond of NGC

1399.The centroid of the light pro?le along the slit (taken to be the nucleus)is roughly in between two pixels allowing for the VPs to be constructed at approximately equidistant inter-vals to the left and right of the nucleus,without interpolation.For every point along the slit,two VPs are plotted:black is the Gauss-Hermite VP (9)and blue is the non-parametric VP.The velocity axis of the VPs has been corrected to the mean sys-tematic velocity of the system.Accompanying every spectrum is the best-?t broadened template (green)reconstructed from the Gauss-Hermite VP.

scale on the image and that of the kinematics may not be perfect as the zero points were determined independently.However,is likely to be no worse than 0.′′05.

EW measurements for the two kinematic templates and the galaxy are shown in Fig.4.The template EWs (orange for the M0III and blue for the K4III)allow for ±20km s ?1errors in systemic velocity and instrumental dispersion.The corrected EW of the galaxy is shown in black while the e?ect of ±20km s ?1errors in velocity and dispersion are shown in red.The asymmetry in the error is associated with the error in velocity:large positive or negative changes in veloc-ity move the CO (2-0)feature out of the prede?ned wave-length limits Origlia,Moorwood and Oliva (1993),thus re-ducing the EW.The kinematics (v,σ)used to correct the galaxy EW are those shown in Fig.4.Systematic di?erences in (v,σ)from using di?erent templates in kinematic extrac-tion are less than 20km s ?1in each case.These EW mea-surements indicate that our kinematic templates are well matched to the luminosity weighted stellar population of the galaxy.With reference to Origlia,Moorwood and Oliva (1993),one can see that the corrected EW measurements of the two templates (11.3and 12.5for the K4III and M0III,respectively)are typical of their spectral classi?cations.The galaxy EW (average value of 12.1)is closer to that of the M0III template;this is also re?ected by the favouring of the M0III template in the kinematic extraction (Fig.4).How-ever,one should be apprehensive about the variations of the galaxy’s EW with radius:the dispersion correction assumes a Gaussian VP which we have seen is not always the case.Overall,the analysis of EWs indicates that the templates are well matched to the galaxy population,so we should not expect problems associated with template mismatch to be present in the kinematics.Additionally,previous (seeing limited)data on the central velocity dispersion of NGC 1399agrees with our data (Fig.4),given the di?erent resolutions.Saglia et al.(2000)report slightly negative h 4towards the galaxy centre which we con?rm although our rise in h 4in the very central 0.′′5is beyond the (seeing limited)spatial resolution of their data.

To maintain a high SNR at larger radii (|r |>0.′′3),the spectra were binned into pairs prior to kinematic extrac-tion.However,after binning,χ2S (7)rose,on average,from ～160to ～230(an approximate change of

√

11

174data points.This indicates that systematic errors are starting to become comparable to the random errors.There

are many possible causes of systematic errors.The optimal

template,convolved with the best VP will not reproduce all the features in the galaxy spectrum;although the galaxy

and template spectra have comparable CO(2-0)EWs,there

may be absorption or even emission features in the galaxy spectra that are not obvious and are not accounted for in

eq.(7),but still contribute to the di?erences between the

galaxy and templates and to the systematic‘noise’.Telluric correction is only estimated to be accurate to1%,increas-

ing to2%-3%at regions where the telluric absorption shows

sharp prominent features.In addition,the stars used to ob-tain a telluric spectra are unlikely to have featureless spectra

at high SNRs.The SNR of the data binned at0.′′3over the

wavelength range used to extract kinematics falls from110 (2.25μm)to30(2.45μm)per pixel due to the thermal back-

ground(the SNR at the unbinned galaxy centre falls from

90to20).This SNR(and corresponding error estimates) quantify random error alone and do not account for system-

atics introduced by telluric correction.Binning will reduce

the random noise but not the systematic noise.Thus asχ2S (7)is weighted by the random noise estimates,when the

random error in each spectrum is reduced,it will increase

so long as the systematic noise persists.

5DISCUSSION

The interpretation of the results is discussed below in ap-propriate sections.

5.1Decoupled Kinematics

There is a strong rotation gradient in v within a radius of

0.′′5(48pc)which is clearly decoupled from the kinematics at larger radii.The magnitude of the central rotation(taking

the di?erence of the maximum velocity in each direction)is

～70km s?1.The absence of high resolution data perpendic-ular to our long slit position prevents us from concluding

if the system is truly counter rotating,or just decoupled.

Previous publications considered the possibility of a decou-pled system(D’Onofrio et al.1995;Saglia et al.2000),but the poorer spatial resolution of the data prevented a reliable detection.Kinematically decoupled cores are often(but not exclusively)found in spherical,high dispersion systems with core-like photometry(Emsellem et al.2004).

5.2The Central Dispersion Pro?le

The drop inσhalf a pixel o?the galaxy centre is ac-companied by less convincing dips inγand the CO(2-0) EW.These features are close to the limit of the spatial resolution and thus may be unresolved.It is well known that random errors inγandσare statistically correlated (Efstathiou et al.1980)and these features are only clearly seen when continuum correction is incorporated into the minimisation ofχ2S(7);if continuum correction is left out of the minimisation(i.e.calculated once from the contin-uum shortward of the CO(2-0)feature),γshows no ob-vious change(and is noisier in general)and the dispersion becomes?at at the centre.

The decrease inγandσcould be correlated e?ects from ?tting constrained parametrised VPs to more complicated pro?les;γis expected to fall at the centre(van der Marel 1994).However,the non-parametric VP at r=0.′′03matches the parametric form well.Alternatively,the dip inσcould re?ect a genuine fall in the dispersion of the stars at the centre of the galaxy,but this is the exact opposite to what one would expect when approaching the BH.

An alternative explanation is the presence of young stel-lar population at the centre,althoughγandσmay not necessarily decrease in this case:a young stellar population may include younger MK types with reduced CO(2-0)EWs, but it would also have an increased fraction of supergiants, which have a higher CO(2-0)EW than giants for the same MK spectral type(Origlia,Moorwood and Oliva1993).In addition,one would expect an correlated change in the op-timal template used to extract the kinematics,which is not the case here.One would expect a di?erent stellar popula-tion to produce a colour gradient but unfortunately there is insu?cient homogeneous data at the required resolution to con?rm this.

Central dips inσhave been seen in other core galax-ies such as M87and NGC4649(van der Marel1994; Pinkney et al.2003),which both have nuclear activity.If NGC1399contained a central nucleus of light from a non-thermal,non-stellar object(van der Marel1994),one would expect the CO bands to show a lowerγrelative to the con-tinuum level which may introduce correlated dips inγand σ.New photometry by Lauer et al.(2005)shows a slight excess of light at the galaxy centre after subtraction of the best-?t Nuker pro?le and therefore suggests the presence of a very faint nucleus.However,the authors emphasise the possible dangers and unknown systematics of extrapolating the Nuker?t into the central regions.

5.3O?set Photometry and Peculiar VPs

The Ks band image of NGC1399shows a departure from spherical symmetry in the central0.′′5with an elongation of the surface brightness towards the east-south-east,ap-proximately0.′′2(19pc)in length.The image is a relatively short exposure and the SNR is low.However,the elonga-tion of the central isophotes covers more than16pixels; the probability of?nding16neighbouring pixels above the mean,given the noise statistics of the image,is negligible so we can rule out the possibility of random error causing such an artifact.Obvious systematic errors which may cause elongation of the central isophotes would be either an error in frame alignment,or an unusual PSF from the AO cor-rection.However,both these e?ects can be dismissed:the globular cluster found1.′′15from the galaxy centre is cir-cular and well?t by a2D Gaussian of equal width in each dimension;no signi?cant residual is seen after subtraction. We note that HST H-band images exists for NGC1399,but owing to a poorer spatial resolution(0.′′13)and PSF sam-pling,these structures are not observed.The non-spherical isophotes at the centre of NGC1399are a genuine feature of the galaxy.There appear to be peculiar kinematic features associated with the photometric anomaly:at r=?0.′′08,the non-parametric VP contains substantial high velocity wings and at r=0.′′14,the non-parametric VP is asymmetric with

12R.C.W.Houghton,J.Magorrian,M.Sarzi,N.Thatte,R.L.Davies,D.Krajnovi′c

an excess of receding velocity structure.We discuss possible explanations for these features below.

Isophote twists are common in core-regions of core galaxies(Lauer et al.2005).However,such twists are gen-erally smooth and seen on large scales,which is not the case here.The elongation could be a projection e?ect from the obscuration by a non-uniform distribution of dust.However, no obvious dust signature can be seen in either this Ks band image or archival HST V-band images.The density distri-bution of the dust would need to vary rapidly to cause such a sudden e?ect in the NIR on such small spatial scales(0.′′3 or29pc).However,neither of these explanations would give rise to the associated kinematic features.

We have already seen that globular clusters(GCs)have been resolved near the centre of NGC1399.Although the elongation of the nucleus does not appear to be separate to the central maximum,it is plausible that a GC happens to lie in the line of sight between ourselves and the galaxy centre and is unresolved from the central photometric max-imum of the galaxy.There would be a kinematic e?ect from such a chance alignment:one would expect a‘spike’in the VP at the systematic velocity of the GC caused by the low dispersion velocity structure of the globular cluster.How-ever,we see high velocity wings in the VP at r=?0.′′08 and a lopsided structure at r=0.′′14which is not the same.

The elongation of the central isophotes is more likely to be an o?set centre as seen in a handful of other core galax-ies(Lauer et al.2005).In fact,Lauer et al.(2005)argue that the o?set centres detected in new WFPC2photometry are all eccentric disks,analogous to that of M31(Tremaine 1995).Certainly,this would produce a strong kinematic sig-nature in the form of high velocity wings if our slit bisected the eccentric disk.Our slit does not completely bisect the peculiar photometry and nor does it coincide with the PA of the eccentric isophotes.However,the alignment of the AO reference star in the slit varied by around1pixel and this translates to a potential wandering of the slit,perpendicular to its length,by two image pixels.Thus,there is likely to be considerable contribution from the elongated photometry in the spectroscopic data.

Keeping in mind the problems with registration between the image and kinematics,an eccentric disc at the centre of NGC1399would create high velocity wings at r=?0.′′08, where the slit is closest to bisecting the anomaly.If the non-parameteric VP at this point is to be trusted,the velocity dispersion of just the wings(ignoring the central bulk)is in excess of1000km s?1.Furthermore,if the stars in the eccentric disc rotate in the same direction,one would expect a lopsided VP at the pericentre of the ellipse if the disk was viewed near edge on.We do see a strong excess or‘hump’of structure receding away from the observer at r=0.′′14which is possibly the closest pixel to the pericentre of the elliptical photometry.Although,Lauer et al.(2005)report no o?set photometry in NGC1399with new WFPC2photometry, the isophote ellipticity is seen to jump from approximately 0to0.2between0.′′09

We note that this eccentric disk hypothesis cannot ex-plain the decoupled kinematics discussed in Sec.5.1.A co-herent disk can only survive well within the SoI of the BH, which we estimate to be～0.′′3.6DYNAMICAL MODELLING

To obtain preliminary constraints on the mass of any BH in NGC1399,we ignore the mild rotation gradient and elon-gated central isophotes and?t spherical dynamical mod-els to our central kinematics combined with Graham et al. (1998)’s velocity dispersion pro?le.The latter extends to70 arcsec and provides important constraints on the galaxy’s mass-to-light ratio.

We assume that mass follows light,except at the galaxy centre where there can be a BH.The mass density distribu-tion is then

ρ(r)=M?δ(r)+Υj(r).(16) Our goal is to?nd the range of BH masses M?and stel-lar mass-to-light ratiosΥthat are consistent with our kine-matics.We take j(r)from the models of Magorrian et al. (1998),which was obtained by deprojecting a composite sur-face brightness pro?le constructed by combining HST and ground-based photometry.Having this j(r)it is straight-forward to calculate the gravitational potentialψ(r)corre-sponding to(16)for any choice of M?andΥ.Throughout this paper all mass-to-light ratios are in the V band.

Our modelling procedure is a straightforward adapta-tion of the extended Schwarzschild method described by Rix et al.(1997)and Cretton et al.(1999):

(i)choose trial values for M?andΥand calculate the corresponding potentialψ(r);

(ii)follow a representative sample of orbits in thisψ(r); (iii)?nd the weighted combination of orbits that min-imises theχ2of the?t between the model and the observa-tions,subject to the constraint that each orbit carries non-negative weight;

(iv)assign the likelihood exp(?1

13

decade in radius.For each E i there are n J values of angular

momentum,with J 2ij running linearly between 0and J 2

c (E i ),the angular momentum of a circular orbit of energy E i .To avoi

d a rash of indices w

e also write the double sum (17)

as

a single sum over n ≡n E ×n J points:f (E ,J 2

)=

n k =1

f k δ(E ?E k )δ(J 2?J 2

k ).

(18)

6.2Observables

Having a trial potential ψ(r )and a set of DF components

(E k ,J 2

k ),we calculate the unnormalised,psf-convolved VP histogram of each component at the projected radius R i of each of our n AO =31kinematical data points:

L (k )ij

=

v j +1

v j

d v z

d x d y psf(R i ?x,?y )

×

d v x d v y δ(E ?E k )δ(J 2?J 2

k ).

(19)

Here we use a rectangular co-ordinate system (x,y,z )with

origin O at the galaxy centre and Oz -axis parallel to lines of sight.For our standard models each histogram has n v =24bins of width v j ?v j ?1=50km s ?1,with innermost bin edge at v 1=0km s ?1,the galaxy’s systemic velocity,and outermost edge at v 25=1200km s ?1.The function psf(?x,?y )is the two-dimensional o?-source point-spread function of Sec.2.5and Table 1.Armed with the L k ij ,the unnormalised VP histogram of a model with DF (f 1,...,f n )is simply

L (R i ;v j ,v j +1)=L ij =

n k =1

f k L (k )

ij .

(20)

The normalisation constant is the psf-convolved surface

brightness,I (R i )=I i =

n k =1

f k I (k )

i ,

(21)

where,by analogy with (19),

I (k )i

=

d x d y psf(R i ?x,?y )

d 3v δ(E?E k )δ(J 2?J 2

k

).(22)We explain how we evaluate the multiple integrals (19)

and (22)in Magorrian et al.(2005).The spherical symme-try of our models means that we can a?ord to calculate these projection coe?cients for each of our 31observed radii directly.More sophisticated axisymmetric models (e.g.,(Cretton et al.1999;Gebhardt et al.2003))usually resort to introducing subgrids to store intermediate quantities in this calculation.

Our treatment of Graham et al’s velocity dispersion pro-?le is similar.We assume that each of their data points measures the second moment of the VP convolved with a Gaussian PSF with FWHM 2arcsec.The (unnormalised)second moment of each DF component is given by (22)with

an extra factor of v 2

z inside the innermost integral.

6.3Fitting models to observations 6.3.1

Gauss–Hermite coe?cients

Calculating the Gauss–Hermite coe?cients of our models is simple:the (unnormalised)contribution of the k th DF com-ponent to the i th VP is given by L (k )

ij ,with j =1,...,n v ,from which equation (11)allows us to calculate this com-ponent’s contribution h (k )

ij to {h j }.However,as we have explained in section 3.1,the observed Gauss–Hermite co-e?cients are not independent.It takes only a little e?ort to include the e?ects of the covariances among the {h j }into the modelling.

Let us de?ne the column vector h ≡(h 0,...,h N )T and let ?h be the vector of coe?cients that minimise the χ2s of equation (7).For ?xed normalisation k and continuum pa-rameters c i ,this χ2s is a quadratic form in the h i :χ2(h )

?

χ2min +

1

2

N

i =0

λi [e i ·(h ??h

)]2,(23)

where the Hessian M ij ≡?2χ2s /?h i ?h j has eigenval-ues λ0,...,λN with corresponding eigenvectors e 0,...e N .Therefore,the new parameters h ′j ≡e j ·h

(24)

have independent errors ?j ≡

?I i

γi

n

k =1

f k h ′(k )

ij

?I i

2

(26)

to (25),somewhat arbitrarily assigning errors ?I i =10?3?I

i .Finally,we add one more term,χ2

G ,to measure how well our models ?t both the unnormalised second moments from Gra-ham et al.’s data and the corresponding psf-convolved sur-face brightnesses,again assuming fractional errors of 10?3in the latter.Adding all these measurements together,the ?nal χ2of our model,

χ2m [ψ,f k ]=χ2AO +χ2I +χ2

G ,

(27)

which,for ?xed potential ψ,depends quadratically on the orbit weights f k .We use a standard non-negative least-squares algorithm (Lawson &Hanson 1974)to ?nd the non-negative set of f k that minimise it.Unlike most other orbit-superposition methods,we do not include the luminosity density j (r )in this ?t explicitly,but use the surface bright-ness pro?le I (R )instead.Our reason for this is that,un-

14R.C.W.Houghton,J.Magorrian,M.Sarzi,N.Thatte,R.L.Davies,D.Krajnovi′c like j(r),I(R)is(in principle)directly measurable and can

therefore be assigned meaningful error bars.We do not ex-

pect real galaxies to have perfectly constant mass-to-light

ratios and,in the absence of anything better,use j(r)merely

to estimate the stellar contribution to the overall potential.

6.3.2VP histograms

Unlike the Gauss–Hermite coe?cients above,there is no

simple way to compare the VP histograms found in§3.2

against models:in addition to the unavoidable correlations

among the L i,there are complicated biases introduced by

the penalty function(15).So,we simply re?t the VP his-

tograms using the same n v velocity bins for which we cal-

culate the models’VPs in§6.2,re?ecting the VPs about

v=0in order to make them symmetric.The procedure is

as follows:

(i)First?nd the best-?t template fraction,continuum

level and normalisation:?nd the f,c i and smoothed L i that

minimise the penalisedχ2p=χ2s+P[L i]for k=1,where

χ2s and the penalty function P are given by equations(7)

and(15)respectively.The best-?t normalisation factor is

then k?1= i(v i+1?v i)L i.

(ii)Holding k,the c i and f?xed,?nd the formal best-?t

VP histogram(?L1,...,?L n

v

)to the unpenalisedχ2s(7).

The best-?t VP?L is in general unphysical with many?L i<

0,but we use it only because it locates the minimum of

theχ2s quadratic form.Let us write our formal best-?t VP

histogram as the column vector?L≡(?L1,...,?L n

v

)T and

consider another L≡(L1,...,L n

v

)T.Once k,f and the c i

have been?xed,equation(7)becomes

χ2s(L)=χ2min+

n v

i=1 e i·L?e i·?L

2/λi.So by taking the projections

L′i≡e j·L,(29) of L along the full set of“eigenVPs”e j,one test directly how well it reproduces the observed galaxy spectrum.We note that our method is essentially a restatement of the work of De Rijcke&Dejonghe(1998)using the language of eigenVPs.Figure6plots the?rst few eigenVPs of one of our spectra.Unlike the terms in a Gauss–Hermite expansion(9), they do not taper o?rapidly at high velocities.

Summing the results of(28)for each VP and neglecting theχ2min terms,a model with DF(f1,...,f n)has

χ2H=n AO

i=1n v j=1 ?L′ij?1?ij 2,(30)

where L′(k)

ij is obtained from L(k)ij through(29).Apart from

replacingχ2GH in equation(27)by(30),our procedure for?t-ting models to VP histograms is identical to that for Gauss–Hermite coe?cients.Figure6.Plot of the?rst four eigenVPs of one of our spectra obtained by diagonalizingχs of eq.(7).Any(symmetric)VP can be expressed as a weighted sum of eigenVPs(eq.29),in which case the errors in the weights are independent.We have divided each of the VPs plotted here by its corresponding eigenerror so that the scale of each gives a direct indication of the uncertainty in its weight.

6.4Results

We have calculated the projection coe?cients L(k)

ij

(ψ)and

I(k)

i

(ψ)for n E×n J=200×40DF components(eq.17)in a range of potentialsψ.Our main results below are obtained using coe?cients calculated for BH masses M?/109M⊙= 0,0.25,...,3.75with a single mass-to-light ratio,Υ0= 8.5Υ⊙.Since the projection coe?cients scale straightfor-wardly with mass,we can use the method of§6.3to?t mod-els with other values ofΥprovided we remember to scale M?byΥ/Υ0and the bins v j of the velocity histograms(19) by

15 Figure7.χ2m(M?,Υ)contours(eq.27)obtained by?tting dynamical models to Gauss–Hermite parametrisations(Sec.3.1)of our spectra and to the velocity dispersion pro?le measured by Graham et al.(1998).Successive contour levels have?χ2m=1.The models have200×40DF components(17)and?t(h0,h2,...,h N)extracted from spectra under the assumption that h i=0for i>N.The left panel shows results for N=4,the right for N=6.The x-axis is both cases is the BH mass scaled byΥ8.5≡Υ/8.5Υ⊙.Mass-to-light ratiosΥare V

band.

Figure9.Anisotropy parameterβ≡1?σ2

θ/σ2r of a typical model

(M?=109M⊙,Υ=8.5Υ⊙),averaged over?ve shells per decade in radius.

drop the Gauss–Hermite parametrisation and turn to?t-

ting eigenVPs.Fitting models to the full set of eigenVPs yields the result plotted in the bottom-right of?g.8.The best-?t BH mass,1.2+0.5

?0.6×109M⊙,is consistent with the results from the Gauss–Hermite?ts,but has smaller error bars.The other panels on the?gure show the e?ect of rear-ranging the eigenVPs in order of increasing eigenerror and ?tting only the?rst N for each spectrum.The case N=3 yields results that look qualitatively similar to our sixth-order Gauss–Hermite?t,while adding one more eigenVP introduces a useful lower-bound on the BH mass.The re-sults for N=5are very similar to?tting the full N=24: in fact,it is impossible to distinguish between N=6and N=24by eye.

Since the the N=24model is essentially a direct ?t to the galaxy spectrum we adopt its best-?t M?=

1.2+0.5

?0.6×109M⊙as our best estimate of the BH mass in NGC1399.Figure9plots the anisotropy parameter of one of the best-?t models in this range.Our models have mod-erate radial anisotropy(β≈0.3)between2arcsec and30 arcsec,similar to the results found by Saglia et al.(2000)in their models of NGC1399.At larger radii the orbit distri-bution in our models becomes tangentially biased,which is what one expects from?tting a constant mass-to-light ra-tio model to a galaxy with a massive dark halo.Much more surprisingly,however,our models become extremely tangen-tially biased in the innermost arcsec,which is where we?nd the interesting kinematic and photometric features.

6.5Tests

Our best-?t BH mass remains unchanged if we change the number of DF components we use in the models:models us-ing100×20components instead of200×40yield the same result.We?nd no evidence for the?at-bottomedχ2m pro-?les claimed by Valluri,Merritt,&Emsellem(2004)(albeit for the axisymmetric case),even though our models sample (E,J2)phase space a factor～20more densely than theirs. We have tried varying the extent and widths of the velocity bins we use when we?t VPs,but the BH mass is unchanged whether we take24?ne25km s?1-wide bins extending to 1200km s?1or20coarse100km s?1-wide bins extending to 2000km s?1.

The assumed width of the velocity bins does a?ect the models in a slightly more subtle way,however.When we rescale our baseΥ0=8.5Υ⊙models to a new mass-to-light ratioΥ,we scale their velocity bins by an amount

16R.C.W.Houghton,J.Magorrian,M.Sarzi,N.Thatte,R.L.Davies,D.Krajnovi′c

Figure8.As for?g.7,but?tting to eigenVPs(§6.3.2)instead of Gauss–Hermite coe?cients.The?rst three panels show the results of ?tting only to the?rst3,4and5eigenVPs of each spectrum.The last shows the results of?tting to the full spectrum by using all24 eigenVPs.

plots in?g.8is that the range of acceptableΥincreases to (9±1)Υ⊙.The BH mass is una?ected.

6.6Caveats on BH mass

Despite these reassurances,there nevertheless are some shortcomings of the models and data upon which this BH mass is based:

(i)We follow the usual“extended-Schwarzschild”proce-dure from which the majority of existing stellar-dynamical BH masses have been obtained and consider only the very best orbit distribution f k for each potential.This best-?t distribution is typically very spiky,with only～140non-zero f k out of200×40!One way around this would be to apply some kind of regularization to the f k(Thomas et al. 2005),but the biases introduced by this procedure are not well understood(Valluri,Merritt,&Emsellem2004). (ii)Because of the huge freedom in?tting the f k,the best-?t model should have a very lowχ2m.For example,in Monte Carlo experiments with synthetic datasets of toy galaxy models we typically?ndχ2m～N data/3,where N data is the number of data points?t in the models.For our real NGC1399data,however,our best-?tχ2GH andχ2H are usu-ally at least as big as the number of parameters we use to describe our kinematics.This is probably due to our neglect of the systematic errors(Sec.2.3and4.2).In contrast,the?t to the full surface brightness pro?le and to the outer disper-sion pro?le is astonishingly good,withχ2I+χ2G～3,making our totalχ2m relatively low.

(iii)Finally,our models assume that galaxy is spherical and non-rotating,despite the clear evidence to the contrary. Nevertheless,the models do seem to require a strong bias towards circular orbits in the central0.5arcsec.

7CONCLUSIONS

Using NAOS-CONICA at the VLT,we have successfully measured the central kinematics within the SoI of NGC1399

17

(r～0.′′34)with a resolution(FWHM)of～0.′′15(14pc,Sec 2.5)using adaptive optics correction on a bright reference star17.′′5away.

Alone,the kinematics extracted from the CaI feature at 2.26μm and the CO bands after2.3μm establish the pres-ence of velocity gradient within a radius of～0.′′5(48pc), suggestive of a kinematically decoupled core.

Ks band imaging reveals o?set and elongated isophotes within a radius of0.′′2(19pc)that are not visible in H-band HST images.Such inner structure is reminiscent of that seen in other core ellipticals.The non-parametric VPs corresponding to this region also show an unusual velocity structure that may be consistent with the presence of an eccentric disk around the BH,akin to that of M31.

We have demonstrated that errors in the Gauss–Hermite coe?cients h j extracted from real galaxy spectra are not independent,and have shown a simple way of taking the covariances among the h j into account when?tting dy-namical models.The VPs near the nucleus of NGC1399are strongly non-Gaussian,however,and are not well described by a low-order Gauss–Hermite expansion.We show that the “eigenVPs”obtained by diagonalizing(7)are a more useful way of describing VPs,at least for numerical purposes.

Subject to the caveats of§6.6,our best estimate for the mass of the BH in NGC1399is1.2+0.5

?0.6×109M⊙,ob-tained by?tting spherical dynamical models directly to our observed spectra.The models are based on the usual exten-sion of Schwarzschild’s method to the problem of potential estimation.Taken at face value,they place this galaxy on the M?-σplane mid-way between the predictions of T02and FF05,being consistent with both.The best-?t model also becomes extremely tangentially anisotropic in the innermost 0.′′5.

We have demonstrated that AO observations are a viable alternative to HST to when measuring black-hole masses and can break the mass-anisotropy degeneracy even in the most massive,non-rotating elliptical galaxies.The di?culty in interpreting these long-slit data emphasises the need for high SNR AO assisted integral-?eld observations to further understand the kinematic and photometric features discovered here.

ACKNOWLEDGEMENTS

We thank the referee,Laura Ferrarese for her constructive comments.This research is based on observations collected at the European Southern Observatory,Chile(ESO Pro-gram072.B-0763).We acknowledge use of the SIMBAD As-tronomical Database and the HyperLeda database.Authors are funded by the Particle Physics and Astronomy Research Council(PPARC)and the Royal Society.

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高二《甜美纯净的女声独唱》教案

高二《甜美纯净的女声独唱》教案 一、基本说明 教学内容 1）教学内容所属模块：歌唱 2）年级：高二 3）所用教材出版单位：湖南文艺出版社 4）所属的章节：第三单元第一节 5）学时数： 45 分钟 二、教学设计 1、教学目标： ①、在欣赏互动中感受女声的音域及演唱风格，体验女声的音色特点。 ②、在欣赏互动中，掌握美声、民族、通俗三种唱法的特点，体验其魅力。 ③、让学生能够尝试用不同演唱风格表现同一首歌。 ④、通过学唱歌曲培养学生热爱祖国、热爱生活的激情。 2、教学重点： ①、掌握女高音、女中音的音域和演唱特点。 ②、掌握美声、民族、通俗三种方法演唱风格。 3、教学难点： ①、学生归纳不同唱法的特点与风格。

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适合女生KTV唱的100首好听的歌别吝色你的嗓音很好学 1、偏爱----张芸京 2、阴天----莫文蔚 3、眼泪----范晓萱 4、我要我们在一起---=范晓萱 5、无底洞----蔡健雅 6、呼吸----蔡健雅 7、原点----蔡健雅&孙燕姿 8、我怀念的----孙燕姿 9、不是真的爱我----孙燕姿 10、我也很想他----孙燕姿 11、一直很安静----阿桑 12、让我爱----阿桑 13、错过----梁咏琪 14、爱得起----梁咏琪 15、蓝天----张惠妹 16、记得----张惠妹 17、简爱----张惠妹 18、趁早----张惠妹 19、一念之间----戴佩妮 20、两难----戴佩妮 21、怎样----戴佩妮 22、一颗心的距离----范玮琪 23、我们的纪念日----范玮琪 24、启程----范玮琪 25、最初的梦想----范玮琪 26、是非题----范玮琪 27、你是答案----范玮琪 28、没那么爱他----范玮琪 29、可不可以不勇敢----范玮琪 30、一个像夏天一个像秋天----范玮琪 31、听，是谁在唱歌----刘若英 32、城里的月光----许美静 33、女人何苦为难女人----辛晓琪 34、他不爱我----莫文蔚 35、你是爱我的----张惠妹 36、同类----孙燕姿 37、漩涡----孙燕姿 38、爱上你等于爱上寂寞----那英 39、梦醒了----那英 40、出卖----那英 41、梦一场----那英 42、愿赌服输----那英

43、蔷薇----萧亚轩 44、你是我心中一句惊叹----萧亚轩 45、突然想起你----萧亚轩 46、类似爱情----萧亚轩 47、Honey----萧亚轩 48、他和他的故事----萧亚轩 49、一个人的精彩----萧亚轩 50、最熟悉的陌生人----萧亚轩 51、想你零点零一分----张靓颖 52、如果爱下去----张靓颖 53、我想我是你的女人----尚雯婕 54、爱恨恢恢----周迅 55、不在乎他----张惠妹 56、雪地----张惠妹 57、喜欢两个人----彭佳慧 58、相见恨晚----彭佳慧 59、囚鸟----彭羚 60、听说爱情回来过----彭佳慧 61、我也不想这样----王菲 62、打错了----王菲 63、催眠----王菲 64、执迷不悔----王菲 65、阳宝----王菲 66、我爱你----王菲 67、闷----王菲 68、蝴蝶----王菲 69、其实很爱你----张韶涵 70、爱情旅程----张韶涵 71、舍得----郑秀文 72、值得----郑秀文 73、如果云知道----许茹芸 74、爱我的人和我爱的人----裘海正 75、谢谢你让我这么爱你----柯以敏 76、陪我看日出----蔡淳佳 77、那年夏天----许飞 78、我真的受伤了----王菀之 79、值得一辈子去爱----纪如璟 80、太委屈----陶晶莹 81、那年的情书----江美琪 82、梦醒时分----陈淑桦 83、我很快乐----刘惜君 84、留爱给最相爱的人----倪睿思 85、下一个天亮----郭静 86、心墙----郭静

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聚合诸侯捍卫中原，匡正天下功业千秋。号令诸侯以匡周室，主要靠的不是 武力。 行为磊落不欺诈，美德流传于身后。孔子赞美齐桓公，也称赞管仲。 百姓深受恩惠，天子赐肉与桓公，命其无拜来接受。桓公称小白不敢，天子 威严就在咫尺前。 晋文公继承来称霸，亲身尊奉周天王。周天子赏赐丰厚，仪式隆重。 接受玉器和美酒，弓矢武士三百名。晋文公声望镇诸侯，从其风者受尊重。 威名八方全传遍，名声仅次于齐桓公。佯称周王巡狩，招其天子到河阳，因 此大众议论纷纷。 赏析 《短歌行》 (“周西伯昌”)主要是曹操向内外臣僚及天下表明心 迹，当他翦灭群凶之际，功高震主之时，正所谓“君子终日乾乾，夕惕若 厉”者，但东吴孙权却瞅准时机竟上表大说天命而称臣，意在促曹操代汉 而使其失去“挟天子以令诸侯”之号召， 故曹操机敏地认识到“ 是儿欲据吾著炉上郁!”故曹操运筹谋略而赋此《短歌行 ·周西伯 昌》。 西伯姬昌在纣朝三分天下有其二的大好形势下， 犹能奉事殷纣， 故孔子盛称 “周之德， 其可谓至德也已矣。 ”但纣王亲信崇侯虎仍不免在纣王前 还要谗毁文王，并拘系于羑里。曹操举此史实，意在表明自己正在克心效法先圣 西伯姬昌，并肯定他的所作所为，谨慎惕惧，向来无愧于献帝之所赏。 并大谈西伯姬昌、齐桓公、晋文公皆曾受命“专使征伐”。而当 今天下时势与当年的西伯、齐桓、晋文之际颇相类似，天子如命他“专使 征伐”以讨不臣，乃英明之举。但他亦效西伯之德，重齐桓之功，戒晋文 之诈。然故作谦恭之辞耳，又谁知岂无更讨封赏之意乎 ?不然建安十八年(公元 213 年)五月献帝下诏曰《册魏公九锡文》，其文曰“朕闻先王并建明德， 胙之以土，分之以民，崇其宠章，备其礼物，所以藩卫王室、左右厥世也。其在 周成，管、蔡不静，惩难念功，乃使邵康公赐齐太公履，东至于海，西至于河， 南至于穆陵，北至于无棣，五侯九伯，实得征之。 世祚太师，以表东海。爰及襄王，亦有楚人不供王职，又命晋文登为侯伯， 锡以二辂、虎贲、斧钺、禾巨 鬯、弓矢，大启南阳，世作盟主。故周室之不坏， 系二国是赖。”又“今以冀州之河东、河内、魏郡、赵国、中山、常 山，巨鹿、安平、甘陵、平原凡十郡，封君为魏公。锡君玄土，苴以白茅，爰契 尔龟。”又“加君九锡，其敬听朕命。” 观汉献帝下诏《册魏公九锡文》全篇，尽叙其功，以为其功高于伊、周，而 其奖却低于齐、晋，故赐爵赐土，又加九锡，奖励空前。但曹操被奖愈高，心内 愈忧。故曹操在曾早在五十六岁写的《让县自明本志令》中谓“或者人见 孤强盛， 又性不信天命之事， 恐私心相评， 言有不逊之志， 妄相忖度， 每用耿耿。

2008年浙师大《外国文学名著鉴赏》期末考试答案

（一）文学常识 一、古希腊罗马 1．（1）宙斯(罗马神话称为朱庇特)，希腊神话中最高的天神，掌管雷电云雨，是人和神的主宰。 （2）阿波罗，希腊神话中宙斯的儿子，主管光明、青春、音乐、诗歌等，常以手持弓箭的少年形象出现。 （3）雅典那，希腊神话中的智慧女神，雅典城邦的保护神。 （4）潘多拉，希腊神话中的第一个女人，貌美性诈。私自打开了宙斯送她的一只盒子，里面装的疾病、疯狂、罪恶、嫉妒等祸患，一齐飞出，只有希望留在盒底，人间因此充满灾难。“潘多拉的盒子”成为“祸灾的来源”的同义语。 （5）普罗米修斯，希腊神话中造福人间的神。盗取天火带到人间，并传授给人类多种手艺，触怒宙斯，被锁在高加索山崖，受神鹰啄食，是一个反抗强暴、不惜为人类牺牲一切的英雄。 （6）斯芬克司，希腊神话中的狮身女怪。常叫过路行人猜谜，猜不出即将行人杀害；后因谜底被俄底浦斯道破，即自杀。后常喻“谜”一样的人物。与埃及狮身人面像同名。 2．荷马，古希腊盲诗人。主要作品有《伊利亚特》和《奥德赛》，被称为荷马史诗。《伊利亚特》叙述十年特洛伊战争。《奥德赛》写特洛伊战争结束后，希腊英雄奥德赛历险回乡的故事。马克思称赞它“显示出永久的魅力”。 3．埃斯库罗斯，古希腊悲剧之父，代表作《被缚的普罗米修斯》。6．阿里斯托芬，古希腊“喜剧之父”代表作《阿卡奈人》。 4．索福克勒斯，古希腊重要悲剧作家，代表作《俄狄浦斯王》。5．欧里庇得斯，古希腊重要悲剧作家，代表作《美狄亚》。 二、中世纪文学 但丁，意大利人，伟大诗人，文艺复兴的先驱。恩格斯称他是“中世纪的最后一位诗人，同时又是新时代的最初一位诗人”。主要作品有叙事长诗《神曲》，由地狱、炼狱、天堂三部分组成。《神曲》以幻想形式，写但丁迷路，被人导引神游三界。在地狱中见到贪官污吏等受着惩罚，在净界中见到贪色贪财等较轻罪人，在天堂里见到殉道者等高贵的灵魂。 三、文艺复兴时期 1．薄迦丘意大利人短篇小说家，著有《十日谈》拉伯雷，法国人，著《巨人传》塞万提斯，西班牙人,著《堂?吉诃德》。 2．莎士比亚，16-17世纪文艺复兴时期英国伟大的剧作家和诗人，主要作品有四大悲剧——《哈姆雷特》、《奥赛罗》《麦克白》、《李尔王》，另有悲剧《罗密欧与朱丽叶》等，喜剧有《威尼斯商人》《第十二夜》《皆大欢喜》等，历史剧有《理查二世》、《亨利四世》等。马克思称之为“人类最伟大的戏剧天才”。 四、17世纪古典主义 9．笛福，17-18世纪英国著名小说家，被誉为“英国和欧洲小说之父”，主要作品《鲁滨逊漂流记》，是英国第一部现实主义长篇小说。10．弥尔顿，17世纪英国诗人，代表作：长诗《失乐园》，《失乐园》，表现了资产阶级清教徒的革命理想和英雄气概。 25．拉伯雷，16世纪法国作家，代表作：长篇小说《巨人传》。 26．莫里哀，法国17世纪古典主义文学最重要的作家，法国古典主义喜剧的创建者，主要作品为《伪君子》《悭吝人》（主人公叫阿巴公）等喜剧。 五、18世纪启蒙运动 1）歌德，德国文学最高成就的代表者。主要作品有书信体小说《少年维特之烦恼》，诗剧《浮士德》。 11．斯威夫特，18世纪英国作家，代表作：《格列佛游记》，以荒诞的情节讽刺了英国现实。 12．亨利·菲尔丁，18世纪英国作家，代表作：《汤姆·琼斯》。 六、19世纪浪漫主义 （1拜伦， 19世纪初期英国伟大的浪漫主义诗人，代表作为诗体小说《唐璜》通过青年贵族唐璜的种种经历，抨击欧洲反动的封建势力。《恰尔德。哈洛尔游记》 （2雨果，伟大作家，欧洲19世纪浪漫主义文学最卓越的代表。主要作品有长篇小说《巴黎圣母院》、《悲惨世界》、《笑面人》、《九三年》等。《悲惨世界》写的是失业短工冉阿让因偷吃一片面包被抓进监狱，后改名换姓，当上企业主和市长，但终不能摆脱迫害的故事。《巴黎圣母院》 弃儿伽西莫多，在一个偶然的场合被副主教克洛德.孚罗洛收养为义子，长大后有让他当上了巴黎圣母院的敲钟人。他虽然十分丑陋而且有多种残疾，心灵却异常高尚纯洁。 长年流浪街头的波希米亚姑娘拉.爱斯梅拉达，能歌善舞，天真貌美而心地淳厚。青年贫诗人尔比埃尔.甘果瓦偶然同她相遇，并在一个更偶然的场合成了她名义上的丈夫。很有名望的副教主本来一向专心于"圣职"，忽然有一天欣赏到波希米亚姑娘的歌舞，忧千方百计要把她据为己有，对她进行了种种威胁甚至陷害，同时还为此不惜玩弄卑鄙手段，去欺骗利用他的义子伽西莫多和学生甘果瓦。眼看无论如何也实现不了占有爱斯梅拉达的罪恶企图，最后竟亲手把那可爱的少女送上了绞刑架。 另一方面，伽西莫多私下也爱慕着波希米亚姑娘。她遭到陷害，被伽西莫多巧计救出，在圣母院一间密室里避难，敲钟人用十分纯朴和真诚的感情去安慰她，保护她。当她再次处于危急中时，敲钟人为了援助她，表现出非凡的英勇和机智。而当他无意中发现自己的"义父"和"恩人"远望着高挂在绞刑架上的波希米亚姑娘而发出恶魔般的狞笑时，伽西莫多立即对那个伪善者下了最后的判决，亲手把克洛德.孚罗洛从高耸入云的钟塔上推下，使他摔的粉身碎骨。 （3司汤达，批判现实主义作家。代表作《红与黑》，写的是不满封建制度的平民青年于连，千方百计向上爬，最终被送上断头台的故事。“红”是将军服色，指“入军界”的道路；“黑”是主教服色，指当神父、主教的道路。 14．雪莱，19世纪积极浪漫主义诗人，欧洲文学史上最早歌颂空想社会主义的诗人之一，主要作品为诗剧《解放了的普罗米修斯》，抒情诗《西风颂》等。 15．托马斯·哈代，19世纪英国作家，代表作：长篇小说《德伯家的苔丝》。 16．萨克雷，19世纪英国作家，代表作：《名利场》 17．盖斯凯尔夫人，19世纪英国作家，代表作：《玛丽·巴顿》。 18．夏洛蒂?勃朗特，19世纪英国女作家，代表作：长篇小说《简?爱》19艾米丽?勃朗特，19世纪英国女作家，夏洛蒂?勃朗特之妹，代表作：长篇小说《呼啸山庄》。 20．狄更斯，19世纪英国批判现实主义文学的重要代表，主要作品为长篇小说《大卫?科波菲尔》、《艰难时世》《双城记》《雾都孤儿》。21．柯南道尔，19世纪英国著名侦探小说家，代表作品侦探小说集《福尔摩斯探案》是世界上最著名的侦探小说。 七、19世纪现实主义 1、巴尔扎克，19世纪上半叶法国和欧洲批判现实主义文学的杰出代表。主要作品有《人间喜剧》，包括《高老头》、《欧也妮·葛朗台》、《贝姨》、《邦斯舅舅》等。《人间喜剧》是世界文学中规模最宏伟的创作之一，也是人类思维劳动最辉煌的成果之一。马克思称其“提供了一部法国社会特别是巴黎上流社会的卓越的现实主义历史”。

2019-2020年高一音乐 甜美纯净的女声独唱教案

2019-2020年高一音乐甜美纯净的女声独唱教案 一、教学目标 1、认知目标：初步了解民族唱法、美声唱法、通俗唱法三种唱法的风格。 2、能力目标：通过欣赏部分女声独唱作品，学生能归纳总结出她们的演唱 风格和特点，并同时用三种不同风格演唱同一首歌曲。 3、情感目标：通过欣赏比较，对独唱舞台有更多元化的审美意识。 二、教学重点：学生能用三种不同风格演唱形式演唱同一首歌。 三、教学难点：通过欣赏部分女声独唱作品，学生能归纳总结出她们的演唱 风格和特点。 四、教学过程： （一）导入 1、播放第十三界全国青年歌手大奖赛预告片 （师）问：同学们对预告片中的歌手认识吗 （生）答： （师）问：在预告片中提出了几种唱法？ （生）答：有民族、美声、通俗以及原生态四种唱法，今天以女声独唱歌曲重点欣赏民族、美声、通俗唱法，希望通过欣赏同学们能总结出三种唱法的风格和特点。 （二）、音乐欣赏

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外国名著赏析论文

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