*(Helpful Resource: https://www.cv.nrao.edu/~sransom/web/Ch2.html)* ![[flux.png|align:center|450]] **Flux** ($F$) is generally defined as a measurable quantity of "something" ($\mathrm{d} N$) traveling through or moving past a surface or substance ($\mathrm{d} A$) over some period of time ($\mathrm{d} t$). $ F \propto \frac{\mathrm{d} N}{\mathrm{d} A \ \mathrm{d} t} \hspace{1cm} \text{where} \hspace{1cm} [ \, F \, ] \equiv \left(\frac{{\rm quantity}}{{\rm area} \times {\rm time}}\right) $ For astronomy, we typically use **radiant flux** when observing an astronomical source, making energy or light ($\mathrm{d} N \equiv \mathrm{d} E$) the measurable quantity. With this in mind, we can also describe flux as *power per area*; however, we have to be careful on how we express radiant flux in terms of other quantities when comparing to [[Intensity|intensity]]. $ F \propto \frac{\mathrm{d} E}{\mathrm{d} A \ \mathrm{d} t} \hspace{1cm} \text{where} \hspace{1cm} [ \, F \, ] \equiv \left(\frac{{\rm energy}}{{\rm area} \times {\rm time}}\right) = \left(\frac{{\rm power}}{{\rm area}}\right) = MT^{-3} $ > [!note] Confusion about the $F$ vs $\mathrm{d} F$? > Because we are physicist and not mathematians, we try to use math notation in the most confusing way possible. Here is a great example of that! I don't know if it should be $F$ or $\mathrm{d} F$, but it may make more sense if you imagine everything as $\Delta X$ vs $\mathrm{d} X$? ## Specific Flux *(Also known as the **flux density**)* Beginning with the fundamental [[Intensity#Specific Intensity|specific intensity]] relationship... $\mathrm{d} E_{\nu} = I_{\nu} \cos \theta \; \mathrm{d} A \; \mathrm{d} \Omega \; \mathrm{d} \nu \; \mathrm{d} t$ ...we can rearrange the factors such that we can define the specific flux. $ F_{\nu} = \int_{\Omega} \frac{\mathrm{d} E_{\nu}}{\mathrm{d} A \, \mathrm{d} t \, \mathrm{d} \nu} = \int_{\Omega} I_{\nu} \cos \theta \; \mathrm{d} \Omega \hspace{1cm} \text{where} \hspace{1cm} [ \, F_{\nu} \, ] \equiv \left(\frac{{\rm energy}}{{\rm area} \times {\rm time} \times {\rm hertz}}\right) = MT^{-2} $ The **specific flux** ($F_\nu$) represents the flux received from a source over some solid angle and defined frequency band from $[\nu,\nu + d\nu]$. *(Since [[Intensity#Specific Intensity|specific intensity]] tends to be the quantity we measure, we define specific flux in terms of $I_{\nu}$ .)* Our frequency space is usually restricted from our observational equipment; therefore, this quantity becomes is more practical, relative to the [[#Bolometric Flux]] ($F$). > [!math] Comment on Perpendicular Area & Flux > In vector calculus, flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface. > $ > F = \iint_{S} \mathbf{\vec{V}} \cdot \mathbf{\vec{n}} \; \mathrm{d} S = \iint_{S} |\mathbf{\vec{V}}| \cos \theta \; \mathrm{d} S > \hspace{1cm} \text{where} \hspace{1cm} > \begin{align} > F &= \text{flux} \\ > \mathbf{\vec{V}} &= \text{quantity vector field} \\ > \mathbf{\vec{n}} &= \text{normal vector to $\mathrm{d} S$} > \end{align} > $ > In the definition of radiant flux... > $F = \int_{\Omega} \frac{\mathrm{d} E_{\nu}}{\mathrm{d} A \, \mathrm{d} t \, \mathrm{d} \nu}$ > ...$\mathrm{d} A$ is **not** the perpendicular area ($\mathrm{d} A_{\perp} = \mathrm{d} A \cos \theta$). This is because the definition of the radiant flux comes from the more foundational relationship of [[Intensity#Specific Intensity|specific intensity]] that already includes the $\cos \theta$ factor. Therefore, in this definition, we are instead looking at the **perpendicular component of energy**, such that $\mathrm{d} E \equiv \mathrm{d} E_{\perp}$. > $F = \int_{\Omega} \frac{\mathrm{d} E_{\nu, \perp}}{\mathrm{d} A \, \mathrm{d} t \, \mathrm{d} \nu} \hspace{1cm} \text{where} \hspace{1cm} \mathrm{d} E_{\nu, \perp} \equiv \mathrm{d} E_{\nu} = I_{\nu} \cos \theta \; \mathrm{d} A \; \mathrm{d} \Omega \; \mathrm{d} \nu \; \mathrm{d} t$ > [!measure] Spherically Symmetric Scaling > Through the conservation of energy (through [[Luminosity]]) between surfaces of two concentric spheres with radii ($r_{1}$, $r_{2}$), we find that flux follows the inverse-square law. > > $ > \begin{align} > L_{1} = F_{1} \left( 4 \pi r_{1}^{2} \right) &= F_{2} \left( 4 \pi r_{2}^{2} \right) = L_{2} \\ > \frac{F_{1}}{F_{2}} &= \left(\frac{r_{2}}{r_{1}} \right)^{2} \hspace{1cm} \Rightarrow \hspace{1cm} F \propto \frac{1}{r^{2}} > \end{align} > $ > > ![[Inverse_square_law.svg.png|align:center|400]] > > This shows that **flux is not an intrinsic property** of source since is dependent on parameters of observation. ^fdb0f0 ## Bolometric Flux The **bolometric flux** is the measure of a radiant flux output across all wavelength/frequency bands of light. Relating to the [[#Specific Flux|specific flux]]... $F = \int_{0}^{\infty} F_{\nu} \; \mathrm{d} \nu$ > [!note] A step further? Not sure if mathematically valid... > $ > F = \int_{0}^{\infty} F_{\nu} \; \mathrm{d} \nu = \int_{0}^{\infty} \int_{\Omega} \left(\frac{\mathrm{d} E_{\nu}}{\mathrm{d} A \, \mathrm{d} t \, \mathrm{d} \nu}\right) \; \mathrm{d} \nu = \int_{\Omega} \int_{0}^{\infty} \left(\frac{\mathrm{d} E_{\nu}}{\mathrm{d} A \, \mathrm{d} t}\right) \; \frac{\mathrm{d} \nu}{\mathrm{d} \nu} = \int_{\Omega} \frac{\mathrm{d} E_{\nu}}{\mathrm{d} A \, \mathrm{d} t} > $ > $\Downarrow$ > $ > F = \int_{0}^{\infty} F_{\nu} \; \mathrm{d} \nu = \int_{\Omega} \frac{\mathrm{d} E_{\nu}}{\mathrm{d} A \, \mathrm{d} t} > $