Back-of-the-envelope CO: It is well-known (certainly amongastronomers) that the Einstein A-coefficient for the Lyman alpha (n= 2 to n = 1) transition in atomic hydrogen is of order 10^9 s^−1(actually, 5 × 10^8 s^−1). As we briefly discussed in class, thisresult could be estimated to order-of-magnitude by considering an(accelerating) electron on a spring that displaces a Bohr radius,a0, and has natural angular frequency ω and then taking the inverselifetime of the excited state as A ∼ P/ω, where P is the powerradiated by an accelerating charge. The ubiquitous carbon monoxidemolecule, 12CO, is used by astronomers to trace the presence andmeasure the temperature of molecular gas in various astrophysicalenvironments. We would rather try to detect H2 directly, but sadlyH2 has no permanent electric dipole moment because of its symmetry,while Carbon monoxide does have a permanent dipole moment, so iseasier to detect.

a)Estimate the wavelength of the lowest energy, rotationaltransition in CO (J = 1 to J = 0, where J is the rotational quantumnumber). Do this by considering a barbell spinning about its axisof greatest moment of inertia and recognizing that angular momentumcomes quantized in units of ?h. Compare your estimate
to the true answer of 2.6 mm.

b)Use scaling relations to estimate the Einstein A coefficientof this transition, i.e., the inverse lifetime of the excited J = 1state. You can use that the dipole moment (that is difficult toguess from first principles) of the CO molecule is 0.1 Debyes, not∼ 1 Debye, as one might have guessed naively (1 Debye= 10^−18cgs.Note that ea0 = 2.5 Debyes, where e is the electron charge).Compare your estimate to the true answer A10 = 7.4 × 10^-8s^-1.

Note: This smaller-than-expected dipole moment of CO is aconsequence of the strong double bond connecting C to O. Most othermolecules common in astrophysics – e.g., H2O, CS, SiS, SiO, HCN,OCS, HC3N – have dipole moments that are all of order 1 Debye.