1. Cross-correlation Cross-correlation function (CCF) is defined byscaling a basis function P(X i ), shifted by an offset X, to fit a set of data D i with associated errors  i measured at positions X i : The width of this profile is well matched to the data. Noisy data D i ±  i Shifted basis functions P(X) P(X) scaled to data at various shifts: CCF(X):    min ~ N 1  error bar:   =   -   min = 1

2. Mismatched profiles–1 If basis function has form: 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. Noisy data D i ±  i Profile too narrow P(X) scaled to data at various shifts: CCF(X):    min > N Wider 1  error bar:   =   -   min = 1 Note local minimum causedby nonlinear model

3. Mismatched profiles–2 If basis function has form: 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. Noisy data D i ±  i Profile too wide P(X) scaled to data at various shifts: CCF(X):    min > N Wider 1  error bar:   =   -   min = 1

4. 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.

5. Wavelength and velocity shifts Target spectrum D i 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. For small velocity shift  v: ii i D P Note that since D is redshifted relative to P in this example, CCF (  v) will produce a peak at positive  v. Interpolate

6. 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: 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.

7. 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.