# AST 341: Stars and Radiation

## Lecture Notes

### 3. Some Definitions: Intensity and Flux

The **monochromatic intensity** is the brightness of a beam of radiation.
It is defined by the equation
dE_{ν} = I_{ν} dA dT dν dω
where dE_{ν} is the amount of energy passing through an area dA in
time dT, over a frequency interval dν, into a solid angle dω.
The units of I_{ν} are ergs cm^{-2} s^{-1}
Hz^{-1} sr^{-1}.
Note that I_{λ}, the monochromatic intensity per
unit wavelength, is equal to I_{ν}c/λ^{2}.
In the limit, I_{ν} is the brightness of a photon. Since photons
do not decay, I_{ν} must not depend on distance. This is implicit in
the definition that I_{ν} is per steradian
(the beam can spread out - but it doesn't lose energy).

The **total intensity I** is the monochromatic intensity
I_{ν} integrated over all frequencies.
The units of I are ergs cm^{-2} s^{-1} sr^{-1}.

The **specific intensity I**_{ν}(θ,φ) is the
monochromatic intensity observed from some direction (θ,φ).
In polar coordinates,
dω=sin(θ)d(θ)d(φ). θ runs 0-π, while
φ runs 0-2π. generally, θ=0 along the axis of symmetry (e.g, the
polar axis of a rotating star). We often set μ=cos(θ),
so dω=-2π dμ.
The units of I_{ν}(θ,φ)
are ergs cm^{-2} s^{-1} Hz^{-1}.

In stellar atmospheres, we often make the assumption that the
atmosphere is **plane-parallel**. That is, rather than examining the whole
spherical star, we examine a small region that is locally flat (like the
surface of the Earth appears flat over small distances). In this case, there
is azimuthal symmetry and the Intensity will be a function only of
θ (or μ).
When integrating over solid angle, the integral over φ is merely 2π.

The **mean intensity J**_{ν} is the monochromatic intensity
I_{ν} integrated over all solid angles. This is the brightness,
irrespective of direction. Use J_{ν} when only the total number of
photons matters.
The units of J_{ν} are ergs cm^{-2} s^{-1}
Hz^{-1}.

The **monochromatic flux F**_{ν} is the integral of the
product I_{ν}cos(θ) over all solid angles. This is the
net flow of energy perpendicular to some surface dA whose normal makes an
angle θ with respect to the observer. Note that flux is a vector. It
depends on θ. Intensity is a scalar. We often separate the flux into
two parts, an ingoing and an outgoing part. F=F^{+}-F^{-}.
In the case of stellar atmospheres, F^{-}, the incoming part, is
negligable.
The units of F_{ν} are ergs cm^{-2} s^{-1}
Hz^{-1}.

For an intensity I_{ν}, the flux emitted by a
stellar disk F_{ν}=πI_{ν}. Often astrophysicists will
define the astrophysical flux = F/π, such that the numerical values of
I and F are the same.

The brightness **R** (also simply called the flux)
is the flux observed from an object
with an radiant flux **F**. By the inverse square law,
**R** = (R/D)^{2}**F**, where R is the radius of the
emitter and D is the distance to the emitter. We often speak of
**F** as the **surface flux**, the flux emitted by each square
centimeter of the surface.

The luminosity **L** is the brightness integrated over all angles and
wavelengths. The units are energy/time. **L = 4πd**^{2}F.
This is the total energy emitted by the object per unit time.

To next topic: Optical depth.

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