Stars and Planets

Radio continuum
Synchroton emission (fast electrons)

Cas(iopeia) A
supernovae remnant

Atomic hydrogen

Radio continuum
synchroton emission (fast electrons) and free-free emission (hot ionized regions)

J1→0 CO line
molecular gas and star formation regions

far-IR continuum
re-radiated thermal emission (warm dust)

mid-IR continuum
diffuse PAH emission and point source (RGB, planetary nebula, SF-regions)

near-IR
cool K-stars (some absorption by dust)

optical
stars (much absorption by dust)

X-rays
collisions by cosmic rays and pulsars

gamma rays
collisions by cosmic rays and pulsars

Crab Pulsar

A photon gas obeys Bose-Einstein statistics, the spectral energy density is

that is, the product of the density of states, the energy, and the probability that it is occupied. where is frequency, is Planck's constant, temperature, c the speed of light, and k Boltzmann constant. The derivation follows from statistical mechanics principles.

the spectral intensity with which the energy escapes to space is :

This is the Planck function (black-body radiation): the amount of energy radiated per unit area, per unit frequency, per unit solid angle (spatial angle)

Integrated over frequency, we obtain:

• The total photon energy density:

  where is Stefan-Boltmann constant.

• The radiated power per unit area
• The total power radiated by a spherical body (Stefan-Boltzmann law)

  where L is luminosity and R the radius.
  It is customary to define the effective temperature in this way:

  even when the radiation does not follow a blackbody.

Properties Planck function

• The peak of the blackbody spectrum shifts to shorter wavelength for hotter blackbodies. It is known as Wien's displacement law:
• Rayleigh-Jeans law at long wavelengths:
  
• Wien's approximation at high frequencies

Distance measurements — Parallax

In Astronomy, it is challenging to measure distances.

The parallax method is an astrometric method to obtain distances to relatively close stars. Over the course of a solar orbit, they will change their position on the sky. The definition of parallax is such that a shift of 1 arcsec ('') corresponds to a distance of 1 parsec (pc).

→ Definition of parallax (c) Wikipedia

→ top: period-luminosity relationship of Cepheids, (c) ATNF
bottom: kappa Pavonis light curve with TESS. (c) Warrick Ball -- Wikipedia

In astronomy we often express brightness on a logarithmic —magnitude— scale

• magnitudes are always defined relative to a reference flux:

  Brighter stars have lower magnitude! The key significance of magnitude lies in its relative scaling, not in the absolute value . Often the star Vega — or rather the emission corresponding to an 11,000 K blackbody — is used for the reference magnitude

  ---

  Or, between two stars a difference in flux amounts to a different in magnitude of

  where is the magnitude of star 1, the magnitude of star 2, the brightness of star 1, the brightness of star 2.
  Stars that differ by a factor 100 in brightness, differ by 5 magnitudes with the brightest star having the lowest value!

In astronomy we often express brightness on a logarithmic —magnitude— scale

• magnitudes are always defined relative to a reference flux:
• Absolute (intrinsic) and apparent magnitudes:
  • absolute magnitude M expresses the brightness of a star as seen from a distance of 10 pc
  • apparent magnitude m expresses the brightness as we see it from Earth
  They can be converted once we know the distance:

  where m-M is called the distance modulus

In astronomy we often express brightness on a logarithmic —magnitude— scale

• magnitudes are always defined relative to a reference flux:
• Absolute (intrinsic) and apparent magnitudes:
• A difference in magnitudes between two wavelengths filters, is called a color:

Arguably the most important diagnostic tool in Astronomy! One identifies:

• Main-sequence (MS): diagonal line; increase for stars with different mass

  MS-stars are stars that fuse hydrogen into helium

• Red giant branch (RGB)

  Stars ascend it after H-burning stops in the core

• Supergiant branch

  Massive stars that have left the MS

• White Dwarf branch

  "Dead" remnant of stellar cores

• Hayashi track

  nearly vertical line in HR-diagram; stars (in equilibrium) cannot cross it

Note: In all HR-diagrams (or its observational equivalent, the color-magnitude diagram ) brigh(ter) stars can be found at the top, while red(der) stars are at the right.

Color-magnitude diagram (CMD)

Note: In all HR-diagrams (or its observational equivalent, the color-magnitude diagram ) brigh(ter) stars can be found at the top, while red(der) stars are at the right.

← An example of a color-magnitude diagram from the GAIA satellite, plotting absolute magnitude vs color.