364
THE OUTSIDE OF A STAR
We write g 2 A?i = <1- There are two 4 X orbits and six 4 2 orbits (four belonging
to the K line and two to H ); hence q = 3.
It is found that e hv ^ RTe is large so that we have with sufficient accuracy
n 2 /% = \qre -hvi°JRT e%
In this form the result is easily seen to be the appropriate modification of
Boltzmann’s formula. If the n 2 atoms were excited by full radiation at
temperature T e we should have щ/щ = qe~ hvi *l RTe -, but n 2 is reduced pro
portionately to the diminished intensity of the radiation, so long as n 2 is
small compared with n x .
Let t x be the average duration of a normal state, t 2 of an excited state.
^ en t 2 /t x = n 2 )n x = %qre~ hvi *l RTe (253-4).
The atom will be excited (t x + i 2 ) _1 times per second and absorb
hv l2 . (t x + i 2 ) -1 ergs per second. For each erg absorbed 1 /с units of momentum
will be absorbed; this momentum is in all directions over the outward
hemisphere and the average component in the radial direction will be
l/2c. (Allowing for the law of darkening the result should strictly be
4/7c.) For equilibrium this outward momentum must balance the
momentum communicated by gravity, viz. Анд, A being the atomic weight
of calcium. Hence ,
hv X2
or
Анд —
+ t 2 =
2c (t x + t 2 )
liv X2
2c Анд
.(253-5).
By (253-4) and (253-5) to our order of approximation
qrhv
t 2 = ; . 12 e - hv ”! RT ° (253-6).
4 cAHg
Inserting numerical values for the sun the results are
t 2 /t x = r x 3-54.10- 3 ,
t x + t 2 = 4-6.10- 5 ,
so that t 2 = r x 16-2.10- 8 .
Since r is necessarily less than unity an upper limit to the duration
of the excited state giving the H and K emission is 16-2.10 -8 seconds. (If
this limit were exceeded there could be no calcium chromosphere on the
sun.) By accurate measurement of the residual intensity at the centre of
the H and K lines we could determine definitely this physical constant
of the calcium atom. Provisionally r is believed to be about 0-1 so that
t 2 — 1-6.10~ 8 secs.
Values of t 2 have been determined in the laboratory by experiments
on canal rays for some elements (but not for Ca + ) and these give results
of the order 10~ 8 secs.; so that the astronomical determination is probably
near the truth.
