### Read the Paper by E. Keto (2002)

**Summary by: Yucong Zhu**

### Abstract

*The hypothesis that massive stars form by accretion can be investigated by simple analytical calculations that describe the effect that the formation of a massive star has on its own accretion flow. Within a simple accretion model that includes angular momentum, that of gas flow on ballistic trajectories around a star, the increasing ionization of a massive star growing by accretion produces a three-stage evolutionary sequence. The ionization first forms a small quasi-spherical H II region gravitationally trapped within the accretion flow. At this stage the flow of ionized gas is entirely inward. As the ionization increases, the H II region transitions to a bipolar morphology in which the inflow is replaced by outflow within a narrow range of angle aligned with the bipolar axis. At higher rates of ionization, the opening angle of the outflow region progressively increases. Eventually, in the third stage, the accretion is confined to a thin region about an equatorial disk. Throughout this early evolution, the H II region is of hypercompact to ultracompact size depending on the mass of the enclosed star or stars. These small H II regions whose dynamics are dominated by stellar gravitation and accretion are different than compact and larger H II regions whose dynamics are dominated by the thermal pressure of the ionized gas.*

**Historical Context**

Prior to this paper, analytic models ( Keto 2002 ) for the structure and evolution of HII regions were built on the simplifying assumption of spherical accretion (i.e. in the absence of an accretion disk – in other words, in the absence of angular momentum). On the other hand, other star formation models that included an accretion disk were unable to (in their models) produce the outflows that are necessary to sustain large scale HII regions. This paper is the first attempt (for an analytic model) to bridge this gap in the theory of massive star formation.

**Key Idea of the Paper**

The growth of massive stars and their resultant **HII regions are modelled analytically, without assuming spherical symmetry(!)**. As the star grows, an increasing amount of ionizing UV photons are released, which leads the surrounding HII region to go through the following 3-stage evolutionary sequence:

**Quenched Phase**

- If the
**advection rate of gas particles exceeds the rate of production of ionizing photons**, the HII region is quenched completely since the ionized gas is quickly accreted onto the star - Once the production rate of ionizing photons exceeds the advection rate, a permanent ionized region can form (with a small spatially fixed ionization front that is set by the balance between recombination, advection, and ionization)

**Bipolar Outflow Phase**

- Due to inherent angular momentum of the star’s surrounding molecular cloud, gas preferentially collects in a midplane
- The hot HII gas takes the path of least resistance, and
**expands through the lower density polar region to form a bipolar HII region** - If this polar ionization front passes sufficiently far out (so that the thermal velocity exceeds the escape velocity), a bipolar wind forms

**Spherical Outflow Phase**

- This is an extreme case of the Bipolar Outflow phase (i.e. with a very large opening angle)
- Once photon production becomes sufficiently large (to heat up all the gas so it is no longer gravitationally bound), the
**ionized gas expands hydrostatically leading to an almost spherical HII region**(leaving only a thin accretion disk)

**Physical Principles of the Model**

The major idea (to make the problem tractable analytically) is to ** solve the fluid equations in 1D along different radial rays **. The accreting system is assumed to be axisymmetric, so we only have to solve the equations for different choices of polar angle, .

**Fluid Dynamics**

For each radial ray, the ** Euler equation ** is solved to obtain the hydrodynamic structure of the gas surrounding a star. In spherical coordinates, the radial equation becomes:

For fluids with a polytropic equation of state , there exist two classes of solutions corresponding to ** either accretion ** (Bondi 1952) ** or a wind **(Parker 1958).

The name of the game now is: for a given polar angle, identify which family of solutions applies (i.e. whether we want to solve for an inflow or outflow solution).

**Inspiration from Spherical Solutions**

To determine if (for a given polar angle) we have inflow or outflow, we ** compare the radius of ionization equilibrium , with the Bondi-Parker transonic radius **. The criterion (motivated from previous studies with spherical accretion – Keto 2002) used to determine the flow is:

- If then we have inflow (accretion)
- If then we have outflow (wind)

where physically represents how far the ionization region extends, and represents the distance at which the gas thermal velocity exceeds the escape velocity from the star. The justification for this criterion is: if the ionization front passes this transonic point, the gas is no longer gravitationally trapped to the star, and thus freely escapes from the system via hydrodynamic pressure (yielding an outflow).

There is a bit of a chicken and egg problem at this point: you need to know the gas density everywhere before you can determine the ionization radius . However, to get the gas density, you must solve the Euler equation (which we can’t solve unless we first know if we should look for the accretion or wind solution). To get around this issue, the author uses the density profile from a simple model of the accretion disk ( Ulrich 1975 – see below) in order to compute the ionization radii before solving the Euler equation.

**Modelling the Accretion Disk**

There are currently two popular models for accretion disks: **the Shakura-Sunyaev (SS) model **(which is hydrostatically stable), and ** a ballistic rotating gas cloud model ** ( Ulrich 1975 ). The advantage of the Ulrich model is that it conserves the initial angular momentum of a collapsing gas cloud, whereas the SS model assumes a perfectly Keplerian rotation profile (so the angular momentum accounting is not consistent in the SS model). Since we want to test the effect that angular momentum has on the structure of HII regions, the ** Ulrich disk model was chosen **(E. Keto – personal communication).

The infalling regions (where ) are directly modeled using the Ulrich model, which completely specifies the velocity field everywhere. As a consequence of the (time-independent) continuity equation , this also pins down the density field everywhere (Mendoza 2004).

For the outflowing regions, the parker wind solution of the Euler equation (Parker 1958) was used. To pin down the unique solution corresponding to the accretion model, the Parker wind solution was matched to the Ulrich density at the stellar surface as a boundary condition (E. Keto – personal communication).

**Results and Predictions**

The right panel shows plots from the analytic model discussed in the paper (Fig 1a-1c). The model parameters were set to correspond to a massive accreting star at various times in its lifecycle (The free parameters are the rate of ionizing photon production, stellar mass, and angular momentum of the initial gas cloud). They depict the transition from a quenched B1 star (Fig. 1a) to a hot O9 star encased in a (nearly) spherical HII region envelope (Fig. 1c). The white circle depicts the Bondi-Parker Radius , while the white bipolar ovals show the ionization radius . The ** intersection between the ovals and the circle signify a transition between inflow and outflow ** in this model (since these intersections mark where ).

This plot sequence suggests the following physical interpretation:

- As the star accretes matter and becomes more massive, it releases much more ionizing radiation.
- This increase in ionization rate creates a larger bubble of hot HII gas, that expands through the path of least resistance (the low density poles).
- As the hot gas passes through the transonic point, it is suddenly able to escape the gravitational pull of the star, creating an outflow.
- Eventually, the outflow region extends to such a large range of polar angles that the resultant outflowing HII region looks close to spherical.

**Caveats of the Model**

One major problem is that **the model is inconsistent**(!). One problem is that the ** solution to the spherically symmetric Euler equation ** (the Parker wind solution) is used to model the wind in this **non-spherical model**. Another issue is that the assumed Ulrich velocity field (which is used to set the gas density and to determine whether we use the inflow or outflow solution of the Euler equation) ** ignores hydrodynamics**. However this (hydrodynamically unstable) Ulrich density is used to solve the Euler equation, ** which hinges upon hydrostatic balance **! (the Ulrich density is used to set the inner boundary condition of the Parker wind solution)

Despite these inconsistencies, the **predictions of this simple model ** (namely, the evolutionary sequence for HII regions) ** hold up to more recent work **. The current state-of-the-art on the topic of HII region evolution involves the use of fully self-consistent 3D hydrodynamic codes (including a full treatment of radiative transfer – Peters 2010). The (qualitative) results from the Peters simulation agrees well with the analytic model presented in this paper.

**Conclusion**

The author is able to model a large diversity of HII region geometries using a simple analytic model. By looking at several models that correspond to stars of different masses, an evolutionary sequence of ultracompact HII regions is suggested that goes through three distinct phases: 1) a quenched phase where the HII region is trapped gravitationally, 2) a bipolar phase where the ionization front breaks out at the polar regions, resulting in bipolar winds, and 3) a spherical phase where the ionized gas is able break away from the star in almost every direction.