Useful formulas and constants

Flux densities to the brightness temperature

The brightness temperature of a source is defined as T,

T=λ2S2kΩ.

Here S, λ and Ω are the flux density, wavelength of observation and angular extent of the resolution. Note that all units are in MKS.
k=1.38×10−23J/K
Units of S is Jm−2s−1
Units of λ is m
Units of Ω is str
We rewrite the Ω in terms of major and minor axis of the beam as, Ω=πθminθmaj4ln(2). Now, we simplify equation for T keeping it in MKS units. T(K)=λ(m)2S(Jm−2s−1)2×1.38×10−23(J/K)Ω(str)T(K)=λ(m)2S(Jm−2s−1)2×1.38×10−23(J/K)×4ln(2)πθmin(rad)θmaj(rad)T(K)=λ(m)2S(Jm−2s−1)θmin(rad)θmaj(rad)×(4ln(2)π×2×1.38×10−23)T(K)=(0.32×1023)×λ(m)2S(Jm−2s−1)θmin(rad)θmaj(rad)

Note that all units are MKS. In a radio telescope when observing through a beam, the flux density changes to flux density per beam. The 'pear beam' is mentioned to specify that you are looking through a beam. If you convert units of Ω from str to arcsec2, the units of flux density per beam remain same, i.e S(Jm−2s−1)→S(Jm−2s−1/beam). Now we simplify more, rad→arcsec, (Jm−2s−1)→Jy, m→cm.
1Jy=10−26Jm−2s−1Hz−1. (Note we use spectral flux density in radio astronomy)
1arcsec=4.85×10−6rad T(K)=(0.32×1023)×10−4×λ(cm)210−26S(Jy/beam)4.85×10−6θmin(arcsec)×4.85×10−6θmaj(arcsec)T(K)=((0.32×1023)×10−4×10−26(4.85×10−6)2)×λ(cm)2S(Jy/beam)θmin(arcsec)×θmaj(arcsec)T(K)=1.36×103λ(cm)2S(Jy/beam)θmin(arcsec)×θmaj(arcsec)T(K)=1.36λ(cm)2S(mJy/beam)θmin(arcsec)×θmaj(arcsec) If we express in frequency ν(GHz), we get T(K)=1.36(c(cm/s)ν(Hz))2S(mJy/beam)θmin(arcsec)×θmaj(arcsec)T(K)=1.369×1020×S(mJy/beam)1018×ν(GHz)2θmin(arcsec)×θmaj(arcsec)T(K)=1224S(mJy/beam)ν(GHz)2θmin(arcsec)×θmaj(arcsec)

Math in TeX notation

When a≠0, there are two solutions to ax2+bx+c=0 and they are x=−b±√b2−4ac2a. y=x2+bx+c=x2+2×b2x+c=x2+2×b2x+(b2)2⏟−(b2)2+c(x+b2)2=(x+b2)2−(b2)2+c|+(b2)2−cy+(b2)2−c=(x+b2)2|(vertex form)y−yS=(x−xS)2S(xS;yS)orS(−b2;(b2)2−c)