140
SOLUTION OF THE EQUATIONS
range of temperature and the radiation of different wave-lengths suffers
different amounts of absorption in passing outwards. The general effect
is that the quality of the radiation corresponds to a rather higher effective
temperature than the quantity.
In this book T e stands for the effective temperature corresponding to
the quantity of the radiant energy. A temperature TJ corresponding to
the quality is usually defined as follows. In a grating spectrum equal
lengths of spectrum correspond to equal steps of wave-length SA. Ex
pressing Planck’s Law (40-7) in terms of A, we have
V (A, T) dA = i 102 ' 1 »-
The maximum intensity in the grating spectrum occurs at the value of A
which makes 1' (A, T ) a maximum, i.e. when
X~5 ( e x _ 2)
is a minimum, where x = hcJXRT. The minimum condition gives
x = 4-965, XT = 0-288 (102-2).
Thus the temperature can be deduced by measuring the wave-length for
maximum energy; and when the radiation is not black we define an
effective temperature T/ by the same formula
A max . Te = 0-288 cm. deg (102-3).
This is the basis of practical methods of determining the effective
temperatures of stars by Wilsing and Schemer, Rosenberg, Sampson,
E. S. King and others*. Their results therefore refer to TJ rather than the
T e of our theory. For the sun T/ is about 4 per cent, higher than T e
(approximately 6000° against 5740) and the same ratio may be expected
to hold for all stars, at any rate as a first approximation*}".
It might therefore be appropriate to increase our effective temperatures
by 4 per cent, before taking out the correction Am in Tables 15 and 16.
But study of the sun’s spectral energy-curve indicates that the slight
displacement of the maximum ordinate is not the significant feature of
the sun’s deviation from a black body, and it is doubtful whether the
proposal would be an improvement (§ 228).
In the cooler stars further errors in Am will arise from the absorption
lines. Especially in types M, N and S the band spectra of chemical com
pounds occupy a considerable part of the spectrum. Of course, the
radiation which is blocked by the bands must squeeze through the gaps,
since L is determined by the internal conditions of the star and not by
surface conditions; but unless the bands are uniformly spread over the
* As the range of observation does not always include the wave-length of
maximum intensity, the procedure may be modified in detail.
t The increase of 4 per cent, for the sun is, however, purely empirical; so that
it may be risky to generalise from it.
