Our galaxy contains numerous \(molecular\) \(clouds\), regions many lightyears in
extent in which the density is high enough and the temperature low enough for
atoms to form into molecules. Most of the molecules are H\(_2\), but a small
fraction of the molecules are carbon monoxide (CO). Such a molecular cloud in
the constellation Orion is shown in Fig. P42.42. The upper image was made with
an ordinary visiblelight telescope; the lower image shows the molecular cloud
in Orion as imaged with a radio telescope tuned to a wavelength emitted by CO
in a rotational transition. The different colors in the radio image indicate
regions of the cloud that are moving either toward us (blue) or away from us
(red) relative to the motion of the cloud as a whole, as determined by the
Doppler shift of the radiation. (Since a molecular cloud has about 10,000
hydrogen molecules for each CO
molecule, it might seem more reasonable to tune a radio telescope to emissions
from H\(_2\) than to emissions from CO. Unfortunately, it turns out that the
H\(_2\) molecules in molecular clouds do not radiate in either the radio or
visible portions of the electromagnetic spectrum.) (a) Using the data in
Example 42.2 (Section 42.2), calculate the energy and wavelength of the photon
emitted by a CO molecule in an \(l\) \(=\) 1\(\rightarrow\) \(l\) \(=\) 0 rotational
transition. (b) As a rule, molecules in a gas at temperature \(T\) will be found
in a certain excited rotational energy level, provided the energy of that
level is no higher than \(kT\) (see Problem 42.39). Use this rule to explain why
astronomers can detect radiation from CO in molecular clouds even though the
typical temperature of a molecular cloud is a very low 20 K.
