Download –1– 28. HIGH-MASS STAR FORMATION: THEORY 28.1. The Effects

–1–
28.
HIGH-MASS STAR FORMATION: THEORY
28.1.
The Effects of Radiation Pressure
There are two critical differences between low and high-mass star formation: First, low-mass
stars form in a time t∗f ∼ m∗ /ṁ∗ that is short compared to the Kelvin-Helmholtz time,
tKH = Gm2∗ /R∗ L∗ ,
(1)
whereas high-mass stars generally have tKH . t∗f (Kahn 1974). For example, for m∗ = 31M¯ , we
have R∗ = 9.3R¯ and L∗ = 105.24 L¯ (Sternberg et al. 2003), corresponding to tKH = 18, 000 yr;
as we shall see below, the formation time for such a star is of order 10 5 yr. As a result, whereas
low-mass stars undergo extensive pre-main sequence evolution, high-mass stars accrete a significant
amount of mass while on the main sequence.
The second critical difference follows from the first: because high-mass stars are accreting
while on the main sequence, they are hot and luminous, and this implies a radiation pressure large
enough to overcome the force of gravity. The force per unit mass due to radiation pressure is
Z ∞
κF F
κF L
κP (T )L
κν Fν
dν =
=
'
,
(2)
frad =
2c
c
c
4πr
4πr2 c
0
where κF is the flux-mean opacity and κP (T ) is the Planck-mean opacity. The approximation
κF ' κP becomes exact as the frequency distribution approaches that of a blackbody. We can now
define an Eddington limit for dust, LE, d , based on the condition that the radiative force match the
gravitational force,
κP (T )LE, d
Gm∗
4πcGm∗
= 2
⇒ LE, d =
.
(3)
2
4πr c
r
κP
Pollack et al. (1994) calculated κP (T ) for a dust model with an MRN distribution (dngr /da ∝
extending to 1 µm and a steeper distribution (∝ a−5.5 ) to 5 µm. The dominant constituents of
the grains are olivine, which vaporizes at about 1200 K; refractory organic material, which vaporizes
at about 600 K; and water ice, which vaporizes at about 140 K (all vaporization temperatures are
evaluated at ρ = 10−12 g cm−3 ). The Planck opacity rises from 0 at T = 0 K to 3 cm2 g−1
at T = 100 K, and reaches a maximum of 8 cm2 g−1 at T ' 600 K. At higher temperatures, the
opacity drops as the grains are vaporized. The effective Eddington limit is defined by the maximum
opacity, which sets the minimum luminosity:
¶
µ
m∗
4
LE, d = 4.9 × 10
L¯ .
(4)
30 M¯
a−3.5 )
By comparison, the standard Eddington limit is based on opacity due to electron scattering, κ es =
0.4 cm2 g−1 , which gives LE ' 106 L¯ at m∗ = 30 M¯ . The luminosity of a star increases rapidly
with mass, and LE, d exceeds L∗ for m∗ & 15M¯ . How can stars more massive than this form?
–2–
Several mechanisms have been suggested to overcome the radiation pressure problem. The first
is ram pressure. Wolfire & Cassinelli (1987) pointed out that the infalling gas could overcome the
radiation pressure provided that the ram pressure exceeded the momentum flux in the radiation
field. This is valid in the limit in which each photon transfers its momentum only once to the
infalling gas; if the gas is sufficiently opaque, multiple scattering can occur and the energy rather
than the momentum of the gas can be tapped. However, at the dust destruction front, UV radiation
is absorbed at a high opacity (κP ∼ 200 cm2 g−1 ) and reradiated into the infrared, where the opacity
is much lower. As a result, the approximation that the momentum transfer per emitted photon is
hν/c is greatly improved. With this approximation, the ram pressure condition is
2
ρvin
=
ṁ∗ vin
F
>
4πr2
c
⇒ ṁ∗ >
L
,
vin c
(5)
where vin = (Gm∗ /r)1/2 is the infall velocity.
This condition must be evaluated at the dust destruction front at a radius r dd , which is given
by the condition that the equilibrium temperature of a grain exposed to the stellar radiation field
is just high enough to vaporize the grain, Tgr = Tdd . Recall that the grain absorption and emission
cross sections are of the form Qπa2gr , where agr is the grain radius. Balancing absorption in the UV
and emission in the IR, we have
¶
µ
L∗
2
4
πagr QUV
= 4πa2gr QIR σSB Tdd
.
(6)
2
4πrdd
Draine’s website has data on the values of Q and indicates that QUV ' 50QIR , so that
rdd =
µ
QUV L∗
4
QIR 16πσSB Tdd
¶1/2
1/2
= 1.8 × 1015 L∗5
cm,
(7)
where L∗5 = L∗ /(105 L¯ ). Inserting this result into equation (5), we find that the accretion rate
needed to overcome radiation pressure is
ṁ∗ > 1.4 × 10−4
Ã
5/4
L∗5
m∗ /30 M¯
!1/2
M¯ yr−1 .
(8)
Note that this is considerably greater than the typical accretion rates for low-mass stars.
Other mechanisms that have been suggested to overcome the effects of radiation pressure are
- Rotation: Nakano (1989) pointed out that since the infalling gas is rotating, it would fall into
a disk, where it is shielded from the stellar radiation. Jijina & Adams (1996) worked out the
dynamics of rotating, dusty gas falling towards a massive protostar, and concluded that this
effect was adequate to permit the formation of stars up to about 100 M ¯ . Their calculations
assumed that only the momentum of the emitted photons could reverse the infall.
–3–
- Beaming: Krumholz et al. (2005) showed that the cavities produced by protostellar outflows
would allow the reprocessed radiation to escape, reducing the radiation pressure in the infalling gas and permitting accretion over a substantial range of solid angle. Their calculations
were based on the solution of the radiative transfer equation and therefore allowed for tapping
the energy of the emitted photons.
- 3D effects: Krumholz et al. (PPV poster and in prep.) have shown that in 3D numerical
simulations, the radiation is able to escape through low-column regions while the gas can
accrete in high-column regions.
- Stellar coalescence: Bonnell et al. (1998) made the radical suggestion that the effects of
radiation pressure could be overcome by building up massive stars through the direct collision
of low-mass stars (see below).
28.2.
High-Mass Star Formation as Scaled-Up Low-Mass Star Formation
The conventional view is that high-mass star formation is a scaled up version of low-mass
star formation: The accretion rate is ṁ∗ ' c3 /G, where the effective sound speed c includes the
effects of thermal gas pressure, magnetic pressure, and turbulence (Stahler et al. 1980, although
they did not address the issue of high-mass star formation). Wolfire & Cassinelli (1987) found that
accretion rates of order 10−3 M¯ yr−1 are needed to overcome the effects of radiation pressure for
the highest stellar masses, and attributed this to the high values of c in high-mass star forming
regions. By modeling the spectral energy distributions (SEDs) of high-mass protostars, Osorio et
al. (1999) inferred that high-mass stars form in somewhat less than 10 5 yr. Behrend & Maeder
(2001) assumed that the accretion rates are proportional to the observed protostellar outflows, and
inferred a rapidly accelerating accretion (i.e., m∗ reaches infinity in a finite time in the absence of
other effects) that produced a massive star in ∼ 3 × 105 yr; this phenomenological model has an
accretion rate that can exceed the value ṁ∗ ∝ c3 that is expected on dynamical grounds, however.
The most detailed model of high-mass star formation is the turbulent core model (McKee & Tan
2002; 2003; hereafter MT03). They assumed that high-mass stars form in turbulent, gravitationally
bound cores (virial parameter αvir ∼ 1). They assumed that the turbulence is self-similar on all
scales above the Bonnor-Ebert scale, where thermal pressure dominates. They also assumed that
the star-forming clump and the protostellar cores within it are centrally concentrated. The pressure
and density therefore have a power-law dependence on radius, p ∝ r −kρ , ρ ∝ r−kρ . It follows that
the cores are polytropes with
¶
µ
1
γp
.
(9)
p = Kρ , γp = 2 1 −
kρ
Since the Bonnor-Ebert scale is small, the core is approximately singular.
–4–
The regions of high-mass star formation studied by Plume et al. (1997) have surface densities
Σcl ∼ 1 g cm−3 , corresponding to visual extinctions AV ∼ 200 mag. By contrast, regions of
low-mass star formation have Σ ∼ 0.03 g cm−2 , corresponding to AV ∼ 7 mag (Onishi et al.
1996). MT03 showed that observed star clusters in the Galaxy have surface densities comparable
to those of high-mass star forming regions, with values ranging from about 0.2 g cm −3 in the Orion
Nebula Cluster to about 4 g cm−2 in the Arches Cluster. The typical Galactic globular cluster has
Σ ' 0.8 g cm−2 .
We can express the properties of the star-forming clump and core in terms of the surface
density of the clump by recalling that in a self-gravitating gas p ∼ GΣ2 . Since the pressure in the
core is comparable to that in the clump in which it is embedded, it follows that Σcore ' Σcl . Under
the assumption that half the mass of the core is incorporated in the final star of mass m ∗f , MT03
found that the radius of the core is
µ
¶
m∗f 1/2 1/2
Rcore = 0.06
Σcl
pc,
(10)
30M¯
where Σcl is the surface density (in g cm−2 ) of the several thousand M¯ clump in which the star
is forming.
As discussed in Lecture 24, such a core accretes onto the central protostar at a rate
ṁ∗ = φ∗
m∗
,
tff
(11)
where tff is the initial free-fall time of the gas that is just accreting onto the star and
φ∗ ' 1.62 − 0.48kρ = O(1).
(12)
MT03 adopted kρ = 23 , comparable to the values inferred for the star-forming clumps studied
by Plume et al. Recall from Lecture 24 that a freely falling core with this density gradient has
φ∗ = 2[3(3 − kρ )]1/2 /kρ = 2.8, so the turbulent pressure slows the collapse by a factor of about
3. Larson-Penston accretion can have much larger values of φ∗ , but this is unlikely to occur in a
turbulent medium since such a flow is highly organized on the scale of the entire core. Numerically,
the typical accretion rate and the corresponding time to form a star of mass m ∗f is
µ
¶
¶
m∗f 3/4 3/4 m∗ 0.5 M¯
Σcl
ṁ∗ ' 0.5 × 10
,
30M¯
m∗f
yr
¶1/4
µ
m∗f
−3/4
5
Σcl
yr.
t∗f ' 1.3 × 10
30 M¯
−3
µ
For typical values of Σcl ∼ 1 g cm−2 , the star formation time is of order 105 yr and the accretion rate
is of order 10−3 M¯ yr−1 . This accretion rate is large enough to overcome the effects of radiation
pressure at the dust destruction front, thereby addressing one of the key theoretical difficulties for
models of high-mass star formation.
–5–
There are several theoretical difficulties with this model. First, it treats the turbulence in the
core and clump as microturbulence, which acts as a local pressure. As we discussed in connection
with the virial theorem, this is approximately valid when averaging over an ensemble, but it can be
in significant error when looking at a particular core at a particular time. Numerical simulations
will test the validity of this approximation.
Second, why doesn’t the core simply fragment into smaller stars? Note that such fragmentation
is allowed by the observations only if the mass distribution of cores does not determine the IMF,
contrary to current observations. Dobbs et al. (2005) simulated the collapse of a high-mass core
similar to that considered by MT03. In the isothermal case, they found that the core fragmented
into many pieces, which is inconsistent with the formation of a massive star. With a more realistic
equation of state, however, only a few fragments formed. Krumholz (2006) has shown that heating
due to the central protostar almost completely suppresses fragmentation and concludes that a
high-mass core will form one or two high-mass stars.
Finally, the model neglects feedback due to the ionizing radiation of the star itself and due to
other stars in the cluster.
Nonetheless, the turbulent core model is consistent with several set of observations: (1) Existing observations are consistent with having the IMF determined by the core mass distribution, as
assumed in the model. (2) The model is scale free for masses above the Bonnor-Ebert mass and
distances greater than stellar radii, so there is no difficulty with having star formation occur in the
crowded environment observed in regions of high-mass star formation. (3) Since it is an extrapolation of theories of low-mass star formation, it includes accretion disks and outflows. As discussed
in the PPV review by Cesaroni et al. (2006), accretion disks are observed around protostars with
m∗ ' 10 − 20 M¯ (note that Jim Moran does not believe that these observations are conclusive,
however); to date such disks have not been seen around O4-O8 protostars. Outflows from massive
protostars are common, although they are less collimated for L & 10 5 L¯ (Arce et al. 2006; Beuther
et al. 2006).
28.3.
High-Mass Star Formation as Different from Low-Mass Star Formation
Several models of high-mass star formation have been proposed that are qualitatively different
from the canonical inside-out collapse theory for low-mass star formation. Keto (2002, 2003)
modeled the growth of massive stars as being due to Bondi accretion, so that the accretion rate
ṁ∗ ∝ m2∗ under his assumption that the ambient medium has a constant density and temperature.
As with the Behrend-Maeder model, the fact that the accretion rate grows faster than linearly
implies that the stellar mass would reach an infinite value in a finite time in the absence of other
effects. As Keto points out, the Bondi accretion model assumes that the self-gravity of the gas is
negligible. The condition that the mass within the Bondi radius Gm∗ /c2 be much less than the
stellar mass can be shown to be equivalent to requiring ṁ∗ . c3 /G; for the value of c ' 0.5 km
–6–
s−1 considered by Keto, this restricts the accretion rate to ṁ∗ . 3 × 10−5 M¯ yr−1 , smaller than
the values he considers. (One can show that when one generalizes the Bondi accretion model to
approximately include the self gravity of the gas, the accretion rate is indeed about c 3 /G.)
Bonnell and collaborators have proposed two alternative models. In the competitive accretion
model (Lecture 24), small stars (m∗ ∼ 0.1M¯ ) form via gravitational collapse, but then grow by
gravitational accretion of gas that was initially unbound to the star–i.e., by Bondi-Hoyle accretion.
This model naturally results in segregating high-mass stars toward the center of the cluster, as observed. Furthermore, it gives a two-power law IMF that is qualitatively consistent with observation
(Bonnell et al. 2001). Simulations by Bonnell et al. (2004) are consistent with this model. However, there are two significant difficulties: First, radiation pressure disrupts Bondi-Hoyle accretion
once the stellar mass exceeds ∼ 10M¯ (Edgar & Clarke 2004), so it is unlikely that competitive
accretion can operate at masses above this. There is no evidence for a change in the IMF in this
mass range, however, which suggests that competitive accretion does not determine the IMF at
lower masses either. Second, competitive accretion is effective only if the virial parameter is much
less than observed, as discussed in Lecture 24.
The most radical and imaginative model for the formation of high-mass stars is that they form
via stellar collisions during a brief epoch in which the stellar density reaches ∼ 10 8 stars pc−3
(Bonnell et al. 1998; Bonnell & Bate 2002), far greater than observed in any Galactic star cluster.
This model also results in an IMF that is in qualitative agreement with observation, although it
must be borne in mind that the simulations to date have not included feedback. In their PPIV
review, Stahler et al. (2000) supported the merger model, emphasizing that gas associated with
protostars could increase the effective collision cross section and permit merging to occur at lower
stellar densities. More recently, Bonnell & Bate (2005) have suggested that binaries in clusters will
evolve to smaller separations due to accretion, resulting in mergers. However, a key assumption
in this model is that there is no net angular momentum in the accreted gas, which makes sense
in the competitive accretion model but not the gravitational collapse model. Stellar dynamical
calculations by Portegies Zwart et al. (2004), which did not include any gas, show that at densities
& 108 stars pc−3 it is possible to have runaway stellar mergers at the center of a star cluster, which
they suggest results in the formation of an intermediate mass black hole (m BH À 102 M¯ ). It
should be noted that they inferred that this could have occurred based on the currently observed
properties of the star cluster (although with the assumption that the tidal radius is greater than
100 times the core radius), not on a hypothetical ultra-dense state of the cluster. Bally & Zinnecker
(2005) discuss observational approaches to testing the merger scenario, and suggest that the wideangle outflow from OMC-1 in the Orion molecular cloud could be due to a protostellar merger that
released 1048 − 1049 erg. While it is quite possible that some stellar mergers occur near the centers
of some star clusters, the hypothesis that stellar mergers are responsible for a significant fraction
of high-mass stars faces several major hurdles: (1) the hypothesized ultra-dense state would be
quite luminous, yet has never been observed; (2) the mass loss hypothesized to be responsible for
reducing the cluster density from ∼ 108 stars pc−3 to observed values must be finely tuned in order
–7–
to decrease the magnitude of the binding energy by a large factor; and (3) it is difficult to see how
this model could account for the observations of disks and collimated outflows.
28.4.
High-Mass Star Formation in Clusters
High-mass stars form in clusters. Observations of other galaxies by Kennicutt, Edgar & Hodge
(1989) show that typically
dNa
∝ L−1 ,
(13)
d ln L
where Na is the number of OB associations and L is the Hα luminosity. From an analysis of
observations of large H II regions in the Galaxy, McKee & Williams (1997) showed that dN a /d ln S ∝
S −1 , where S is the ionizing photon luminosity, consistent with the extragalactic observations of
Kennicutt et al. This is valid only at large values of S, where the number of high-mass stars,
N∗h (defined as stars more massive than 8M¯ ) is large enough to fairly sample the IMF. MW97
−1
assumed that, since S ∝ N∗h at large S, the relation dNa /d ln N∗h ∝ N∗h
would extrapolate down
to low values of the ionizing luminosity. Furthermore, since OB associations are observed via their
ionizing luminosity, they are counted only when they are young; the count is thus proportional to
their birthrate. They found that in the Galaxy,
dṄa (N∗h )
2400
=
d ln N∗h
N∗h
Myr−1 .
(14)
Extending this down to N∗h ' 1 and assuming that each association lives for about 20 Myr
(corresponding to several generations of active star formation), this birthrate is equivalent to the
total star formation rate in the Galaxy. In other words, it is consistent to assume that most stars
form in associations that have at least one high-mass star.
28.4.1.
Mass Segregation
High-mass stars are observed to be concentrated in the centers of clusters. In the Orion
Nebula Cluster, Hillenbrand & Hartmann (1998) find that, for a cluster radius of 2 pc, half the
0.3 − 1 M¯ stars lie within a projected radius of about 0.8 pc of the center. On the other hand,
half the stars more massive than 5 M¯ lie within 0.25 pc. A more dramatic example of mass
segregation occurs in NGC 3603: Sung & Bessell (2004) find that the slope of the high-mass IMF,
Γ ≡ −dN∗ /d ln m∗ , ranges from 0.5 inside 6 00 to 1.2 ± 0.2 beyond 12 00 . Stolte et al. (2006) find that
the ratio N∗ (m∗ > 4 M¯ )/N∗ (< 4 M¯ ) drops by almost a factor 5 between 700 − 2000 and 2700 − 3300 .
The issue raised by these observations is whether this mass segregation primordial or is it due to
dynamical effects?
A star of mass ms moving through a cluster containing stars of mass hm∗ i ¿ ms experiences
–8–
a drag force due to dynamical friction
F = −0.43 ln Λ
µ
Gm2s
r2
¶
,
(15)
where r is the distance to the center of the cluster and where the numerical coefficient has been
evaluated for a singular isothermal sphere (Binney & Tremaine 1987, p. 427). Here ln Λ is the
Coulomb logarithm and
Rvc2
M
bmax v 2
'
=
,
(16)
Λ=
G(ms + hm∗ i)
Gms
ms
where bmax ' R is the maximum impact parameter, M is the mass of the cluster, and where
we have assumed that the velocity of the star v is about equal to the circular velocity given by
vc2 = GM (r)/r. Note that the drag force depends on the mass of the cluster only through the
Coulomb logarithm. Following Binney & Tremaine, we estimate the time for the star to sink to the
center by assuming that it spirals in with a velocity ' vc , which is constant in an SIS. The specific
angular momentum is L = rvc , and the rate of change of the specific angular momentum is
dL
dr
Fr
= vc
=
dt
dt
ms
⇒ r
dr
Gms
= −0.43
ln Λ.
dt
vc
(17)
Integrating this, we find that the time for the star to sink to the center of the cluster is
tsink = 1.17
2 v
rpc
r 2 vc
c5
= 2.68 × 108
ln Λ Gms
(ms /M¯ ) ln Λ
yr → 7 Myr,
(18)
where the final numerical evaluation is for a 30 M¯ star at an initial radius of 1 pc in the ONC,
which has a mass of 1800 M¯ and a 1D velocity dispersion of 2.24 km s−1 for 1 − 3 M¯ stars
√
(Hillenbrand & Hartmann 1998; note that vc = 2σ). This is significantly greater than the age of
the ONC, which is . 3 Myr, suggesting that the mass segregation is primordial. Bonnell & Davies
(1998) carried out a number of numerical simulations of star cluster evolution and concluded that it
was not possible for stellar-dynamical processes to produce the observed degree of mass segregation.
The effects of gas on the dynamical friction were not included, however, and this can increase the
rate of dynamical friction somewhat (Ostriker 1999).