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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 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

Õ

Brightness and Flux http://www.cv.nrao.edu/course/astr534/Brightness.h

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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 if the

normal to the film is tilted by an angle from the ray direction. This is just the same

projection effect that reduces the amount of water collected by a tilted rain gauge by .

Likewise at the source, such as the spherical Sun, the projected area perpendicular to the line

of sight scales as .

Using the ray-optics approximation, we can define the specific intensity (sometimes called

spectral intensity or spectral brightness, spectral radiance, or loosely, just brightness) in

terms of

= infinitesimal surface area (e.g., of a detector)

= the angle between a "ray" of radiation and the normal to the surface

= infinitesimal solid angle measured at the observer's location

Specific intensity measured by a detector whose normal is an angle from the line of sight.

The surface containing 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

flows through in time in the frequency range to within the solid angle

on a ray which points an angle away from the surface normal, then

Since power is defined as energy per unit time, the power received in the solid angle

and in the frequency range to is

cos ÒÒ

cos Ò

cos Ò

I ·

dÛ Ò dÛ d Ê

Ò

dÛ

dE · dÛ dt · · · + d d Ê Ò

dE dÛd Êdtd· · = I · cos Ò

dP d Ê · · · + d


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Two ways of looking at brightness.

The total intensity , the specific intensity integrated over all frequencies,

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.

I d· Ñ

Z 0

1

I ·


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This is usually the case for astronomical sources, and astronomers almost never use flux

densities to describe sources so extended that the factor must be retained (e.g., theemission 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 Hz , are much too big for practical astronomical use, so

we define smaller ones:

S (Ò; )d Ê · Ù

Z source

I · ¶ (2A3)

cos Ò

À2 À1

1 Jansky Jy 0 W m Hz = 1 Ñ 1 À26 À2 À1


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This spectral luminosity is an intrinsic property of the Sun; it does not depend on the distance

to the Sun.

L :5 0 erg s Hz · = 3 Â 1 10 À1 À1


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