4) (10 points) The figure to the right shows the Jupper = 1 to J1o...

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4) (10 points) The figure to the right shows the Jupper = 1 to J1ower = 0 rotational (quantum) transition of the 12C16O (carbon monoxide) 0010 spectrum observed in a galaxy using a 40 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 20 called the "brightness temperature" and, at radio wavelengths, is related to the Planck function and the spectral line constants via: Bv = 500 1000 Furthermore, the galaxy contains a Cepheid Velocity (km/s) 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 level via: 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: in = 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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