Definition of microwave bands
	Estimation of CO line optical depth in the LVG regime

	HF band: 3-30 MHz (10-100 m, high frequency band) popular for amateur radio operators
	VHF band: 30-300 MHz (1-10 m, very high frequency band) broadly used by FM radio broadcast, television broadcast, land mobile stations (emergency, business, and military), Amateur Radio, marine communications, air traffic control communications and air navigation systems
	UHF band: 300MHz-3GHz (10-100cm, ultra high frequency band) used for transmiting televison signal
	L band: 1-2 GHz (15-30 cm, radar long band) used by some communication satellites used by some terrestrial Eureka 147 digital audio broadcasting (DAB): at some frequencies between 1452.960-1478.640 GHz (T-DAB) and 1480.352-1490.624 GHz (S-DAB). used by amateur radio service between 1.24-1.30 GHz used by Global Positioning System carriers such as GPS, Galileo Navagation System and GLONASS. They are centered at 1176.45 MHz (L5), 1227.60 MHz (L2), 1381.05 MHz (L3), and 1575.42 MHz (L1). GSM mobile phones operates at 800–900 and 1800–1900 MHz. Iridium satellite phones operates at 1610-1625MHz. used for radio astronomy  (e.g., hyperfine structure line of neutral H gas around 21cm or 1.5 GHz)
	S band: 2-4 GHz  (7.5-15 cm) used for weather radar and some commnunication satellites used for Digital Audio Radio Satellite (DARS), from 2.31 to 2.36 GHz used by Mobile Satellite Services (MSS) at some portions between 2.0-2.2 GHz used by wireless network equipments compatible with IEEE 802.11b and 802.11g standards (2.4 GHz) or IEEE 802.16a and 802.16e standards (3.5 GHz)
	U-NII band: 5.15-5.825 GHz (~5.5 cm, Unlicensed National Information Infrastructure) used by IEEE- 802.11a devices and many wireless ISPs.
	C band: 4-8 GHz  (3.75-7.5 cm) A typical C-band satellite uses 3.7-4.2 GHz for downlink and 5.925-6.425 GHz for uplink. The 5.4 GHz band (5.15–5.35 / 5.47–5.725 / 5.725–5.875 GHz) is used for IEEE 802.11a Wi-Fi and cordless phone applications.
	X band: 7-12.5 GHz (2.4-4.3 cm) heavily used by communication satellites, with standard uplink band 7.25-7.75 GHz and downlink band 7.9-8.4 GHz. used by 3-cm radar in 5.2-10.9 GHz for weather monitoring, air traffic control, maritime vessle trafic control, defence tracking, and vehicle speed detection, etc. used for terrestrial communication and networking in some countries. used by radio amateurs in many countries between 10-10.5 GHz.
	Ku band: 12-18 GHz (1.67-2.5 cm, Ku refers to K-under) It's primarily used for satellite communications.  used for radio astronomy
	K band: 18-27 GHz (~1.5 cm, "K" stems from German word "kurz" which means short.) Between 18-26.5 GHz, it is easily absorbed by water vapor resonance peak around 22.24 GHz (=1.35cm).
	Ka band: 26.5-40 GHz (7.5-11.3 mm, Ka refers to K-above.) The so-called 30/20 GHz band is used in communication satellites (uplink in either 27.5 or 31 GHz band) and high resolution close range targeting radar aboard millitary airplanes. Some frequencies are also used for vehicle speed detection.
	Q band: 33-50 GHz (6-9 mm) used for radio astronomy  used by automotive radar and radar investigating the Earth's surface.
	V band: 40-75 GHz (4-7.5 mm) used for radar and scientific research used by short distance high capacity unlicensed point-to-point fixed wireless systems (not very common)
	W band: 75-111 GHz (2.7-4 mm) used for astronomy used by millitary radar targeting and tracking and some non-millitary applications

Here we take the example of H2O fountain star IRAS 16342-3814:
------ Part I ------
The number density of CO molecule in the CO emission region in this star is

N(CO) = Mlr / (4PI * r^2 * Vexp * mH2) * fco * fp

where
Mlr    = 6x10^-6 Msun/yr (current mass loss rate estimated from our 12CO line)
r        = 1.5x10^16 cm (radius of CO emission region, derived from beam dilution factor and distance of 2 kpc)
Vexp = 15 km/s (AGB wind speed, enhancement of density on bubble wall will be accounted for by factor fp, see below)
mH2  = 3.2x10^-24 g (mass of a H2 molecule)
fco     = 10^-4 (CO/H2 ratio for an AGB wind)
fp       = 2.0 (assume CO density on the bubble wall is enhanced by a factor of 'fp' w.r.t. the slow AGB wind)

Then, we have N(CO) ~ 5.6 cm^-3.
------ Part II ------
In the regime of large velocity filed (LVG) approximation, the sobolev length within which photons interact with the particles can be estimated as 

dr = dV / (dV/dr) = dV / (dV/dtheta) * dist

where
dV           = 1 km/s (the assumed local line width: thermal width + turbulence)
dV/dtheta = 91.7 km/s/arcsec (the velocity gradient of the interaction wind measured from the OH and H2O masers)
dist           = 2 kpc (luminosity distance of the star)

Then we have dr = 21.8 AU = 3.27 x 10^14 cm
(note: 1 pc * 1 arcsec = 1 AU)
------ Part III ------
In the regime of large velocity filed (LVG) approximation, the optical depth of CO 2-1 line can be calculated as

Tau(CO)  = k * dr = c^2/(8PI * nu^2) * n2 * A21 * (exp(h*nu/kT) - 1) * phi(nu) * dr
                = c^2/(8PI * nu^2) * (Nco/Q) *g2 * A21 * (exp(h*nu/kT) - 1) * (dr/dnu)
where
k -- is absorption coefficient in unit of cm^-1
dr = 3.27x10^14 cm (the sobolev length, see in Part II above)
c = 3x10^10 cm/s (the speed of light)
nu = 2.30538x10^11 Hz (the frequency of the 12CO, 2-1 transition)
Nco ~ 5.6 cm^-3 (number density of CO molecules, see in Part I above)
Q = 26.45 (partition function of CO at a assumed thermodynamical temperature of T=100 K, from CDMS)
g2 = 2J+1 = 5 (statistic weight of upper level J=2)
A21 = 7x10-7 s^-1 (spontaneous transition probability, calculated from the line intensity in CDMS database)
h = 6.626x10-27 erg s (Planck constant)
k = 1.38x10-16 erg K^-1 (Boltzmann's constant)
T = 100 K (assumed thermodynamical temperature for the CO gas)
dnu = 7x10^7 Hz (CO line width, corresponding to Vexp = 46 km/s, as measured from our observation)

Then we get Tau(CO) ~ 2.73x10^-4. It is an accelerated optically thin gas on the wall of bipolar lobes.