Kagoshima Univ./ Ehime Univ.
Galactic radio astronomy
Lesson5
Einstein coefficient & HI line
Toshihiro Handa
Dept. of Phys. & Astron., Kagoshima University
Mellinger
Energy level
▶ Schrödinger equation of an electron
▶ Steady state = energy eigen value
■
Parameter separation
▶ Schrödinger equation for steady state
Mellinger
Energy eigenvalue & bound states
▶ Solution depends on the boundary condition
■
■
Continuous E is possible if E>0.
Discreet values if E<0 (bound state)
▶ Continuous solution in quantum mechanics!
▶ In many cases, we consider bound states.
■
Discreet eigenvalues and eigenfunctions
Mellinger
Electron in an atom/molecule
▶ Electron in an atom/molecule
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Bound state → discreet energy levels
▶ Energy state
Mellinger
Energy transition and matter
▶ Electron transition between energy levels
■
■
Emission & absorp of EM wave
DE=hn
▶ Structure of energy levels=matter identify
■
■
Wavelength of emission & absorption lines
Matter identification with spectrum
Mellinger
Einstein coefficient(1)
▶ Emission & absorption: 2 level model
■
Transition probability of emission A
 Independent
■
of input intensity
Transition probability of absorption B
 Proportional
to input intensity
dI = n2 A-n1 B I
Mellinger
Einstein coefficiant(2)
▶ Steady state dI=0
dI = n2 A-n1 B I  n2 A = n1 B I  I
𝑛2 𝐴
=
𝑛1 𝐵
▶ Thermal equilibrium b/w matter & radiation
■
Therm. eq.→energy is in Boltzmann distribution
𝑛2
𝑛1
■
= 𝑒 −Δ𝐸/𝑘𝑇 =𝑒 −ℎ𝜈/𝑘𝑇
Therm. eq.→I must be blackbody radiation
I = Bn (T) =
■
2ℎ𝜈 3
1
𝑐 2 𝑒𝑥𝑝ℎ𝜈/𝑘𝑇 −1
Impossible to become a blackbody!?
Mellinger
Stimulated emission(1)
▶ Add another process
■
Difficult to see the existence
▶ Introduction of “stimulated” emission
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spontaneous emission A21, absorption prob. B12
stimulated emission B21
 Emission
proportional to input intensity!
dI = n2 A21-n1 B12 I+n2 B21 I
= n2 A21-(n1B12-n2B21) I
 Seems
to reduce the effective absorption coeff.
Mellinger
Stimulated emission(2)
▶ Steady state dI=0
n2 A21=(n1B12-n2B21) I  I = 𝑛1
𝐴21
𝐵 −𝐵21
𝑛2 12
■
Therm. eq.→energy is in Boltzmann distribution
𝑛2
𝑛1
■
■
=
𝑒 −Δ𝐸/𝑘𝑇 =𝑒 −ℎ𝜈/𝑘𝑇 ,
(RH)=
𝐴21
1
𝐵21 𝐵12 𝑒𝑥𝑝ℎ𝜈/𝑘𝑇 −1
𝐵21
Therm. eq.→I must be I = Bn (T) =
We got that
𝐵12
𝐵21
=1,
𝐴21
𝐵21
=
2ℎ𝜈 3
𝑐2
Mellinger
2ℎ𝜈 3
1
𝑐 2 𝑒𝑥𝑝ℎ𝜈/𝑘𝑇 −1
Relation between Einstein coff.
𝐵12
𝐵21
=1,
𝐴21
𝐵21
=
2ℎ𝜈 3
𝑐2
▶ In the case with statistical weight g1, g2
■
Boltzmann dist.
𝑛2
𝑛1
=
𝑔2
𝑔1
𝑒 −ℎ𝜈/𝑘𝑇
g1B12=g2B21, A21=
2ℎ𝜈 3
𝑐2
B21
▶ A21, B12, B21 are fixed for matter.
■
■
Relation is valid if thermal non-equilibrium
A21, B12, B21 : Einstein coefficients
Mellinger
Maser(1)
▶ If stimulated emission truly exist, then
dI = n2 A21-(n1B12-n2B21) I
 Effective
absorption increases. We cannot know it?
▶ What happens, if n2> (B12/B21) n1?
■
𝑛2
𝑛1
■
Negative temperature i.e. inverse population
=
𝑔2
𝑔1
𝑒 −Δ𝐸/𝑘𝑇 . Therefore, it means T<0.
▶ Stimulated emission can arise!
Mellinger
Maser(2)
▶ We can make T<0 for 3 level system
pomping
inverse population
seed photon
maser
Mellinger
Maser(3)
First detected molecule with 3 atoms in space
▶ MASER
Microwave Amplification by Stimulated Emission of Radiation
■
Developed by Towns in 1954
 He
found ammonia in space, too.
▶ LASER
■
Microwave →Light
▶ Characteristics of stimulated photon
■
Same freq., phase, polarization as the seed photon
Mellinger
Excitation temperature
▶ In general, emission is
■
I=(2hn 3/c2) [(n1/n2) -1]
▶ Thermal non-equilib.: n1/n2 is not Boltzmann
■
But convenient expressed by “temperature”
▶ Excitation temperature Tex
■
■
Define as
Tex= -
𝑔2
𝑔1
𝑒 −Δ𝐸/𝑘𝑇𝑒𝑥
Δ𝐸
𝑘 𝑙𝑛
𝑔 1 𝑛2
𝑔 2 𝑛1
=
𝑛2
𝑛1
Mellinger
Emissivity
▶ Describe emissivity e by Einstein coeff.
■
■
For isotropic radiation…
Radiation energy dEn for dV dt dW
dEn =hn j(n) n2 A21 dV dt
=
■
ℎ𝜈
4𝜋
j(n) n2 A21 dS dx dt dW
In the case of radiation only
dIn =
■
It gives en =
ℎ𝜈
4𝜋
𝑑Ω
4𝜋
𝑑𝐸𝜈
𝑑𝑡 𝑑𝑆 𝑑Ω
j(n) n2 A21
Mellinger
= en dx
Absorption coefficient(1)
▶ abs. coeff. k described with Einstein coeff.
■
■
For isotropic absorption…
Radiation energy dEn into dV dt dW
dEn = -hn j(n) (n1 B12 -n2 B21) In dV dt
=■
ℎ𝜈
4𝜋
j(n) (n1 B12 -n2 B21) In dS dx dt dW
In the case that absorption only
dIn =-
■
𝑑Ω
4𝜋
It gives kn =
ℎ𝜈
4𝜋
𝑑𝐸𝜈
𝑑𝑡 𝑑𝑆 𝑑Ω
=-kn In dx
j(n) (n1 B12-n2 B21)
Mellinger
Absorption coefficient(2)
■
(continued)
ℎ𝜈
kn = 4𝜋 j(n) (n1 B12-n2 B21)
ℎ𝜈
𝑔1 𝑛2
= j(n) n1 B12 1−
4𝜋
𝑔2 𝑛1
=
=
=
ℎ𝜈
4𝜋
𝑐2
8𝜋𝜈 2
𝑐2
8𝜋𝜈 2
j(n)
𝐵12
n1 B21
𝐵21
j(n)
𝑔2
n1A21
𝑔1
j(n)
𝑔2
n1A21
𝑔1
−
𝑛2
𝑛1
𝑔1 𝑛2
1−
𝑔2 𝑛1
[1-𝑒 −ℎ𝜈/𝑘𝑇𝑒𝑥 ]
Mellinger
Source function
▶ Source function Sn=
Sn =
𝑛2 𝐴21
𝑛1 𝐵12 −𝑛2 𝐵21
=
𝜀𝜈
𝜅𝜈
2ℎ𝜈 3
1
𝑔2 𝑛1
𝑐2
−1
𝑔1 𝑛2
=
2ℎ𝜈 3
1
𝑐 2 𝑒 ℎ𝜈/𝑘𝑇𝑒𝑥 −1
▶ In therm. eq. Tex=T (temperature in eq.)
▶ LTE: Local Thermal Equilibrium
■
Tex’s are the same between all levels.
Mellinger
Neutral atomic hydrogen
▶ proton + electron
■
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Proton is a particle with spin 1/2  2 values
electron is a particle with spin 1/2  2 values
 spin
of a particle=should be related with mangetizm
 Interaction between two spins
A10=2.86888×10-15 [s-1], n =1.420405751786[GHz]
Mellinger
HI emission(1)
▶ A10=2.86888×10-15 [s-1]
 Enough
slow transition to excite under ISM density
 Show maser if poping  hydrogen maser clock
▶ Absorption coefficient
kn =
■
𝑐2
8𝜋𝜈 2
j(n)
𝑔1
n0A10
𝑔0
g0=1←no degenerate,
1− 𝑒 −ℎ𝜈/𝑘𝑇𝑒𝑥
g1=3←F=+1,0,-1 degen.
𝑔1 −ℎ𝜈/𝑘𝑇
𝑠
𝑒
𝑔0
■
nH=n0+n1= n0 1+
■
For HI, Tex=Ts (spin temperature)
Mellinger
HI emission(2)
▶ Approximation as

■
■
ℎ𝜈
𝑘
ℎ𝜈
≪1
𝑘𝑇𝑠
=0.07 [K]≪Ts~100 [K]
First order approximation
 1-𝑒 −ℎ𝜈/𝑘𝑇𝑠
≅
 nH= n0
𝑔1
𝑔0
kn =
𝑐2
1+
3
8𝜋𝜈 2 4
ℎ𝜈
𝑘𝑇𝑠
𝑒 −ℎ𝜈/𝑘𝑇𝑠 =4n0
ℎ𝜈
nH A10
𝑘𝑇𝑠
Mellinger
j(n)
𝑛𝐻
-15
=2.6×10
𝑇𝑠
j(n) [cgs]
HI emission(3)
▶ The column density can get through
■
NH=∫nH dx =
1
2.6×10−15
∫ Tstn dn [cgs]
▶ In optically thin case, TB= Ts(1-e-t)= Tstn
■
NH=
1
2.6×10−15
∫ TB dn [cgs]
𝜈
𝑐
▶ Use Doppler velocity dn = dv,
NH[cm-2]=1.8224×1018 ∫ TB dv [K km s-1]
▶ Caution: this equation is valid only for
■
Optically thin
Mellinger
What is the natural width j(n)?
▶ Duration in quantum transition Dt
■
■
Not 0 ∵emitted EM wave is not d(t) time profile
Not ∞ ∵finish the transition in finite time span
▶ Gradually increase and gradually decrease
▶ Give a width after Fourier transformation
wave
wave packet
particle
Mellinger
report
▶ Attach to your e-mail. Deadline : 13 Nov.
■
Submit to [email protected]
▶ Questions
1. Show relations between Einstein coeff.
2. How long the mean time to transit a neutral
hydrogen atom?
Mellinger