Section 4: Spectroscopic Measurements Flashcards
what are the quantities we obtain from a spectrum
-precise wavelength determination of features ie lines and edges
-spectral energy density of the continuum
-spectral line shape (intensity, width(
quantities obtained from a spectrum are interpreted using
physical models for the production of radiation
or laboratory or other standard measurements to derive parameters like density, temperature, velocity etc
most spectrometers in use are based on
diffraction gratings or interference filters
(fourier transform spectrometers based on michelson are also used for high spectral-resolution work, especially in the infrared)
Modern spectrometers often imaging spectrometers, called Integral Field Units, giving spectrum for
multiple x.y positions in image
grating spectrometer results
spectral resolving power
angular dispersion
grating response width
spectral resolving power
Rg = λ/delta λ = Ngm
angular dispersion
d theta /dλ = m/acos theta
grating response width
Wg=λ/Nga
lower limit to the angular width of lines produced by spectrometer
problems with raw spectra
usual CCD problems (cosmic ray hits, flatfield…)
spectral lines tilted on detector
no wavelength identification
For ground-based instruments, wavelength calibration for gratings or prisms is sometimes done using
absorption lines from the Earth’s atmosphere (telluric lines)
alternative method for ground-based instruments, wavelength calibration is done using
reference spectral emission lamp
(the spectrum from the source and from the calibration lamps is imaged onto the same CCD)
standard reference lamps include
Thorium/Argon and He/Ne/Ar
x is the position on the CCD, expressed in
‘pixel number’
the dispersed reference spectrum or the reference telluric lines provide calibration as follows
- strong lines/patterns of lines from reference spectrum identified using spectral atlas
- fit spectral profile (eg gaussian) on CCD to find x position of central λ of ref lines
- a curve is fitted to these identifications, giving λ(x)
CCD-based spectroscopes provide a
2D image of the spectrum
the dispersed emission line spectrum consists of
multiple images of the spectroscope slit, spread in colour/position along the CCD
ideally, slit images should be as
narrow as possible and aligned with CCD columns
to get a spatially integrated spectrum, sum
down columns to produce spectrum
In practice, there might be slight mis-alignments of the optics, which
will result
in tilted slit images
in a ground-based instrument, tilted slit images could, in principle, be corrected by
rotating the detector
in a space-based instrument, tilted slit images need to be correct in the
data analysis phase
ie rotation of the spectral lines so they are aligned with the y-axis
however this remapping can cause other issues
what if the remapped pixel values are not integers, how to we assign the counts to actual pixel positions
this is done by interpolating the data to the integer pixel positions
(consider the 1D problem and assume that the data follows a piecewise linear cure)
interpolation - the gradient between i1’ and i2’ is
m=N(i2’)-N(i1’) / i2’-i1’
(denominator is approx 1)
interpolation - the value of N(i’) at pixel i’ is
starting point:
N(i’)=N(i1’) +m(1- deltai’)
interpolation - the counts value at this pixel is the
distance weighted mean of the counts at the two neighbouring locations
the distance weighted mean of N(i1’) and N(i2’) gives us
N(i’)
linear interpolation is only an example, can also
fit a quadratic, or a polynomial through several neighbouring points depending on how rapidly the data is changing from point to point
Sources of instrumental noise should be reasonably well removed by
subtracting dark current, flat-fielding etc
even after noise removed, the source of interest will be seen against
the background (or foreground) contribution of the sky
if optically thin emission from the source of interest, would have to consider
emission from behind
this all has to be removed to get at the source spectrum
usually background is not constant - how to deal with
define two background regions, on either side of the feature and then do a linear interpolation using data points in these two regions
let background = Nb(λ) =
mλ+c
(background subtraction) the total real counts in the source are
sum from λ=λ1 to λ2 of
(N(λ)-Nb(λ))
Whether subtracting background from an image or a spectrum, we deal with errors
in the same basic way
Ns+b=
Ns +Nb
total counts in the source is, Ns=
Ns+b - Nb
=(Ns+Nb)-Nb
the combined errors (assuming poisson noise) are σs^2=
σs+b^2 +σb^2
=…
=Ns+2Nb
combined errors assumes gain and quantum efficiency are
1
otherwise would need to take into account (see section 2)
uncertainty in N=
σ= sqrt (Ng/Q) (Q/g)
=sqrt (NQ/g) [in DN]
The “significance” of an observation is defined as
the signal expressed as a number of standard deviations
aka SNR
if Ns is 4 times σs, then this is called
a 4σ measurement
if we consider a situation where the background is low compared to the signal then Ns/σs=
sqrt (Ns)
if Ns=rate x delta t and rate=const the significance
increases as sqrt(delta t)
ie 10x more signficance = 100 x longer exposure time
Properties of spectral lines can be analysed to understand the
conditions in the emitting medium. Because this is done “remotely” without affecting the astronomical
objects, the analysis is said to
provide ‘diagnostics’
An absorption line is formed by a
cooler less-dense gas overlying a
hotter dense one
In general, the more atoms there
are in the correct state to absorb
the radiation from below, the
deeper the (absorption) line profile will be
absorption lines - also have to take into account the effect of
convolution with instrumental line profile
The equivalent width 𝑾 is a way to
measure the total absorption in a spectral line
W such that the area of rectangular region = area of the spectral line
equivalent width - let the line intensity be described by
B_λ which is a function of wavelength
equivalent width - let the continuum intensity be
Bc which is taken as a constant
area of absorption line=
integral form 0 to inf of (Bc-Bλ)dλ
=WBc
equivalent width, W=
integral from - to inf of
Bc-Bλ / Bc dλ
The absorption of photons is described by
the optical depth
τ_λ
in the absorption line, photons are taken out of the continuum (in the simplest case) and this is described by
Bλ/Bc = e^-τ_λ
if τ_λ is large then
Bλ is small relative to Bc (strong absorption)
for a weak spectral line (τ_λ«1), what do you use for expression for W
the series expansion for the exponential function to simplify the expression for W
The optical depth is related to
the atomic properties of the absorbing
material, and the density of absorbers
the atomic properties are encapsulated in the
absorption crosssection, 𝝈𝝀, which is a function of wavelength
A narrow slab of absorbing gas, with thickness 𝚫𝒔, number density 𝒏 [m-3], has optical depth
𝜏𝜆 = 𝑛𝜎𝜆Δ𝑠 = 𝜎𝜆N
column density
N= 𝑛Δ𝑠
But it is clear that from 𝑾 we can obtain the
total amount of absorbing matter 𝑵, hence the usefulness of W
strong lines
𝝉𝝀 ≫1
Wλ prop to sqrt N
intermediate lines
𝝉𝝀 ≈ 𝟏
wλ prop to sqrt lnN
curve of growth
shows how the equivalent width varies with density
curve of growth is calculated theoretically and the measured equivalent width can be linked directly to N
Spectral lines are produced by
a hot, tenuous gas
eg coronae, gas clouds
lyman break
everything at shorter wavelengths completely absorbed
due to neutral H along LOS in IGM
Atomic line diagnostics often involve
measuring the ratio of
intensities between two emission lines from the same element
The possible transitions in an element are illustrated by diagrams
called
Term or Grotrian diagrams
Grotrian diagrams for neutral Fe, once-ionised and twice ionised Fe show
the richness of spectra from heavy ions
so many possible transitions
Most common spectroscopic diagnostics rely on assuming that
the radiating gas/plasma is in local thermodynamic equilibrium
In local thermodynamic equilibrium (LTE) the matter is
in thermal equilibrium in some small neighbourhood around a point in space
can be heating/cooling but doing so slow
matter and radiation do not have to be in equilibrium
- For a gas/plasma, LTE means that the particle distribution function
𝒇(𝒗) is
a Maxwell-Boltzmann distribution
LTE also means that the system reaches equilibrium via
collisions between particles
In many cases we can also assume that interaction with radiation is
negligible
the radiation escapes the gas/plasma without interaction -
optically thin approximation
most of the collisional excitation is by
free electrons
Being faster than ions or atoms at the same temperature, the free
electrons have more interactions per second
a collision -
a free electron perturbs an orbital electron
In LTE, the probability 𝒑𝒊 of finding an atom in state 𝒊 is determined by
the boltzmann factor, pi
boltzmann factor is proportional to
po, gi (degeneracy) and energy
what is po in the boltzmann factor
constant found by normalising the probability over all possible
states of energy 𝑬𝒊
For two levels 𝒊 and 𝒋 (𝒋 = upper level and 𝒊 = lower level) the ratio
of the number of atoms in each level is
ni/ni =gi/gi exp (-(Ej-Ei)/kT)
Ej-Ei is the
transition energy between levels
the population ratio of two levels depends on
the temperature
In an optically thin gas/plasma, the excitation to the upper level is by
collisions
The average number 𝒏𝒊𝒋 of transitions per second per unit volume,
from state 𝒊 to state 𝒋, due to collisions is
nij = ni ne Cij
Cij
the collisional rate coefficient, and depends on the electron speed,
and the atomic properties
ne
the number of electrons per unit volume
ni
the number of atoms in state 𝒊 per unit volume
In a true equilibrium, the average state does not change, so for…
every upwards transition 𝒊 → 𝒋 must be accompanied somewhere by
a downwards transition 𝒋 → 𝒊.
this is called detailed balance
In the case where upwards transitions are collision-dominated, and downward transitions are due to spontaneous radiation (the “coronal approximation”) we have
ni ne Cij = nj Aji
the coronal approximation
the case where upwards transitions are collision-dominated, and
downward transitions are due to spontaneous radiation
einstein coefficient
the average number of
transitions per second per atom
Iji
photon emission rate per unit volume
=nj Aji
Consider two levels of an atom - call them 1 and 2. Assume that they
are both excited by collisions from the ground level, and decay by
radiation back to the ground state. Then:
I2g/I1g = Cg2/Cg1
the ratio of intensity of these 2 lines gives a measure of the
gas temperature
Collisions between fast electrons and atoms in a hot gas can result
in
the ejection of a bound electron into a free state (ie collisional ionisation)
Atoms with more than one electron can exist in different stages of
ionisation. Removing successive electrons takes
increasing amounts of energy
ionisation potential
the energy required to remove an electron
represented by Chi
Consider the situation where the upper level is an ion plus a free
electron with speed 𝒗. The energy of this level is
E= X +1/2me v^2
To calculate the number of e-i pairs in this ionisation stage, we
calculate the number, 𝜟𝒏(𝒗) of states for an ion and a free electron
with velocity 𝒗, and then integrate over all possible v
The statistical weight of a particle is
the
number of states that it can occupy
Electrons
are fermions, so obey
Fermi-Dirac statistics
at most only 2 e- per state
saha ionisation equation relates
the number of ions in ionisation stages r+1 and r
population of different ionisation stages of the same species
depends sensitively on
the plasma temperature
calculations of the saha equation for various ion pairs as a function of temperature results in
ionisation balance curves
ionisation balance curves
ratio will peak at some temp and then fall off
contribution function
g(t)
calculated from atomic physics and peaks strongly at some T
By plotting the ratio of two spectral lines from the same ion, the 𝚺
and 𝒏𝒆 terms
cancel and we are left with the ratio of the two g(t) functions
this ratio depends on temperature
Assuming emission from isothermal plasma, the temperature can be
determined from
the observed intensity ratio
contribution function sensitive only over a limited temperature tange, depending on
the lines used
If we allow also for collisional de-excitation from level 𝒋 to level 𝒊,
then the expression for detailed balance becomes
same as before but with an additional njneCij term
collisional de-excitation from level j to level i might happen if
Aji is very small
ie the natural (radiative) decay time of level j is very long
Aji=
1/τ
where τ is the radiative lifetime
forbidden transition - the upper level is called a
metastable level
de-population of metastable levels can be largely due to
collisions which depend on the electron density
ie this gives diagnostic on electron density
intensity of spontaneously de-excited allowed line 2 –> 1
I21=n1neC12
metastable level m can be depopulated by
radiation or by collisions C* either to the ground state or other states
n1neC1m=
nm Am1 + Σ nm ne C*
the sum is over all possible states accessible by collisions
Only that fraction of the excited metastable population that decays by radiation contributes to
the line intensity
the observed ratio is dependent on
electron number density ne
(derivation on slides 150 and 151)
The width of the observed spectral line is due to
a number of different processes and hence a diagnostic
finite lifetime of electron in the excited sates (lorentzian) causes
natural broadening and collisional broadening
doppler motion on micro or macroscale (gaussian) causes
thermal, turbulent and rotational broadening
Overall line we observe is a
convolution of
these processes
lorentzian * gaussian ==> voigt
(and convolved with the ILP)
this course focuses on which types of broadening
thermal and turbulent
this keeps everything gaussian
The emission lines from a hot gas are broadened due to
random particle motions
λ-λ0/λ0 = v/c
line profile Φ(λ) is given by
Φth(λ) dλ = f(v)dv
f(v) is the 1D Maxwellian distribution so Φth is proportional to
exp(-mv^2/2kT)
Using our other temperature diagnostics (line ratios), we can
determine
T and hence the expected thermal line width and vth
Sometimes the observed spectral line width exceeds this thermal
value, even once the ILP is removed. One cause of this is
the additional speed can be turbulence
i.e. macroscopic motions of different parcels of emitting gas/plasma in
addition to the microscopic thermal motions
Let the emitting gas have a distribution of macroscopic speeds
p(vl)dvl=
the probability that gas is moving at turbulent LOS speed vl –> vl+dvl
Ignoring other broadening for the moment, the shape of the spectral
line profile 𝝓𝒕𝒖𝒓𝒃 𝝀 emitted by this gas is given by:
𝜙𝑡𝑢𝑟𝑏 (𝜆) 𝑑𝜆 = 𝑝 (𝑣𝐿) 𝑑𝑣
To obtain the overall line profile, we need to evaluate
the probability that a particle ahs a total LOS speed vtot=vl+v
The functions 𝒑(𝒗𝑳) and 𝒇(𝒗) are both probability density functions
and the LOS speeds 𝒗𝑳 and 𝒗 can be assumed to be
independent
the probability of obtaining 𝒗𝑻𝑶𝑻 is obtained by
multiplying the pdfs and then integrating over all values of 𝒗, subject to 𝒗𝑻𝑶𝑻 = 𝒗𝑳 + v
total line profile is thus given
by
the convolution of the individual line profiles
line emitted by the thermal plasma convolved with turbulent broadening =
overall emitted line
the total line profile is the convolution of
the thermal Gaussian
profile and the turbulent line profile
Assume that the turbulent line profile is also Gaussian and define a
“turbulent broadening” such that
(Δλturb/λ0)^2 = (vturb/c)^2
We can evaluate the resulting overall line profile using
the convolution theorem
So the total line profile is also a Gaussian, with line width 𝚫𝝀𝑻𝑶𝑻,
from the combined effects of
turbulence and thermal broadening
