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--- un:brightness [2024/06/02 06:10] – asad
+++ un:brightness [2024/06/02 06:36] (current) – asad
@@ Line 1: / Line 1: @@
-====== Brightness and flux ======
+====== Brightness ======
-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 **brightness** and the distance-dependent quantity **flux**. If you take three pictures of the Sun from Venus, Earth and Mars, you will find the same brightness across the images even though the flux was different. Brightness is an //absolute// measure and flux only an //apparent// one.
+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.((Main reference of the article: Condon & Ransonm, //Essential Radio Astronomy//, Princeton University Press.))
-===== Brightness =====
+===== Intensity =====
-**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.
+**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.
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 ===== Flux =====
-From the unit of **flux density** (or just flux) W m$^{-2}$ Hz$^{-1}$, we can write the definition as
+From the unit of **flux density** W m$^{-2}$ Hz$^{-1}$, we can write the definition as
-$$ S_\nu = \int_\nu \frac{dP}{d\sigma\ d\nu} = \int_\nu I_\nu \cos\theta\ d\Omega. $$
+$$ \frac{dP}{d\sigma\ d\nu} = I_\nu \cos\theta\ d\Omega. $$
 {{https://www.cv.nrao.edu/~sransom/web/x219.png?nolink&400}}
+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.
+===== Luminosity =====
+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.
-----
-Main reference: Condon and Ransonm, //Essential Radio Astronomy//, Princeton University Press.