Table Of Content**VolumeTitle**
ASPConferenceSeries,Vol.**VolumeNumber**
**Author**
c**CopyrightYear**AstronomicalSocietyofthePacific
(cid:13)
Momentum-driven feedback andthe M-σrelationin
non-isothermal galaxies
2
1
0 RachaelC.McQuillinandDeanE.McLaughlin
2
Astrophysics Group,LennardJonesLaboratories, KeeleUniversity, Keele,
n
Staffordshire, ST55BG,UK
a
J
2 Abstract. We solve for the velocity fields of momentum-conserving supershells
1 drivenbyfeedbackfromsupermassiveblackholesornuclearstarclusters(centralmas-
sive objects: CMOs). We treat, for the first time, the case of CMOs embedded in
] gaseousprotogalaxieswithnon-isothermaldarkmatterhaloeshavingpeakedcircular-
O
speedprofiles. WefindtheCMOmassthatissufficienttodriveanyshelltoescapeany
C suchhalo. Inthelimitoflargehalomass, relevantto realgalaxies, thiscriticalCMO
. massdependsonlyonthepeakcircularspeedinthehalo,scalingasM V4 .
h crit ∝ c,pk
p
-
o
r
t 1. Introduction
s
a
[ TheM–σrelationbetweensupermassiveblackhole(SMBH)mass(e.g.,Gu¨ltekinetal.
1 2009) or nuclear star cluster (NC) mass (Ferrareseetal. 2006) and the stellar velocity
v dispersion of galaxy bulges can be understood as a result of momentum-conserving
4 feedback (e.g., King 2005; McLaughlinetal. 2006). Recently, Volonterietal. (2011)
3
havearguedthatSMBHmassalsocorrelateswiththeasymptoticcircularspeedatlarge
5
radii inbulges. Suchan M–V relation should presumably also berelated tofeedback,
2 c
. apointthatweinvestigate inMcQuillin&McLaughlin(2012)andsummarizehere.
1
Accretion at near- or super-Eddington rates onto an SMBH in a gaseous proto-
0
2 galaxy is expected to result in a fast wind driven back into the galaxy. Similarly, the
1 windsandsupernovaefrommassivestarsinayoungNCwilldriveasuperwindintoits
v: hostgalaxy. Inasphericalapproximationtoeithercase,thewindfromthecentralmas-
i sive object (CMO)sweeps ambient gasinto ashell that, at least initially, cools rapidly
X
and is therefore momentum-driven (King 2003; McLaughlinetal. 2006). For a CMO
r
a larger than some critical mass, the wind thrust ( M ) can overcome the gravity of
∝ CMO
the host dark matter halo (measured by σ)and the shell can escape, cutting off fuel to
theCMOandlockinginan M –σrelation andassociated scalings.
CMO
King(2005)showedthattheCMOmassrequiredtodriveashellatlargeradiusin
asingular isothermalspherewithvelocitydispersion σ is
0
M f κσ4 (πG2λ) = 4.56 108 M σ4 f λ 1 , (1)
σ ≡ 0 0 × ⊙ 200 0.2 −
where f is a gas mass fracti(cid:14)on (assumed to be spatially constant, and with f 0.2
0 0
so f = f /0.2); κ is the electron scattering opacity; and σ = σ /200kms 1≈. The
0.2 0 200 0 −
parameterλisrelatedtothefeedbackefficiencyofeachCMOtype: λ 1forSMBHs,
≈
and λ 0.05 for NCs (McLaughlinetal. 2006). In McQuillin&McLaughlin (2012),
≈
we derive a version of eq. (1) for non-isothermal dark matter haloes. In this more
realistic andgeneralcase,thecriticalCMOmassdepends onapeakcircularspeed.
1
2 R.C.McQuillinandD.E.McLaughlin
Figure 1. Velocity fields, v2 versus r, for momentum-drivenshells in a singular
isothermal sphere with spatially constant gas fraction (eq. [4]). The long-dashed
curve in each panel is the solution with C = 0 for the CMO mass indicated. The
physical(v2 0)partsofsolutionswithC , 0areshownassolidlines. Themass
unitM isde≥finedineq.(1);theradiusunitisr GM /σ2 49pcσ2 .
σ σ ≡ σ 0 ≃ 200
2. VelocityFieldsofMomentum-DrivenShells
We assume that the CMO wind thrust is dp /dt = λL /c (King&Pounds 2003;
wind Edd
McLaughlinetal. 2006). The equation of motion for amomentum-driven shell in any
darkmatterhaloprofile M (r)istherefore(seealsoKing2005)
DM
d 4πGλM GM (r)
g
M (r)v = CMO M + M (r) . (2)
dt g κ − r2 CMO DM
h i (cid:2) (cid:3)
Hereristheinstantaneousradiusoftheshell;v= dr/dt isthevelocityoftheshell;and
M (r) is the ambient gas mass originally inside radius r (i.e., the mass that has been
g
swept up into the shell when it has radius r). Then, letting M (r) f h(r)M (r),
g ≡ 0 DM
where h(r) is a function describing how the gas traces the dark matter [i.e., h(r) 1
≡
whenthegastracesthedarkmatterdirectly],were-writeeq.(2)tosolveforthevelocity
fieldv(r)ratherthanr(t)directly:
d 8πGλ 2M2 (r)h2(r)
M2 h2v2(r) = M h(r)M (r) DM M + M (r) . (3)
dr DM f κ CMO DM − r2 CMO DM
h i 0 (cid:2) (cid:3)
3. SingularIsothermalSphere
We first revisit the singular isothermal sphere, in which M (r) = 2σ2r/G. With this
DM 0
massprofileandh(r) 1,sothegastracesthedarkmatterdirectly,eq.(3)hassolution
≡
M 2GM C
v2 = 2σ2 CMO 1 CMO + , (4)
0" Mσ − #− r r2
where M is defined in eq. (1) and the constant of integration, C, represents the shell
σ
momentum at r = 0. This solution is only physical if v2 > 0. At arbitrarily large
TheM-σrelationinnon-isothermalgalaxies 3
radius, the shell tends to a constant coasting speed, v2 2σ2 M /M 1 , so that
M > M is required for shells to move at all at la∞rg≡e r. T0his CisMOfundσam−entally the
CMO σ (cid:2) (cid:3)
result of King (2005). But at small radius, the termC/r2 in eq. (4) dominates and the
behaviour ofashellisdetermined byitsinitialmomentum.
Figure1plotsv2versusrfromeq.(4)for M /M =0.3,1.01and3,witharange
σ
CMO
of C values in each case. When M < M (left panel), no shell can ever escape.
σ
CMO
All solutions either have the shell stalling (i.e., crossing v2 = 0) at a finite radius, or
are unphysical (v2 < 0) at all radii. When M = 1.01M (middle panel), only a
σ
CMO
few solutions formally allow shells to reach large r without stalling. However, these
potential escapes require initial(r = 0)shellmomenta,C,solargeastogivevelocities
v > 0.2cσ atradiir 1pcσ2 ;andeventhen,theytendtoalarge-rcoastingspeed
200 ∼ 200
of∼just v = 0.14σ . Thus, M > M is anecessary but not sufficient condition for
0 CMO σ
theescap∞eofmomentum-drivenfeedbackfromanisothermalsphere. ForM = 3M
σ
CMO
(rightpanel),theabilitytoreachlargeradiistilldependsontheinitialmomentumofthe
shell. Shells decelerating from smallradius require v > 0.05cσ atr 1pcσ2 to
200 ∼ 200
avoidstalling. Launchsolutions(thoseacceleratingou∼twardsfromv2 = 0atanon-zero
radius)arealsopossible, butonlystartingfromradiir > 40pcσ2 .
200
Inthecase M = 3M ,theshells thatreachlarg∼eradiieventually coast atv =
σ
CMO
2σ , which is the escape speed from an isothermal sphere. We conclude that at l∞east
0
M 3M is required (along with a large shell momentum at r = 0) for escape.
σ
CMO ≥
The objection by Silk&Nusser (2010) to momentum-conserving black hole winds as
thesole source ofthe SMBH M–σrelation ultimately traces back toourresult forv2 .
However,theproblem isspecifictotheassumptionofanisothermaldarkmatterhalo∞.
4. Non-IsothermalDarkMatterHaloes
More realistically, dark matter haloes have density profiles that are shallower than
isothermal atsmallradii and steeper than isothermal atlarge radii. Thecircular speed,
V2 = GM (r)/r, corresponding to any such density profile increases outwards from
c DM
thecentre, hasawell-defined peak, and thendeclines towards larger radii. Asaresult,
momentum-driven shells always accelerate at large radii (McQuillin&McLaughlin
2012) and are guaranteed to exceed the halo escape speed eventually, just so long as
theydonotstallbeforereachingradiiwheretheycanbegintoaccelerate.
The natural velocity unit in such non-isothermal haloes is the peak value of the
circularspeed, V ,andwedefinetheassociated mass
c,pk
M f0κ Vc4,pk = 1.14 108 M Vc,pk 4 f λ 1 . (5)
σ ≡ λπG2 4 × ⊙ 200kms−1! 0.2 −
An obvious choice of fiducial velocity dispersion here is σ V /√2, for which
0 c,pk
≡
eq. (5) reduces to eq. (1). In either form, given our results for the isothermal sphere
aboveandforotherhalomodelsbelow(e.g.,Figure3), M isbestviewedasjustaunit.
σ
A general analysis of equation (3) shows that, in any dark matter halo with a
single-peakedcircular-speed curve,thereisacriticalCMOmass, M ,thatissufficient
crit
toguaranteetheescapeofshellswithanyinitialmomentum(McQuillin&McLaughlin
2012). This M depends primarily onV andonthedark mattermass, M ,within
crit c,pk pk
the peak of the circular-speed curve. In the limit that M M , which is most
pk σ
≫
4 R.C.McQuillinandD.E.McLaughlin
Figure2. Velocityfieldsformomentum-drivenshellsinaMilkyWay-sized,NFW
halo with spatially constant gas fraction. Long-dashed curves represent solutions
with zero momentumat r = 0. Velocities are putin units of σ V /√2. The
0 c,pk
≡
massunitM isdefinedineq.(5). Theradiusunitontheloweraxisistheradiusat
σ
whichthedarkmattercircular-speedcurvepeaks;intheMilkyWay,r 50kpc.
pk
≈
Ontheupperaxis,r GM /(V2 /2) 24pc(V /200kms 1)2.
σ ≡ σ c,pk ≃ c,pk −
relevant for real galaxies, wefind that M tends to avalue that isindependent ofany
crit
otherdetails(i.e.,theexactshape)ofthedarkmattermassprofile:
M M 1 + M M + M2 M2 M . (M M ) (6)
crit ≃ σ σ pk O σ pk −→ σ pk ≫ σ
h (cid:14) (cid:16) (cid:14) (cid:17)i
We have applied our general analysis to the specific dark matter halo models
of Hernquist (1990), Navarroetal. (1997, NFW), and Dehnen&McLaughlin (2005).
Figure 2, from McQuillin&McLaughlin (2012), shows numerical solutions ofeq. (3)
forthevelocityfieldsofmomentum-drivenshellsinanNFWhalowithparameters ap-
propriate to an L galaxy: a dark matter circular-speed curve peaking at r = 50 kpc
⋆ pk
with V = 200 kms 1, and thus a dark matter mass of M 4.7 1011 M inside
c,pk − pk
r . Equations (5)and(6)thengive M 1.00024M forthe≃critica×lCMOm⊙ass.
pk crit σ
≃
The left panel of Figure 2 shows v2 versus r for shells with different initial mo-
menta,given M = 0.3M . Shellsthatdecelerate fromlargevelocityatsmallradius
σ
CMO
hit v2 = 0 and stall, unless they have an impossible v > 106c at r 1 pc. Shells
∼
launched from radii r < r go on to stall at some large∼r radius, which is still inside
pk
r . Launches from r > r accelerate monotonically outwards and therefore always
pk pk
escape, buttheyonly everstartfromimplausibly large r > 500kpc. Bycontrast, when
M = M (middle panel), all the solutions shown are∼able to escape, regardless of
σ
CMO
their initial velocities or momenta—as expected, since M is so near M here. The
crit σ
right-hand panelofthefigureconfirmsthatallshellsescapeforlargerCMOmasses.
The solid line in Figure 3 shows the CMO mass M that is sufficient for the
crit
escape of any momentum-driven shell from an NFW halo, as a function of the halo
mass M . As expected from eq. (6), M M for large M M . The dashed
pk crit σ pk σ
→ ≫
line in the figure shows the CMO mass that is necessary for the escape of shells with
zero momentum at r = 0 specifically. This isslightly smaller than the sufficient CMO
M atanyhalomass,tending toavalueof 0.94M for M M .
crit σ pk σ
≃ ≫
TheM-σrelationinnon-isothermalgalaxies 5
Figure 3. Solid line: the CMO mass (in units of M defined by eq. [5]) that is
σ
sufficientfortheescapeofanymomentum-drivenshellfromanNFWhalowithdark
mattermassM insidethepeakofitscircularspeedcurve. Dashedline: theneces-
pk
saryCMOmassfortheescapeofshellswithinitialmomentumzerospecifically.
5. Conclusion
The critical CMO mass of King (2003, 2005; eq. [1]) is necessary but not sufficient
for the escape of momentum-driven feedback from isothermal protogalaxies. Bycon-
trast, in non-isothermal dark matter haloes with realistic, single-peaked circular-speed
curves, equations (5) and (6) above give a CMO mass that is sufficient for the escape
ofany momentum-driven shell: M 1.14 108 M (V /200 kms 1)4 f λ 1,if
crit c,pk − 0.2 −
thehaloes aremassive (essentially, M≃ M× )and⊙havespatially constant gasmass
pk crit
≫
fractions. Ourfullanalysis canbefoundinMcQuillin&McLaughlin (2012).
Ourresults directly predict an M –V relation, and thus provide abasis from
CMO c,pk
whichtoaddressclaimedrelationsbetweenSMBHmassandasymptoticcircularspeed
in bulges (e.g., Volonteri etal. 2011). The characteristic galaxy velocity dispersion
thatisrelevanttoobserved M –σrelations isthenσ V /√2—although itisan
CMO 0 ≡ c,pk
openissuehowthisrelatesgenerallytomeasuredvelocitydispersionswithinthestellar
effective radii ofbulges. Meanwhile, working intermsofV seems anatural wayto
c,pk
extend the analysis of galaxy–CMO correlations to systems with significant rotational
support as well as pressure support. This could be of particular interest in connection
withNCsinintermediate-mass early-type galaxies, andeveninbulgeless Sc/Sddisks.
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