and Cosmology

A.4 The Magnitude Scale
depends on the bolometric luminosity L of a source via
M bol =−2.5 log L + const.
(A.30)
The constant can be fixed, e.g., by using the parameters
of the Sun: its apparent bolometric magnitude is
m ⊙bol =−26.83, and the distance of one Astronomical
Unit corresponds to a distance modulus of μ =−31.47.
With these values, the absolute bolometric magnitude
of the Sun becomes
M ⊙bol = m ⊙bol − μ = 4.74 ,
(A.31)
so that (A.30) can be written as
( ) L
M bol = 4.74 − 2.5 log , (A.32)
L ⊙
and the luminosity of the Sun is then
L ⊙ = 3.85 × 10 33 erg s −1 .
(A.33)
The direct relation between bolometric magnitude and
luminosity of a source can hardly be exploited in practice,
because the apparent bolometric magnitude (or the
flux S) of a source cannot be observed in most cases. For
observations of a source from the ground, only a limited
window of frequencies is accessible. Nevertheless,
in these cases one also likes to quantify the total luminosity
of a source. For sources for which the spectrum
is assumed to be known, like for many stars, the flux
from observations at optical wavelengths can be extrapolated
to larger and smaller wavelengths, and so m bol
can be estimated. For galaxies or AGNs, which have
a much broader spectral distribution and which show
much more variation between the different objects, this
is not feasible. In these cases, the flux of a source in
a particular frequency range is compared to the flux
the Sun would have at the same distance and in the
same spectral range. If M X is the absolute magnitude of
a source measured in the filter X, the X-band luminosity
of this source is defined as
L X = 10 −0.4(M X −M ⊙X ) L ⊙X .
(A.34)
Thus, when speaking of, say, the “blue luminosity of
a galaxy”, this is to be understood as defined in (A.34).
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