A&A 427, 299-306 (2004)
DOI: 10.1051/0004-6361:20041131
A. P. Whitworth^{1} - H. Zinnecker^{2}
1 - School of Physics & Astronomy, Cardiff University,
5 The Parade, Cardiff CF24 3YB, Wales, UK
2 -
Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam,
Brandenburg, Germany
Received 21 April 2004 / Accepted 22 July 2004
Abstract
We explore the possibility that, in the vicinity of an OB star, a prestellar
core which would otherwise have formed an intermediate or low-mass star may
form a free-floating brown dwarf or planetary-mass object, because the outer
layers of the core are eroded by the ionizing radiation from the OB star before
they can accrete onto the protostar at the centre of the core. The masses of
objects formed in this way are given approximately by
,
where
is the isothermal sound speed in the neutral gas of the core,
is the rate of emission of Lyman continuum photons
from the OB star (or stars), and
is the
number-density of protons in the HII region surrounding the core. We
conclude that the formation of low-mass objects by this
mechanism should be quite routine, because the mechanism operates over
a wide range of conditions (
,
,
)
and is very effective. However, it is also a rather wasteful way of forming
low-mass objects, in the sense that it requires a relatively massive initial
core to form a single low-mass object. The effectiveness of photo-erosion
also implies that that any intermediate-mass protostars which have
formed in the vicinity of a group of OB stars must already have been well on
the way to formation before the OB stars switched on their ionizing radiation;
otherwise these protostars would have been stripped down to extremely low mass.
Key words: stars: formation - stars: low-mass, brown dwarfs - ISM: HII regions
The existence of brown dwarves was first proposed on theoretical grounds by Kumar (1963a,b) and Hayashi & Nakano (1963), but more than three decades passed before observations were able to establish their existence unambiguously (Rebolo et al. 1995; Nakajima et al. 1995; Oppenheimer et al. 1995). Now they are routinely observed (e.g. McCaughrean et al. 1995; Luhman et al. 1998; Wilking et al. 1999; Luhman & Rieke 1999; Lucas & Roche 2000; Martín et al. 2000; Luhman et al. 2000; Béjar et al. 2001; Martín et al. 2001; Wilking et al. 2002; McCaughrean et al. 2002). It appears that a significant fraction of brown dwarves in clusters are not companions to more massive stars, but rather they are free-floating. Reipurth & Clarke (2001) have suggested that this is because brown dwarves are ejected from nascent multiple systems before they can acquire sufficient mass to become Main Sequence stars, and this suggestion is supported by the numerical simulations of Bate et al. (2002), Delgado Donate et al. (2003) and Goodwin et al. (2004a,b). Here we explore an alternative possibility, namely that in the central regions of clusters like the Orion Nebula Cluster, isolated brown dwarves can form from pre-existing prestellar cores which are eroded by the ambient ionizing radiation field. Similar explanations for the formation of free-floating low-mass objects have been proposed by Zapatero Osorio et al. (2000) and Kroupa (2002), but they do not appear to have been explored in detail before now.
If a dense neutral core is immersed in an HII region, the ambient Lyman continuum radiation drives an ionization front into the core. The ionization front is normally preceded by a compression wave, and this may render the core gravitationally unstable. At the same time, the flow of ionized gas off the core reduces its mass and injects material and kinetic energy into the surrounding HII region. This situation was analyzed by Dyson (1968; see also Kahn 1969) who argued that it could explain the broadening and splitting of emission lines from the Orion Nebula. Dyson also considered the circumstances under which such a core might become gravitationally unstable.
Bertoldi (1989) and Bertoldi & McKee (1990) have developed semi-analytic models for larger cometary globules created by the interaction of ionizing radiation with a pre-existing cloud on the edge of an HII region. They use these semi-analytic models to evaluate the acceleration of the globule (due to the rocket effect; cf. Kahn 1954; Oort & Spitzer 1955) and its gravitational stability. Their results agree broadly with the numerical simulations of Sandford et al. (1982) and Lefloch & Lazareff (1994). More recently Kessel-Deynet & Burkert (2002) have modelled the radiation driven implosion of a pre-existing large cloud having internal structure. They use Smoothed Particle Hydrodynamics to handle the three-dimensional, self-gravitating gas dynamics, and find that small-scale internal structure delays, and may even inhibit, gravitational collapse.
However, in this paper we are concerned - like Dyson and Kahn - with small-scale neutral cores immersed well within an HII region, in the vicinity of the exciting stars, rather than large cometary globules at the edge of an HII region. In particular, we wish to evaluate the competition between erosion of material from the outer layers of such a core (due to the ionization front which eats into the core), and the formation and growth of a central protostar (triggered by the compression wave driven into the core ahead of the ionization front).
In order to keep free parameters to a minimum in this exploratory analysis, we make a number of simplifying assumptions. We assume that there exists a prestellar core in equilibrium, and that a nearby massive star (or group thereof) then switches on its ionizing radiation instantaneously, and ionizes the surroundings of the prestellar core, on a time-scale much shorter than the sound-crossing time for the core (we justify this assumption later). The resulting increase in external pressure triggers the collapse of the core, from the outside in, and at the same time an ionization front starts to eat into the core, thereby eroding it from the outside, as it contracts. The final stellar mass is therefore determined by a competition between the rate of accretion onto the central protostar, and the rate of erosion at the boundary. Erosion ceases to be effective when the ionization front has eaten so far into the collapsing core that the newly ionized gas flowing off the ionization front is - despite its outward kinetic energy - still gravitational bound to the protostar.
The evolution consists of three phases, starting from the instant t=0 when the surroundings of the prestellar core first become ionized. At this juncture a compression wave is driven into the core setting up a uniform, subsonic inward velocity field. When this wave impinges on the centre at time , the first phase terminates and a central protostar is formed, which then grows steadily by accretion. At the same time an expansion wave is reflected and propagates outwards leaving an approximately freefall velocity field in its wake. At time this outward propagating expansion wave encounters the inward propagating ionization front, and the second phase terminates. Thereafter the ionization front is eroding material which is falling ever faster inwards. Eventually, at time , the ionization front encounters material which is falling inwards so fast that it cannot be unbound by the ionization front, and the third phase terminates. The final stellar mass is the mass interior to the ionization front at time .
In Sect. 3 we derive an expression for the diffuse ionizing radiation field in an HII region, and in Sect. 4 we formulate the speed with which the ionization front eats into the core. In Sect. 5, we describe the initial prestellar core and formulate the time it takes for the compression wave to propagate to its centre. In Sect. 6 we describe the modified density and velocity field set up in the inner core by the outward propagating expansion wave, and formulate the time at which the outward propagating expansion wave meets the inward propagating ionization front. In Sect. 7 we formulate the time at which the ionization front has eaten so far into the modified density field inside the expansion wave that it has encountered material which is infalling too fast to be unbound by the act of ionization. The material interior to the ionization front at time is presumed to constitute the final star. We can therefore obtain an expression for its mass.
We shall assume that the HII region which engulfs the pre-existing
prestellar core is excited by a compact cluster of OB stars which
emits Lyman continuum (i.e. hydrogen ionizing) photons at a constant
rate
.
We shall also assume that most of the
surrounding space is occupied by a uniform gas in which the density
of hydrogen nuclei in all forms is
.
The radius of the
initial HII region is therefore
If we neglect the shadows cast by other prestellar cores and attenuation by
dust inside the HII region, the radial - or direct - number-flux of Lyman
continuum photons
(i.e.
the number of ionizing photons crossing unit area in unit time) varies with
distance R from the OB cluster as
(2) |
(4) |
We denote the instantaneous radius of the neutral core by , and the density of hydrogen nuclei in all forms in the neutral core by , where r is distance measured from the centre of the core. We assume that the ionization front eats into the core at speed , where is the isothermal sound-speed in the neutral gas. Consequently the ionization front is not preceded by a strong shock front, just a weak compression wave. This is a reasonable assumption, except near the beginning, when the ionization front advances very rapidly into the core. We can only allow for this by performing numerical simulations. However, this initial phase of rapid erosion does not last long, and will not greatly affect the overall phenomenology of core erosion or the final protostellar mass. It will simply shorten the time interval, t_{1}, before the protostar forms. Thus the expressions for the final protostellar mass which we derive later (Eqs. (40) and (41)) should be taken as indicative rather than precise.
Conservation of mass across the ionization front
requires
To proceed, we assume that the ionized gas flows away from the ionization
front at constant speed
(8) |
For the purpose of this exploratory calculation, we assume that the pre-existing prestellar core is a singular isothermal sphere. Generalizing the treatment to a non-singular (i.e. Bonnor-Ebert) isothermal sphere should not change the result greatly, but would necessitate a detailed numerical formulation of the problem; it would tend to reduce the mass of the final protostar, and hence assist in producing low-mass final objects.
In a singular isothermal sphere, the density of hydrogen nuclei
in all forms,
and the mass interior to radius r,
M(r), are given by
When this core is first overrun by the HII region, at time t = 0,
the resulting increase in the external pressure drives a
compression wave into the core, thereby triggering its collapse.
We assume that the compression wave travels ahead of the ionization front.
It leaves in its wake a subsonic - and approximately uniform -
inward velocity field. The compression wave reaches the centre
at time
(17) |
At
a central protostar forms and subsequently grows in
mass at a constant rate
At the same time that the compression wave converges on the centre,
an expansion wave is launched outwards at speed
,
relative to the gas. If we neglect the small
inward velocity already acquired by the gas due to the inward propagating
compression wave, the radius of the expansion wave is given by
= | (20) |
= | (21) |
Interior to the expansion wave, material flows inwards, to feed
the constant accretion rate onto the central protostar, and quickly
approaches a freefall-like profile. Therefore the density and velocity
can be approximated by
Substituting for
in Eq. (12)
from Eq. (13), we obtain
(26) |
The ionization front meets the expansion wave at time
when
,
or in terms of the
dimensionless variables, at
when
.
Equating Eqs. (27) and (22), we obtain
(29) |
Once the inward propagating ionization front passes the outward
propagating expansion wave, the ionization front encounters
infalling gas with density and velocity given by Eq. (23).
Equation (9) therefore has to be adjusted to account for the
inward motion of the neutral gas:
To follow the ionization front into the third phase, we pick ever smaller values
of
and calculate
from
Eq. (34). The radial velocity of the ionization front is then given by
,
where
can be calculated from Eq. (33). The
specific kinetic energy of the newly ionized gas, relative to the central protostar
is therefore
Figure 1: For a) ; b) ; and c) , the solid lines show the loci on the -plane where free-floating objects of mass (one Jupiter mass), (the deuterium-burning limit) and (the hydrogen-burning limit) form due to photo-erosion of pre-existing cores. The dotted lines show the predictions of the approximate Eq. (40). The dashed (dash-dot) line is the locus below which the volume of the initial core is less than 0.001 (0.01) of the volume of the whole HII region. | |
Open with DEXTER |
The procedure for obtaining the final protostellar mass is therefore as follows. (i) Choose and calculate from Eq. (25). (ii) Solve Eq. (30) for . (iii) Solve Eq. (34) for , ( ). (iv) Identify satisfying Eq. (37). (v) Compute according to Eq. (38).
Figure 1 shows, for three representative values of (i.e. (a) ; (b) ; and (c) ) loci of contant on the -plane. These values of are chosen because they represent, respectively, one Jupiter mass, the deuterium-burning limit, and the hydrogen-burning limit. Notionally objects in the mass range are referred to as brown dwarves, whilst those in the mass range are referred to as planetary-mass objects. We see that for a broad range of values of , and , there exists the possibility to form free-floating brown dwarves and planetary-mass objects, provided suitable pre-existing cores are available to be photo-eroded.
In estimating the radiation field incident on a core, we have assumed that
the volume occupied by the core is much smaller than the volume of the HII region
around it (see Sect. 3). It is therefore necessary to check that this is the case.
If we require the ratio between the initial volume of a core and the volume of the
whole HII region to be less than some fraction ,
then substituting from
Eqs. (1) and (15) we obtain the condition
(39) |
Since the protostar does not grow hugely during the Third Phase,
we can approximate the results by adopting the mass at the end of the
Second Phase. If in addition we assume
,
so that from Eq. (31)
,
we can write
Equation (40) is plotted as a dotted line in Fig. 1. We see that it is a very good approximation to the numerical results. Given the approximate nature of the whole analysis, Eq. (40) can be viewed as our best estimate of the final protostellar mass.
This exploratory analysis suggests that it is possible for a relatively massive prestellar core which suddenly finds itself immersed in an HII region to be stripped down to the mass of a brown dwarf, or even a planet, by photo-erosion, before it can collapse to form a star. To improve on this analysis, it is probably necessary to resort to numerical modelling. One could then, in principle, (i) use a more realistic configuration for the pre-existing prestellar core; (ii) include the development of the HII region as it overruns the core; (iii) treat properly the early phase when the ionization front advances very rapidly into the outer layers of the core, and the compression wave which is driven into the core; (iv) attempt to capture effects due to departures from spherical symmetry, in particular the radiative transfer aspects; (v) include self-gravity explicitly in the equations of motion. We suspect that the photo-erosion mechanism may be somewhat more effective than our analysis suggests, because firstly, with a prestellar core which is not as centrally condensed as a singular isothermal sphere (e.g. Ward-Thompson et al. 1999; Bacmann et al. 2000), the protostar will not form so quickly, giving the ionization front more time to erode; and secondly, the ionization front will penetrate at higher speed into the less dense central regions.
We can express the final protostellar mass
in terms of
the initial core mass
,
by combining Eqs. (40), (16) and (1) to obtain
First, the ratio of the final protostellar mass to the initial core mass is
given by
(42) |
Second, if the number of cores in the mass interval
(i.e. the core mass function) has the form
(43) |
(44) |
We have omitted from our analysis the possible effect of a stellar wind
impinging on a core (A. Burkert, private communication). A simple estimate
suggests that a stellar wind would need to be very powerful to have
a significant effect. Suppose that the central OB star blows a wind with
mass-loss rate
,
and speed
.
The resultant
ram pressure acting on a core at radius R is
(47) |
The analysis we have presented in this paper is based on several
assumptions and approximations, which, whilst reasonable, are
idealizations of what is likely to occur in nature. With this proviso,
we infer that, in the central regions of large clusters, free-floating
brown dwarves and planetary-mass objects may form when more massive
pre-existing cores are overrun by HII regions excited by newly-formed
massive stars. The sudden ionization of the surroundings of
a core drives a compression wave into the core, creating at its
centre a protostar which then grows by accretion. At the same time an
ionization front starts eating into the core, thereby removing its
outer layers. The final mass of the protostar is determined by a
competition between the rate at which it can accrete the infalling envelope, and
the rate at which the ionization front can erode the envelope. The simple
analysis presented here suggests that in large, dense clusters this will
result in the formation of free-floating brown dwarves and/or planetary-mass
objects. The process is both robust, in the sense that it operates
over a wide range of conditions, and also very effective, in the sense that
it strips off most of the mass of the initial core. This means that it is a
wasteful way of creating low-mass protostars, because a relatively massive
initial core is required to produce a single final brown dwarf or
planetary-mass object. It also means that any intermediate-mass protostars which
have formed in the vicinity of a group of OB stars must already have been
well on the way to formation before the OB stars switched on their ionizing
radiation; otherwise these protostars would have been stripped down to
extremely low mass.
Note added in proof. The idea that in the vicinity of OB stars final stellar masses may be determinated by photo-erosion (in the manner we have analysed here), and the idea that this is a mechanism for forming free-floating brown dwarves, are originally due to Hester (Hester et al. 1996; Hester 1997).
Acknowledgements
We thank the referee, Doug Johnstone, who drew our attention to a fundamental error in the original version of this paper, and made several other important suggestions which we have incorporated in the final version. We gratefully acknowledge the support of a European Commission Research Training Network, awarded under the Fifth Framework (Ref. HPRN-CT-2000-00155).