Measures of Temperature

Effective Temperature: Teff  or Te -  is the temperature that a black body the size of the star would have if its luminosity were equal
to that of the star.  Hence L = 4πR2σTe4 and therefore Te = [L/(4πR2σ)]¼Te is the most useful and most used
temperature for astrophysical purposes.  If an author gives an unspecified temperature, chances are he/she means Te.  Color Temperature:  Tc - is the temperature of a blackbody which best
approximates the stellar energy distribution over a given
interval of wavelength
λ1 to λ2.  Different wavelength intervals
generally yield slightly different Tcs for the same star.

Brightness Temperature:  TB(λ) - is the temperature of a blackbody
whose surface brightness (intensity) matches that of the observed body at some specified wavelength.  TB is widely
used by radio astronomers, but seldom is used by optical astronomers, since it can only be determined if the intensity
of and therefore the solid angle subtended by the body is known.  (Stars generally subtend unknown solid angles so,
even though their fluxes are measured, their intensities are not measurable.)

Ionization Temperature:  Tion - is determined from the state of ionization of an environment, which is determined from the spectrum
by application of the Saha equation, (Nn+1/Nn)Pe = f (Tion, constants).  States of ionization of different elements or
molecules generally yield similar, but slightly different, Tions for the same star.

Excitation Temperature:  Tex - is determined from the state of excitation of an environment, which is determined from the spectrum
by application of the Boltzmann equation, Nj,i/Nj = f (Tex, constants).  States of excitation of different elements, ions
or molecules, generally yield similar, but slightly different, Texs for the same star.

Kinetic Temperature:  Tkin or Tk - is the temperature measured on earth by an ordinary thermometer.  It is the quantity which
describes and therefore is determined by the velocity distribution of particles.  A Maxwellian velocity distribution,
characterized by Tk is assumed.  Under these circumstances, the average particle kinetic energy is related to the kinetic
temperature by the relationship < ½mv2 > = (3/2)kTk , where k is Boltzmann's constant.  Tk for a stellar atmosphere
can be spectroscopically inferred since thermal motions produce measurable Doppler broadening of spectral lines.

Since strong absorption lines are associated with high opacity
Tk,strong line is usually less than Tk,weak line. *******************************************************************************************************

Since atmospheres are not in thermodynamic equilibrium, in general none of the six temperatures are equal.  Usually the following inequalities hold:
Te > Texc,  Te > Tion,  Te > Tk.         