A New Way to Measure Time Delays for Reverberation Mapping

A New Way to Measure Time Delays for Reverberation Mapping William F. Welsh San Diego State University REVERBERATION TESTS WITH REAL DATA The new LCCF was computed and compared against the standard CCF on two Seyfert 1 systems, NGC 5548 (Fausnaugh, et al. 2016) and IC 4329A (Bentz, et al. 2023). A randomwalk was used to patch data gaps and create equally-sampled light curves. In general, the same lags are recovered, but the LCCF produces more sharply peaked correlation functions. This is particularly true for the ACF case (upper right panels). DISCUSSION These and other tests show that the resonance-based Cauchy-Lorentz correlation function provides higher resolution than the standard CCF. The higher resolution should translate to higher precision and a smaller uncertainty in the lag. Note that the undesirable “blurring” in a standard CCF comes from the convolution inherent in its construction and cannot be reduced. Thus the resonance LCCF hold promise as a quick and easy alternative method for measuring lags and shifts; it is certainly no worse. For true AGN reverberation mapping cases where the lag is often a sizeable fraction of the duration of the observations, the usual trouble holds. We find that while the LCCF beats the standard CCF, it does not have a significant advantage over the “local” CCF used on detrended data to measure the lag. The advantage is in the higher precision, not in the accuracy. An analytic expression can be derived for the uncertainty in the lag, and the result is analogous to the reduced chi-square statistic. However, the estimate seems too small in some cases. This is currently being investigated. INTRODUCTION The time lag between AGN continuum and broad emission-line light curves gives us an estimate of the light-travel time across the Broad Line Region (BLR), and thus its physical size. The cross correlation function (CCF) is the standard way to estimate the time-delay lags. In this work, we present a new functional form for measuring the lag that offers several advantages over the venerable CCF. This technique is based on the concept of a physical resonance to estimate the lag - it is not an improved numerical method to compute the standard CCF nor a new way to handle data gaps. Its main strength is that it offers higher resolution than the standard CCF. It can also provide uncertainties in the lag, and it is very simple to compute. While it is a general technique that can be used to measure the shift between any two time series, the method works well for the reverberation mapping case where the AGN continuum light curves contain significant red-noise and thus the CCF is very broad. The higher resolution yields sharper 2-d velocity-delay maps. A VELOCITY-RESOLVED TEST Since the CCF and the new LCCF usually yield the same lags, the real benefit of the LCCF lies in its higher resolution. This is made clear in a velocity-resolved reverberation example shown below. Simulations intended to mimic the proposed Kronos Mission data (Horne, et al. 2004; Peterson, et al. 2004) were used. The 2-d image of time-lag versus velocity is shown in the upper panel; the lag at each velocity in the middle panel; and the correlation coefficient in the bottom panel (scaled for comparison). While the LCCF image is inferior to the actual transfer function (v,) and cannot capture the spiral arms in this simulation, the double-peaked structure that indicates a Keplerian accretion disk is seen. And the LCCF correlation strength clearly shows a much stronger double-peaked line profile than the standard CCF. A TRANSFER FUNCTION TEST CASE A simulated emission-line light curve was generated by convolving a randomwalk continuum light curve with the transfer function show below (in blue). The LCCF (red) has substantially higher resolution than the traditional CCF (black). Furthermore, the LCCF remains positive just as does. We present a new method for measuring time lags that has advantages over the standard cross correlation function (CCF). This method uses the concept of a resonance and is based on the Cauchy/Lorentz function. It provides higher resolution than the venerable CCF and is very simple to compute. The method works especially well for AGN 2-d velocity-resolved reveberation. In general, this new Lorentz-Cauchy correlation function (LCCF) holds promise for any case where a lag or shift needs to be measured (e.g. radial velocity Doppler shifts, image registration, etc). ABSTRACT CAUCHY-LORENTZ CORRELATION We propose a new form for the correlation function, based on the intuitive concept of a physical resonance. A spike in the correlation strength will occur when the two data sets are most similar as defined by a Cauchy / Lorentz function. Thus we have the resonance Cauchy-Lorentz Correlation Function: LCCF(t): To simplify, we have subtracted off the means and normalized by the standard deviations. The 𝛾is the half-width of the Lorentzian and the peak amplitude is 1/(𝝿𝛾). The value for 𝛾is arbitrary, but a good choice is the average noise value (i.e. error bar, or the RMS if uncertainties are not known). This gives a normalization that is easy to interpret: a peak of 1 for perfect correlation, a value of 0.5 for a statistically good correlation, and value close to zero for a very low correlation. Note that unlike the standard CCF, the Lorentz-Cauchy CF is always > 0. We thank John Hood, Jr for his support of our research