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--- un:brightness [2024/06/02 06:27] – 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.((Main reference of the article: Condon & Ransonm, //Essential Radio Astronomy//, Princeton University Press.))
+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** (we will just call it '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
 $$ \frac{dP}{d\sigma\ d\nu} = I_\nu \cos\theta\ d\Omega. $$
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 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.