Transcript Cross-correlation - University of St Andrews

Noisy data Di ± i
Cross-correlation
• Cross-correlation function (CCF)
is defined byscaling a basis
function P(Xi ), shifted by an
offset X, to fit a set of data Di with
associated errors i measured at
positions Xi :
CCF(X) 
2
P(X

X)D
/

 i
i
i
Shifted basis functions P(X)
P(X) scaled to data at various
shifts:
i
2
2
P
(X

X
)
/


i
i
i
CCF(X):
1
Var(CCF(X)) 
2
2
P
(X

X)
/


i
i
i
• The width of this profile is well
matched to the data.

min ~ N
1 error bar:  =  - min = 1
Noisy data Di ± i
Mismatched profiles–1
• If basis function has form:
X 2 
P(X)  exp 2 
2 
Profile too narrow
• then when width  of profile is
well matched to the data,
correlation length ~ FWHM of 
minimum ~ .
• If profile is too narrow:
• CCF has
– larger error bars
– shorter correlation length
• The minimum in

is shallow.
Note local minimum
causedby nonlinear model
P(X) scaled to data at various
shifts:
CCF(X):

min > N
Wider 1 error bar:  =  - min = 1
Noisy data Di ± i
Mismatched profiles–2
• If basis function has form:
X 2 
P(X)  exp 2 
2 
Profile too wide
• then when width  of profile is
well matched to the data,
correlation length ~ FWHM of 
minimum ~ .
• If profile is too wide:
• CCF has
– smaller error bars
– longer correlation length
– lower peak value
• The minimum in  is wider, but
not as deep.
P(X) scaled to data at various
shifts:
CCF(X):

min > N
Wider 1 error bar:  =  - min = 1
Radial velocities from cross-correlation
• Data: spectrum of black-hole binary
candidateGRO J0422+32
• Basis function: “template” spectrum of normal
K5V star of known radial velocity.
• Cut out H alpha emission line, fit 5 splines to
continuum with ±2 clipping to reject lines.
Wavelength and velocity shifts
• Target spectrum Di is measured at wavelengths
i and has associated errors i .
• Template spectrum P is measured on same (or
very similar) wavelength grid. Errors assumed
negligible.
i v
• For small velocity shift v: i 
c
Interpolate
CCF(v) 
 P(
i
D
 i )Di / 
2
i
i
2
P(



)
/

 i
i 
i
i
2
P
i
Note that since D is redshifted relative
to P in this example, CCF (v) will
produce a peak at positive v.
i
Practical considerations
• P(X) and D(X) are usually on slightly different
wavelength scales so we can’t just shift the spectra
bin-by-bin.
• We want to measure the CCF as a function of relative
velocity, not wavelength.
• Velocity shift per bin varies with wavelength:

v  c
 c ln 
0
• For each velocity shift in CCF:
– Loop over selected range of pixels in target spectrum D(X).
– Use pixel wavelength and CCF velocity shift to compute
corresponding wavelength in template spectrum P(X)
– Use linear interpolation to get flux and variance in template spectrum
at this wavelength.
– Increment summations and proceed to next pixel in target spectrum.
Radial velocity
of GRO J0422+32
• Subtract continuum fit.
• Cross-correlate on 6000 A to
6480A subset of data.
• Compute CCF for shifts in
range ± 1800 km s–1.
• CCF shows peak between 500
and 600 km s–1.
• Use =1 to get 1 error bar
on radial velocity.