Estimate initial globule parameters (placement in Fig 1 of Bertoldi 1989)

[will-henney]

Bertoldi uses a two-dimensional parameter space for classifying the initial stages of evolution of photoevaporated globules.

We would like to estimate which regime the WR globules are in.

However, this is not so simple since we need to decide what the initial density and size of the globules was.

[will-henney]

We can either use the parameter pair $(\eta, \Gamma)$ or $(\delta, \psi)$ to characterise the cloud. We will start with the first:

$\eta$ is the dimensionless column density

$\Gamma$ is the usual ionization parameter

[will-henney]

Estimating the initial separation and size

These are the easiest to deal with.

Separation $R_0$

Upper limit is the current separation -- 5 to 30 arcsec, or 0.25 to 1.5 parsec.

Lower limit is not so clear.

Size $r_0$

It would be easiest to work with the relative size of the globules $x_0 \equiv r_0 / R_0$

A reasonable lower limit is the current value of 0.01 to 0.02 (need to check this)

A reasonable upper limit would be the typical separation between globules, about 0.03 to 0.1 I thinkl

[will-henney]

Estimating the initial density

One approach would be to estimate the initial mass of the globules and combine this with our estimate of the initial size.

Lower mass limit is current mass

This can be estimated by calculating the density of the neutral globule, assuming it to be in pressure equilibrium with the photoevaporation flow.

Assume that neutral density is a factor $c_s^2 / v_A^2$ larger than the ionized density at the base of the flow. Where $c_s$ is the ionized sound speed and $v_A$ is the Alfven speed in the neutral globule (assume 1 to 3 km/s)

[will-henney]

Upper mass limit is lower limit plus mass of ionized nebula

The mass of the ionized nebula should be dominated by the diffuse gas. We can calculate it in two stages:

1. Calculate the volumetric emission measure VEM from the total H alpha flux in a similar way to #4 and #20
2. Use the mean electron density to convert to $M_i = \mathrm{VEM} \times m_p / \bar{n}_e$. Note that $n_e$ is the average electron density in the diffuse nebula, which is much smaller than at the base of the globule evaporation flows

Also note that this would be for the entire system of globules. We must divide by the number of globules to get the mass of an individual globule.

[will-henney]

Estimating the typical separation from surface density of knots

For each group of knots, we can measure the area on the sky $A$ that they occupy and how many knots $N$ are in the group. Then the average projected separation is given by $(A / N)^{1/2}$.

Alternatively, if we can estimate the LOS depth of the group (for example, assume it is the average of size in radial and PA directions), then we can find the volume $V$ of the group. Then the mean 3D separation is given by $(V / N)^{1/3}$. This has the advantage of being free from projection effects.

Edit: This is ignoring geometrical factors of order unity

[RobeReyes]

Map of globules distance, the distance are in arcsec

[will-henney]

OK, so what are you going to do next with this? Remember, we just want a rough estimate of the typical nearest-neighbor separation to get an upper limit for the initial relative size of the globules $x_0 = r_0 / R_0$

[will-henney]

Summary of Roberto's findings on upper/lower limits of globule parameters

For more details, see Roberto's notebook

1. Distance from star $R$. Upper limit (current separation) = 0.5 to 1.0 pc. Lower limit is technically zero, but we could consider down to 0.1 times the current value.
2. Fractional radius $x = r / R$. Lower limit (current value of radius) = 0.02 +/- 0.01. Upper limit (current values of typical interglobule separation) = 0.1
3. Mass of globule $m$. Lower limit (current value, assuming magnetic support with Alfven speed of 3 km/s) = 0.002 Msun. Upper limit (assuming the current ionized mass of the nebula was originally in the globules) = 0.02.

The density can be deduced from these.