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.

Note: In astronomical nomenclature the subscript notation of frequency or wavelength (e.g., , , ) typically indicates a spectrum!

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

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.

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

• Electronic. Following Bohr's classical atomic model, energy levels for the electron are discretized as

  where with the Bohr radius. This correspond to energy levels with transitions occurring at near-UV/visible wavelengths.

• Electronic. Following Bohr's classical atomic model, energy levels for the electron are discretized as

  where with the Bohr radius. This correspond to energy levels with transitions occurring at near-UV/visible wavelengths.

• Electronic. Following Bohr's classical atomic model, energy levels for the electron are discretized as

  where with the Bohr radius. This correspond to energy levels with transitions occurring at near-UV/visible wavelengths.

Transitions

The transition strength is determined by:

• the extent the upper level is populated; and
• the rate at which the excited state will decay by photon emission (Einstein-A)

  (for an emission line).

• Selection rules. Not all transitions are allowed.

  Dipole transitions require (i) a permanent dipole moment; and (ii) . Symmetric molecules — notably H₂ — do not have a permanent dipole moment! They can emit only through much weaker quadrapule transitions . In cold disk, the gas is molecular but H₂ is essentially invisible

Question:
How then do astronomers measure the mass of cold, molecular gas?

Transitions

The transition strength is determined by:

• the extent the upper level is populated; and
• the rate at which the excited state will decay by photon emission (Einstein-A)

  (for an emission line).

• Selection rules. Not all transitions are allowed.

  Dipole transitions require (i) a permanent dipole moment; and (ii) . Symmetric molecules — notably H₂ — do not have a permanent dipole moment! They can emit only through much weaker quadrapule transitions . In cold disk, the gas is molecular but H₂ is essentially invisible

→ energy levels of the CO molecule. At high temperatures — e.g., room temperatures — all levels will be occupied. But at lower T — e.g., ~10 K for molecular clouds or the interior regions of protoplanetary disks — only the lowest J levels will be.

Given the level populations and the transition rates — the Einstein-A coefficient — the line strength can be calculated.

• (semi-classical) Bohr atomic model

  Bohr postulated that angular momentum of the electron is quantized in units of . The corresponding energy levels are

• For the hydrogen atom, the resulting transitions give rise to lines
  • n=1: Lyman series. Transitions in UV
  • n=2: Balmer series. Transitions at visible wavelengths
• The transition is further identified by a greek letter, α, β ...

  For example for the transition, for the , etc.

• In stellar atmospheres, these transitions appear in absorption

• (semi-classical) Bohr atomic model

  Bohr postulated that angular momentum of the electron is quantized in units of . The corresponding energy levels are

• For the hydrogen atom, the resulting transitions give rise to lines
  • n=1: Lyman series. Transitions in UV
  • n=2: Balmer series. Transitions at visible wavelengths
• The transition is further identified by a greek letter, α, β ...

  For example for the transition, for the , etc.

• In stellar atmospheres, these transitions appear in absorption

Question:
Why in absorption?

The Boltzmann equation determines the occupation levels within an ionization state

where E_i denotes the energy level with respect to the ground state (E₁=0), g_i the statistical weight, and Z the partition function:

The Boltzmann equation determines the occupation levels within an ionization state

The Saha equation determines the occupation levels among ionization states:

for the reaction where E_ion is the ionization (or dissociation) energy, and the Z_int are internal partition functions.

The Boltzmann equation determines the occupation levels within an ionization state

The Saha equation determines the occupation levels among ionization states:

for the reaction where E_ion is the ionization (or dissociation) energy, and the Z_int are internal partition functions. Applied to ionization of hydrogen this reads:

where AB=HI, A=HII, B=e, , , and is known as the ionization fraction or degree of ionization. For the internal partition function of HI only the ground state is considered. (Note. Accounting for the spin of the nucleus would increase both and by a factor 2 but leave the equation unchanged.)

Question: Do we see Balmer lines in the Sun's atmosphere?

The Boltzmann equation determines the occupation levels within an ionization state

The Saha equation determines the occupation levels among ionization states:

Applied to ionization of hydrogen this reads:

The Boltzmann equation determines the occupation levels within an ionization state

The Saha equation determines the occupation levels among ionization states:

Applied to ionization of hydrogen this reads:

Pressure ionization becomes important at high densities

This occurs when with a₀ the Bohr radius. Or Above these densities, everything is ionized.