**Luminosity** is an intrinsic property of a source measuring the total amount of radiant power (electromagnetic energy emitted per unit of time) by an astronomical object. Luminosity is quoted in units of - solar luminosity ($L_{\odot}$) - Watts ($W = J/s$) for the SI unit system - erg per sec (${\rm erg/s}$) in the CGS unit system - electron volts ($eV$) for high energy physics/phenomenology ## Relationship to Flux The relationship of luminosity is closely related to the [[Flux#Specific Flux|flux density]] of a source. If a source emits isotropically with a constant luminosity ($L$), then as the distance to the observing surface increases, there is a decrease in observed [[Intensity|surface brightness/flux]]. $L = \int F \; \mathrm{d} A$ If you take the observed [[flux]] to be constant from an isotropically emitting source... $ L = (F \cdot A) = F \left[ 4 \pi D_{\rm L}^{2} \right] = F \left[ 4 \pi (a_{0} r (1 + z))^{2} \right] \hspace{1cm} \Rightarrow \hspace{1cm} F = \frac{L}{4 \pi D_{\rm L}^{2}} $ where $\begin{aligned} D_{\rm L} &\equiv \text{Luminosity Distance} = a_{0} r (1 + z) \\ a_{0} &\equiv \text{Friedmann Scale Factor (present day)} \end{aligned}$ ...then we can express the luminosity in terms of the [[Distances#Luminosity Distance|luminosity distance]], where the correction is related to the conversion of [[Distances#Comoving Distance|comoving distance]] to the [[Distances#Proper Distance|proper distance]]. ## Relationship to Intensity In terms of specific intensity, we can define this quantity as an integral. $ L = \frac{\mathrm{d} E}{\mathrm{d} t} = \int I_{\nu} \; \mathrm{d} A \, \mathrm{d} \Omega \, \mathrm{d} \nu \hspace{1cm} \text{where} \hspace{1cm} \left[ L \right] \equiv {\rm erg \; s^{-1}} = {\rm Watts} $ Intrinsic property of a source. ## Relationship to Magnitude The difference in [[Magnitude#Bolometric Magnitude|bolometric magnitude]] between two sources is related to their luminosity ratio according to: $M_{\rm bol,1}-M_{\rm bol,2}=-2.5\log _{10}{\left(\frac{L_{1}}{L_{2}}\right)}$ ## Eddington Limit The **Eddington Limit** (or **Eddington Luminosity**) is the maximum luminosity a spherical astrophysical body can achieve when in [[Stellar Structure#Hydrostatic Equilibrium|hydrostatic equilibrium]], where the gravitational pressure is balanced with the radiation pressure. $ L_{\rm Edd} = \frac{4 \pi G M c}{\kappa} \approx 3.2 \times 10^{4} \; \left(\frac{M}{M_{\odot}}\right) \left(\frac{\kappa_{\odot}}{\kappa}\right) \; L_{\odot} $ When a star exceeds the Eddington luminosity, it develops very intense, radiation-driven, stellar wind on its outer layers. > [!derivation]- > > To derive the limit, we assume the luminous mass ($M$) is composed purely of ionized hydrogen and that the opacity ($\kappa = \sigma_{\rm T} / m_{\rm p}$) is constant (originating entirely from [[Thomson Scattering]] with cross-section $\sigma_{\rm T}$). > > Beginning with the [[Stellar Structure#Hydrostatic Equilibrium|hydrostatic equilibrium]] relationship and the radiation pressure gradient, we will equate the two to find the maximum luminosity when the two forces are balanced. > > $ > \frac{\mathrm{d} P_{\rm grav}}{\mathrm{d} r} = - \frac{G M \rho}{r^{2}} \hspace{2cm} \frac{\mathrm{d} P_{\rm rad}}{\mathrm{d} r} = - \frac{\rho \, \kappa}{c} \, F_{\rm rad} = - \frac{\rho \, \kappa}{c} \, \left(\frac{L}{4 \pi r^{2}}\right) > $ > $ > \begin{align} > \frac{\mathrm{d} P_{\rm rad}}{\mathrm{d} r} &= \frac{\mathrm{d} P_{\rm grav}}{\mathrm{d} r} \\ > - \frac{\rho \, \kappa}{c} \, \left(\frac{L}{4 \pi r^{2}}\right) &= - \frac{G M \rho}{r^{2}} \\ > L &= 4 \pi r^{2} \left(\frac{G M \rho}{r^{2}}\right) \left(\frac{c}{\rho \, \kappa}\right) \\ > L &= \left(\frac{4 \pi G M c}{\kappa}\right) \\ > \end{align} > $ > > Converting to [[Sun|solar]] units, we define the **Eddington Luminosity**. > > $ > L_{\rm Edd} \equiv \underbrace{\left(\frac{4 \pi G M_{\odot} c}{\kappa_{\odot} \, L_{\odot}}\right)}_{\rm constant} \left(\frac{M}{M_{\odot}}\right) \left(\frac{\kappa_{\odot}}{\kappa}\right) \, L_{\odot} \; \approx \; 3.2 \times 10^{4} \left(\frac{M}{M_{\odot}}\right) \left(\frac{\kappa_{\odot}}{\kappa}\right) \, L_{\odot} > $ > > Any luminous source that exceeds this limit ($L > L_{\rm Edd}$) will start loosing mass through radiation. If the luminosity is less than this limit ($L < L_{\rm Edd}$) will being gravitationally contracting until hydrostatic equilibrium is reached.