352 
THE OUTSIDE OF A STAR 
245. By the usual quantum condition, viz. 
pdq = mh, 
we have for quantisation of angular momentum 
= mh (245*1), 
where J is the moment of inertia of the molecule. 
The rotational energy is 2 
l J co 2 = (245*2). 
Accordingly if J x and J 2 are the moments of inertia of the molecule in the 
states n x and n 2 , the frequency of the line + m is given by 
(“«■>. 
and of the line — m by 
These formulae give a parabolic spacing of the lines (v = A + Bm + Cm 2 ) 
which agrees well with observation. We can determine J x from observation 
since by (245-31) ^ ^ = 4A2/8 
or Av = hj^TT 2 J 1 (245*4), 
where Av is the spacing between consecutive lines at m = + 1. 
Considering molecules in a state n x (i.e. ready to absorb); the number 
with angular momentum corresponding to m will be proportional to 
q m e~ Xm l RT (245*5), 
where by (245*2), = m 2 h 2 /&ir 2 J x , and q m is the weight of states corre 
sponding to m. For a diatomic molecule the vector of angular momentum 
is restricted to the plane perpendicular to the line joining the two atoms 
so that for reasonably large values of m the weight q m is proportional to 
m*. (Compare § 42, where each value of n' represents n' + 1 orbits corre 
sponding to the possible values of n".) 
Since then q m oc m oc the number of molecules in state m is pro- 
portional to x J e - xm/ BT t 
which is a maximum when Xm/RT = I j or 
rffi 2L 2 
^ = \BT (245-6). 
Hence by (245*4) w max . = (245*7). 
* Representing the angular momentum by a point in the plane, the classical 
weight of a range of values is proportional to the area (since the components of 
momentum are Hamiltonian coordinates, § 48). In the quantum theory the weight 
of the annulus between (m ± |) hj2rr is appropriated to the quantised circle mh/2-rr, 
and its area is nearly proportional to m.