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.