4) (10 points) The figure to the right shows the
Jupper = 1 to J1ower = 0 rotational (quantum)
transition of the 12C16O (carbon monoxide)
spectrum observed in a galaxy using a
telescope with a 23" beam (i.e. resolution)
in which the galaxy is not resolved. This
transition is denoted as CO J = 1-0. TB is
called the "brightness temperature" and, at
radio wavelengths, is related to the Planck
function and the spectral line constants via:
Furthermore, the galaxy contains a Cepheid
variable with an observed V-band apparent
magnitude (mv) of 26.85 and a period of 12.55 days.
a) (6 points) Similar to what we did for HI, derive an expression for the TOTAL CO
column density in the form of:
Ntot (CO) = constant / TBdV
where TBdV is in units of K km/s (i.e. Kelvin cilometers/second). You will need the
following identities/formulae and constants for your derivation:
i. The TOTAL column density is related to N1 the column density in the J = 1
Ntot = N1 91
ii. gi is the statistical weight of the J = 1 level: gi = 2Jupper + 1 = 3
iii. f(T) is the partition function which can be approximated by:
iv. Tx is the kinetic temperature of the gas: Tk = 20 K.
v. B is the rotation constant for CO: B = 57.635968 GHz
vi. The frequency of any CO transition is given by Vul = Jupper (Jupper +1) B
vii. A10 is the Einstein A coefficient for spontaneous emission: A10 = 7.2x10-8 s-1
b) (4 points) From your expression and the figure above, calculate the CO mass of this
galaxy (in Mo)
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