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Basic Definitions:-

The Saha Equation

The Saha Equation

• Just as the Boltzmann Equation describes the relative populations between energy levels, the Saha Equation describes the relative population of different ions (i.e., the ionisation fraction) of two adjacent ionisation stages of the same element:

• where and are the energies of levels in each ionisation stage, ,ε represents the energy of the free electrons, and , are their statistical weights. We can now write the Saha equation as:

,

The Saha Equation

• If the ionic populations are in their electronic ground states, then we can use the partition functions of the ground states (i.e., and ), we can simplify the problem and obtain:

where: which is the ionisation energy to move from the ground state of an ion, , to the adjacent higher ionisation stage.

The Saha Equation • Using the free electron partition function from

before, we can rewrite the equation at the top of the previous slide as:

• In astrophysics, we are usually interested in the most abundant ionisation stage of a particular element, X, i.e. the ratio:

• Note two important points:

– the ratio of statistical weights between the ground states

of the adjacent ionisation states is typically of ~1;

and:

– close to the maximal ionisation level, the ratio of the

numbers of ions in adjacent ionisation states is also ~1

The Saha Equation

• Referring back to the equation before last, we can now rewrite it as:

• From before, recall that for non-degenerate electrons the number of available states is >> number of electrons, so we find the energy difference between adjacent excitation states:

• In practice this difference is , showing that in LTE the photon field can populate atomic/ionic levels, but will not fundamentally alter the ionisation balance of a given element.

Spectral appearance

• As a result of this, the spectra of astrophysical objects, such as stars, can be quite complicated due to the presence of many lines from high-level energy states, even though the ionisation level of the plasma is relatively low

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Saha’s Equation

• We can generalise the Saha equation to give the proportion of ions in two adjacent ionisation

states (e.g., between He0 and He+, or between

He+ and He2+) in terms of the electron pressure,

using the identity pe = n k T :

• This representation of Saha’s ionisation equation

is useful in understanding the behaviour of spectra for stars of similar temperature, but different size.

* See Collins on-line book, Chapter 11 for

discussion of the Saha equation

Classifying Stellar

spectra

The stellar spectral sequence

• Since a star’s spectral type is determined from its spectrum, we need to determine how the observed spectrum relates to the conditions in the stellar atmosphere

The stellar spectral sequence

• Sun (G5V)

• Vega (A0V)

Stellar spectra • There are ~ 1011 stars in our Galaxy and a large

number are accessible to spectroscopic study

– the Sloan Digital Sky Survey contains spectra for > 71,000 stars

– the Simbad stellar database contains information on

3,488,728 objects

• Historically, spectral classification has been based upon the optical spectra of stars particularly in the spectral regions of:

– the visual ( 500 – 700 nm)

– photographic blue ( 380 – 500 nm)

• For classification, a resolution of 0.2 nm (2 Å) has been found to be best ( trade-off between

resolution and S/N => minimum observable line strength)

Spectral types

• Spectral classification attempts to group stars with similar spectra

– this is primarily for convenience

• Based on absorption line strengths

• Despite the limitations, ~99% of all stars fit into a system of cool to hot stars, dwarfs to giants (so, primarily 2-dimensional)

• the grouping into classification systems generally ignores individual differences

– it is not unusual to discover that a particular star has

been classified with a different spectral types at different

times (sometimes by the same people)

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Problems • Each spectral type is a simplified grouping that

covers a range of properties

• Within each bin, the spectral behaviour is assumed to depend on a limited number of spectral properties (e.g., temperature, surface gravity,

surface composition)

• Classification can be compromised by additional material (e.g., circumstellar)

• Historically classification has been based on optical spectra

– there may be surprises/insights in other spectral bands

Stellar Spectral Sequence •If we see a line of hydrogen, does that mean that the star contains hydrogen?

•If we DON'T see a line of hydrogen, does that mean that the star DOESN'T contain hydrogen?

•If we do see an absorption line of hydrogen, how much hydrogen must there be in the photosphere?

•If a line due to hydrogen is stronger than one due to iron, does that mean there are more H than Fe atoms in the photosphere?

Need to understand spectra quantitatively How does observed spectrum relate to temperature, pressures, gas composition in stellar atmosphere?

Stellar Spectral Sequence

•Father Angelo Secchi (1860’s): Stars classified by eye using colours + abs. lines

•Henry Draper: Classified alphabetically using strength of Hydrogen features - physical principles underlying the spectral differences were not well known

•Harvard scheme (1912): Annie Jump Cannon re-arranged scheme based on Saha-Boltmann predictions for H line strengths from alphabetical to form: O B A F G K M Often still called early-type (O,B) and late-type (K,M) stars,

this again of historical origin

Stellar Spectral Sequence

O B A F G K M

The Morgan-Keenan scheme • The Morgan-Keenan (MK) scheme is a two-

dimensional scheme that was developed in the 1940s and allows for both temperature and luminosity effects

• Improvements have led to decimal places being used to subdivide (and sub-subdivide) spectral classes, though not all decimal subdivisions are used for every class

• The classification scheme has shown itself to be both useful and adaptable in practice, and will be with us for a long time to come (despite its idiosyncrasies!)

Spectral sequence

Today, further refinements, i.e.: •L, S & T (brown dwarfs)

•Unusual abundances (p)

•Wolf-Rayet stars (W)

•Emission-line stars (e)

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Derivation of temperature • Stellar surface temperature helps to decide the

spectral class where a star will be placed

• This makes use of both the Saha and Boltzmann equations

• From the Saha equation we can estimate the ionic balance at a range of temperatures and compare this with observations e.g., consider the ionic abundance of helium:

He2+ He+ He0

Derivation of temperature • Absorption strength will depend on:

– the number of ions of a species

– the population of the energy levels within a species

• We thus also need to include the effects of Boltzmann statistics

• Example - Balmer absorption by hydrogen

1. need the amount of neutral hydrogen (as determined

from the Saha equation)

2. the proportion of electrons in the n = 2 level (from the

Boltzmann equation)

Derivation of temperature • From a combination of the two effects we find

that Balmer absorption peaks around spectral class A-F

Stellar spectral sequence- continuum

shape and Balmer α line strength

Theoretical curves of growth for HI for FUSE A9V

O5V

G8IV

A7IV

K5V

F5V

Balmer alpha absorption marked

Derivation of temperature • We can determine the mean ionisation level from

a spectrum and use this to determine the temperature of the star:

The Cooler End – Molecular Spectra

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Derivation of luminosity • We can determine the luminosity (or size) of a

star with reference to the Saha equaton

B2

B5

B8

Luminosity Classes

All A0 spectral class

but differences in line widths

*Luminosity (radius) differences*

Saha’s Equation

• We can generalise the Saha equation to give the proportion of ions in two adjacent ionisation

states (e.g., between He0 and He+, or between

He+ and He2+) in terms of the electron pressure,

using the identity pe = n k T :

• This representation of Saha’s ionisation equation

is useful in understanding the behaviour of spectra for stars of similar temperature, but different size.

* See Collins on-line book, Chapter 11 for

discussion of the Saha equation

Derivation of luminosity

• We can also differentiate Saha’s equation w.r.t. density to determine how the ionisation conditions depend on both density and temperature

• We find that, for the ionisation conditions of a dwarf star to be similar to those in a giant star, the temperature in the giant star must be lower by an amount:

where we note that Im is not the same for all ions

Derivation of luminosity • As a result, we find that lines of slightly higher

ionization tend to be stronger in giants relative to dwarfs of the same spectral class.

• Examples of such sensitive lines are:

– SrII 4077, 4215ÅÅ (407.7 nm, 421.5 nm)

– TiII 4161, 4399ÅÅ (416.1 nm, 439.9 nm)

– FeII 4233Å (423.3 nm)

• A particularly useful line ratio is that of SrII 4215Å / FeI 4260Å which increases linearly with the absolute magnitude of the star.

* Can also use pressure broadening of lines where gravity strong

Stellar spectral sequence

Two dimensional spectral sequence incorporating abundances

WN

WC

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Example Spectrum 1: White Dwarf:

Example Spectrum 2:

Symbiotic Binary Star (EG And):