Velocity Coordinate Spectrum
from Doppler-broadened
emission and absorption lines
D.Pogosyan (U. Alberta, Canada)
with: Alex Lazarian (UW Madison, USA)
ApJ 2000, 2004, 2006, 2008
and contributions from A. Esquivel (UNAM), A. Chepurnov (UW), J. Cho
Observing turbulence statistics
Y
y
x
Fluctuating density
Turbulent velocity
v
z
X
?
Intensity I(X,Y,v)
Basics of turbulence statistics

Fluctuating density
r(x)=<r>+dr (x)
d(r) ~ r –g, g < 0

shallow
steep
Turbulent velocity u(x) – considered Gaussian isotropic
m=2/3 – Kolmogorov
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
Minimal goal – determine m and g
On the back of our minds: anisotropic underlying spectra,
separate determination of solenoidal and potential velocity parts,
different thermal phases of the gas, especially temperature
fluctuations
Density in PPV cube
PPV density correlation function
consists of two contributions,
(one purely due to motions,
the other – due to density dr )
It is highly anisotropic,
however, defining linear scale as
so that 3D PPV positions is
leads to “isotropic” representation . Can we use it to determine velocity scaling m ?
where pure velocity term is obtained by taken g = 0
Velocity Channel Analysis Summary
(Lazarian & Pogosyan, 2000,2004)
Thin slice:
g > 0: 2D intensity spectrum is
dominated by density
inhomogeneities
D(R)~R1-g-m/2 : P(K)~K-3+g+m/2
g < 0: 2D intensity spectrum is defined
by random motions
D(R)~R1-m/2
DV
:
P(K)~K-3+m/2
Thick slice:
Velocity effects disappear,
2D intensity spectrum reveals
underlying density statistics xr
D(R)~R1-g : P(K)~K-3+g
Velocity Coordinate Statistics (VCS)
(Lazarian & Pogosyan, 2006,2008)
VCS –
looking
directly along
velocity
coordinate
emission
I(n)
Low self-absorption,
linear density response-
I(v)=e r s (v)
VCS behaviour
 Slope break at resolution scale directly reflects velocity statistics – slop change=4/m.
 Position of the slope break gives turbulent velocity amplitude at resolution scale:
 If both shallow-steep and steep-shallow changes are present, density spectral index g
can be found.
Effect of (self)absorption
<
VCS at large scales is corrupted by absorption
VCS for absorption lines
VCS does not require continuous map of an extended region !
I(n)
VCS for staturated absorption lines




Is it possible to extract the
information about turbulence
from saturated absorption lines ?
This information is in the wings
of a line !
Extend of the saturated regions is
determined by the central optical
depth and the noise level.
Width of the wing DW is set by t,
turbulent and thermal broadening,
and intrinsic width.
DW
D’Odorico et al. 02
Wings of saturated lines
u(x)-turbulent velocity
H(v-u) – shifted Voigt profile
Jenkins 07
Prospect: using multiple lines from the same point source
VCS from the wings

Challenges:



It is difficult to determine P(k) directly due to power aliasing
due to narrow width of the wings.
Even structure functions are not defined for 1D spectra steeper
than kv-3 which includes Kolmogorov case.
Possible solution:
Higher order structure function
This is still quadratic statistics, not be confused with higher moment, e.g bispectrum
Concluding Remarks



Velocity Coordinate Spectrum contains wealth of information about the turbulent
medium
VCS as a technique for the analysis of PPV space introduces new range of
datasets. Both emission and absorption lines can be used.
VCS has particular advantages over other techniques of looking at PPV space:




VCS does not require mapping of extended regions with sufficient spatial resolution.
Absorption lines from unresolved point sources can be used.
WithinVCS, thermal effects can be controlled, for example by using different spectral
lines from the same source.
Asymptotical scaling regimes provide a guidance where to look for the most
revealing turbulence signature, but the detailed comparison of the models with
data will have to account for a range of observational and theoretical limitations,
including, e.g,, finite dynamical range, noise and error properties of the data,
sampling, aliasing and bias of statistical estimators, etc. To achieve the status of
‘precision’ studies of magnetized turbulence in ISM and IGM, sophisticated
interface between the models and the data have to be created (see Chepurnov &
Lazarian, 2007, for first steps).
VCA Applications