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
dEν = Iν dA dT dν dω
where dEν 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 Iν are ergs cm-2 s-1 Hz-1 sr-1. Note that Iλ, the monochromatic intensity per unit wavelength, is equal to Iνc/λ2.

In the limit, Iν is the brightness of a photon. Since photons do not decay, Iν must not depend on distance. This is implicit in the definition that Iν is per steradian (the beam can spread out - but it doesn't lose energy).

The total intensity I is the monochromatic intensity Iν integrated over all frequencies. The units of I are ergs cm-2 s-1 sr-1.

The specific intensity Iν(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 Iν(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 Jν is the monochromatic intensity Iν integrated over all solid angles. This is the brightness, irrespective of direction. Use Jν when only the total number of photons matters. The units of Jν are ergs cm-2 s-1 Hz-1.

The monochromatic flux Fν is the integral of the product Iνcos(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 Fν are ergs cm-2 s-1 Hz-1.

For an intensity Iν, the flux emitted by a stellar disk Fν=πIν. 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)2F, 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 = 4πd2F. This is the total energy emitted by the object per unit time.

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