*(Helpful Resource: https://www.cv.nrao.edu/~sransom/web/Ch2.html)* ![[intensity_angles.png|align:center|400]] ![[intensity_perspectives.png|align:center|400]] **(Radiant) Intensity** or **surface brightness** ($I$) is the [[Flux|radiant flux]] per steradian, where the area is measured on the plane is perpendicular to the direction of energy propagation and the emitting source is contained in the observable solid angle. $ I \propto \frac{\mathrm{d} E}{\mathrm{d} A_{\perp} \, \mathrm{d} t \, \mathrm{d} \Omega} \hspace{1cm} \text{where} \hspace{1cm} [ \, I \, ] \equiv \left(\frac{{\rm energy}}{{\rm area} \times {\rm time} \times {\rm steradian}}\right) = \left(\frac{{\rm flux}}{{\rm steradian}}\right) = MT^{-3} $ ## Specific Intensity *(Also known as **spectral intensity** or **intensity density**)* More often in astronomy, we discuss the **specific intensity** that refers to the **intensity density** in frequency space. Similar to the [[Flux#Specific Flux|flux density]], we use $I_{\nu}$ to represent the intensity in the frequency band from $[\nu,\nu + d\nu]$. $ I = \int_{0}^{\infty} I_{\nu} \; \mathrm{d} \nu \hspace{1cm} \text{where} \hspace{1cm} [ \, I_{\nu} \, ] \equiv \left(\frac{{\rm energy}}{{\rm area} \times {\rm time} \times {\rm steradian} \times {\rm Hertz}}\right) = M T^{-2} St^{-1} $ From an observational perspective, we can write the follow *foundational equation* for deriving the other intensity/flux relationships. This expression is composed by simply trying to account for the major contributions that can affect specific intensity. $\mathrm{d} E_{\nu} = I_{\nu} \; ( \mathrm{d} A \, \cos \theta ) \; \mathrm{d} \Omega \; \mathrm{d} \nu \; \mathrm{d} t \hspace{1cm} \text{where} \hspace{1cm} \mathrm{d} A_{\perp} = \mathbf{\mathrm{d} A} \cdot \mathbf{\hat{n}} = \mathrm{d} A \cos \theta$ Carefully rearranging, we can easily define a relationship between [[Flux#Specific Flux|specific flux]] and specific intensity. $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 $ > [!measure] Intensity Scaling > After showing that [[Flux#^fdb0f0|flux scales with the inverse-square law]], we can use this result to show that **intensity is an intrinsic property** of a source, such that it is not dependent on the parameters of the observer. > > $ > F_{\nu} \propto \frac{1}{r^2} > \hspace{1cm} > \mathrm{d} \Omega = \frac{\mathrm{d} A}{r^2} > \hspace{1cm} \Rightarrow \hspace{1cm} > I_{\nu} = \frac{F_{\nu}}{\mathrm{d} \Omega} \propto {\rm constant} > $ > > In other words, we can state that $I_{\rm \nu}$ is conserved along any ray in empty space. > $\frac{\mathrm{d} I_{\nu}}{\mathrm{d} s} = 0$ > *(See the [[Radiative Transfer Equation]] for non-empty space)* ## Related Definitions ### Specific Flux Density $F_{\nu} = \int_{\Omega} \frac{\mathrm{d} E_{\nu}}{\mathrm{d} A \, \mathrm{d} t \, \mathrm{d} \nu} = \int_{\Omega} I_{\nu} \cos \theta \; \mathrm{d} \Omega \hspace{1cm} \text{where} \hspace{1cm} [ \, F_{\nu} \, ] \equiv \left(\frac{{\rm energy}}{{\rm area} \times {\rm time} \times {\rm hertz}}\right) = MT^{-2}$ *(See [[Flux#Specific Flux]] for more details...)* ### Specific Mean Intensity The **specific mean intensity** ($J_{\nu}$) is the average intensity density in all directions from an observing surface. To calculate this quantity, we integrate over the solid angle and divide by $4\pi$ (since there are $4\pi$ steradians on a sphere). $J_{\nu} = \frac{\int_{\Omega} I_{\nu} \; \mathrm{d} \Omega}{\int_{\Omega} \mathrm{d} \Omega} = \frac{1}{4 \pi} \int_{\Omega} I_{\nu} \; \mathrm{d} \Omega$ If a source has an isotropic emission: $J_{\nu} = I_{\nu}$. ### Specific Energy Density The **specific mean energy** ($U_{\nu}$) is the average energy density in all directions from an observing surface. To calculate this quantity, we integrate over the solid angle and divide by the speed of light ($c$). $U_{\nu} = \int_{\Omega} \frac{\mathrm{d} E_{\nu}}{(\mathrm{d} A \, \cos\theta) \, (c \, \mathrm{d} t) \, \mathrm{d} \nu} = \frac{1}{c} \int_{\Omega} I_{\nu} \; \mathrm{d} \Omega$ If a source has an isotropic emission: $U_{\nu} = \frac{4 \pi}{c} I_{\nu}$ ### Specific Radiation Pressure The **specific radiation pressure** ($P_{\nu}$) is pressure of the radiation field over a surface. It is given by the perpendicular force ($\mathrm{d} E_{\nu} \cos \theta / (c \, \mathrm{d} t)$) over the observing surface ($\mathrm{d} A$) in the specific frequency band ($\mathrm{d} \nu$). After integrating over all directions... $ \underbrace{\left(\frac{\mathrm{d} E_{\nu} \, \cos \theta}{c \, \mathrm{d} t}\right)}_{\rm force} \cdot \underbrace{\left(\frac{1}{\mathrm{d} A}\right)}_{\rm area} \cdot \frac{1}{\mathrm{d} \nu} = \frac{I_{\nu} \cos^{2} \theta}{c} \; \mathrm{d} \Omega \hspace{1cm} \Rightarrow \hspace{1cm} P_{\nu} = \frac{1}{c} \int_{\Omega} I_{\nu} \, \cos^{2} \theta\; \mathrm{d} \Omega $ If a source has an isotropic emission: $P_{\nu} = \frac{4 \pi}{3 c} I_{\nu} = \frac{1}{3} U_{\nu}$ ## Frequency-Wavelength Conversion $\nu = \frac{c}{\lambda} \hspace{1cm} \Rightarrow \hspace{1cm} \mathrm{d} \nu = -\frac{c}{\lambda^{2}} \; \mathrm{d} \lambda$ We can convert all of the prior relationships into terms of wavelength by using the above relationship between $\lambda$ and $\nu$. To do the conversion, we need to remember that to convert *densities*, such that we perform a change of variables under an integral sign. For example, here we convert the specific flux under the integral of the associated frequency band. $ \begin{aligned}[b] F &= \int_{\nu_{\rm A}}^{\nu_{\rm B}} F_{\nu}(\nu,T) \; \mathrm{d} \nu \\ &= \int_{\nu_{\rm A}}^{\nu_{\rm B}} F_{\nu}(\nu,T) \; \left(\frac{\mathrm{d} \nu}{\mathrm{d} \lambda}\right) \; \mathrm{d} \lambda \\ &= \int_{\lambda_{\rm A}}^{\lambda_{\rm B}} F_{\nu}(\nu,T) \left(-\frac{c}{\lambda^{2}}\right) \; \mathrm{d} \lambda \\ &= - \int_{\lambda_{\rm A}}^{\lambda_{\rm B}} F_{\lambda}(\lambda,T) \; \mathrm{d} \lambda \\ &= \int_{\lambda_{\rm B}}^{\lambda_{\rm A}} F_{\lambda}(\lambda,T) \; \mathrm{d} \lambda \\ \end{aligned} \hspace{1cm} \Rightarrow \hspace{1cm} F_\lambda(\lambda,T) = \frac{c}{\lambda^{2}} F_{\nu} \left(\frac{c}{\lambda},T\right) $ > [!note] > We drop the minus sign because we integrate in the opposite direction for $\lambda$ so the definition of $X_\lambda$ is nice.