Interpreting Light from the
Atmosphere of the Sun
A primer on how spectra make pretty pictures, and how
we can use them to get physical quantities
Interpreting Light from the
Atmosphere of the Sun
“What do you think you’re looking at??”
A primer on how spectra make pretty pictures, and how
we can use them to get physical quantities
Light
Light
•
Almost everything we know about the Sun comes
from light
Light
•
Almost everything we know about the Sun comes
from light
•
So why do we spend so much time getting it wrong
Light
•
Almost everything we know about the Sun comes
from light
•
So why do we spend so much time getting it wrong
•
I’m projecting: why have *I* spent so much time
getting it wrong?
Solar Spectrum
Solar Dynamics Observatory: images in the UV to EUV
SolarDynamicsObservatory:
imagesintheUVtoEUV
Where did the light come from?
•
What we’re looking at (measuring) is where the
photon came from
•
This might seem philosophical, but it’s a crucial
question
•
The photon was created or scattered by the
conditions at that point.
•
So what are the chances that it can make it to our
eye or telescope?
Journey of a photon through gas
Photons can be
absorbed by ions
with bound electrons
Journey of a photon through gas
incident beam
Photons can be
absorbed by ions
with bound electrons
emergent
beam
And they can be
scattered (mostly by free
charges like electrons)
Journey of a photon through gas
incident beam
Photons can be
absorbed by ions
with bound electrons
emergent
beam
And they can be
scattered (mostly by free
charges like electrons)
So how far can they get, on average, before one of these
processes removes them from the beam, Iν ?
Mean-free-path of a photon
So how far can they get, on average, before one of these
processes removes them from the beam, Iν ?
•
This isn’t such a hard question to make a stab at,
once we know two basic quantities
•
•
What’s the number density n of the particles?
•
We have to treat all the different absorbers and scatterers, separately
•
e.g., e–, H I, He II, Si VII, Fe XIII, …
What’s the probability that a photon will interact with any
individual particle?
•
This will depend on wavelength or frequency, ν
Probability of removing photons from the
beam
•
The probability that a single particle will absorb a photon can be
expressed in terms of how large that particle “looks” to the photon.
•
This is given as a “cross-section” σν, given in units of area
•
The more particles there are, the greater the total “cross sectional area”
there is to intercept it.
side view
view along
beam
each particle has an
apparent cross-sectional
area for interaction σν at
frequency ν
•
The probability for a photon to
be removed from the beam is
going to be proportional to
•
The probability of interaction
(apparent area) for each type of
particle, σν
•
the number density of each type of
particle, np
•
•
more particles will occupy more area
perpendicular to the beam
the distance the beam travels along
the column, s
•
The probability for a photon to
be removed from the beam is
going to be proportional to
•
The probability of interaction
(apparent area) for each type of
particle, σν
•
the number density of each type of
particle, np
•
•
more particles will occupy more area
perpendicular to the beam
the distance the beam travels along
the column, s
the view along the beam is
going to start looking very
crowded!
!
Changing n or σν
column density (n ✕ s)
cm-2ij (⌫
⌫ //
Same, but with more
absorbers
1
2
3
⌫0/
)
⌫ / ij (⌫ -⌫
1
if σ depends on ν (or λ) then
the effective size of the
absorbers can change as
we tune ν:
2
if σ depends on ν (or λ) then
the effective size of the
absorbers can change as
we tune ν. If the scatterers
0are free electrons, for
example, then this crosssection won’t change
3
ij (⌫ - ⌫ )
We’re only changing the frequency/wavelength of the light
between these three situations:
n remains the same, only σν decreases for the absorbers
- ⌫0 )
projecting whole column
onto its face area
For a given particle p (e.g., Fe IX ion, e–) and a given frequency
ν, this will produce a probability Premν,p of a photon being
absorbed or scattered:
[cm2 cm-3
rem
P⌫,p (s)
=
cm] = dimensionless
⌫ np (s)s
The fraction of the beam δIν/Iν removed by particles of type p
will be equal to this probability:
I⌫ = -I⌫ ⇥
⌫ np (s)s
where δIν has a negative sign because we reduce the intensity
of the beam.
•
In reality, though, np is not necessarily constant along
our line of sight, s
we need to integrate its effects along the beam direction
•
dI⌫
=I⌫
•
⌫ np (s)ds
= -⌫ ds
The quantity κν is called the linear extinction coefficient, since it
represents the removal (“extinction”) probability per unit distance
along the beam
•
κν is sometimes called the “opacity” of the plasma, but watch out for other
defitions: e.g., per unit mass
Mean-free-path at last…!
•
The inverse of κ…
number of photons removed per unit length
•
•
–
…is the mean free path, sν
length between photon removals!
•
•
You can also think of it like this:
•
•
distance travelled
s⌫ =
=
number of particles encountered
s
=
⌫s · n
1
⌫n
where σνs sweeps out a volume, which you multiply by the
number density n to get a number of particles encountered.
κν & τν
dI⌫
= -⌫ ds
I⌫
Z
)
Z
dI⌫
= - ⌫ ds
I⌫
The integral on the right hand is one of the most
fundamental concepts in all astronomy: the optical
Z s0
depth, τν
•
⌫ ds
0
⌧⌫ (s ) = -
0
What’s the τ?
•
•
So what’s the physical meaning of
τ?
Let’s use a meaningful length scale
for s:
•
•
•
in this case, we’ll use the mean-free-path,
s–ν
We can see that τ=1 corresponds
to one photon mean-free-path
into the plasma!
Meaning that on average,
photons tend to come from τ=1
-⌧⌫ (s) =
=
Zs
Zs
⌫ ds
0
⌫ n ds
0

= s
s
⌫n
0
= s⌫ ⌫ n
1
=
⌫n
⌫n
-⌧⌫ (s⌫ ) = 1
Optically thin or thick?
•
•
Typically, if a block of plasma has τν >> 1
•
it would take a photon several mean free paths to make it to the front
•
the back of that plasma can therefore rarely be seen
•
This case is called optically thick
and if if τν << 1
•
then the almost all photons initially headed towards the observer just
carry on along the same direction
•
The mean free path is much longer than the physical depth of the
plasma
•
This case is called optically thin
Effect of τν on a beam of light
The form of our simple differential
equation means that a beam of
intensity Iν is attenuated by a factor that
is exponential with distance s along
the beam:
•
Z
h
ln
•
This does assume only extinction and
not emission within the plasma column,
of course.
ln I⌫
✓
iI(s 0 )
= -⌧⌫ (s 0 )
◆
= -⌧⌫ (s 0 )
I(0)
0
I⌫ (s )
I⌫ (0)
0
dI⌫
= -⌧⌫
I⌫
-⌧⌫ (s 0 )
I⌫ (s ) = I⌫ (0)e
Dependence of τν on ν
⌧⌫ (s 0 ) =
Because
Z0
n(s)
s0
⌫
ds
the optical depth, and therefore the mean free path, can
depend strongly on the wavelength we look at.
If we consider bound-bound absorption
ij (⌫
- ⌫0 )
where ψ is the profile of the probability for absorbing the photon around
ν = ν0, where hν0 = Ej – Ei .
ij (⌫
/
/
⌫
⌫
⌫
then
- ⌫0 )
photon is absorbed and excites an electron from state i to state j
/
ij (⌫
-⌫⌫0/
)
ij (⌫
-⌫
0
⌧⌫ (s ) =
⌫
Z0
•
•
Column density N(s) is the
number of absorbing
particles we can see along a
column of unit crosssectional area A and length s
s0
=
•
n(s) ds
0
N(s
)
⌫
The integral here N(s’) is the column density
(number of absorbers per unit area along the line
of sight) between s=0 and s=s’
σν
As a consequence, an τν = 1 requires a far smaller
number of particles along the line of sight at ν = ν0
than it does at frequencies (wavelengths) away
from the centre of the absorption line.
We therefore tend to see photons from higher in
the atmosphere at the centre of the line because
the mean free path is a shorter distance at
the central frequency/wavelength
•
we see deeper into the atmosphere for
wavelengths/frequencies away from line
centre.
Α = 1 cm2
⌫
/
⌫
/
ij (⌫
ij (⌫
- ⌫0 )
-⌫⌫0/
)
ij (⌫
-
Varying visible depth with λ or ν
Varying visible depth with λ or ν
Varying visible depth with λ or ν
Varying visible depth with λ or ν
https://svs.gsfc.nasa.gov/cgi-bin/details.cgi?aid=11708
Varying visible depth with λ or ν
Credit: NASA/IRIS/T. Pereira
At the centre of the line, we see the
plasma closest to us (highest)
ν0
https://svs.gsfc.nasa.gov/cgi-bin/details.cgi?aid=11708
Varying visible depth with λ or ν
Credit: NASA/IRIS/T. Pereira
At the centre of the line, we see the
plasma closest to us (highest)
ν0
https://svs.gsfc.nasa.gov/cgi-bin/details.cgi?aid=11708
Varying visible depth with λ or ν
Credit: NASA/IRIS/T. Pereira
At the centre of the line, we see the
plasma closest to us (highest)
ν0
Different equilibria in the Sun
Thermodynamic equilibrium is a
complete interaction of material and
radiation – a black body. The
radiation intensity is given by the
Planck function, Bν(T). It holds in
the solar interior.
Different equilibria in the Sun
Thermodynamic equilibrium is a
complete interaction of material and
radiation – a black body. The
radiation intensity is given by the
Planck function, Bν(T). It holds in
the solar interior.
Local thermodynamic equilibrium
(LTE) is where matter and radiation
almost completely interact, but with
a small escape of radiation. The
equilibrium is defined in terms of the
local temperature. It holds in the
solar photosphere. Iν = Bν(T) is still
a good approximation
Photon mean free path is very
short in both cases with respect to
any scales of change in
temperature, pressure, etc.
So the radiation field is a black
body one, characterised by the
same temperature as the particles’
Maxwellian distribution
Different equilibria in the Sun
Non-local thermodynamic
equilibrium (NLTE) is where there is
incomplete interaction of matter with
radiation, with radiation freely
escaping from the region. It holds in
the solar chromosphere.
Mean free path starts to be larger than
size of the system. Radiation can
therefore escape the chromosphere
and is not returned, so Iν starts to
depart from Bν(T)
Different equilibria in the Sun
Non-local thermodynamic
equilibrium (NLTE) is where there is
incomplete interaction of matter with
radiation, with radiation freely
escaping from the region. It holds in
the solar chromosphere.
Mean free path starts to be larger than
size of the system. Radiation can
therefore escape the chromosphere
and is not returned, so Iν starts to
depart from Bν(T)
Coronal equilibrium is an
equilibrium between the numbers of
ionisation and recombination
processes per unit volume. The
radiation from the photosphere
passes through the material without
any appreciable effect. It holds in
the solar corona.
Mean free path is now typically
several solar radii. As a result, there is
basically no connection between the
radiation field and the populations of ions
or excited states within those ions.
Population balance almost completely
determined by collisions with
electrons.
Radiative transfer equation
This is a way of expressing how a beam’s intensity is
modified by both subtraction of photons from
(absorption + scattering) = κν
and addition by
emission = jν
into a line-of-sight along s.
Radiative transfer equation
This is a way of expressing how a beam’s intensity is
modified by both subtraction of photons from
(absorption + scattering) = κν
and addition by
emission = jν
into a line-of-sight along s.
⌫ =
X
np ( (⌫)bb
p +
p
ff
Th
(⌫)bf
+
(⌫)
+
(⌫)
p
p
p + . . .)
Radiative transfer equation
This is a way of expressing how a beam’s intensity is
modified by both subtraction of photons from
(absorption + scattering) = κν
⌫ =
and addition by
emission = jν
into a line-of-sight along s.
X
np ( (⌫)bb
p +
p
ff
Th
(⌫)bf
+
(⌫)
+
(⌫)
p
p
p + . . .)
here the superscripts indicate:
•
bb – bound-bound absorption of a photon
•
bf – bound-free absorption of a photon
•
ff – free-free absorption
•
Th – Thomson scattering of a photon from
a free electron
Specific Intensity
•
We can define the intensity of the beam as a specific
intensity
E⌫ = I⌫ d! d⌫ dA dt
•
•
…which takes account of the energy emitted:
•
at frequency ν
•
in frequency range dν
•
through area dA
•
in time dt
•
into a solid angle dω
Typically this takes units of erg cm-2 s-1 sr-1 (Å-1 | Hz-1)
Changes to Iν along the beam direction, s
dE⌫
+
dE⌫
= -⌫ E⌫ ds = ⌫ I⌫ ds d! d⌫ dA dt
= j⌫ ds d! d⌫ dA dt
+
dE⌫ = dE+
dE
⌫
⌫
Changes to Iν along the beam direction, s
dE⌫
+
dE⌫
= -⌫ E⌫ ds = ⌫ I⌫ ds d! d⌫ dA dt
= j⌫ ds d! d⌫ dA dt
+
dE⌫ = dE+
dE
⌫
⌫
dI⌫ d! d⌫ dA dt = -⌫ I⌫ ds d! d⌫ dA dt + j⌫ ds d! d⌫ dA dt
1
ds
:
⇠
(
(
⇠
(
(
(
(
⇠
(
(
⇠dA dt = -⌫ I⌫ (
(d⌫ dA dt + j⌫ (
(d⌫ dA dt
dI⌫ ⇠
d!
ds(
d!
ds(
d!
⇠d⌫
Changes to Iν along the beam direction, s
dE⌫
+
dE⌫
= -⌫ E⌫ ds = ⌫ I⌫ ds d! d⌫ dA dt
= j⌫ ds d! d⌫ dA dt
+
dE⌫ = dE+
dE
⌫
⌫
dI⌫ d! d⌫ dA dt = -⌫ I⌫ ds d! d⌫ dA dt + j⌫ ds d! d⌫ dA dt
1
ds
:
⇠
(
(
⇠
(
(
(
(
⇠
(
(
⇠dA dt = -⌫ I⌫ (
(d⌫ dA dt + j⌫ (
(d⌫ dA dt
dI⌫ ⇠
d!
ds(
d!
ds(
d!
⇠d⌫
dI⌫
= -⌫ I⌫ + j⌫
ds
Changes to Iν along the beam direction, s
dE⌫
+
dE⌫
= -⌫ E⌫ ds = ⌫ I⌫ ds d! d⌫ dA dt
= j⌫ ds d! d⌫ dA dt
+
dE⌫ = dE+
dE
⌫
⌫
dI⌫ d! d⌫ dA dt = -⌫ I⌫ ds d! d⌫ dA dt + j⌫ ds d! d⌫ dA dt
1
ds
:
⇠
(
(
⇠
(
(
(
(
⇠
(
(
⇠dA dt = -⌫ I⌫ (
(d⌫ dA dt + j⌫ (
(d⌫ dA dt
dI⌫ ⇠
d!
ds(
d!
ds(
d!
⇠d⌫
dI⌫
= -⌫ I⌫ + j⌫
ds
This is the equation of
radiative transfer
Radiative transfer equation in s
dI⌫
= -⌫ I⌫ + j⌫
ds
j⌫
= S⌫
⌫
•
The ratio of emission to absorption is called the source function, Sν
•
In the s co-ordinate, along the direction of the beam, we would
write
dI⌫
= ⌫ (-I⌫ + S⌫ )
ds
•
This says that intensity will decrease along s if Sν < Iν, and
vice versa. (Note that Iν has a negative sign in this arrangement.)
Radiative transfer equation in τ
dI⌫
= -⌫ I⌫ + j⌫
ds
÷
d⌧⌫
= -⌫
ds
dI⌫
j⌫
= I⌫ d⌧⌫
⌫
•
So, if increase in intensity is positive as we go deeper
into the Sun…
•
the source function is not as large as the incident radiation field
from beneath it.
What about the direction of s?
•
If we have a particular reference
direction, r
•
•
e.g. a direction specified because we
have gravity
then we have to think about Iν
as a function of r and θ
•
θ = cos-1μ is the angle between our
beam direction s, and r.
E⌫
=
I⌫
d!
d⌫
dA
dt
What about the direction of s?
We can think of the optical depth
with respect to an imposed
direction, r
•
•
dI⌫
µ
= I ⌫ - S⌫
d⌧⌫
for which a general solution is
Z1
- ⌧µ⌫ ⌧⌫
S⌫ (⌧⌫ )e
d
I⌫ (µ, 0) =
µ
0
•
(not for today)
E⌫
=
I⌫
d!
d⌫
dA
dt
Radiative transfer equation in τ
•
But, what falls out of that general
solution is more specific solutions:
I⌫ (µ = 1, 0) = S⌫
Z ⌧2
0
•
e.g., if Sν is constant then it
comes outside the integral
•
If the plasma is optically thick,
then the back of the plasma is at
τ2 >> 1
•
so radiation emerging from the
surface at the normal is Iν(1,0) = Sν

- ⌧⌫
1
e ✓d
µ
⌧⌫
1
µ
7
◆
◆
⌧2
= S⌫ -e-⌧⌫
0

= S⌫ -e-⌧2 - (-1)
I⌫ (µ = 1, 0) = S⌫ 1 - e-⌧2
Approximation can be used for
the photosphere where Iν = Sν = Bν(T) i.e., giving temperature
Radiative transfer equation in τ
•
But, what falls out of that general
solution is more specific solutions:
I⌫ (µ = 1, 0) = S⌫
Z ⌧2
0
•
e.g., if Sν is constant and plasma
is optically thin, then the back of
the plasma is at τ2 << 1
series representation of exponential function
for small τ:
1-e
-⌧
=1-
z }| {
1
X
(-⌧)k
0
k!
⌧2 ⌧3
=1-1+⌧+
+ ...
2
6
I⌫ (µ = 1, 0) = S⌫ ⌧2

- ⌧⌫
1
e ✓d
µ
⌧⌫
1
µ
7
◆
◆
⌧2
= S⌫ -e-⌧⌫
0

= S⌫ -e-⌧2 - (-1)
I⌫ (µ = 1, 0) = S⌫ 1 - e-⌧2
This approximation is useful in the
corona and transition region
where total emission can be used
to quantify emitting material
Optically thin line emission
•
So what does it mean to
see an optically thin
emission line?
•
Why does it form?
•
What does its brightness
tell us about the plasma
there?
•
Conditions: ne, T
•
Dynamics: vflow
Example spectrum of the Sun in the extreme UV
(100 < λ < 912 Å), showing spectral lines from
different Fe ions
Optically thin line emission
•
•
What does its brightness
tell us about the plasma
there?
•
Conditions: ne, T
•
Dynamics: vflow
How do we turn the
pretty pictures into
quantitative information
about the plasma?
Sample image of the Sun, made from adding up all
the light in an emission line of Fe X
How does an emission line happen?
•
An ion doesn’t have all its electrons in the lowest-energy
(ground) configuration
•
could be there for a number of reasons: collisional or radiative excitation;
dropped down from a higher level or recombined from free state…
•
One of those excited electrons can spontaneously drop to
the lowest available energy state
•
in doing so, it gives up a photon
•
spontaneous emission
•
also called bound-bound emission because it transitions between two
bound orbits within the atom/ion
A radiating volume of plasma
•
•
How to interpret the
brightness of a spectral
line from transition j→i ?
Energy emitted from
unit volume containing
nj ions in upper state j is
the radiance εji
ions in state j
electrons
ions not in state j
A radiating volume of plasma
•
If measure the brightness of a volume of plasma, ΔV
•
Power radiated in that spectral line is integral of radiance
Pji =
•
Z
h⌫ji Aji nj dV
V
Problem is… we can’t directly measure the
population density, nj !
ions in state j
electrons
ions not in state j
•
However, we can get nj via a bunch of other ratios
that we can calculate / estimate
Z
= h⌫ji Aji
Pji = h⌫ji Aji
nj dV
V
Z
nj n+q nX nH
n
e
+q
nX nH ne
V n
fraction of ions of
charge +q that
have electrons in
excited state j
fraction of element X
with ion charge +q
number density of
electrons (multiple ways to
estimate this)
0.83
relative abundance
of an element X
compared to
hydrogen
Coronal Temperatures
Fractions of Fe ions as a
function of T (plotted
logarithmically)
n+q
nX
Ion notation in these graphs:
Mazzottaetal.(1998)
9 = Fe+9 = Fe X
19 = Fe+19 = Fe XX, etc.
Note: for corona, Fe+8 to Fe+14 are the
ionisation stages we usually
come across
A more exhaustive list of ionisation and Recombination and
Excitation and De-excitation processes
Aschwanden, ‘Physics of the Solar Corona’
A more exhaustive list of ionisation and Recombination and
Excitation and De-excitation processes
Aschwanden, ‘Physics of the Solar Corona’
Coronal equilibrium: de/excitation
Aschwanden, ‘Physics of the Solar Corona’
•
We assume that only collisions with electrons are important for excitations
•
…and that almost all de-excitation happens spontaneously, i.e., by
radiation (and not by collisions)
•
-1
typically trad = Aji << time between electron collisions
Simple level diagram for 2-level atom
Level j
Collisional
excitation
Radiative de-excitation
Rate = ni ne Cij
cm-3 s-1
Rate = nj Aji
cm-3 s-1
Level i
ne ni Cij = Aji nj
*
2-level atom: assumptions
• no radiative excitations
Ni Ne Cij = Nj Aji
j
i
• radiation field is mostly photons of too low an
energy (only several eV) compared to energy
gap between i and j
• no population of j from higher levels
• there is no higher level
• no stimulated emission (lasing)
• no recombination of electrons from
outside into j
Coronal approximation
1. Equilibrium: number of collisional excitations and radiative
de-excitations per unit time and volume is the same
•
volume doesn’t get brighter or dimmer over time
•
no other processes matter
2. Optically thin emission
j
3. Two-level atom
i
•
see previous slide for assumptions
4. Maxwellian distribution of particle energies
-1
5. If trad = Aji << tcoll then almost all ions have the lowest
+q
energy configuration – i.e., nj / n ~ 1
Coronal equilibrium: excitation of ion with 2 levels
Collisional excitation (rate coefficient Cij ): X i+q + e– → X j+q + e–
is balanced by radiative de-excitation (rate Aji):
X j+q → X i+q + hν
ne ni Cij = Aji nj
where Aji is the radiative transition probability (in s-1) from j to i
ne, ni, nj = number densities of electrons and of ions in states i and j (cm-3). (Unlike LTE, there is in general no collisional de-excitation Cji.)
So photon emission rate = collisional excitation rate in a simple 2-level
ion. This is very useful…
Cij
nj
ne
=
<1
Aji
ni
in coronal equilibrium (usually much less)
“Simplifying”…
Z
nj n+q nX nH
Pji = h⌫ji
Aji +q
ne dV
n
nX nH ne
V
Z
1 n+q nX nH
= h⌫ji
Aji nj +q
ne dV
n
nX nH ne
V
AX 0.83
1
Z
+q
nX◆
ni✓n
7 nH◆
7
= h⌫ji ◆ ◆
Cij ne +q
ne dV
nH ◆ne
n
nX
V
◆
1
Fji =
h⌫ji 0.83 AX
2
4⇡d
Z
+q
n
2
Cij
ne dV
nX
V
Dependence of the line brightness on T
collision rate coefficient Cij(T)
Cij =
r
8⇡
me kTe
a20
✓
⌦ij
Eij
exp wi
kTe
◆
Saha equation is also f(T)
q
n+(q+1)
Z+(q+1) 2
3/2 kT
=
(2⇡me kT ) e
+q
+q
3
n
Z
ne h
1
Fji =
h⌫ji 0.83 AX
2
4⇡d
•
Z
+q
n
2
Cij
ne dV
nX
V
G(T )
The temperature-dependent terms can be combined
into a contribution function, G(T)
•
strictly, G(T, ne), but ne can usually be independently estimated
Contribution functions G(T, ne)
Contribution functions for two of those
coronal-temperature EUV resonance lines
Temperature(K)
Some coronal-temperature EUV lines
•
Each line lights up under different temperature conditions
•
there is therefore a mix of plasma at different temperatures in this solar
active region
Courtesy: Hinode/EUV Imaging Spectrometer team
Contributionfunctionsforsomehigh-temperaturecoronal
X-rayresonancelines
Temperature(106K)
58
Emission Measure (EM)
Z
1
n+q 2
Fji =
h⌫
0.83
A
C
n
dV
ji
X
ij
e
4⇡d2
nX
V
Z
2
EM =
ne dV
V
1
Fji =
h⌫
0.83
A
ji
X
2
4⇡d
Z
V
Gji (T ) n2e dV
The brightness of a line is the integral over the emitting
volume of the contribution function multiplied by the
emission measure
Emission Measure (EM)
Z
1
Fji =
h⌫
0.83
A
ji
X
2
4⇡d
Z
1
=
h⌫
0.83
A
a
ji
X
pix
2
4⇡d
V
V
2
Gji (T ) ne
dV
2
Gji (T ) ne
dh
If we know how large a pixel area apix is, we can separate the
volume into an area and a distance along a column dh
The second term in theZ integrand is now the column
emission measure
2
V
ne dh
Differential emission measure φ(T)
•
We know that in the Sun, not all the plasma is going
to be at the same temperature:
•
and we have optically thin emission so our line-of-sight will pass
through these multiple temperatures
Z
1
2
Fji =
h⌫ji 0.83 AX apix
Gji (T ) ne dh
2
4⇡d
V
Z
2 dh
...
Gji (T ) ne
dT
dT
V
Z
φ(T) is known as the
differential emission
...
Gji (T ) (T )dT
V
measure
Differential emission measure φ(T)
•
The shape of this φ(T)
is useful for
understanding the
temperature distribution
of plasma from a set of
observable fluxes {Fji}
•
Its shape can tell us
about heating and
cooling processes
In summary
•
Be careful to find out where the light you’re
measuring really comes from
•
Spectroscopy is incredibly powerful for understanding
the Sun
•
Once you know what the energy output truly is, you
can start thinking about the energy input
•
…and do some quantitative physics! "