## Wednesday, March 12, 2014

### astronomy units in cosmology

can you express the flux density of the cosmic microwave background in Jansky per squared arcminute?

#### 1 comment:

1. the Planck-spectrum in CMB science is usually written down using the dimensionless frequency $x=h\nu/(k_BT)$,

S(x) = 2\frac{(k_BT)^3}{(ch)^2}\frac{x^3}{\exp(x)-1},

and the prefactor $S_0$ has units of power per unit area, unit frequency and unit solid angle. converting the unit solid angle $pi$ into squared arcminutes yields a conversion factor $(180 \times60 /\pi)^2$. the unit Jansky is defined as $10^{-26}$ Watts per unit area, unit frequency and solid angle, so one obtains:

S_0 = 2\frac{(k_BT)^3}{(ch)^2}\times\left(\frac{180\times60}{\pi}\right)^2\times10^26 = 22.99527 \mathrm{Janksy}/\mathrm{arcmin}^2

for a CMB temperature of $T=2.726$ Kelvin.