Massive stars in sub-parsec rings around galactic centers

Abstract

We consider the structure of self-gravitating marginally stable accretion disks in galactic centers in which a small fraction of the disk mass has been converted into proto-stars. We find that proto-stars accrete gaseous disk matter at prodigious rates. Mainly due to the stellar accretion luminosity, the disk heats up and geometrically thickens, shutting off further disk fragmentation. The existing proto-stars however continue to gain mass by gas accretion. As a results, the initial mass function for disk-born stars at distances R ~ 0.03-3 parsec from the super-massive black hole should be top-heavy. The effect is most pronounced at around R ~ 0.1 parsec. We suggest that this result explains observations of rings of young massive stars in our Galaxy and in M31, and predict that more of such rings will be discovered. Comment: Figure 1 replaced (the one supplied in the previous version was for a different SMBH mass than intended)

Full text

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arXiv:astro-ph/0512255v2 12 Dec 2005
Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 5 February 2008(MN LATEX style file v2.2)
Massive stars in sub-parsec rings around galactic centers.
Sergei Nayakshin
1Dept. of Physics & Astronomy, University of Leicester, Leicester, LE1 7RH, UK
5 February 2008
ABSTRACT
We consider the structure of self-gravitating marginally stable accretion disks in galac-
tic centers in which a small fraction of the disk mass has been converted into proto-
stars. We find that proto-stars accrete gaseous disk matter at prodigious rates. Mainly
due to the stellar accretion luminosity, the disk heats up and geometrically thickens,
shutting off further disk fragmentation. The existing proto-stars however continue to
gain mass by gas accretion. As a results, the initial mass function for disk-born stars at
distances R ∼ 0.03−3 parsec from the super-massive black hole should be top-heavy.
The effect is most pronounced at around R ∼ 0.1 parsec. We suggest that this result
explains observations of rings of young massive stars in our Galaxy and in M31, and
predict that more of such rings will be discovered.
Key words: Galaxy: centre – accretion: accretion discs – galaxies: active – stars:
formation
1 INTRODUCTION
Accretion disks around super-massive black holes (SMBHs)
have been predicted to be gravitationally unstable at large
radii where they become too cool to resist self-gravity and
can collapse to form stars or planets (Paczy´ nski, 1978;
Kolykhalov & Sunyaev, 1980; Lin & Pringle, 1987; Collin &
Zahn, 1999; Gammie, 2001; Goodman, 2003). There is now
observational evidence that the two rings of young massive
stars of size ∼ 0.1 parsec in the centre of our Galaxy were
formed in situ (Nayakshin & Sunyaev, 2005; Paumard et al.,
2005), confirming the theoretical predictions. In our neigh-
bouring Andromeda Galaxy (M31), Bender et al. (2005) re-
cently discovered a population of hot blue stars in a disk
or ring of similar size, i.e. with radius of ∼ 0.15 parsec.
The significance of this discovery is that SMBH in M31 is
determined to be as massive as MBH ≈ 1.4 × 108M⊙, or
about 40 times more massive than the SMBH in the Milky
Way. This fact alone rules out (Eliot Quataert, private com-
munication) the other plausible mechanism of forming stel-
lar disks around SMBHs, e.g. the massive cluster migration
scenario (e.g., Gerhard, 2001), because the shear presented
by the M31 black hole is much stronger than it is at same
distance from Sgr A∗, and its hard to see how a realistic
star cluster would be able to survive that (G¨ urkan & Rasio,
2005).
In this paper we shall attempt to understand what hap-
pens with the gaseous accretion disk around a SMBH when
the disk crosses the boundary of the marginal stability to
⋆E-mail: Sergei.Nayakshin at astro.le.ac.uk
self-gravitation (Toomre, 1964) and forms first stars. We
find that in a range of distances from SMBH, interestingly
centered at R ∼ 0.1 parsec, creation of first low-mass proto-
stars should lead to very rapid accretion on these stars. The
respective accretion luminosity greatly exceeds the disk ra-
diative cooling, thus heating and puffing the disk up. The
new thermal equilibrium reached is that of a disk stable
to self-gravity where further disk fragmentation is shut off.
Star formation is however continued via accretion onto the
existing proto-stars, which then grow to large masses. We
therefore predict that stellar disks around SMBHs should
generically posses top-heavy IMF, as seems to be observed
in Sgr A∗(Nayakshin & Sunyaev, 2005; Nayakshin et al.,
2005). In the discussion section we note three main differ-
ences between star formation process in a “normal” galactic
environment and that in an accretion disk near a SMBH.
2PRE-COLLAPSE ACCRETION DISK
In this section we determine the structure of the marginally
stable accretion disk, Q ≈ 1, i.e. the disk structure just be-
fore first gravitationally bound objects form. We envisage a
situation in which the disk of a finite radius has been created
by a “mass deposition event” on a time scale much shorter
than the disk viscous time, but much longer than the local
dynamical time, 1/Ω (see below). Such an event could be a
collision of two large gas clouds al larger distances from the
SMBH, which cancelled most of the angular momentum of
the gas, or cooling of a large quantity of hot gas that already
had a specific angular momentum much smaller than that

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2S. Nayakshin
of the Galaxy (hot gas can be supported by its pressure in
addition to rotation). In these conditions, it is reasonable to
expect that the disk will settle into a local thermal equilib-
rium, in which the gas is heated via turbulence generated by
self-gravitation (Gammie, 2001) and is cooled by radiation.
The magnitude of viscosity α-parameter, and the disk cool-
ing time, tcool, are then coupled by (Gammie, 2001; Levin,
2003; Rice et al., 2005):
tcool=4
9
1
γ(γ − 1)αΩ
(1)
where γ is the adiabatic index of gas. As we shall see be-
low, for the parameters of interest, the evolution of the disk
after star formation is turned on proceeds on a time scale
again shorter than the local viscous time. Therefore, below
we assume that the disk is in the hydrostatic and thermal
equilibrium, but not in a steady accretion state, when the
accretion rate
˙ M(R) = const. We now estimate the condi-
tions in the disk (as a function of radius R) when it reaches
surface density large enough to suffer local gravitational col-
lapse. Star formation is a local process in this approach, and
different rings in the disk could become gravitationally un-
stable at different times.
The appropriate accretion disk equations for Q ∼ 1
have been discussed by many authors (see references in the
Introduction). The hydrostatic balance condition yields
c2
s≡P
ρ= H2Ω2, (2)
where cs is the isothermal sound speed, P and ρ are the
total pressure and gas density, H is the disk scale height
and Ω2= (GMBH/R3+ σ2
frequency at radius R from the black hole. σv here is the
stellar velocity dispersion just outside the SMBH radius of
influence, i.e. where the total stellar mass becomes larger
than MBH. Using equation 2, the disk midplane density is
determined by inversion of the definition of Toomre (1964)
Q-parameter:
v/R2) is the Keplerian angular
ρ =
Ω2
√2πGQ
. (3)
To solve for temperature of the disk, we should specify
heating and cooling rates per unit area of the disk. The
former is coupled to the rate of the mass transfer through
the disk, ˙ M:
Q+
d=3Ω2 ˙ M
8π
. (4)
The accretion rate is given by
˙ M = 3πνΣ , (5)
where Σ = 2Hρ is the disk surface density. The kinematic
viscosity ν in terms of the Shakura & Sunyaev (1973) pre-
scription is ν = αcsH. Marginally stable self-gravitating
disks are believed to have α ∼ 1 (Lin & Pringle, 1987;
Gammie, 2001; Rice et al., 2005) generated by spiral density
waves.
The cooling rate of the disk (per side per unit surface
area) is given by
Frad=3
8
σT4
(τ + 2/3τ), (6)
Figure 1. Disk mass (solid curve), Md= πΣR2, and midplane
temperature (dashed) as a function of distance from the SMBH
are shown in the upper panel. The SMBH mass is that of Sgr A∗.
The lower panel shows two estimates of the mass of the first
fragments forming in the disk. The realistic value of the fragments
mass is likely to be in between these two curves.
where τ = κΣ/2 is the optical depth of the disk. This ex-
pression allows one to switch smoothly from the optically
thick τ ≫ 1 to the optically thin τ ≪ 1 radiative cooling
limits. We approximate the opacity coefficient κ following
Table 3 in the Appendix of Bell & Lin (1994). For the prob-
lem at hand, it is just the first four entries in the Table are
important as disk solutions with T
unstable(see also Appendix B in Thompson et al., 2005)
since opacity rises as quickly as κ ∝ T10in that region. This
rather simple approximation to the opacities is justified for
the order of magnitude parameter study that we intend to
perform here. In addition, we set a minimum temperature
of T = 40 K for our solutions. Even without any gas accre-
tion, realistic gas disks near galactic centres will be heated
by external stellar radiation to effective temperatures of this
order or slightly larger. The main conclusions of this paper
do not sensitively depend on the exact value of the minimum
temperature or exact opacity law.
>
∼2000 K are thermally
2.1Masses of first stars in the disk
The upper panel of Figure 1 shows the resulting disk “mass”
defined as Md= πΣR2and the midplane temperature (mul-
tiplied by 103). The lower panel of the Figure shows two es-
timates of mass of the first fragments in the disk. Different
authors estimate the volumes of the first unstable fragments

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3
slightly differently, but the reasonable range seems to be
from H3to 2H ×(2πH)2. The two curves in the lower panel
of Figure 1 should then encompass the reasonable outcomes,
from Mfrag = ρH3to Mfrag= ρ8π2H3. From the Figure, the
fragment mass is, in the observationally interesting range of
radii, i.e. R ∼ 0.1 − 1 pc, Mfrag
were to rapidly and completely collapse into clumps of mass
of this order, one would expect low-mass stars or even giant
planets to dominate the mass spectrum of collapsed objects.
Numerical simulations with a constant cooling time
show (e.g., Gammie, 2001) that if the disk cooling time is at
the threshold for the fragmentation to take place, then the
first gas clumps will grow very rapidly by inelastic collisions
with other clumps, possibly until they reach the isolation
mass Miso ∼ (πR2Σ)3/2/M1/2
case, then the main point of our paper – that stars born in
an accretion disk near a SMBH are massive on average – is
proven, because the isolation mass can be hundreds to as
much as 104Solar masses (Goodman & Tan, 2004). How-
ever we suspect that Gammie (2001) simulations yielded no
further gravitational collapse of the gas clumps precisely be-
cause the cooling time were kept constant. As the clump den-
sity increases, the clump free-fall time decreases as ∝ ρ−1/2,
and hence the clumps could not collapse as they could not
cool rapidly enough. It is quite likely that had the cool-
ing time inside the clumps were allowed to decrease as the
clumps get hotter, the clumps would collapse before they
agglomerate into larger ones.
<
∼M⊙, and hence if disk
BH(Levin, 2003). If this is the
3 EFFECTS OF FIRST STARS ON THE DISK
We shall now assume that gravitational instabilities in the
Q ≈ 1 disc resulted in the formation of first proto-stars. Ac-
cording to the discussion in §2.1, we conservatively assume
that these proto-stars are low mass objects, and show that
in certain conditions even a small admixture of these to the
accretion disk may significantly affect its evolution.
3.1Coupling between stellar and gas disks
As the stars are born out of the gas in a turbulent disc, we
assume that the initial stellar velocities are the sum of the
bulk circular Keplerian velocity vK in the azimuthal direc-
tion and a random component with three dimensional dis-
persion magnitude σ0 ≈ cs. This also implies that at least
initially stellar disk height-scale, H∗, is roughly the same as
that of the gas disk, H. Proto-stars would interact by direct
collisions and N-body scatterings between themselves and
also via dynamical friction with the gas. The rate for proto-
stellar collisions, 1/tcoll, is the sum of two terms, the geomet-
ric cross-section of the colliding stars and the gravitational
focusing term (e.g., see Binney & Tremaine, 1987). One
can show that 1/tcollΩ ≃ max[Σ∗R2
from which it is obvious that collisions are unimportant as
long as the collision radius, Rcoll ∼ 2Rproto (the proto-star
radius), is much smaller than the disk height scale. In all of
the cases considered below this will be satisfied by few orders
of magnitude, therefore we shall neglect direct collisions.
The N-body evolution of the system of stars immersed
into a gas disk is described by (Nayakshin & Cuadra, 2005)
coll/M∗,(Σ∗/Σ)Rcoll/H],
dσ
dt∼ 4πG2M∗
?
ρ∗lnΛ∗
σ2
−
ρCdσ
s + σ2)3/2
(c2
?
(7)
where lnΛ∗ ∼ few is the Coulomb logarithm for stellar col-
lisions, Cd ∼ few is the drag coefficient for star-gas inter-
actions (Artymowicz, 1994), σ is one-dimensional velocity
dispersion, and ρ∗ = Σ∗/2H∗ is the stellar surface density.
Therefore, as long as the gas density ρ>
dispersion cannot grow as it is damped by interactions with
the gas too efficiently. Recalling that ρ∗ = Σ∗/2H∗, we find
that in this situation
?1/4
Thus, initially, when Σ∗ ≪ Σ, stars are embedded in the
gaseous disk and form a disk geometrically thinner than
that of the gas. However, if stellar surface density grows and
approaches that of the gaseous component, then the stel-
lar velocity dispersion will run away. The stars then form a
geometrically thicker disk (numerical simulations, to be re-
ported in a future paper, confirm these predictions). Galaxy
disks apparently operate in this regime, with molecular gas
having a much smaller scale height than stars.
∼ρ∗, stellar velocity
σ
cs
≈H∗
H
∼
?Σ∗
Σ
< 1 .(8)
3.2 Accretion onto proto-stars
Since the stars remain embedded in the disk, they will con-
tinue to gain mass via gas accretion. Such accretion has been
previously considered by many authors (e.g., Lissauer, 1987;
Bate et al., 2003; Goodman & Tan, 2004). We assume that
the accretion rate is
˙ M∗ = min?˙ MBondi,˙ MHill,˙MEdd?
where the accretion rates in the brackets are the Bondi, the
Hill and the Eddington limit, respectively (e.g. Nayakshin,
2005). The latter is calculated based on the Thomson opac-
ity of free electrons instead of dust opacity because we as-
sume that the cooler regions of accretion flow onto the star
are shielded from the stellar radiation by the inner, hotter
accretion flow (Krumholz et al., 2005):
year−1, where r∗ = R∗/R⊙.
, (9)
˙ MEdd = 10−3r∗M⊙
3.3 Heating of the disc by proto-stars
Presence of the stars will lead to additional disc heating via
radiation and outflows, and N-body scattering. The energy
liberation rate per surface area due to N-body interactions
is given by
Q+
∗N∼ Σ∗σ
?dσ
dt
?
∗∼ 4πG2M∗Σ∗ρ∗lnΛ∗
σ
,(10)
where (dσ/dt)∗ stands for the first term only in equation 7.
Internal disk heating with the α-parameter equal to unity is
Q+
d=9
4Σc2
sΩ . (11)
Comparing the stellar N-body heating with that of the in-
ternal disk heating, we have
Q+
Q+
∗N
d
∼
?Σ∗
Σ
?3/2M∗
MBH
?R
H
?3
(12)
We have assumed above that Toomre-parameter Q ∼ 1
when the stars just appear in the disk, and that Ω2=

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4 S. Nayakshin
GMBH/R3, i.e. that we are within the SMBH sphere of influ-
ence. Considering this expression for typical numbers, one
notices that N-body heating is never important for large
black holes and disks with finite disk thickness, i.e. at dis-
tances of tens of parsec and further away, but it may become
important for smaller SMBH such as Sgr A∗and sub-parsec
distances.
Radiative internal output of the stars is another source
of disk heating. We shall use a very simple parameterisa-
tion for the internal stellar luminosity as a function of mass,
L∗ ∝ M3
accretion luminosity:
∗. To this radiative output we should also add the
Lacc =GM˙ M∗
R∗
. (13)
The sum should not exceed the Eddington limit, LEdd =
4πGM∗mpc/σT ≈ 1038m∗ erg s−1, and hence our prescrip-
tion is
?
L∗ = minL⊙
M3
M⊙3+ Lacc,LEdd
∗
?
, (14)
where LEdd = 4πGM∗mpc/σT is the Eddington limit with
Thomson opacity σT. The radiative disk heating per unit
surface area is then
Q+
∗rad=Σ∗
M∗L∗ .(15)
4 ANALYTICAL ESTIMATES
As equation 3 suggests, accretion disks on parsec scales
are as dense as 1012particles cm−3, which is multiple or-
ders of magnitude denser than the densest gas in molecu-
lar clouds far from galactic centers. Therefore it is not sur-
prising that the first proto-stars will be accreting at super-
Eddington rates, typically. The corresponding accretion lu-
minosity heating (equation 15 with L∗ = LEdd) is
Q+
∗rad∼ 105Σ∗r∗ erg s−1cm−2.
At the same time, the disk intrinsic heating at Q ≈ 1 is,
from equation 11,
(16)
Q+
d≈ Σ T2
MBH
3 × 106M⊙
?0.1pc
R
?3/2
erg s−1cm−2, (17)
where Σ and Σ∗ are in units of g cm−2, and T2 is the disk
temperature in units of 100 K. We see that even a very small
admixture of proto-stars (Σ∗ ≪ Σ) accreting at Eddington
accretion rates will result in stellar heating much exceed-
ing the intrinsic one. Since the disk thermal equilibrium is
established on time scales comparable to the disk dynam-
ical time, the disk will heat up at a constant Σ until its
radiative losses can balance the accretion luminosity. This
will increase the disk sound speed and the Toomre (1964)
Q-parameter above unity. Therefore, the accretion feedback
will stop further fragmentation from happening. The stars
embedded into the disk will however continue to gain mass
at very high rates. This should lead to a top-heavy initial
mass function for the stars.
Note that a similar conclusion has been already reached
by Levin (2003) who considered rather later stages in the
evolution of a more massive AGN disk, when the high mass
stars were turned into stellar mass black holes. He pointed
Figure 2. Temperature (upper panel) and Q-parameter (lower
panel) versus radius in parsec for the marginally self-gravitating
disk with (dashed) and without (solid) proto-stars embedded in
the disks. The proto-stars have masses M∗= 0.1M⊙and surface
density Σ∗= 0.001Σ.
out that accretion onto these embedded black holes will
likely heat the disk, driving the Toomre (1964) Q-parameter
above unity for radii somewhat smaller than a parsec.
5NUMERICAL RESULTS
We shall first consider the case of Sgr A∗for which the
mass of the SMBH is estimated to be MBH ≃ 3.5 × 106M⊙
(Sch¨ odel et al., 2002; Ghez et al., 2003). For the particular
example, we shall accept that Σ∗ = 0.001Σ and that the ini-
tial masses of proto-stars are M∗ = 0.1M⊙. The upper panel
of Figure 2 shows the disk midplane temperature before the
stars are introduced (solid) and after (dashed) versus radius
R. Temperature of the gas increases for radii 0.03<
parsec. Toomre (1964) Q-parameter after the stars are intro-
duced is plotted in the bottom panel of Figure 1. Q indeed
becomes greater than unity in the same radial range, thus
shutting off further gravitational collapse.
The radial range where the proto-stars shut off further
fragmentation is rather insensitive to assumptions of our
model. Figure 3 shows the Q-parameter after stars are in-
troduced into the Q = 1 gaseous disk for Sgr A∗case but
with varying assumptions. In particular, the solid curve is
the same as that in Figure 2, lower panel; the dotted one is
calculated for the opacity coefficient κ multiplied arbitrarily
by 3, whereas for the dashed one κ was divided by√T. These
∼R<
∼0.2

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5
Figure 3. Toomre Q-parameter versus radius for the self-
gravitating disk in which first stars were born. The solid line is
the same as that in Figure 2, lower panel. The other curves are
obtained by varying assumption of the model to test sensitivity
of the results (see text).
arbitrary changes were introduced to estimate the degree to
which the results are dependent on the (uncertain) opacity
detail. Finally, the dot-dashed curve is calculated assumed
the standard opacity but increasing the proto-stellar mass
to 1M⊙ and stellar surface density Σ∗ to 0.01, respectively.
The stellar heating is then more pronounced and a larger
area of the marginally stable disk can be affected.
We also consider the case of a more massive black hole,
in particular we set MBH = 1.4 × 108M⊙, as thought to
be the case for M31. Figure 4 shows the disk temperature
structure (upper panel) before the collapse (dashed) and af-
ter the collapse. Note that these curves are almost identical
to those for Sgr A∗case except for a general shift to larger
radii. This shift is about a factor of 3 only, which should
not be surprising given that in the standard accretion disk
theory (Shakura & Sunyaev, 1973) the midplane disk tem-
perature is a very weak function of the central object mass.
6 DISCUSSION
In this semi-analytical paper, we studied the “first minutes”
of an accretion disk around a super-massive black hole af-
ter the disk became unstable and formed first stars. We as-
sumed that the disk accumulated its mass over time scales
much longer than the local dynamical time, and is thus in
a thermal equilibrium before the gravitational collapse. In
this case, irrespectively of the typical mass of the first proto-
stars, even a 0.1% admixture (by mass) of these significantly
alters the thermal energy balance of the disk. The proto-
stars accrete gas from the surrounding disk at very high
(super-Eddington) rates at a range of disk radii. The accre-
tion luminosity of these stars is sufficient to heat the disk up
in that range of radii to the point where it becomes stable
to self-gravity (Q > 1), which then shuts off further frag-
mentation of the disk. The proto-stars already present in
the disc would however continue to gain mass at very high
rates. Quite generally, then, an average star created in such
Figure 4. Same model as that used for Figure 1, except for a
higher SMBH mass, MBH= 1.4 × 108M⊙. Note that, compared
with Figure 1, the radial range where proto-stars succeed in heat-
ing the disk up has shifted to slightly larger radii.
a disk will be a massive one, in stark contrast to the typical
galactic star formation event.
Significance of accretion feedback onto embedded stellar
mass black holes for accretion disks near galactic centres was
pointed out by Levin (2003). He noted that the accretion
disks can be stabilised by the feedback out to radius of about
one parsec, in good agreement with our results. Since the
Eddington luminosity depends only on the disk opacity and
the mass of the central object, it is then not really surprising
that the feedback is effective for accretion onto stars as well.
The range of radial distances from the SMBH where
this effect operates is a slow function of the SMBH mass,
and is typically from a fraction of 0.1 parsec to a few par-
sec, with the peak of the effect taking place at R ∼ 0.1
parsec. The reason why the feedback is only effective in a
range of radii is that at large radii, i.e. tens of parsec, the
gas density in the disk drops significantly so that accretion
rates onto the proto-stars become much smaller than the
respective Eddington-limited rates. At radii much smaller
than ∼ 0.1 parsec, the intrinsic disk heating (equation 17)
becomes very large. A related point is that steady-state con-
stant accretion rate disk models show that there is always
the minimum radius where star formation becomes impossi-
ble as Q > 1 there (e.g. Goodman, 2003; Levin, 2003; Nayak-
shin & Cuadra, 2005). The value of the minimum distance
where star formation should be expected is comparable to
the minimum radius for which we predict favourable condi-
tions for development of a top-heavy IMF.