Astronomy Principles and Practice Fourth Edition.pdf

218 The radiation laws
where σ is known as Stefan’s constant. It appears that Stefan made a lucky guess at the law in 1879;
it was deduced theoretically by Boltzmann in 1884. The value of σ may be evaluated by integration of
the black body curve and is given by
15.5.3 Wien’s displacement law
σ = 2π 5 k 4
15c 2 h 3 = 5·67 × 10−8 Wm −2 K −4 . (15.7)
The second important law concerning the black body radiation curves is that known as Wien’s
displacement law, proposed in 1896, which states that the product of the wavelength at which the
maximum radiation is liberated and the temperature of the black body is a constant. This may be
expressed by
λ max T = constant (15.8)
the numerical value of the constant being equal to 2·90 × 10 −3 m K. Wien’s law, therefore, provides
a means of determining the temperature of a body at any distance without measuring the absolute
amount of energy that it emits. By investigating the spectrum and assessing the wavelength at which
the maximum of energy is being liberated, knowledge of Wien’s constant allows calculation of the
temperature of the body. In astronomy, however, this method is limited to a fairly small range of
temperature for which λ max occurs in the visible part of the spectrum.
15.6 Magnitude measurements
15.6.1 The stellar output
If a star is considered simply as a spherical source radiating as a black body, its total energy output can
be determined by equation (15.6), according to its surface temperature and its surface area. This total
output is referred to as the stellar luminosity, L, and may be expressed as
L = 4π R 2 σ T 4 W (15.9)
where R is the radius of the star.
A typical value of stellar luminosity may be of the order of 10 27 W, that of the Sun being
3·85 × 10 26 W. The power received per unit area at the Earth depends on the stellar luminosity and on
the inverse square of the stellar distance. If the latter is known, the flux provided by the source may be
readily calculated and expressed in terms of watts per square metre (W m −2 ). More usually, the flux
density from a point source such as a star is defined as the power received per square metre per unit
bandwidth within the spectrum, i.e. W m −2 Hz −1 , with the bandpass expressed in terms of a frequency
interval, or W m −2 λ −1 , with the selected spectral interval expressed in terms of wavelength (m). If
an extended source is considered, then this pair of expressions would be rewritten as W m −2 Hz −1 sr −1
and W m −2 λ −1 sr −1 , respectively.
15.6.2 Stellar magnitudes
Certainly, in the optical region of the spectrum, it is not normal practice to measure stellar fluxes
absolutely. In section 5.3.2, a preliminary description was given of the magnitude scale as proposed
by Hipparchus whereby the brightnesses of stars are compared in a relative way. This scheme has
perpetuated through the subsequent centuries.
In the late 18th and 19th centuries, several astronomers performed experiments to see how the
magnitude scale was related to the amount of energy received. It appeared that a given difference
in magnitude, at any point in the magnitude scale, corresponded to a ratio of the brightnesses which