3.5 Filters 3 PHOTOMETRY
instance, the color is directly related to the temperature of the object. As such, quite often HR
diagrams (typically a plot of the luminosity against its effective temperature) will be represented
with a color on the x-axis and an absolute magnitude (or at least a distance-normalized magnitude)
on the y-axis. This is more the “observer” picture of an HR diagram, whereas the more traditional
L − T
eff
diagram is more of a “theorist” view. This is because we measure filter magnitudes (and
thus colors) directly, and we infer luminosities and temperatures.
Despite its impossibility of being directly measured, we still would like to determine an object’s
bolometric magnitude. For objects with known spectra (F
λ
), often we have a bolometric correc-
tion available. The bolometric correction of an object is the quantity that needs to be added to the
visual (V -band) magnitude to get what the bolometric magnitude would be. Recall that if we have
the spectrum, we can deduce what the bolometric magnitude would be (if we already know the
distance and size). We’ll figure out how to use this information in just a moment. Mathematically,
the bolometric correction is
BC ≡ m
bol
− m
V
= M
bol
− M
V
(3.37)
Astronomers have large tables that give pre-calculated bolometric corrections for stars of various
spectral classes. In general, though, finding the bolometric correction is not an obvious task.
Example: Color Indices and Bolometric Corrections Sirius, the brightest-appearing star
in the sky, has U, B, and V magnitudes of m
U
= −1.47, m
B
= −1.43, and m
V
= −1.44. Thus for
Sirius,
U − B = −1.47 − (−1.43) = −0.04 (3.38)
and
B − V = −1.43 − (−1.44) = 0.01 (3.39)
The bolometric correction for Sirius is BC = −0.09, so its apparent bolometric magnitude is
m
bol
= m
V
+ BC = −1.44 + (−0.09) = −1.53 (3.40)
To perform such a calculations, and many like them, we must first talk about sensitivity func-
tions. Sometimes these are called the response function, the transmission function, or any number
of things. The idea, though, is that the sensitivity function of a filter determines what fraction
of photons of a given wavelength pass through the filter to a detector. We already saw these in
Figures 4 and 5. It’s important to notice that these are very dependent on wavelength, especially
at the fringes of sensitivity. We will denote the sensitivity of the ith filter (no filter in particular)
as S
i
(λ). S
i
(λ) is always between 0 and 1. When S
i
(λ) = 0, the the filter is opaque to that
wavelength, and if S
i
(λ) = 1, then the filter is transparent to that wavelength. For the purposes
of this discussion we will be neglecting attenuation due to interstellar reddening, the atmosphere,
and intrinsic inefficiencies in the telescope/CCD.
With this machinery, we may determine what the flux through any given filter could be, in a fashion
similar to (3.35). Through a given filter i, the flux through that filter from an object with specific
flux F
λ
would be
F
i
=
Z
dλ S
i
(λ)F
λ
(3.41)
28