Table Of ContentMon.Not.R.Astron.Soc.000,1–8(2001) Printed1February2008 (MNLATEXstylefilev1.4)
Triaxial stellar systems following the r1/n luminosity law:
an analytical mass–density expression, gravitational
torques and the bulge/disc interplay
I. Trujillo1⋆, A. Asensio Ramos1, J. A. Rubin˜o-Mart´ın1, Alister W. Graham1,2,
J. A. L. Aguerri3, J. Cepa1 and C.M. Guti´errez1
1 Instituto de Astrof´ısica de Canarias, E-38205 La Laguna, Tenerife, Spain
2 2 Department of Astronomy, Universityof Florida, Gainesville, Florida, USA
0 3 Astronomisches Institut der Universit¨at Basel, Venusstrasse 7, CH-4102 Binningen, Switzerland
0
2
n Accepted 0000December 00.Received0000December00;inoriginalform0000October00
a
J
9 ABSTRACT
2
We have investigated the structural and dynamical properties of triaxial stellar
1 systems whose surface brightness profiles follow the r1/n luminosity law – extending
v the analysis of Ciotti (1991) who explored the properties of spherical r1/n systems.
5
A new analytical expression that accurately reproduces the spatial (i.e. deprojected)
8
luminosity density profiles (error less than 0.1%) is presented for detailed modelling
4
of the S´ersic family of luminosity profiles. We evaluate both the symmetric and the
1
0 non–axisymmetric components of the gravitational potential and force and compute
2 the torques as a function of position. For a given triaxiality, stellar systems with
0 smaller values of n have a greater non–axisymmetric gravitational field component.
/ We alsoexplorethe strengthofthe non–axisymmetricforcesproducedbybulges with
h
differing n and triaxiality on systems having a range of bulge–to–disc ratios. The
p
increasing disc–to–bulge ratio with increasing galaxy type (decreasing n) is found to
-
o heavily reduce the amplitude of the non–axisymmetric terms, and therefore reduce
r the possibility that triaxial bulges in late–type systems may be the mechanism or
t
s perturbation for non–symmetric structures in the disc.
a Using seeing–convolvedr1/n–bulge plus exponential–disc fits to the K–band data
:
v fromasample of80nearbydisc galaxies,weprobe the relationsbetweengalaxytype,
i S´ersic index n and the bulge–to–disc luminosity ratio. These relations are shown to
X
be primarily a consequence of the relation between n and the total bulge luminosity.
r Inthe K–band,thetrendofdecreasingbulge–to–discluminosityratioalongthespiral
a
Hubble sequence is predominantly,althoughnotentirely,aconsequenceofthe change
in the total bulge luminosity;the trend betweenthe totaldisc luminosity and Hubble
type is much weaker.
Key words: celestial mechanics: stellar dynamics – galaxies: structure of – galax-
ies: photometry – galaxies: elliptical and lenticular, cD– galaxies: spiral – galaxies:
kinematics and dynamics
1 INTRODUCTION 1988; Caon, Capacciolli & D’Onofrio 1993; Young & Cur-
rie 1994; Graham et al. 1996), and for the bulges of spi-
As the quality of photometric data has improved over the
ral galaxies (Andredakis, Peletier & Balcells 1995; Seigar
years (largely due to the use of CCDs), the applicabil-
& James 1998; Moriondo, Giovanardi & Hunt 1998; Khos-
ity of a fitting-function which can account for variations
roshahi, Wadadekar & Kembhavi 2000; Prieto et al. 2001;
in the curvature of a light profile has been demonstrated
Graham2001;M¨ollenhoff&Heidt2001).Thesesystemsare
for elliptical galaxies (Capaccioli 1987, 1989; Davies et al.
not universally described with either an exponential profile
or an r1/4 law (de Vaucouleurs 1948, 1959), but rather a
continuousrange of light profileshapes exist which arewell
⋆ [email protected] described bytheS´ersic (1968) r1/n model.
(cid:13)c 2001RAS
2 I. Trujillo et al.
In ellipticals, the shape parameter n from the S´ersic ple of 80 spiral galaxies, the physical basis to the n–T (or
model is strongly correlated with the other global proper- n–B/D) relation is investigated in Section 4.
ties derived independentlyof the r1/n model, such as: total
luminosity and effective radius (Caon et al. 1993; Young &
Currie1994, 1995; Jerjen &Bingelli 1997; Trujillo, Graham
2 THE r1/n MODEL
&Caon2001),centralvelocitydispersion(Graham,Trujillo
&Caon2001)andalsocentralsupermassiveblackholemass The projected, elliptically symmetric S´ersic r1/n intensity
(Graham et al. 2001). Additionaly, the spiral Hubble type distribution I(r) can be written in terms of the projected,
has been shown to correlate with the bulge index n such elliptical radial coordinate ξ (see details in Trujillo et al.
that early–typeSpiral galaxy bulgeshavelarger valuesof n 2001) such that:
thanlate–typespiralgalaxybulges(Andredakisetal.1995;
Graham2001).Thiscorrelationarisesfromthefactthatthe I(ξ)=I(0)e−bn(ξ/re)(1/n), (1)
index n is well correlated with the bulge–to–disc luminos-
where I(0) is the central intensity, and r is the effective
ityratio(B/D;see,e.g.Simien&deVaucouleurs1986) and e
radiusoftheprojected major–axis. Curvesofconstant ξ on
thisisoneoftheparametersusedtoestablishmorphological
theplane of thesky are the isophotes. The quantityb is a
type(Sandage 1961). n
functionoftheshapeparametern,andischosensothatthe
Given theabundanceof observationalwork andpapers
effectiveradiusencloseshalfofthetotalluminosity.Theex-
now using theS´ersic model, it seems timely that a theoret-
actvalueisderivedfromΓ(2n)=2γ(2n,b ),whereΓ(a)and
ical study is performed on realistic, analytical models fol- n
lowingther1/n law.Structuralanddynamicalpropertiesof γ(a,x) are thegamma function and the incomplete gamma
isotropic, spherical galaxies following r1/n models have al- function respectively (Abramowitz & Stegun 1964, p. 260).
The index n increases monotonically with thecentral lumi-
readybeenstudiedindetailintheinsightfulpaperofCiotti
nosity concentration of the surface brightness distribution
(1991).However,mostellipticalgalaxiesandbulgesofspiral
(Trujillo, Graham, & Caon 2001).
galaxies are known to be non–spherical objects. Typically,
The total projected luminosity L associated with this
themassmodelswhichhavebeenusedforthestudyoftriax-
model is given by
ial galaxies havefollowed analytical expressions which were
selected to reproduce the properties of the de Vaucouleurs 2πn
L=I(0)r2(1−ǫ) Γ(2n), (2)
r1/4 profile (e.g. Jaffe 1983; Hernquist 1990; Dehnen 1993), e b2n
n
or more recently the modified Hubble law (Chakraborty &
whereǫis theellipticity of theisophotes. Forahomologous
Thakur 2000). For that reason, previous studies based on
triaxial ellipsoid, the spatial (deprojected) luminosity den-
these kinds of analytical models, although certainly useful,
sityprofileν(ζ)canbeobtainedbyanAbelintegralequation
are however unable to probe the full range of properties
(Stark 1977):
which are now observed in real galaxies. Consequently,it is
ofimportancetoknowhowmuchroomforimprovementex- ν(ζ)=−f1/2 ∞ d I(ξ) (ξ2−ζ2)−1/2dξ, (3)
its in the studyof triaxial objects following the r1/n family π dξ
of profiles. Zζ (cid:20) (cid:21)
Due to the fact that the observed r1/n luminosity pro- where f1/2 is a constant that depends on the three-
files cannot be deprojected to yield analytical expression dimensional spatial orientation of the object (Varela,
for the spatial density†, the r1/n law has been considered Mun˜oz-Tun˜oz & Simonneau 1996; Simonneau, Varela &
less useful for studies of detailed modelling. An analytical Mun˜oz-Tun˜oz 1998) and ζ parametrizes the ellipsoids of
representation(approximation)forthemassdensityprofiles constant volume brightness. f1/2 equals 1 when the proper
which accurately reproduces the observed r1/n luminosity axisframeoftheobject hasthesameorientation astheob-
profiles would be of great interest for simulations of real server axis frame (i.e. when the Euler angles between the
galaxies. We have therefore derived just such an analytical two frames equalzero).
expressionforthemassdensityprofilesoftheS´ersicfamilyof
models. Our approximation surpasses the accuracy of both
theDehnen models for representing thespecific r1/4 profile 2.1 Mass density profiles
and also their extension tothedoublepower–law models of
Assumea triaxial object whose mass is stratified overellip-
Zhao (1997). soids with axis ratios a:b:c (a≥b>c) and the x– (z–) is the
In this paper we present a detailed study of how the
long(short)axis(seeFig.1).Thesymmetryoftheproblem
physical properties of triaxial stellar systems change as a
motivates usto work with ellipsoidal coordinates where:
function of the index n. An accurate analytical expres-
sion for modelling the spatial density is presented in Sec- x = ζsinψcosθ
tion 2. In Section 3 we explore the axisymmetric and the y = αζsinψsinθ
non–axisymmetric components of the potential, forces and
z = βζcosψ (4)
torques associated with a S´ersic light distribution. Finally,
byusingliterature availableK–band observationsof asam- and whereα=b/a and β=c/a. Themass models considered
in this study are the triaxial generalizations of the spheri-
cal models discussed in detail by Ciotti (1991). The math-
† Recently, Mazure & Capelato (2002) have provided an exact ematical singularities present in Eq. 3 were considered and
solutionforthis,andotherrelatedspatialproperties,intermsof solved by Simonneau & Prada (1999, Eq. 16). Substituting
theMeijerGfunctionswhentheS´ersicindexnisaninteger. Eq.1intoEq.3,lettingξ=ζcoshϕ,andmultiplyingbythe
(cid:13)c 2001RAS,MNRAS000,1–8
Triaxial stellar systems following the r1/n luminosity law 3
Figure2.a)Thedimensionlessmassdensityprofiles(seeSection
2.1)for the values of n=0.5,1, 2, 4 and10. b)the relativeerror
betweenthetheanalyticalapproximationproposedinEq.7and
theexactsolutionareshownforthepreviousvalues ofn.
Figure 1.Asurfaceofconstantdensityforthetriaxialellipsoid
describedinEq.(5)and(6).
non–axisymmetricgravitationalfieldoverthez=0plane(i.e.
thedisc plane when studyingspiral galaxies).
mass–to–lightratioΥ≡M/L,weobtainasimilarexpression
to theone found by these authors:
ρ(ζ)= f1/2I(0)bnΥ ∞e−bn ζcorsehϕ 1/n(ζcoshϕ)1/n−1dϕ,(5) 3.1 pNootne–nstpiahleirnictahlecopmlapnoenze=nt0of the gravitational
πnr1/n
e Z0 (cid:0) (cid:1)
Weevaluate this quantitybycalculating:
with
x2+ y 2+ z 2 =ζ2. (6) G(r)≡ ΦΦ2((rr)), (8)
α β 0
(cid:16) (cid:17) (cid:18) (cid:19)
where Φ (r) and Φ (r) are the m=2 and m=0 component
The dimensionless mass density profiles ρ(ζ) ≡ r3/Mρ(ζ), 2 0
e ofthegravitational potential, suchthatthenth–orderterm
whereM isthetotalmass, areshown fordifferentvaluesof
Φ (r) is evaluated from the gravitational potential on the
n in Fig. 2a. It should be noted that the inner density pro- m
e z=0 plane Φ(r,θ) by using the Fourier decomposition (see,
filedecreasesmoreslowlywithincreasingradiusforsystems
e.g. Combes & Sanders 1981). Gravitational potential and
havinglower values of n.
gravitational force expressions are shown on AppendixB.
Themassdensityprofilesofther1/n family(Eq.5)can
TheprofilesofG(r)fordifferenttriaxialitiesandvalues
be extremely well approximated by the analytical expres-
of n are shown in Fig. 3. As it is expected, as the triaxial-
sion:
ity increases the non–spherical nature of the gravitational
ρ (ζ)= f1/2I(0)bn2(n−1)/2nΥhp(1/n−1)Kν(bnh1/n), (7) field increases. Also, we highlight that for a given triaxial-
app r nπ 1−C(h) ity, smaller values of n (i.e less concentrated mass distri-
e
bution) give greater non-spherical gravitational fields. The
where h = ζ/r , C(h) = h (logh)2+h logh+h and K
e 1 2 3 ν maximum non–axisymmetrical behavior of the potential is
is the νth–order modified Bessel function of the third kind
obtained at radial distances less than 2 r . This radial dis-
e
(Abramowitz&Stegun1964,p.374).IntheAppendixAwe
tance is also a function of the index n, decreasing a n in-
showthevaluesoftheparameters(ν,p,h ,h ,h )asfunction
1 2 3 creases and remains quite independent of the triaxiality of
oftheindexn.Thisapproximationcontainstwoexactcases:
the object. For a moderately triaxial object with n=1, the
n=0.5 and n=1, and provides relative error less than 0.1%
non–axisymmetrical component of the potential can vary
for therest ofthecases (Fig2b) intheradial range10−3 ≤
some 6% between r=0 and r=2r , and varies some 15% for
ζ/r ≤ 103. This approximation surpasses (by a factor of e
e ourhighly triaxial model.
102–104) the expression presented in Lima Neto, Gerbal &
For an n=1 model, and starting from our moderately
M´arquez (1999).
triaxial case (α=0.75, β=0.50), we increased the value of β
to 0.75. The results are shown in Fig. 3c and reveal that
G(r) varied only mildly. This figure shows that the non–
3 NON–AXISYMMETRIC PERTURBATIONS axisymmetric effect (along the radial distance) in the z=0
DUE TO A TRIAXIAL r1/n STRUCTURE plane is mainly due to how the mass of the bulge is dis-
tributed in this plane.
Forthreedifferenttriaxialitymassdistributions:a)spherical
(α=β=1);b)moderatelytriaxial(α=0.75,β=0.5);c)highly
triaxial (α=0.5, β=0.25), we have explored, in detail, the
(cid:13)c 2001RAS,MNRAS000,1–8
4 I. Trujillo et al.
3.2 Non–spherical component of radial
gravitational forces in the z=0 plane
The non–spherical component of the radial gravitational
forces in thez=0 can beestimated by:
∂Φ (r)/∂r
N(r)≡ 2 . (9)
∂Φ (r)/∂r
0
In Fig. 3 the N(r) profiles (Eq. 9) are evaluated for the
same cases as was the G(r) profiles. A remarkable point is
that N(r) reaches its maximum value in the radial range
2 r <r<4 r . For a spiral bulge structure, this means that
e e
the most important non–axisymmetric effects take place in
a zone which is dominated by the disc. As with the G(r)
parameter, stronger distortions occur as the triaxiality in-
creases and the index n decreases. The mechanism which
controls this distortion is basically determined by the mass
distribution in the z=0plane (Fig 3c).
It is noted that the relative (i.e. % change) non–
axisymmetric effects on the radial forces are larger than
the relative distortion on the potential. As an example,
for a moderately triaxial structure with n=1 the non–
axisymmetric component of theradial forces can reach 8%.
3.3 Torques on z=0 plane
Figure 3. The parameters G(r), N(r) and Π(θ) are shown for
differentvaluesofnandtriaxiality:a)FirstRow:G(r),N(r)and
The torques provoked by the triaxial structures along the
Π(θ) for a moderately triaxial object and different n; b) Second
angularcoordinateareevaluatedaroundthecircleofradius
Row:G(r),N(r)andΠ(θ)forahighlytriaxialstructure;c)Third
r wherethemaximumnon–axisymmetricaldistortionof
max Row: G(r), N(r) and Π(θ) for three moderately triaxial objects
the radial forces is produced (i.e. at the peak of the N(r)
withn=1andthesameaxisratioalongtheyandxaxis.
profile).GiventhegravitationalpotentialΦ(r,θ)inthez=0
plane, we haveat theradius r :
max
Aswiththepreviousparameters,fortherangeoftriaxi-
F (r )
Π(θ)≡ T max , (10) alitiesinvestigatedandagivenn,varyingthemassdistribu-
F (r )
R max tion along the z–axis (i.e. varying the triaxiality parameter
where F (r ) = [∂Φ(r ,θ)/∂θ]/r represents the β) only results in a slight change to Π(θ) (see Fig. 3c). For
T max max max
amplitude of the tangential force along the angular coor- a spherical distribution all aboveparameters are 0.
dinate at radius r , and F (r )=(∂Φ(r ,θ)/∂r) is
max R max max
the radial force at this radius. Due to the symmetry of the
ellipsoid, the values of Π(θ) need only be plotted for one
◦ ◦ 4 LINKING THEORY AND OBSERVATIONS:
quadrantinthez=0plane;weuse0 <θ<90 (Fig.3).De-
THE CONNECTION BETWEEN n AND THE
pendingonthequadrant,Π(θ)iseithernegativeorpositive
B/D LUMINOSITY RATIO
because thesign of thetangential force changes from quad-
rant to quadrant. The maximum torque around a circle of In the previous section we have seen how the non–
radiusr dependsonthetriaxiality of theobject.Asthe axisymmetrical effects (in the z=0 plane) from a triaxial
max
triaxiality increases themaximum torquetendstobecloser bulge increase as n decreases. Taken with the correlation
to the major axis – as one would expect. The position of betweennandgalaxytype(Andredakis,Peletier&Balcells
this peak is quite independentof the valueof n. 1995) shown in Fig. 4, this invokes the natural question:
The absolute value of the torque for any given triaxi- How, if at all, are the structural properties of the bulges
ality increases as n decreases. For ourhighly triaxial bulge, (i.e. n) related (that is, cause and effect) with the non–
Π(θ) ranges from 0.17 (n=10) to 0.24 (n=1), which would axisymmetricalcomponents(i.e.arms)observedinthedisc?
beconsidereda”barstrength”classof2intheclassification The results obtained in the previous sections were evalu-
scheme of Buta & Block (2001). In the case of our moder- ated without any mention to the relative mass of the bulge
ately triaxial object, the maximum absolute value of Π(θ) and disc. It turns out that the axisymmetrical mass dis-
rangesbetween0.06 and0.09. Thesevaluescorrespond toa tribution of the disc causes a strong softening of the non–
“bar strength” class of 1. Thus, even a moderately triaxial axisymmetrical perturbation caused by the non–sphericity
bulge is capable of provoking non–negligible torques on a of the bulge. The degree of “smoothing” is an increasing
disk – that is to say, a bar is not neccessarily required. A function of theD/B ratio.
detailedstudyseparatingthetorquecontributionfromboth Fig. 5 shows the N(r) profile for a moderately triaxial
barsand bulgeswould of coursebeof interest,and it isour bulge with n = 1 and B/D=0.1 and 0.01, and for a bulge
intention to add a range of bar potentials to our models in with n=4 and B/D=1.0 and 0.1 (Fig. 4). N(r) was evalu-
thefuture. ated here assuming the disc follows an exponential surface
(cid:13)c 2001RAS,MNRAS000,1–8
Triaxial stellar systems following the r1/n luminosity law 5
4.1 Why does the n–Type (or n–B/D) relation
exist?
To explore the connection between bulges and discs in spi-
ral galaxies (see, e.g. Fuchs 2000), we have used the data
from two independent samples of galaxies observed in the
K–band. The K–band provides a good tracer of the mass
due to the near absence of dust extinction and the reduced
biasing effect of a few per cent (in mass) of young stars.
We used the data from Andredakis et al. (1995) and the
structural analysis of the de Jong (1996) data performed
by Graham (2001). Both studies were done fitting a seeing
convolved S´ersic law to the spiral galaxy bulges. In both
samples we have removed those objects which contained a
clear bar structure, leaving a total of 28 objects from An-
dredakis et al. (1995) and 52 objects from Graham (2001).
Figure 4. The best–fitting bulge index n is plotted against the
The relations present in Fig. 4 between n and the B/D lu-
B/D luminosity ratio (left) and versus the galaxy morphologi-
minosity ratio,and nversusthemorphological typeThave
cal type (right). The galaxies come fromthe samples of deJong
a Spearman rank–order correlation coefficient of r = 0.77
(1996) (solid points) and Andredakis et al. (1995) (open dia- s
and r =−0.73, respectively,for thecombined sample.
monds).Seetextfordetails. s
Andredakisetal.(1995)suggest“althoughotherpossi-
bilitiescannotbeexcluded,themoststraightforward expla-
nation for this trend is that the presence of the disc affects
thedensitydistributionofthebulgeinsuchawayastomake
thebulgeprofilesteeperintheouterparts.Onemechanism
toproducesuchaneffectmightbethatastrongerdisctrun-
cates the bulge, forcing its profile to become exponential”.
Following thislineofthought,viacollisionless N–bodysim-
ulations,Andredakis(1998)studiedtheadiabaticgrowthof
the disc onto an existing r1/4 spheroid. He found that the
disc potential modifies the bulge surface brightness profile,
lowering the index n. This decrease was larger with more
massiveandmorecompactdiscs.Thismechanism,however,
saturated at around n=2 and exponential bulges could not
beproduced.
We believe that this line of reasoning is not the most
appropiateexplanationfortherelationbetweennandB/D.
Figure 5.TheN(r)profilesforamoderatelytriaxialbulgewith Firstly, we find that the index n is not only well correlated
n=1and4,andB/D=1,0.1,0.01areshown. with the luminous B/D ratio, but is equally well corre-
lated (r =−0.75) with bulgeluminosity M (bulge)(Fig.
s KT
6a). Additionally, the correlation between n and disc lu-
brightness distribution. The ratio between the length scale minosity M (disc) is relatively poor (r = −0.53). Sec-
ofthediscandtheeffectiveradiusofthebulgeisassumedto ondly,M K(bTulge)ismorestronglycorrelasted(r =−0.86)
‡ KT s
be constant and with a value of h/re=5 . The expressions with the B/D ratio than MKT(disc) and the B/D ratio
forthepotentialandtheradialforceofthesestructurescan (r = −0.50) (Fig. 6b). Hence, it is variations in the bulge
s
befound in Binney & Tremaine (1987, p. 77 and 78). which are predominantly responsible for variations in the
Fig. 5 illustrates that for B/D luminosity ratios typi- B/D ratio.
cal of real galaxies, the non–axisymmetrical effects on the These above two correlations seem to indicate that n
disc largely disappear (< 3%). For B/D=1, the values of may be related directly with the properties of the bulge
the N(r) profile remain basically unchanged to that seen in rather than with the combined B/D ratio. Consequently,
Fig.3,butforB/D=0.1thesevaluesdecreaseapproximately as n is correlated with the total bulge luminosity, the cor-
by a factor 2, and for B/D=0.01 this factor is bigger than relation between n and B/D is a result of the more fun-
10. Thus, although the non–radial effects on the z=0 plane damental correlation between M (bulge) and B/D. That
KT
increaseasndecreases,thesmoothingeffectsoftheincreas- is,it isnottherelativeincrease indisc–to–bulge luminosity
ingly dominant disc are stronger. Bulges with small values which produces bulges with smaller values of n, but simply
of n are unable to produce significant non–axisymmetrical that bulges with larger values of n are more luminous (or
effects on a massive disc. vice–versa)andthisproducesthecorrelationbetweennand
theB/D luminosity ratio.
Favouring this argument, we note that among the El-
liptical galaxies (without theneedtoinvokeanydisc)there
‡ Althoughthereisarangeofbulge-to-discsizeratios,amedian existsastrongcorrelation(Pearson’sr=-0.82;Graham,Tru-
valueforh/re intheK-bandis5(Graham&Prieto1999). jillo & Caon, 2001) between n and the total luminosity of
(cid:13)c 2001RAS,MNRAS000,1–8
6 I. Trujillo et al.
Figure 6.a)TheabsoluteK–bandmagnitudeofthebulge(top Figure7.TheK–bandB/DratioandtheabsoluteK–bandmag-
panel) and the disc (bottom panel) are plotted versus the index nitudeofthebulgeareshownasafunctionofthemorphological
n. The galaxies come from the samples of de Jong (1996) (solid type.ThegalaxiescomefromthesamplesofdeJong(1996)(solid
points) and Andredakis et al. (1995) (open diamonds). See text points) and Andredakis et al. (1995) (open diamonds). See text
fordetails.b)TheabsoluteK–bandmagnitudeofbothbulgeand fordetails.
discareshownasafunctionoftheB/Dratio.Thegalaxiescome
fromthesamplesofdeJong(1996)(solidpoints)andAndredakis
etal.(1995) (opendiamonds).Seetextfordetails. importantasthecombinedchangeinthebulgeanddisclu-
¶
minosity . It would then follow that the luminous mass of
the bulge (i.e. M (bulge)) is related with the spiral arm
KT
theseobjects.Theindexnofpressuresupportedstellarsys- structure.
tems are related with the total luminosity of these struc-
tures. In agreement with this, Aguerri, Balcells & Peletier
(2001) have found (using collisionless N–body simulations)
that the bulges of late–type galaxies can increase their n 5 CONCLUSIONS
values via dense satellite accretions where the new value of
The main results of this work are thefollowing:
nisfoundtobeproportionaltothedevouredsatellitemass.
a)Wehavegeneralisedtheanalysisofthephysicalprop-
ertiesofsphericalstellarsystemsfollowingther1/nluminos-
ity law to a homologous triaxial distribution. The density
4.2 MKT(bulge) versus B/D for classifying distribution,potential,forcesandtorquesareevaluatedand
morphological types compared with the spherical case when applicable (Ciotti
1991).Anextremelyaccurateanalyticalapproximation(rel-
Due to the strong correlation between the B/D luminos-
ative error less than 0.1%) for the mass density profile is
ity ratio and M (bulge), it might be of interest to ask
KT provided.
which one of these quantities is preferred to establish the
b)Wederiveanexactexpression showinghowthecen-
§
morphological type T of a galaxy . Working from B–band
tralpotentialdecreasesastriaxialityincreases.Wealsoshow
images (which is good for observing the young star pop-
thatforafixedtriaxiality,astheindexndecreasesthenon–
ulation, and consequently the spiral arm structure, which
axisymmetricaleffectsinthez=0planeincrease.Evenfora
is one of the basic criteria to the Hubble galaxy classifica-
moderately triaxial object, thenon–axisymmetrical compo-
tion), Simien & de Vaucouleurs (1986) fitted r1/4 profiles
nentofthepotentialandtheradialforcesarenotnegligible
and exponential discs to a sample of 64 spiral galaxies and
for small values of n. These components can range from 6
34 S0 type galaxies. They presented a good correlation be-
to 8%, respectively, compared to the value of the spherical
tweenthebulge–to–disc luminosityandT,butnotbetween
component. For our highly triaxial model, they can range
M (bulge) and T. Consequently, their B–band observa-
BT oversome 20%.
tionssuggestedthattheB/Dluminosityratiowaspreferred
c)Thenon–axisymmetricaleffectsinthediscplanedue
toM (bulge)forestablishingthemorphologicaltypeT.In
BT tothebulgestructurearestronglyreducedwhenanaxisym-
Fig.7weshowtherelationbetweentheB/Dluminosityra-
metrical disc mass is added. For this reason, bulges with
tio and T (r = −0.65) and between M (bulge) and T
s KT smaller values of n appear unlikely to produce any signifi-
(r = 0.67). Thus, from K–band observations, and fitting
s cant non–axisymmetrical effect on their disc, which is typi-
r1/n bulge profile models, the change in the luminous mass
cally 10 to 100 more timesmore massive than thebulge. In
of the bulge along the Hubble sequence appears equally as
this regard, the B/D mass ratio and the triaxiality of the
§ ¶
Wereferheretothemorphologicaltypeestablishedontheba- WewereabletorepeatthisanalysisusingB–banddatafromde
sisofB–bandobservations.Infraredimageshaveshownthatthe Jong (1996) (excluding barred galaxies), which gave rs =−0.64
appearenceofgalaxiescanbesubstantiallydifferent(Blocketal. between the B/D luminosity ratio and T, but only rs = 0.54
1999). betweenMBT(bulge)andT.
(cid:13)c 2001RAS,MNRAS000,1–8
Triaxial stellar systems following the r1/n luminosity law 7
bulge are more important, that is, can dominate over the Seigar,M.&James,P.,1998,MNRAS,299,672
effects of small n. S`ersic J. 1968. Atlas de Galaxias Australes Co´rdoba: Obs. As-
d)ThecorrelationfoundbetweennandtheB/D lumi- trono´mico
nosityratiofoundinspiralgalaxiesisexplainedherenotasa Simien,F.,deVaucouleurs,G.,1986,ApJ,302,564
Simonneau E., Varela A., Mun˜oz–Tun˜oz C., 1998, Il Nuovo Ci-
consequenceoftheinterplaybetweenthebulgeandthedisc,
mento,113B,927
butduetothestrongcorrelationbetweennandM (bulge),
T SimonneauE.,PradaF.,1999,(astro–ph/9906151)
and between M (bulge) and B/D. Also, K–band data do
T StarkA.,1977,ApJ,213,368
not support the idea that the B/D luminosity ratio can be
Trujillo,I.,Aguerri,J.A.L.,Cepa,J.,Gutierrez,C.M.,2001,MN-
preferredoverMT(bulge)asanindicatortoestablishgalaxy RAS,321,269
morphological type (T). Both parameters present equally Trujillo,I.,Graham,A.W.,Caon,N.,2001,MNRAS,326,869
good correlations with galaxy typeT. VarelaA.,Mun˜oz–Tun˜ozC.,SimonneauE.,1996,A&A,306,381
Young,C.K.,Currie,M.J.,1994, MNRAS,268,L11
Young,C.K.,Currie,M.J.,1995, MNRAS,273,1141
ZhaoH.,1997,MNRAS,287,525
REFERENCES
Abramowitz M., Stegun I., 1964, Handbook of Mathematical
Functions. Dover,NewYork
APPENDIX A: MASS DENSITY
Aguerri,J.A.L.,BalcellsM.,PeletierR.,2001, A&A,367,428
APPROXIMATION PARAMETERS
Andredakis Y.C., Peletier R., Balcells M., 1995, MNRAS, 275,
874
ThetableA1showsthevaluesoftheparametersthatappear
Andredakis,Y.C.,1998,MNRAS,295,72
in the mass density approximation (Eq. 7).
Binney,J.,Tremaine,S.:1987,GalacticDynamics,PrincetonUni-
versityPress,Princenton
ButaR.,&BlockD.L.,2001,ApJ,550,243
Block,D.L.,Puerari,I.,FrogelJ.A.,Eskridge,P.B.,StocktonA., APPENDIX B: GRAVITATIONAL POTENTIAL
BurkhardF.,1999,Astroph.andSpaceSci,269–270, 5
AND FORCES OF A TRIAXIAL HOMOLOGOUS
CaonN.,CapaccioliM.,D’OnofrioM.,1993,MNRAS,265,1013
STRUCTURE
Capaccioli M., 1987, in Structure and Dynamics of Elliptical
Galaxies,IAUSymp.127,Reidel,Dordrecht,p.47 B1 Gravitational Potential
CapaccioliM.,1989,inTheWorldofGalaxies,ed.H.G.Corwin,
L.Bottinelli(Berlin:Springer-Verlag),p.208 The gravitational potential at position x=(x,y,z) may be
ChakrabortyD.K.&ThakurP.,2000,MNRAS,318,1273 written as
Chandrasekhar,S.1969,EllipsoidalFiguresofEquilibrium(New ∞
[ψ(a)−ψ(ζ)]dτ
York:Dover) Φ(x)=πGabc (B1)
Ciotti,L.1991,A&A,249,99 Z0 (τ +a2)(τ +b2)(τ +c2)
Combes,F.,&Sanders,R.H.1981,A&A,96,164
Davies J., Philips S., Cawson M., Disney M., Kibblewhite E., (Chandrasekhar 196p9, p. 52, theorem 12), with
1988,MNRAS,232,239
2 ζ ′ ′ ′
Dehnen,W.,1993, MNRAS,265,250 ψ(ζ)= ζ ρ(ζ )dζ . (B2)
deJong,R.S.,1996,A&AS,118,557 a2
Za
deVaucouleursG.,1948, Ann.Astrophys,11,247
It follows from Eq. (B1) that the potential at an internal
deVaucouleursG.,1959, Hand.Phys.,Vol.53,p.311
point is a result of two contributions: that due to the ellip-
Fuchs, B., 2000, in: F. Combes and G. Mamon (eds.), Galaxy
Dynamics:fromtheEarlyUniversetothePresent,ASPCon- soid interior to thepoint x considered, and that dueto the
ferenceSeries,Vol.197 homoeoidal shell exterior to x:
Graham A.W., Lauer T., Colles M.M.,Postman M., 1996, ApJ, 1
Φ(x)=4πGαβ
465,534
1−β2
GrahamA.W.,PrietoM.,1999,ApJ,524,L23
GrahamA.W.,2001,AJ,121,820 ∞
p 1−α2 ′ ′ ′
GrahamA.W.,Trujillo,I.,Caon,N.,2001,AJ,122,1707 F arcsin 1−β2, ζ ρ(ζ )dζ +
1−β2
Graham A.W.,Erwin,P., Caon, N., Trujillo,I., 2001, ApJ, 563, " (cid:18) r (cid:19)Zζ
L11 p
Hernquist,L.,1990,ApJ,356,359 ζ ′ ′ (1−β2)ζ′2 1−α2 ′
ζ ρ(ζ )F arcsin , dζ(B3,)
Jaffe,W.,1983, MNRAS,202,995 ζ′2+λ 1−β2
Jerjen,E.,Bingelli,B.1997,intheNatureofEllipticalGalaxies; Z0 r r ! #
TheSecondStromloSymposium,ASPConf.Ser.,116,239 with F(p,q) the Elliptic integral of the first kind
KhosroshahiH.G.,WadadekarY.,KembhaviA.,2000,ApJ,533, (Abramowitz & Stegun 1964, p. 589), and with the restric-
162 tion
Lima Neto G. B., Gerbal D. & Ma´rquez I., 1999, MNRAS, 309,
481 x2 y2 z2
+ + =1. (B4)
MazureA.&CapelatoH.V.2002,A&A,inpress ζ′2+λ (ζ′α)2+λ (ζ′β)2+λ
Mo¨llenhoffC.,HeidtJ.,2001, A&A,368,16
The strength of the central potential decreases as the
MoriondoG.,GiovanardiC.,HuntL.K.,1998,A&AS,130,81
triaxiality of theobject increases:
Prieto, M., Aguerri, J.A.L., Varela, A.M. & Munoz-Tunon, C.,
2001,A&A,367,405
Φ(0) f1/2αβ 1−α2
Sandage, A., 1961. The Hubble atlas of galaxies, Washington: = F arcsin 1−β2, . (B5)
CarnegieInstitution Φsph(0) 1−β2 (cid:18) r1−β2(cid:19)
p
p
(cid:13)c 2001RAS,MNRAS000,1–8
8 I. Trujillo et al.
Table A1.Parametervaluesofthemassdensityapproximation
n ν p h1 h2 h3 Max.Rel.Error(%)
0.5 -0.50000 1.00000 0.00000 0.00000 0.00000 0.000
1.0 0.00000 0.00000 0.00000 0.00000 0.00000 0.000
1.5 0.43675 0.61007 -0.07257 -0.20048 0.01647 0.100
2.0 0.47773 0.77491 -0.04963 -0.15556 0.08284 0.070
2.5 0.49231 0.84071 -0.03313 -0.12070 0.14390 0.070
3.0 0.49316 0.87689 -0.02282 -0.09611 0.19680 0.060
3.5 0.49280 0.89914 -0.01648 -0.07919 0.24168 0.050
4.0 0.50325 0.91365 -0.01248 -0.06747 0.27969 0.020
4.5 0.51140 0.92449 -0.00970 -0.05829 0.31280 0.020
5.0 0.52169 0.93279 -0.00773 -0.05106 0.34181 0.015
6.0 0.55823 0.94451 -0.00522 -0.04060 0.39002 0.005
7.0 0.58086 0.95289 -0.00369 -0.03311 0.42942 0.005
8.0 0.60463 0.95904 -0.00272 -0.02768 0.46208 0.004
9.0 0.61483 0.96385 -0.00206 -0.02353 0.48997 0.004
10.0 0.66995 0.96731 -0.00164 -0.02053 0.51325 0.005
As reported in Ciotti (1991), the models with low n
have an inner (r <r ) potential which is much flatter than
e
models with high n. As the triaxiality increases there is no
important change to the gradient of the gravitational po-
tential along the semimajor–axis; the main effect is to shift
thegravitationalpotentialprofileinwardsfromthespherical
case, resulting in a lower potential at intermediate radii.
B2 Gravitational Forces
Thegravitationalforcesforatriaxialstructurearegivenby
theexpression:
∂Φ(x)
− =4πGαβx
∂x i
i
ζ 1 ζ′2ρ(ζ′) dζ′
,
Z0 (ζ′ai/a)2+λ (ζ′2+λ)((αζ′)2+λ)((βζ′)2+λ)f(ζ′,ai)
i=1,2,3 (B6)
p
with a ≡ a, a ≡ b, a ≡ c, and
1 2 3
′ x2
f(ζ ,a )= i . (B7)
i [(ζ′a /a)2+λ]2
i
i=1,2,3
X
The restrictions given in Eqs. (6) and (B4) also apply here.
ThispaperhasbeenproducedusingtheRoyalAstronomical
Society/Blackwell Science LATEX stylefile.
(cid:13)c 2001RAS,MNRAS000,1–8
