Astronomers measure the brightness of an astronomical object as a function of direction, frequency, time and polarization. Therefore, it is very important to understand the distance-independent quantity intensity and the distance-dependent quantity flux. If you take three pictures of the Sun from Venus, Earth and Mars, you will find the same intensity across the images even though the flux was different. Brightness or intensity is an absolute measure and flux only an apparent one.¹⁾

Brightness and intensity are sometimes used to mean different things: brightness is the true power emitted by the source per unit area per unit solid angle, and intensity the power per unit area per unit solid angle along the path to a detector. If there is no absorption, emission or scattering along the path, the two quantities are the same, which is the case we are considering.

We will use the ray-optics approximation to define brightness and flux. Here we assume that light travels as bullets of photons along straight lines. The assumption is good only if the emitting source is much larger than the wavelength of the emitted radiation which is almost always true in astronomy.

The figure shows propagation of radiation from a source to a detector from the point of views of the source (right) and the detector (left). In this setup, the specific brightness or intensity

$$ I_\nu \equiv \frac{dE}{dt (\cos\theta\ d\sigma) d\nu d\Omega} $$

where $dP=dE/dt$ is the infinitesimal power, $(\cos\theta \ d\sigma)$ is the area projected perpendicular to the direction of propagation, $d\nu$ is the frequency interval and $d\Omega$ the solid angle. The unit of this quantity is thus W m$^{-2}$ Hz$^{-1}$ sr$^{-1}$. The word specific and the subscript $\nu$ are used to mean that the intensity is measured per unit frequency. The brightness can also be expressed as a function of wavelength:

$$ I_\lambda \equiv \frac{dP}{(\cos\theta\ d\sigma) d\lambda d\Omega} $$

where $|I_\nu\ d\nu|=|I_\lambda\ d\lambda|$ which entails

$$ \frac{I_\lambda}{I_\nu} = \left|\frac{d\nu}{d\lambda}\right| = \frac{c}{\lambda^2} = \frac{\nu^2}{c}. $$

Specific intensity is conserved along the path of a ray and, by extension, total intensity $I$ is also conserved. Brightness is defined within infinitesimal frequency or wavelength ranges because the properties of the source may vary with frequency and most theorems about radiation are true for all narrow ranges.

Conservation of intensity directly entails that brightness does not depend on distance. Intensity or brightness is the same at the source or the detector. Passive optical systems like collectors of light (lens, mirror, dish, antenna) cannot change the brightness of an object, they can only change the angular size. An optical image of the Andromeda galaxy looks dazzlingly bright only because of long exposure times, i. e. because of the detectors, not collectors.

From the unit of flux density W m$^{-2}$ Hz$^{-1}$, we can write the definition as

$$ \frac{dP}{d\sigma\ d\nu} = I_\nu \cos\theta\ d\Omega. $$

Integrating over the solid angle subtended by the source at the detector gives the flux density

$$ S_\nu = \int_{\text{source}} I_\nu(\theta,\phi) \cos\theta\ d\Omega $$

which can be simplified farther noting that if the source is much smaller than 1 rad, then $\cos\theta\approx 1$ and thereby

$$ S_\nu = \int_{\text{source}} I_\nu(\theta,\phi) d\Omega. $$

Astronomical sources usually do not have large angular size, but there are exceptions, for example, the diffuse emission from our Galaxy. So the $\cos\theta$ factor is indeed needed sometimes.

If a source is compact, smaller than the size of the PSF of the telescope, than only its flux can be measured, not the brightness. The red giant star Betelgeuse appears bright to our eyes not because it actually has a high brightness but because it has a high flux. On the other hand if a source is extended, only its brightness at each point can be directly measured, and the flux is obtained by integrating the brightness over a solid angle.

Radio astronomers use another unit for flux because their sources are extremely faint. The unit is called ‘jansky’ (Jy) and 1 Jy = $10^{-26}$ W m$^{-2}$ Hz$^{-1}$. They even need care about mJy and $\mu$Jy sources. Optical astronomers sometimes use magnitudes instead of flux. The most widely used magnitude system in optical astronomy is the AB magnitude system where magnitude is defined as

$$ m_{AB} \equiv -2.5 \log_{10} \frac{S_\nu}{3631 \text{ Jy}}. $$

Flux depends on distance because $S_\nu \propto \int d\Omega \propto 1/d^2$ following an inverse-square law. Similar to total intensity, total flux can be calculated by integrating the flux density over all frequencies.

The spectral (or specific) intensity

$$ L_\nu = 4\pi d^2 S_\nu $$

where $d$ is the distance between the source and the observer. It does not depend on distance because the $d^2$ is cancelled by the $1/d^2$ dependence of $S_\nu$. Total luminosity

$$ L_\nu = \int_0^\infty L_\nu d\nu $$

is nothing but the true power of a source in watts (W). This is also called bolometric luminosity because bolometers are broadband detectors that could measure radiation at all frequencies by measuring the induced heat.