# Chapter 5\* Monochromatic Flux, K-correction

## Monochromatic flux

* Difficult to measure **bolometric luminosity** - Astronomers use filters
* A photon with observed frequency $\nu$ has higher frequency $(1+z)\nu$ when emitted

  $$
  F\_{\nu} \mathrm{d} \nu=\frac{L\_{(1+z) \nu}}{4 \pi D\_{L}^{2}} \mathrm{d}(1+z) \nu=(1+z) \frac{L\_{(1+z) \nu}}{4 \pi D\_{L}^{2}} \mathrm{d} \nu
  $$

  $$
  F\_{\lambda} \mathrm{d} \lambda=\frac{L\_{\lambda /(1+z)}}{4 \pi D\_{L}^{2}} \mathrm{d} \lambda /(1+z)=\frac{1}{1+z} \frac{L\_{\lambda /(1+z)}}{4 \pi D\_{L}^{2}} \mathrm{d} \lambda
  $$

  We see the monochromatic flux is not related to the $\nu$ or $\lambda$ we receive

  Introduce the intrinsic flux

  $$
  F\_{\lambda, \text {intrin}}=\frac{L\_{\lambda}}{4 \pi D\_{L}^{2}}
  $$

  we have

  $$
  F\_\lambda=\frac{F\_{\lambda, \text {intrin}}}{1+z}\frac{L\_{\lambda /(1+z)}}{L\_{\lambda}}
  $$

## K-correction

* Assume we use B band to calculate the distance module (DM) of a star

  $$
  m\_{i n t r i n}=-2.5 \log F\_{\lambda, i n t r i n}+\text {Const }=M+D M
  $$

  But for observation, the apparent magnitude is $m$ rather than $m\_{intrin}$

  $$
  m=M+D M+K
  $$

  where $K$ is the K-correction

  $$
  \begin{array}{c}{K(z, \lambda)=-2.5 \log \left\[\frac{1}{1+z} \frac{L\_{\lambda /(1+z)}}{L\_{\lambda}}\right]} \ {K(z, \nu)=-2.5 \log \left\[(1+z) \frac{L\_{(1+z) \nu}}{L\_{\nu}}\right]}\end{array}
  $$

  (not reliable for $z>1$)

## Surface brightness

* Define surface brightness as

  $$
  \mu=\frac{f}{\pi\theta^2}=\frac{fd\_A^2}{\pi D^2}
  $$

  where $f$ is the observed flux, $\theta$ is the angular extension $D/d\_A$

  Note that

  $$
  f=\frac{L}{4\pi d\_L^2},\quad d\_A=\frac{d\_L}{(1+z)^2}
  $$

  we have

  $$
  \mu=\frac{L}{4\pi^2D^2(1+z)^4}=\frac{\mu\_0}{(1+z)^4}
  $$
* For objects with high redshift, the surface brightness declines rapidly
* Similarly, we ca consider the bolometric and monochromatic fluxes

  $$
  \mu\_{bol,obs}=\frac{\mu\_{bol,em}}{(1+z)^4}
  $$

  $$
  \mu\_{\nu}=\frac{\mu\_{(1+z)\nu}}{(1+z)^3},\quad \mu\_{\lambda}=\frac{\mu\_{\lambda/(1+z)}}{(1+z)^5}
  $$

  This is used in the Tolman Test, which is consistent with the results from the RW metric

## Blackbody radiation

* Consider the emitted and the observed intensity (with is proportional to the surface brightness under isotropic assumption) with temperature $T$

  $$
  I\_{\nu, e m}=\frac{2 h \nu^{3}}{c^{2}} \frac{1}{\exp \left(\frac{h \nu}{k T}\right)-1}
  $$

  $$
  \begin{align\*}
  I\_{\nu, obs}&=\frac{2 h (\nu(1+z))^{3}}{c^{2}} \frac{1}{\exp \left(\frac{h \nu(1+z)}{k T}\right)-1}\frac{1}{(1+z)^3}\\
  &=\frac{2 h \nu^{3}}{c^{2}} \frac{1}{\exp \left(\frac{h \nu}{k T/(1+z)}\right)-1}
  \end{align\*}
  $$
* We see that it is still a blackbody spectrum, while the observed temperature is lower by a factor of $(1+z)$
* For CMB, $T\_{obs}=2.73\text{ K}$, and back in the era of decoupling, $T\sim300\text{ K}$

  $$
  z\_{em}\approx1100
  $$