Spectroscopy Applied to Selected Examples

Spectroscopy Applied to Selected Examples

Radial Velocities Exoplanets λ obs λ rest λ rest = Δλ λ rest v c for z << 1

Cosmological z = λ obs λ rest λ rest = Δλ λ rest v c for z << 1 Note :1+ z = λ obs λ rest z = 1 + v c 1 v c 1

Spectral Line Profiles!

Stellar Line Broadening Radiation Damping Energy levels are fuzzy ΔEΔt! for each energy level The Δt factor is set by the lifetime of an electron in the level in question, i.e. for Aij it is the upper (j) level. Let ΔE =! Δt =! i< j A ji a number of lower levels i. be a measure of the fuzziness of level j which is depopulated via to φ ν = γ r 4π 2 ( ν ν ) 2 0 + γ r 4π 2 where γ r = A jl + A il l< j l<i

Pressure (Collisional) Classically, this can be viewed as the effects of: collisions inducing a phase change during the transition statistical perturbations by the ensemble of nearby particles To first order: φ ν = γ c 4π 2 ( ν ν ) 2 0 + γ c 4π 2, where γ c = 2 MFT collisions Lorentz Profile Functionally, this looks similar to the natural radiation damping, and these are sometimes combined to make the Lorentz Profile = natural + pressure φ ν = γ 4π 2 ( ν ν ) 2 + γ 0 4π, where γ = γ 2 r + γ c

Thermal Doppler Broadening For an observer viewing a gas, the radial component of the maxwellian velocity distribution will be 1 m 2 n v = n 2πkT e mv 2 2kT Using ν ν 0 ν 0 v c φ ν = m 2πkT 1 2 e mc2 ( ν ν 0 ) 2ν2 kt 0 d v dν Define Δν D = ν 0 c 2kT m as the Doppler half-width at 1/e of the maximum. Then we get: φ ν = 1 π 1 e Δν D Δν Δν D 1 2

Voigt Function (natural + pressure + Doppler)

Macroturbulence solar stellar surface granulation Microturbulence stellar surface

Rotational Broadening A rotating star seen equator on will have strips of equal v sin i v sin i This will cause the broadening of the spectral lines

What do you do if the line has: rotational broadening and thermal broadening and macroturbulence and... The various functions must ALL be convolved! Convolution: Given two functions f ( σ ) and g( σ ) Their convolution is K σ shorthand : + ( ) g( σ σ 1 )dσ 1 ( ) = f σ 1 K ( σ ) = f ( σ ) g( σ ) What to do??

transform Fourier Transform to the Rescue! ( ) = F( x) f σ inverse transform + F x e 2πixσ dx ( ) = f ( σ ) + e 2πixσ dσ

some functions their transforms so what?

Convolutions in normal space become simple products in Fourier space!

Fourier Transform in Spectroscopy FT of rotationally-broadened line profile v sin i see David F. Gray s The Observation and Analysis of Stellar Photospheres

rotation profile with v sin i = 2 km/s rotation profile with v sin i = 16 km/s thermally-broadened line profile thermally-broadened line profile convolved with 16 km/s rotation thermally-broadened line profile convolved with 2 km/s rotation Note: a newer powerful technique of wavelets is also in use today...

Equivalent Width & Curve of Growth flat square root linear!!

Interstellar Lines & Elemental Depletions Depletion δ ( x) i N HI all ions N ( x i ) ( ) ( ) + N H 2 ( ) ( ) N x N H!

z=2.8 Lyman α Forest z=2.1 Reminder :1+ z = λ obs λ rest z=2.8 z=1.8 z=2.4 z=3.374 Quasar emission Lyman α at z=3.374 intergalactic cloud at z=3.19 from Bechtold 1994, ApJS, 91, 1

! Molecules Electronic Levels Vibrational Levels Rotational Levels (not shown)

Rotational Transitions Rigid Rotors E rot hc cm 1 = B J ( J + 1) wavenumbers where J=rotational quantum # B=rotational constant for that electronic/vibrational state (In real cases, also need to correct for stretching) For rotational transitions, need a permanent dipole (H2 has none) and the transition rule is: ΔJ = ±1 Traditionally, the upper level is labeled J and the lower level J.

This leads to an emitted spectrum with wavenumbers!ν J ',J '' = BJ '( J '+ 1) BJ ''( J ''+ 1) for a constant value of B for both levels (not always the case). But if J ' = J ''+ 1 for example, we get!ν J ',J '' = B( J ''+ 1) ( J ''+ 2) BJ ''( J ''+ 1) = 2B( J ''+ 1) leading to a series of lines separated by 2B: 2B J' ' 0 1 2 3 ν 2B 4B 6B 8B ν Example: CO B=1.98 cm -1 J ' J ''!ν ( cm 1 ) λ( cm) 1 0 3.96 0.252 2 1 7.92 0.126 3 2 11.88 0.0843

Vibrational E vib hc = ω ( v+ 1 ) 2 harmonic oscillator = ω ( e v+ 1 ) 2 ω e X ( e v+ 1 ) 2 2 + ω e Y ( e v+ 1 ) 3 2 + anharmonic oscillator Δ v = ±1, ± 2, ± 3, ± 4, etc. 3 2 1 0 Example: CO ω e = 2170 cm 1 ω e X e = 13.5 cm 1 ω e Y e = 0.031 cm 1

Combined (rovibrational) E vib + E rot hc = ω ( e v+ 1 ) 2 ω e X ( e v+ 1 ) 2 2 + ω e Y ( e v+ 1 ) 3 2 + B v J ( J + 1) D v J 2 ( J + 1) 2 ( ) ( ) where B v = B e α e v+ 1 2 D v = D e β e v+ 1 2 etc. R branch P branch J -J =+1 J -J =-1 Designation: R(J ) P(J ) J = J = R!ν R = ν 0 + 2B v ' + ( 3B v' B v'' ) J ''+ ( B v' B v'' ) J '' 2 P J '' = 0,1,2,3.!ν P = ν 0 ( B v' + B v'' ) J ''+ ( B v' B v'' ) J '' 2 J '' = 1,2, 3, 4. 3 2 1 0 3 2 1 0 v =1 v =0!!!!2B!!!! R(2)!!R(1)!!R(0)!!ν 0!!!P(1)!!P(2)!!P(3)!

!ν = electronic term Electronic (+ vibrational + rotational) [ ] + [ vibrational term] + [ rotational term] Selection Rules" ( ) electronic only to same spins usually vibrational Δ v = 0,±1,±2,±3, etc. rotational ΔJ = 0, ±1 except 0 0 Note! Not possible for ΔJ = 0 Q branch

Example: The Red Rectangle (HD 44179)

Spectroscopy & Spectrophotometry with SpeX Flats - to get the flat-fielding for each image Arcs - to do the wavelength calibration Science target spectrum - what you want to know Calibration Star - correct for telluric absorption - to provide a flux calibration - usually an A0V star close in airmass and sky position (flexure)

For more detail: http://homepages.uc.edu/~sitkoml/ Mikes_SpeX_Manual_v2.pdf (Warning: 34 MB!!)

And, of course, there is spectropolarimetry... From Harrington & Kuhn 2009, ApJ, 695, 238