the gaseous universe section 3.4 of the text. phases of matter there are four: solid - rare, in...

The Gaseous Universe

Section 3.4 of the text

Phases of Matter

There are four:• Solid - rare, in astronomy• Liquid - rarest in astronomy: examples include

Earth, Mars(?), Europa(?), Titan, Pluto(?)• Gas - ubiquitous• Plasma – ionized gas

In the classroom

1 mole of N2 gas = 6.023 x 1023 molecules occupies 22.4 litres at STP (1 atm pressure, 273 K temperature)

1 litre = 10 x 10 x 10 cm3

So the gas number density is ~ 6 x 1023/22.4 x 103

~ 3 x 1019 per cm3

Consider Water

A molecule of H2O has a mass of2 x 1.67 x 10-24 + 2.66 x 10-23

= 2.89 x 10-23 grams

So 1.0 cm3 (= 1.0 gm) contains ~3.5 x 1022

- about 1000 times the number as in air.

Astrophysical GasesNote the enormous range!

Terminology

Roman numerals represent ionization stages

I = neutral gas (e.g. HI = neutral hydrogen)II = once ionized

(HII = fully ionized hydrogen; HeII = He that’s lost one of its two electrons)

Fe XXVI = iron with 25 electrons removed (i.e. all but one!)

Circumstances!In the center of the sun, all atoms are completely ionized; but in the cool photospheric regions, we see absorption lines of many neutral species. So even within a single body we have to consider ranges of excitation/temperature/etc

The Meaning of TemperatureVarious ways of characterizing/measuring T:

• Consider kinetics (the characteristic velocities of particles in thermal [random] motion)

• Consider ambient radiation (the characteristic colour/energy of a typical photon passing through) e.g. what is the ‘temperature’ of interstellar space?

• Consider the typical radiation emitted by some material (e.g. what radiation is emitted by cool interstellar dust?)

• Consider the stage of excitation of a neutral gas (e.g. how many electrons are in the ground state vs the number in higher orbitals?)

• Consider the stage of ionization of various gases in a plasma• Various other measures involving emission from other allowed or forbidden

transitions (we will come back to this – e.g. the 21 cm radiation from neutral hydrogen)

The Obvious Question

Will these various measures all agree in a given body / locale / medium / circumstance?

For example: in a region of interstellar space, consider the temperatures indicated by the ambient radiation, the excitation/ionization state of the atoms present, and the kinetic motions of any particles. Will they agree?

In General, NO

We have to consider them in turn: what each represents, how they are established, and how to infer the important physics from them.

KINETIC Temperature

Consider a hot gas, with particles colliding elastically (i.e. no loss of energy in the collisions)

(side issue: where might the energy go?)

What spectrum of velocities do you expect to see when it is at equilibrium?

(For example, will all particles have precisely the same velocity??)

Maxwell-Boltzmann Velocity Distributiononce equipartition is reached and a unique equilibrium “T” applies

In the Classroom In an Interstellar Cloud

The Functional Form

Note that <v> = 2.00 vmp

Things to Note• At any given instant, essentially no particles are at rest

• There is a long tail to higher velocities.

• Equipartition of energy means (in a multi-component system) that the lower-mass particles have higher mean velocities

• In gravitationally-bound systems, the light particles can evaporate away

Examples

• Earth’s atmosphere has no free Hydrogen or Helium

• The moon and Mercury have no significant atmospheres at all

Other Applications

Clusters of stars can evaporate. Note: the stars do not suffer direct physical collisions!

Back to the M-B Function

Note the Boltzmann factor: exp (- m v2 / (2kT))

or, equivalently,exp ( - ½ m v2 / (kT))

The Implication

The probability of finding a particle of high kinetic energy falls off exponentially in part (remember the other factors in the equation!), on a scale that is determined by kT.

This determines the long high-energy tail of the distribution.

A Recurrent Theme

We encounter the Boltzmann factor in other definitions of temperature (ionization, excitation,…) and write it more generally as

exp ( - Χ / kT) (where Χ is a “chi”)

Achieving Equipartition (Thermal Equilibrium) in Gases

Imagine merging a hot gas and a cool gas. For them to come to equipartition (the MB distribution) takes some time - many elastic collisions must occur!

See page 86 for various circumstances. In most astrophysical gases, thermal equilibrium is quite quickly established.

But there are important exceptions! (p.87).

Ideal Gases

The Perfect (Ideal) Gas Law (CHEM 101!) is

P = n K T

(where n is the number density of particles)

Think: why is there no dependence on the individual particle mass?

Equivalently

P = ( ρ / μ mh) kT

where ρ is the density in physical unitsμ is the mean molecular weight of the materialmh is the mass of the hydrogen atom

Some Cases

We write composition as X + Y + Z (for H, He, ‘metals’)

In a neutral gas,

In a completely ionized gas

(Why the numerical factors? Consider the electrons!)

More Generally

The ideal gas law applies only to ‘well separated’ particles.

Remember van der Waals?

We have to consider all contributions. In stars, this can include radiation pressure, or the quantum-mechanical pressure provided by degenerate electrons (in white dwarfs) or neutrons (in neutron stars)

Particle Collisions: Mean Free Path

How likely is a particle to collide with a field of other particles through which it is moving?

See the simple derivation on page 90.

Conclusions:

Mean Free Path between collisions = 1 / (n σ)

where σ = particle cross-section n = particle number density

Mean time between collisions (for a single particle)= 1 / (n v σ)

Collision rate (for a single particle)= n v σ

Analogy: Do Bullets Collide?

Yes – But Very Rarely!

Applications

Do stars every suffer physical collisions with one another? (work it out!)

How did the Solar System form?

Will we be hit by asteroids?

Do Galaxies Collide? http://globalnews.ca/video/2359184/animation-of-milky-way-and-andromeda-galaxies-

colliding

Other Considerations

Effective cross-sections may be different from pure physical radii – consider Coulomb forces, gravitational focussing, etc

These can be folded into a hybrid collision rate coefficient that pertains in given physical circumstances.

(See table 3.2, p 92, and consider the various regimes discussed. See also the next section of the notes, pertaining to the inclusion of radiative effects.)