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Astronomy 2 > Theoretical Astrophysics > Flashcards

Theoretical Astrophysics Flashcards

1
Q

spectrum

A

shows emission as a function of wavelength or frequency

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2
Q

a hot, dense object produces

A

continuous spectrum (blackbody)

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3
Q

a hot, tenuous gas produces

A

emission line spectrum

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4
Q

a cooler, tenuous gas overlying a hotter dense object produces

A

an absorption line spectrum

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5
Q

tenuous

A

not dense

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6
Q

example of continuous spectra

A

cosmic microwave background spectrum

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7
Q

example of emission line spectra

A

solar emission spectra

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8
Q

example of absorption line spectrum

A

solar spectrum and Fraunhofer lines

cooler gas in the upper photosphere and lower chromosphere overlying hotter gas in the lower photosphere

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9
Q

a spectral line results from the

A

transition of an electron between two discrete energy states in an atom

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10
Q

chemical composition of an object can be identified from

A

the presence of spectral lines

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11
Q

the shape/profile of spectral lines tell us

A

the properties of the atom
the properties of the emitting or absorbing material

eg: temperature, pressure, speed and magnetic field

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12
Q

an emission line is characterised by:

A

its total (integrated) intensity
its central wavelength
its width (full width at half max/equivalent width)
its shape/profile

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13
Q

examples of emission line shapes

A

gaussian
lorentzian
elliptical

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14
Q

I(λ) describes

A

the intensity of the radiation at wavelength λ

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15
Q

I(λ) d(λ) is

A

the amount of radiation in infinitesimal wavelength range λ to λ+dλ

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16
Q

the total emission in the line is found by

A

integrating

I total = ∫ I(λ) dλ

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17
Q

absorption lines are characterised by

A

central wavelength and shape

line width and depth is combined into ‘equivalent width’

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18
Q

equivalent width of absorption line

A

draw rectangle extending from the continuum to zero intensity, with area equal to the total area of the absorption line

width of rectangle = equivalent width

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19
Q

changing I(λ) to I(v)

A

I(λ)dλ = I(v)dv
I(λ)|dλ/dv| = I(v)

using λ=c/v

dλ/dv = -c/v^2 = -λ^2/c

I(v)=λ^2/c I(λ)

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20
Q

units of I(λ)

A

wm^-2sr^-1m^-1

change m^-1 to Hz^-1 for I(v)

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21
Q

why do we take absolute value of dλ/dv

A

as wavelength increases, frequency decreases

absolute value to ensure energy is positive

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22
Q

a spectral line corresponds to

A

the transition of an electron in the atoms/ions of a gas between two energy levels/states

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23
Q

if energy level is well-defined, we might expect the line to

A

have a very well defined frequency and therefore be infinitely narrow

i.e. photons emitted/absorbed at a single wavelength

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24
Q

what are natural and collisional broadening due to?

A

finite lifetime of electrons in atomic states

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25
Q

what are thermal and rotational broadening due to?

A

motion of atoms, on microscopic and macroscopic scale

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26
Q

heisenberg’s uncertainty principle

A

the time an electron typically stays in a state and the uncertainty in its energy linked by Heisenberg’s uncertainty principle

deltaEdeltat > or = h bar /2

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27
Q

𝜏rad

A

natural or radiative lifetime which depends on the atomic structure and the quantum mechanical selection rules

average time between emission of photons by a single atom

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27
Q

natural broadening

A

from heisenberg approx

deltaE delta t = h bar

sub in E=hv

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28
Q

why is energy uncertainty of lower level neglected

A

lower level always more stable than the upper level

uncertainty much smaller so neglected in comparison

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28
Q

typically an electron sits in an upper state for time 𝜏rad before…

A

making a spontaneous transition downwards, radiating a photon

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28
Q

natural broadening

A

intrinsic property of an atomic transition line

always there but very small in comparison to other broadening effects.

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28
Q

lorentz profile equation

A

A is einstein coefficient
vo is line-centre frequency

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29
Q

what does 𝜏rad determine

A

whether the transition is forbidden or allowed

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29
Q

shape of a naturally broadened line

A

lorentz profile

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29
Q

lorentzian intensity diagram

A

longer wings
pointier peak

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30
Q

the average number of spontaneous transitions per second per atom from level j (higher) to level i (lower)

A

Aji = 1/𝜏rad

units of per second

Aji is the einstein A coefficient

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31
Q

allowed transition

A

transition with 𝜏rad< or = 10^-8s

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32
Q

forbidden transition

A

transition with 𝜏rad>10^-8s

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33
Q

why are allowed reasonance lines strong?

A

many photons emitted per second as the electrons are not in the upper level for long

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34
Q

why are forbidden transitions allowed to happen?

A

other factors come into play that can lead to strong emission of forbidden lines

in particular collisional de-excitation

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35
Q

well-known astrophysical allowed transitions

A

h alpha, Ly alpha
transitions of neutral H

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36
Q

well-known astrophysical forbidden transitions

A

S II, O III

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37
Q

A_ji can be used to calculate

A

power radiated in a spectral line

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38
Q

a downwards transition from level j to level i produces

A

a photon of energy E_ji

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39
Q

there are ____ transitions per second per atom

A

A _ji

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40
Q

energy emitted per second per atom

A

=A_jiE_ji

(where E_ji=hv_ji)

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41
Q

power radiated in spectral line by N_j atoms in state j is

A

P_ji=N_jA_jiE_ji

important that it is not just N

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42
Q

there are continuous excitations between all energy levels by

A

collisions with electrons and the absorption of photons

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43
Q

In a gas of N atoms total, not all atoms can be in an excited state unless

A

artificially kept there, ie in a laser

N_j<N for excited states j

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44
Q

the upper states of forbidden spectral lines are long-lived but can be

A

perturbed by encounters with nearby particles, and de-excite

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45
Q

collisions reduce the lifetime of

A

the upper state compared to its natural lifetime

so the line is broadened

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46
Q

collisional lifetime,

A

average time between collisions that lead to de-excitation

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47
Q

collisional lifetime depends on

A

speed of the perturbing particles
the density of particles
the interaction cross-section

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48
Q

coll lifetime - speed of particles

A

higher speed means more collisions per second

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49
Q

coll lifetime - density

A

denser gas means a higher chance of encountering another particle

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50
Q

coll lifetime - interaction cross-section

A

larger cross section means a larger effective area presented by the particle

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51
Q

increased temp - coll lifetime

A

decreased collisional lifetime

higher T, higher energy of particles, more collisions, less time between

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52
Q

average time between collisions - single particle - visual

A

particle moving in space

area of cross-section is sigma

length of cylinder = vt so in 1 second =v

volume v sigma

number density n so one second collides with nV=nvsigma other particles

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53
Q

to get actual collisional timescale, have to take into account

A

particle is moving randomly among all surrounding particles which are also moving

mean speed <v></v>

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54
Q

collisional broadening is an example of

A

pressure broadening

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55
Q

pressure broadening

A

general term given to line broadening resulting from any kind of interaction between particles

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56
Q

if a radiating particle is moving with some velocity v along the line-of-sight then the observer sees

A

a red-shifted or blue-shifted photon

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57
Q

wavelength or frequency shift is given by

A

delta lambda / lambda0 = deltav/v0=v/c

this is non-relativistic Doppler shift

where v=observed frequency , v0=rest frequency (same for wavelength)

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58
Q

if there is a large number of radiating particles, all moving in different directions at different speeds, then

A

each photon emitted appears in a different part of the line profile

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59
Q

the total line profile is obtained by

A

summing over emission from all contributing particles

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60
Q

thermal broadening summary

A

ensemble of emitting particles with random directions/speeds –> total radiation is a combination of emission from all atoms –> observer sees broadened line

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61
Q

thermal broadening

A

if the emitting gas is hot and the particles have a random, small-scale motion the line will be broadened

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62
Q

thermal broadening - line width depends on

A

average speeds of the particles

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63
Q

doppler broadening

A

thermal and rotational

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64
Q

assumption made in rotational broadening

A

omega same at equator and poles i.e. solid body rotation

omega=constant

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65
Q

z-axis rotational broadening

A

rotation axis

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66
Q

y-axis rotational broadening

A

line of sight direction

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67
Q

the velocity at any point on the surface of the star at a distance r from its rotation axis is

A

v= omega x r

gets (-omegay, omegax,0)

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68
Q

line of sight speed at projected distance x

A

vlos = omega x

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69
Q

strips of the stellar disk at the same x

A

all have the same vlos

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70
Q

vlos along central meridian

A

0

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71
Q

doppler shift from all points at projected distance x from the rotation axis of the star

A

deltavx/v0 = vx-v0/v0 = vlos/c = omegax/c

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72
Q

maximum doppler shift from each end of the equator

A

delta v max/v0 = veq/c = omega Rstar /c

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73
Q

the amount of emission at each frequency v0+deltavx is proportional to

A

the length of the chord at x from the rotation axis

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74
Q

if the star is spherical then

A

there is the same amount of emission at v0-deltavx and v0+deltavx

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75
Q

calculating maximum rotational line broadening

A

need 2delta lambda to account for redshift and blueshift

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76
Q

the line profile, I(v) has the shape of a

A

half-ellipse

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77
Q

if we integrate up all of the intensity in the line profile this gives

A

the line flux
Ftot

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78
Q

integrating intensity is equivalent to

A

finding area under the hlaf-ellipse

area of half ellipse is piab/2

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79
Q

rotational broadening is also detectable in

A

spectral lines from distant galaxies

however profile is not usually simple ellipse due to complicated distribution of material in galaxies

more typical shape has ‘horns’

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80
Q

angular speed of material travelling in own galaxy depends on

A

the distance from the galactic centre

(galaxy does not rotate as a solid body)

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81
Q

at the sun, distance R0 from galactic centre, angular speed is

A

Ω(R0)

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82
Q

at point P, at a distance R, angular speed is

A

Ω(R)

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83
Q

velocity of sun at radius R0

A

Vs=Ω(R0)R0

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84
Q

component of Vs along line of sight

A

Vs sin gamma

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85
Q

LOS speed depends on

A

difference in angular speeds

distance from sun

observing position

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86
Q

if galaxy was a solid body, would have Ω(R)-Ω(R0)=

A

0

and vp=0

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87
Q

if spins in H atom are both in same direction, the energy is

A

slightly higher than if they are anti-parallel

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88
Q

hyperfine splitting

A

the two possible energy levels for the ground state of neutral hydrogen

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89
Q

transition between the two states in ground state of neutral H

A

spin flip transition

90
Q

why do we see loads of neutral H even though forbidden

A

very abundant in the galaxy so 21cm line is very strong

91
Q

doppler shift and rotational broadening of the 21cm line can be measured easily because

A

radiative lifetime is very long so natural broadening is very small

in ISM neutral H is cold so thermal broadening negligible

densities are low in ISM so collisional broadening negligible

interstellar dust does not absorb 21cm radiation (optically thin)

92
Q

a system

A

a fixed group of atoms or molecules under study

this can be a single particle or many particles

it can be closed or open

93
Q

a state

A

a condition of the system at a particular time that can be described by a parameter or set of parameters

94
Q

degenerate states

A

distinct states (ie different quantum numbers) which have the same energy

95
Q

thermodynamic equilibrium

A

the condition of thermal, radiative, mechanical and chemical balance

no net flows of energy, matter or net changes of phase

(ie no large scale changes with time)

96
Q

the probability, p, of a system in thermodynamic equilibrium at temp T and in a state with energy E is proportional to

A

the boltzmann factor

p prop to e^-E/kT

97
Q

if the system can be in one of n different states with equal probability p, then the sum of the probabilities

A

must equal 1

ie is normalised

98
Q

degeneracy is the number of

A

distinct ways that a system can occupy a state of given energy

99
Q

the probability of a system being in state i is directly proportional to

A

the number of ways state i, with energy Ei can be occupied

100
Q

if there are 3 ways to arrange g=

A

3

101
Q

pi=

A

a gi e^-Ei/kT

a=constant of proportionality
gi=degeneracy
e^-Ei/kT = boltzmann factor

102
Q

evaluating a and pi for a simple 2 level system

A

degeneracies g1=g2=1

plug these into p equation

p1+p2=1 and degeneracies same so
a[e^-E1/kT + e^-E2/kT]=1

sub in a into p1, set E2-E1=deltaE and divide top and bottom by exp(-E1/kT)

same for state 2

103
Q

evaluating a and pi for a simple 2 level system

if deltaE»kT then

A

exp(-deltaE/kT) approx = 0

so p1 approx =1, p2 approx=0

(at low temp, system is in its lowest energy state)

104
Q

evaluating a and pi for a simple 2 level system

if deltaE«kT then

A

exp(-deltaE/kT) approx = exp(0) =1

so p1=1/2 and p2=1/2

(at high temps, both states are equally populated)

105
Q

evaluating a and pi for a degenerate 2 level system

A

three ways to arrive at total (p_e) spin of stste E2
g2=3 and g1=1

sub in

summing p1+p2 and rearrange for a

sub a back in for p1 and p2

106
Q

evaluating a and pi for a degenerate 2 level system - at temps characteristic of H I clouds, we have delta E &laquo_space;kT so

A

exp(-deltaE/kT) approx =1

so p1 = 1/4 and p2=3/4

107
Q

evaluating a and pi for a degenerate multi-level system

A

pi has sum on denominator which is the normalisation constant, summed over all possible levels

108
Q

ratio of probabilities

A

ratio of numbers of atoms

makes maths easier as constants cancel

109
Q

the emission from an astrophysical abject is produced by

A

particles - electrons, ions and neutrals

110
Q

way to quantify number of particles at a given position or velocity

A

using a particle distribution function

111
Q

reading particle distribution functions

A

y-axis value is prop to number of particles in the state given on x axis

112
Q

probability density function f(x) is

A

the probability that some property of a particle takes value x

(x could be position, velocity etc)

113
Q

f(x) must be

A

normalised

such that a particle lies somewhere in the range of possible values xmin to xmax

integral between xmin and xmax of f(x)dx =1

114
Q

if f(x) is normalised, the fraction of particles between x1 and x2 is

A

integral between x1 and x2 of f(x)dx

115
Q

the mean value <q(x)> of a property q(x) is calculated from

A

the integral between x min and x max of q(x)f(x)dx

if f(x) is normalised

116
Q

if all particles have equal mass, the mean energy of the particles is

A

<E>=1/2 m <vx^2>
</E>

117
Q

mean square speed normalisation

A

integral between + and - infinity f(vx)dvx

=c integral between + and - infinity of e^-mvx^2/2kT dvx

(note g=1 so not included)

c=normalisation constant

118
Q

to evaluate mean square speed need to set a to

A

m/2kT

119
Q

particle has energy per degree of freedom of

A

1/2kT

120
Q

the more particles in dvx…

A

the more emission in dv

121
Q

gaussian line profile varies like

A

exp(-delta v)^2

122
Q

FWHM is calculated by

A

evaluating (v-v0) where I(v)=I0/2

123
Q

FWHM is also called

A

the thermal width of the line

124
Q

FWHM of line from 1D distribution of atoms with boltzmann distribution of energies is obtained by

A

I(v)=I0/2
which is true if exp(-mc^2/2kt (v-vo)^2/v0^2)=1/2

take logs of both sides and rearrange for (v-v0)

thermal width is 2(v-v0)

125
Q

to determine the distribution function of v we need

A

to find the degeneracy of states with speed v
then normalise

126
Q

degeneracy of velocity states

A

count how many allowed combinations of (vx,vy,vz) correspond to the same total speed v

127
Q

for particle confined in box length L

allowed states of particle correspond to

A

allowed wavelengths that will fit in the box (resonances)

lambda=2L/n correspond to de broglie h/p

128
Q

derivation of vn=n pi hbar/m L

A

pn=h/lambdan = nh/2L

v=p/m so vn=pn/m = nh/2mL

sub in h bar = h/2pi

129
Q

energy of state n

A

sub in vn to 1/2mv^2

can be used to evaluate the boltzmann factor

130
Q

if the 1D box gets bigger such that L tends to infinity

A

spacing in energy between states decreases, tending towards a continuum of states

all values of 1D speed are allowed and each corresponds to a single classical state

131
Q

in 3D velocity space, the velocity states form

A

a regular lattice with a uniform density of states

132
Q

a thin shell in velocity space of speed v, thickness dv has volume

A

dV where
dV=4piv^2 dv

133
Q

the density of states is uniform so the number of states in range v, v+dv is

A

proportional to v^2 dv

134
Q

the number of states with the same speed v(ie same total energy) is

A

the degeneracy of the state of speed v

this is prop to v^2

135
Q

the fraction of particles in each state is prop to

A

the boltzmann factor of that state

136
Q

the number of states in range v to v+dv is prop to

A

v^2 dv

137
Q

the maxwell boltzmann distribution

A

area1=area2 for normalised distribution

very high and very low speeds are improbable

most particles cluster around some characteristic speed

distribution is not symmetric - it has a more extended high energy tail

138
Q

the MB distribution can be written as

A

f(v)dv = normalisation constant x degeneracy x BF

degeneracy=v^2

139
Q

approach for normalising the MB distribution

A
  1. obtain the reduction formula that links In and In-2
  2. evaluate I0 and I1
140
Q

normalising the MB distribution

A

integral is of standard form x^ne^-ax^2

integrate by parts

bit in brackets =0 because at 0 first part=0, at infinity second part=0

plug back into find C

141
Q

most of the particles are at a speed shown by

A

the peak in the distribution function

142
Q

how to evaluate most probable speed

A

finding turning point of MB function = where gradient=0

143
Q

trivial solutions to turning point for most probable speed

A

v=0 and v=infinity

these correspond to the extreme ends of the MB distribution, we want peak in middle

144
Q

the FWHM of a thermally-broadened line is related to the

A

speed that most of the particles have in thermal equilibrium

145
Q

the mean speed is found by evaluating

A

integral between 0 and infinity of v f(v) dv

sub in MB equation

using reduction formula, evaluate I3

sun back in

146
Q

calculating mean square speed

A

similar to mean speed but using I4 in reduction formula

already have I0 from before so sub in

147
Q

rms speed

A

the sqaure root of the mean square speed

148
Q

mean energy

A

<E>=1/2m<v^2> = 3/2kT
</E>

149
Q

working out fraction of particles in tail only works if

A

speed is significantly greater than the most probable speed

150
Q

calculating fraction of particles in tail

A

integral between ve and infinity of v^2exp(-mv^2/2kT)dv

change variable to x=mv^2/2kT

x^1/2varies slowly compared to e^-x for large x so take out as constant

151
Q

lower limit xe=mve^2/2kT is

A

the ratio of the kinetic energy of the particle to a typical thermal energy

152
Q

to use equation for fraction of particles in tail, need to

A

check that we are in the limit of large x

153
Q

a blackbody is an object that

A

absorbs all radiation incident upon it and also re-radiates it all

the matter and radiation are then in thermodynamic equilibrium

154
Q

classical theory of cavity radiation - Rayleigh-Jeans approach

A

cavity with walls at T and small aperture where radiation can enter/exit

radiation inside is emitted/absorbed by walls and has blackbody spectrum

aperture samples the radiation inside cavity and emergent radiation also bbody

155
Q

radiation in cavity vs astrophysical object - walls of cavity

A

lots of particles in thermodynamic equilibrium at T, ie in a MB distribution

156
Q

radiation in cavity vs astrophysical object = heating of the cavity

A

there is a source of radiation at the centre of the object

157
Q

radiation in cavity vs astrophysical object - radiation reflecting from the walls of the cavity

A

photons emitted by the source are absorbed and re-emitted by the particles

158
Q

radiation in cavity vs astrophysical object - radiation leaving aperture in the cavity

A

eventually the photons reach the ‘edge’ of the object and leave

159
Q

equation for the electric field E(x,t) of a standing electromagnetic wave in 1D

A

E(x,t)=E0 sin(2pix/lambda)sin(2pimu t)

160
Q

for reflection, E(x,t) must be

A

zero at the walls

ie sin(2pix/lambda)=0 at the walls

161
Q

setting x=0 and x=a as each end of cavity. Then reflection means

A

2a/lambda = n

162
Q

the integer mode number n can be obtained in terms of

A

frequency and the box size

163
Q

each n corresponds to

A

a different allowed frequency mode

164
Q

a small number dn of modes corresponds to

A

a small range of frequency dv

165
Q

the number of modes in range v to v+dv is given by

A

the frequency range dv divided by the frequency spacing between allowed frequencies

since n is an integer, modes are separated by n=1

166
Q

in 3D, nx,ny and nz define

A

a grid of points in n-space uniformly distributed at integer values

each point corresponds to an allowed 3D standing wave

167
Q

the number of allowed frequencies in v to v+dv is equal to

A

the number of points between shells of radii n and n+dn in n-space

this has volume dV_n

168
Q

modes are separated by integers so density of modes =

A

1

169
Q

the voume dV_n of the spherical shell, radius n is

A

4pi n^2 dn

where n=2av/c

170
Q

there is an independent mode of propagation in which

A

E and B are rotated through 90 degrees

171
Q

any polarisation of the wave can be formed by the

A

sum of these two independent modes of propagation

172
Q

two independent waves per frequency so

A

for each frequency, g=2

photon view: there are two independent photon spin states

173
Q

to find the specific energy density

A

combine equation for number density of states with equation for equipartition of energy and divide by the volume

174
Q

how to find the total energy density emitted by a blackbody

A

integrate specific energy density u(v) over all frequencies

175
Q

ultraviolet catastrophe

A

integrating using the rayleigh jeans law gives infinite energy density

RJ has excellent agreement at low frequencies and high temps but fails towards higher frequencies

176
Q

RJ limit

A

hv &laquo_space;kt

177
Q

specific energy density units

A

Jm^-3 Hz^-1 (or m^-1 if in terms of wavelength)

178
Q

Planck’s theory of cavity radiation: the problem

A

assumption that classical equipartition of energy could describe the average energy for the allowed frequency modes

179
Q

Planck’s theory of cavity radiation: the solution

A

for a blackbody, the average energy of the standing waves is a function of the frequency

180
Q

if the degeneracy in independent of E then the average energy is

A

integral E P dE / integral P dE

181
Q

planck’s idea

A

treat energy as a discrete quantised variable instead of a continuous variable

182
Q

average energy of a mode with frequency v

A

sum of E.BF/sum of BF

insert E-nhv

set x=BF

expand sum

simplify using maclaurin series

183
Q

in the quantum approach, the average energy is a function of

A

T and v instead of being a constant kT

184
Q

can calculate the total energy density of the blackbody radiation using

A

planck’s solution

change variables
use standard integral

185
Q

regimes for planck function: Rayleigh jeans law

A

usually valid at radio an IR wavelengths

BF approx = 1+hv/kT

(set x=hv/kt and use maclaurin expansion for e^x)

186
Q

regimes for planck function: Wien’s law

A

valid at x-ray and gamma wavelengths

1/BF-1 approx = BF

187
Q

if star’s spectrum can be approximated by bb then power per unit area emitted by the star is

A

B=sigma T^4

sb law

188
Q

total power radiated by surface of star is

A

L=4piR^2sigmaT^4

surface area x radiated flux

189
Q

if planet has radius R then the power absorbed by it is

A

P=(1-a)FpiR^2

incoming flux x area

190
Q

a

A

albedo
quantifies the power reflected by the planet
(1-a) gives fraction of absorbed power

191
Q

assuming planet radiates as a BB in equilibrium so Pin=Pout then Pout=

A

4piR^2sigma T^4

surface area x radiated power per unit area

192
Q

temperatures of planets in the solar system

A

venus much higher than expected - runaway greenhouse effect

the moon has big range due to no atmosphere

193
Q

habitable zones

A

distance from the star where the planets could have liquid water

194
Q

it is necessary to assume models for atmospheric composition to account for

A

albedo
greenhouse effect

195
Q

refining the calculation to look at a small patch of the planet’s surface

projected area as seen by the star is

A

dAp=dAcosB

196
Q

local temp on planet surface - 3 assumptions

A
  1. local value of planetary albedo is a so a fraction 1-a of the power arriving is absorbed
  2. heat is not conducted away, and the area re-radiates freely
  3. The area re-radiates as a blackbody.
197
Q

local temp depends on

A

the inclination angle B of the location on the surface of the planet relative to the star

198
Q

an atmosphere can be either

A

transparent (optically thin) or opaque (optically thick)

199
Q

scattering

A

photons change direction
elastic: no change in photon energy
inelastic: change in photon energy

200
Q

refraction

A

the direction of the propagation of the radiation changes

201
Q

absorption

A

photons are absorbed by the medium and their energy is transferred to the medium

202
Q

absorption process

A

any process that removes radiation from the line of sight of the observer

ie scattering

203
Q

how well a target can scatter/absrob EM radiation is often described through its

A

cross-section which represents an effective area presented by the target particle to the radiation

204
Q

cross section usually depends on

A

v, particle properties and the process of interest

205
Q

thomson scattering

A

scattering by free electrons
if hv«mc^2
cross section independent of frequency of photons

206
Q

resonant scattering (spectral lines)

A

photon interacts with a bound electron exciting it to a high level
energy of transition matches the photon energy
excited electron de-excites, re-emitting photon with hv in a random direction

207
Q

rayleigh scattering

A

photon interacts with bound electron but photon energy too small to excite

scattering is by particles smaller than wavelength of photon

causes interstellar reddening

208
Q

thomson scattering - oscillating electric field of incident photon causes

A

free charged particle to oscillate

209
Q

thomson scattering - oscillating particle emits

A

radiation at the same frequency as incident wave so wave scattered

210
Q

thomson cross section is the same for

A

all wavelengths

211
Q

resonant scattering - photon interacts with bound electron and its frequency matches

A

the frequency of a transition

212
Q

resonant scattering - cross section depends on

A

photon energy since this must match energy of transition

213
Q

rayleigh scattering - scattering particle are much smaller than

A

wavelength of radiation
prop to lambda^-4

214
Q

longer wavelengths are scattered

A

less than shorter
(red scattered less than blue)

215
Q

ionisation potential

A

energy necessary to free an electron from the atom

216
Q

kinetic energy of ejected electron

A

E-Ei

217
Q

recombination

A

at later time, electron can be recaptured by another ion

218
Q

an equilibrium is reached in the H cloud when

A

ionisation rate Ni equals recombination rate Nr

219
Q

stromgren radius

A

radius of the sphere around a star inside which the material is kept ionised by the star’s radiation

220
Q

stromgren radius assumptions

A

region in photon-ionisation equilibrium

H plasma fully ionsides

region spherical

221
Q

units of specific intensity

A

Wm^-2Hz^-1sr^-1

222
Q

in free space (ie no processes affect radiation) the specific intensity is

A

conserved along a ray

dI/dr=0

223
Q

absorption coefficient

A

relates interaction cross section and the number density of particles in the medium

av=nsigmav

224
Q

to evaluate the overall absorption coefficient, need to

A

consider all relevant processes

av = n1sigma1 +n2sigma2

225
Q

the total optical depth of a gas measures

A

what fraction of the incoming radiation is removed from the line-of-sight by scattering or absorption

226
Q

optical depth tells us if

A

the medium is transparent or opaque

low optical depth = transparent

227
Q

for a uniform medium the optical depth is

A

Tv=avL

L is distance along the ray path

228
Q

source function

A

Sv=jv/av

229
Q

observed specific intensity has three contributions

A
  1. radiation incident on medium and exponentially attenuated by it by the optical depth
  2. contribution by medium itself from source function
  3. exponential absorption of the medium’s own radiation
230
Q

optically thin - the probability that the photon is absorbed or scattered is

A

very low

typically the distance between scatterings is bigger than the size of the cloud

231
Q

optically thick - the probability that the photon is absorbed or scattered is

A

very high

distance between scatterings much smaller than the size of the cloud

Astronomy 2 (4 decks)

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