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Brightness and Flux
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Brightness and Flux Density
Astronomers learn about an astronomical source by measuring the strength of its radiation as
a function of direction on the sky (by mapping or imaging) and frequency (spectroscopy), plus
other quantities (time, polarization) that we ignore for now. We need precise and quantitative
definitions to describe the strength of radiation and how it varies with distance between the
source and the observer. The concepts of brightness and flux density are deceptively simple,
but they regularly trip up experienced astronomers. It is very important to understand them
clearly because they are so fundamental.
We start with the simplest possible case of radiation traveling from a source through empty
space (so there is no absorption, scattering, or emission along the way) to an observer. In
the ray-optics approximation, radiated energy flows in straight lines. This approximation is
valid only for systems much larger than the wavelength Õ of the radiation, a criterion easily
met by astronomical sources. You may find it helpful to visualize electromagnetic radiation as
a stream of light particles (photons), essentially bullets that travel in straight lines at the speed
of light. To motivate the following mathematical definitions, imagine you are looking at the
Sun. The "brightness" of the Sun appears to be about the same over most of the Sun's
surface, which looks like a nearly uniform disk even though it is a sphere. This means, for
example, that a photograph of the Sun would expose the film equally across the Sun's disk. It
also turns out that the exposure would not change if photographs were made at different
distances from the Sun, from points near Mars, the Earth, and Venus, for example.
The Sun in three imaginary photos taken from a long distance (left), medium distance
(center), and short distance (right) would have a constant brightness but increasing angular
size.
Only the angular size of the Sun changes with the distance between the Sun and the observer.
The photo taken from near Venus would not be overexposed, and the one from near Mars
would not be underexposed. The number of photons falling on the film per unit area per unit
time per unit solid angle does not depend on the distance between the source and the
observer. The total number of photons falling on the film per unit area per unit time (or the
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Brightness and Flux
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total energy absorbed per unit area per unit time) does decrease with increasing distance. Thus
we distinguish between the brightness of the Sun, which does not depend on distance, and
the apparent flux, which does.
Note also that the number of photons per unit area hitting the film is proportional to cos Ò if the
normal to the film is tilted by an angle Ò from the ray direction. This is just the same
projection effect that reduces the amount of water collected by a tilted rain gauge by cos Ò .
Likewise at the source, such as the spherical Sun, the projected area perpendicular to the line
of sight scales as cos Ò .
Using the ray-optics approximation, we can define the specific intensity (sometimes called
spectral intensity or spectral brightness, spectral radiance, or loosely, just brightness) I· in
terms of
dÛ = infinitesimal surface area (e.g., of a detector)
Ò = the angle between a "ray" of radiation and the normal to the surface dÛ
dÊ = infinitesimal solid angle measured at the observer's location
Specific intensity measured by a detector whose normal is an angle Ò from the line of sight.
The surface containing dÛ can be any surface, real or imaginary; that is, it could be the physical
surface of the detector, the source, or an imaginary surface anywhere along the ray. If energy
dE· flows through dÛ in time dt in the frequency range · to · + d· within the solid angle dÊ
on a ray which points an angle Ò away from the surface normal, then
dE· = I· cos ÒdÛdÊdtd·
Since power is defined as energy per unit time, the power dP received in the solid angle dÊ
and in the frequency range · to · + d· is
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http://www.cv.nrao.edu/course/astr534/Brightness.html
dP
=
I· cos ÒdÛdÊd·
2
(Watts)
(m sr Hz)
Thus the quantitative definition of specific intensity or spectral brightness is
I· Ñ
dP
cos ÒdÛ d· dÊ
(2A1)
and the mks units of I· are W m À2 Hz À1 srÀ1 .
The reasons for specifying an infinitesimal frequency range d· are: (1) The detailed spectra of
sources carry astrophysically important information, (2) source properties (e.g., opacity) may
vary with frequency, and (3) most general theorems about radiation (e.g., specific intensity is
conserved along a ray path in empty space) are true for all narrow frequency ranges, so they
are also true for broadband radiation (e.g., the total intensity integrated over all frequencies is
also conserved along a ray path in empty space).
Theorem: Specific intensity is conserved (is constant) along any ray in empty space.
This follows directly from geometry. Consider two surfaces, dÛ1 and dÛ2 along a ray of length
r.
Let dÊ1 be the solid angle subtended by dÛ2 as seen from the center of the surface dÛ1 .
Let dÊ2 be the solid angle subtended by dÛ1 as seen from the center of the surface dÛ2 .
Then
dÊ1 =
cos Ò2 dÛ2
r2
dÊ2 =
cos Ò1 dÛ1
r2
The energy dW1 flowing through the area dÛ1 in solid angle dÊ1 is
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dW1 = (I· )1 cos Ò1 dÊ1 dÛ1 d·
dW1 = (I· )1 cos Ò1
Ò
Ó
cos Ò2 dÛ2
dÛ1 d·
r2
dW2 = (I· )2 cos Ò2
Ò
Ó
cos Ò1 dÛ1
dÛ2 d·
r2
Likewise
Since the radiation energy is conserved in free space (where there is no absorption o r
emission), dW1 = dW2 , so
(I· )1 = (I· )2
QED
This conservation theorem has two important consequences:
(1) Brightness is independent of distance. Thus the camera setting for a good exposure of
the Sun would be the same, regardless of whether the photograph was taken close to the Sun
(from near Venus, for example) or far away from the Sun (from near Mars, for example), so
long as the Sun is resolved in the photograph.
(2) Brightness is the same at the source and at the detector. Thus you can think of
brightness in terms of energy flowing out of the source or as energy flowing into the detector.
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Two ways of looking at brightness.
The total intensity, the specific intensity integrated over all frequencies,
IÑ
Z
1
I· d·
0
is also conserved. The conservation of brightness also applies to any lossless optical system,
a system of lenses and mirrors for example, that can change the direction of a ray. No
passive optical system can increase the specific intensity or total intensity of radiation. If you
look at the Moon through a large telescope, the Moon will appear bigger (in angular size) but
not brighter. Many people are disappointed when they see a large, nearby galaxy (e.g.,
Andromeda) through a telescope because it looks so dim; they expected to see a brilliantly
glowing disk of stars, as in the photograph below. The difference is not in the telescope; it is in
the detector—the photograph appears brighter only because the photograph has summed the
light over a long exposure time.
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No passive optical system (e.g., a telescope) can increase the specific intensity of an
extended source. The Andromeda galaxy (M31) appears much brighter in this photograph
than it does to the eye, either with or without the aid of a telescope, only because a long
exposure accumulates more light. Image credit
If a source is discrete, meaning that it subtends a well-defined solid angle, the spectral power
received by a detector of unit projected area is called the source flux density S· . Equation
(2A1) implies
dP
= I· cos ÒdÊ
dÛd·
so integrating over the solid angle subtended by the source yields
S· Ñ
Z
I· (Ò; ¶) cos ÒdÊ
(2A2)
source
If the source angular size is Ü 1 rad, cos Ò Ù 1 and the expression for flux density is much
simpler:
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S· Ù
Z
I· (Ò; ¶)dÊ
(2A3)
source
This is usually the case for astronomical sources, and astronomers almost never use flux
densities to describe sources so extended that the cos Ò factor must be retained (e.g., the
emission from our Galaxy).
In practice, when do we use spectral brightness and when do we use flux density to describe a
source? If a source is unresolved, meaning that it is much smaller in angular size than the
point-source response of the eye or telescope observing it, its flux density can be measured
but its spectral brightness cannot. Calling the red giant star Betelgeuse a "bright star" is
misleading. The intensity of this relatively cool star is lower than the intensity o f a hotter star
that is scarcely visible to the eye. Betelgeuse appears "brighter" than most other stars
because its flux is higher, and that only because the solid angle subtended by Betelgeuse is
large. If a source is much larger than the point-source response, its spectral brightness at any
position on the source can be measured directly, but its flux density must be calculated by
integrating the observed spectral brightnesses over the source solid angle. Consequently, flux
densities are normally used to describe only relatively compact sources.
This figure illustrates the definition of flux density.
The mks units of flux density, W m À2 Hz À1 , are much too big for practical astronomical use, so
we define smaller ones:
1 Jansky = 1 Jy Ñ 10À26 W mÀ2 HzÀ1
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and 1 milliJanksy = 1 mJy Ñ 10À3 Jy, 1 microJansky = 1 ÖJy Ñ 10À6 Jy. Unlike source
R
brightness, flux density is not independent of source distance d. Since source dÊ / 1=d2 ,
S· / dÀ2
(the inverse-square law). The specific intensity or brightness is an intrinsic property of a
source, while the flux density of a source also depends on the distance between the source
and the observer.
The spectral luminosity L· of a source is defined as the total power per unit bandwidth
radiated by the source at frequency · ; its mks units are W Hz À1 . The area of a sphere of
radius r is 4Ùr2 , so the inverse-square law relation between spectral luminosity and flux
density in free space is
L· = 4Ùd2 S·
(2A4)
where the distance d between the source and the observer is much larger than the dimensions
of the source itself. Note that spectral luminosity is an intrinsic property of the source.
Spectral luminosity does not depend on the distance d between the source and the observer
because the d2 in Eq. 2A4 cancels the dÀ2 dependence of S· . The bolometric luminosity Lbol
of a source is defined as the integral over all frequencies of the spectral luminosity:
Lbol Ñ
Z
1
L· d·
0
Example of spectral brightness, flux density, and spectral luminosity:
What is the specific intensity of the Sun at · = 10 GHz if the Sun is a nearly ideal blackbody
with temperature T Ù 5800 K?
Since
h·
6:63  10À27 erg s  1010 Hz
=
= 8 Â 10À5 Ü 1
À1
À16
kT
1:38 Â 10
erg K
 5800 K
so we can use the Rayleigh-Jeans approximation
I· = B · Ù
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2kT· 2
c2
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Brightness and Flux
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for the spectral brightness of a black body.
I· Ù
2 Â 1:38 Â 10À16 erg KÀ1 5800 K (1010 sÀ1 )2
(3 Â 1010 cm sÀ1 )2
À13
I· Ù 1:78 Â 10
Ò À1 Ó
erg
s
cm2 (sr) Hz
Note that Hz = sÀ1 and that both angles (e.g., rad) and solid angles (e.g., sr) are
dimensionless, so the units in parentheses above can be dropped. Note also that most
astrophysical theory is done in cgs units but radio observations are usually reported in mks
units since engineers use mks. Recall that
1 W = 1 J sÀ1 = 107 erg sÀ1
1 m = 102 cm
so
À13
I· Ù 1:78 Â 10
Ò
10À7 W
(10À2 m)2 sr Hz
I· Ù 1:78 Â 10À16
Ó
W
m2 sr Hz
at · = 10 GHz. This result is a property of the Sun and does not depend on the observer's
distance.
What is the flux density of the Sun at · = 10 GHz, measured at the Earth? The Sun has a
radius RÌ = 7:0 Â 1010 cm, and its distance from the Earth Ñ 1 astronomical unit (AU) is
rÌ Ù 1:49598 Â 1013 cm.
The angular radius of the Sun viewed from the Earth is
À1
ÒÌ = sin
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Ò
RÌ
rÌ
Ó
Ù 4:7 Â 10À3 rad
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Brightness and Flux
http://www.cv.nrao.edu/course/astr534/Brightness.html
S· =
Z
I· cos ÒdÊ
Sun
Remembering that in spherical coordinates an infinitesimal solid angle is dÊ = sin ÒdÒd¶ , we
have
S· = I·
Z
2Ù
¶=0
Z
ÒÌ
cos Ò(sin ÒdÒd¶)
Ò=0
Defining x Ñ sin Ò so dx = cos ÒdÒ we get
Z
S· = 2ÙI·
sin ÒÌ
xdx
0
S· = ÙI· sin2 ÒÌ
Since ÒÌ Ü 1 rad, sin ÒÌ Ù ÒÌ (and cos Ò Ù 1 above can be ignored), leading to the much
simpler
S· Ù
Z
I· dÊ
Sun
2
S· Ù I· ÊÌ Ù ÙI· ÒÌ
S· Ù 1:78 Â 10À16
W
 Ù(4:7  10À3 rad)2
2
m Hz sr
S· Ù 1:24 Â 10À20
W
= 1:24 Â 106 Jy
2
m Hz
at · = 10 GHz. This result is not a property of the Sun alone because it varies with the
distance of the observer from the Sun.
What is the spectral luminosity of the Sun at · = 10 GHz?
Convert the flux density to cgs units ( S· = 1:24 Â 10À17 erg sÀ1 cmÀ2 Hz À1 ) and use the
inverse-square law:
2
L· = 4ÙrÌ
S· = 4Ù(1:5 Â 1013 cm)2 Â 1:24 Â 10À17 erg sÀ1 cmÀ2 HzÀ1
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L· = 3:5 Â 1010 erg sÀ1 HzÀ1
This spectral luminosity is an intrinsic property of the Sun; it does not depend on the distance
to the Sun.
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