1 - Stellar Brightnesses

Spatially resolved source: Most sources are unresolved, however. Observed (apparent) brightness is: (will sometimes call this the “observed flux”)

Okay, what is d ??????? If the luminosity (wattage) of a star is L ν, then If we measure the observed flux we can derive its luminosity, an important intrinsic property of the star:

Stellar Parallax If p=1” = 1 arcsec (1/3600°), then d = AU = 1 parsec = 3.08x10 16 m. In “astronomer’s units”: Unless there is a REALLY good reason for doing so, use “astronomer’s units”, not km, radians, etc. for this.

parallax and uncertainty in milli-arcsec REAL Parallax Measurements Have Uncertainties How Uncertain are the Distances?

Stellar Spectral Energy Distributions are NOT Blackbodies, but often come close enough to utilize the mathematical formulation of a BB.

Blackbody Radiation [Note: the total energy emitted by an isotropically-emitting surface unit area is πB ν. Sometimes astronomers forget when they need the factor of  and when they do not. ] (units renormalized - just to show the functional form)

Wien Displacement Law The peak of B ν occurs at some ν max defined by To solve this, let This must be solved numerically, and has a solution x = …… So

So hν max ≈ 2.82 kT or Doing the same in wavelength units:

Solar spectrum in wavelength and frequency units Note:λ max ν max ≠ c because B λ ≠B ν. It is B λ dλ = B ν dν!! When applying Wien’s Law, you MUST use the formula appropriate for the units you are using.

One can also use photons instead of power! From “The Optics of Life: A Biologist’s Guide to Light in Nature” by Sönke Johnsen.

Stefan-Boltzmann Law Again, letsoand The net energy emitted from a surface is proportional to T 4.

Net Luminosity L ν Total energy emitted per unit area in all directions by an isotropically-emitting blackbody is : Integrating over all frequencies: Integrating over the surface of a (spherical) star: And we actually observe: Stellar Luminosity

Stellar Temperatures Wien or or.... Effective (i.e. T a blackbody of the same integrated flux would have) Brightness Color

Kinetic T - defined by the particle speeds, using the maxwellian velocity distribution Using the rms velocity distribution insures that T is a measure of the mean kinetic energy of the particles

Excitation T - based on the relative population of electronic states in atoms and ions which are excited by collisions from other particles and photons Ionization T - based on the relative populations of ionization states of the atoms and ions are ionized by collisions from other particles and photons

For Molecules: Rotational T Vibrational T Electronic T Because stars are not in perfect thermodynamic equilibrium, all these temperatures may differ from one another! It may be necessary to specify which one you mean.