Table 1.2 from S&G summarizes how magnitudes in the standard photometric filters
ID: 1884450 • Letter: T
Question
Table 1.2 from S&G summarizes how magnitudes in the standard photometric filters correspond to fluxes (find a scanned copy on the course website if you do not own the textbook).
(a) Showing all work, explicitly calculate the zero points ZP for the B, V , and R filters using the values for f (which S&G call F) given in the table. (Your answers should agree with what is given in the table.)
(b) Using absolute magnitudes for the Sun of MB = 5.48, MV = 4.83, and MR = 4.42, calculate the fluxes f (in units of erg cm2 s 1 A1 ) and f (in units of Jy) received at Earth from the Sun when observing with the B, V , and R filters. Ignore any absorption by the Earth’s atmosphere.
(c) The ST magnitude system is designed so that, for any filter, apparent magnitude zero corresponds to f = 3.63 × 109 erg cm2 s 1 A1 . Determine the apparent and absolute B, V , and R magnitudes of the Sun in the ST system.
(d) The AB magnitude system is designed so that, for any filter, apparent magnitude zero corresponds to f = 3.63 × 1020 erg cm2 s 1 Hz1 = 3630 Jy. Determine the apparent and absolute B, V , and R magnitudes of the Sun in the AB system.
Explanation / Answer
“Magnitudes” are an ancient and arcane, but by now unchangeable, way of
characterizing the brightnesses of astronomical sources. They were
introduced by the Greek astronomer Hipparchus ca. 130 BC. He arranged
the visible stars in order of apparent (naked eye) brightness on a scale that
ran from 1 to 6, with stars ranked “1” being the brightest. The ranks were
called magnitudes. The faintest stars visible to the eye under excellent sky
conditions were ranked as sixth magnitude.
Much later it became clear that because of the way the human eye responds
to a stimulus, magnitudes were proportional to the logarithm of the EM
power entering the eye from the source.
The modern magnitude system has been quantified as follows:
m = 2.5 log10
f(i)
Q(i)
where f is the mean spectral flux density (see Lec. 2) from a source at the
top of the Earth’s atmosphere averaged over a defined band and Q is a
normalizing constant for that band.
Although this definition looks peculiar, it offers two important practical
conveniences: (1) Cosmic sources have an enormous range of brightness,
and the magnitude system provides a quick shorthand for expressing these
without referring to exponents. (2) The change of magnitude caused by a
given (small) fractional change in the flux density of a source is numerically
equal to the fractional change. E.g., a 5% error in flux density produces a
0.05 mag change.
Nonetheless, magnitudes are the source of considerable confusion among
professional astronomers because there is not one magnitude system but
instead several. For historical reasons within subfields, the definitions differ
in two ways: (1) The spectral flux density can be expressed either as f()
or f(). (2) The normalizing constant Q(i) differs among the systems; and
even within a given system, it can differ with waveband.
The most widely used magnitude system through the year 2000 was based
on a set of normalizing constants derived from the spectrum of the bright
star Vega. We are now slowly moving to “absolute” systems based on
calibrations in terms of physical flux units.
I. MONOCHROMATIC MAGNITUDE SYSTEMS
In Lecture 2 we introduced a “monochromatic” magnitude system, which is
defined as:
m() 2.5 log10 F() 21.1
where F() is the spectral flux density per unit wavelength of a source
at the top of the Earth’s atmosphere in units of erg s1 cm2 A
1
.
This is also known as the “STMAG” system because it is standard for
the Hubble Space Telescope. For more details, see the Synphot User’s
Guide at STScI.
The corresponding system based on flux per unit frequency is
m() 2.5 log10 F() 48.6
where F() is in units of erg s1 cm2 Hz1
. This is also known as
the “AB” or “AB” system. This system has been adopted by the Sloan
Digital Sky Survey and GALEX. (The resulting magnitudes are therefore
very different from the STMAG system in the UV, for example.)
These are the most intuitively obvious of the various magnitude scales used
by astronomers since the normalizing constants are the same at all
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