# Section 1: The Cross Sections

Here is displayed two cross sections of the clumps within the molecular cloud.  Note that the Z-direction is the observers line of sight into the cloud.  It must first be noted that while the clumps appear completely uniform in these cross-sections, when calculating the line-profile this is not the case.  I have chosen to display uniformly spaced clumps here to convey how varying the filling factor or clump radius affects their spacing.  Arrows are included which indicate the velocity each clump is moving at.  The length of each arrow indicates how large the velocity is.

On top is displayed a constant-Z cross section.  This is essentially a slice of what your radio telescope is actually observing. The large circle is the telescope beam, which may vary in size (see section 2).  Only clumps (or fractions of clumps) appearing within the beam are included in the final line profile calculation.  Below this is a constant-Y cross section. The X-axis range is set to your beam diameter.  The Z-range of this cross section is very small compared to the actual line-of-sight range of your beam.

# Section 2: The Parameters

Here one may vary the various parameters that significantly affect the model.  Any changes appear instantaneously in the cross section displays, but you must click on “Calculate Line Profile” for the changes to be included in the line profile (see section 3).  I will now describe each of the parameters in detail.

## Rcloud: The radius of the molecular cloud

This parameter affects the radius of the whole molecular cloud.  You will notice that as you vary the cloud radius, the cross section plots do not seem to change at all.  This is because the cloud radius does not affect any of the clump properties.  Your line profile, however, will vary dramatically as you increase or decrease the size of the molecular cloud

## Rbeam: The telescope beam’s projected radius

This parameter will change the radius of your radio telescope beam.  Note that this is the projected radius, in parsecs, onto the cloud.  Thus, this projected radius is affected by both your telescope and the distance to the cloud.  Radio telescope beam sizes vary with the wavelength you are observing in and can range from 1 to 600 arcseconds.  The primary beam of the Atacama Large Millimeter Array (ALMA) is 21 arcseconds.  The distance also affects how much of the cloud you actually “get in” your telescope beam.  The projected beam radius may be calculated by $\mathbf{R_{beam} (pc) = \frac{\alpha_{beam} (arcsec)}{206265} \times distance (pc)}$

## Rcl : The radius of each clump

This parameter changes the radius of each spherical clump.  You will notice that as you increase the radius, less clumps are appearing in your telescope beam.  This is because the volume filling factor is remaining constant.  Thus, because the total volume of “clump matter” must remain constant, if you increase the clump radius then fewer whole clumps will be present in the cloud.

## f: The clump volume filling factor

This parameter controls the fractional volume of clumps compared to the entire cloud volume.  For example, if f=.50 then 50% of the cloud will be comprised of clumps.  This affects the total amount of “clump matter”, therefore one clump of radius .02 pc will contribute the same as two clumps of radius ~.016 pc to the total clump volume.  For a given fixed clump radius, however, adjusting the filling factor will change the total number of clumps and their relative spacing.

## σcl: The velocity distribution of clumps

The individual velocity of each clump is determined using a gaussian probability distribution $\mathbf{P(v)=\dfrac{1}{\sigma_{cl}\,\sqrt{2\pi}}\, Exp(\dfrac{-v^2}{2\sigma_{cl}^2})}$.  Therefore σcl is the 1 sigma standard deviation of clump velocities.  That is, 68% of clumps will have a velocity less than σcl and 95% of clumps will have a velocity less than 2σcl.

## Γcl: The relative density of clumps

The number density within a given clump is determined by ncl = Γcl * n0, where n0 is the interclump number density of the cloud.  Thus, Γcl=2 means that clumps are twice as dense as their surrounding medium.

# Section 3: The Line Profile

Here is plotted the synthetic line profile calculated using the clump and cloud parameters set in section 2.  I have chosen to plot the 13CO J=1-0 profile since the 12CO 1-0 is often optically thick and therefore does not convey as much information.  A number of assumptions are made when calculating this profile:

1) A constant excitation temp Tex=8K

2) n(12CO)/n(H) = 7 x 10-5   and  n(13CO)/n(12CO)=1/30  (Pineda et. al. 2011)

3) The 1→0 13CO spontaneous emission coefficient Aul=6.33 x 10-8  (Goorvitch 1994)

4) The rotation constant B0/k = 5.289K for 13CO

5) The Partition Function $\mathbf{\sum_J (2J+1) e^{-B_0 J (J+1)/kT_{ex}} \approx (1 + (kT_{ex}/B_0)^2)^{1/2}}$

6) A cloud temperature Tcloud=100K and ambient number density n0=100 cm-3

7) A constant CMB background temperature of TCMB=2.725

Furthermore, an equivalent width is calculated using the thermal CMB background temperature as the continuum level.  The equivalent width may therefore be interpreted to be the width of a 2.725K high box needed to equal the area under the line profile.  The equivalent width is the best way to measure the effects of doppler broadening seen in the wings of the profile.