A New Way o Measu e Time Delays o Re e be a ion Mapping
William F. Welsh
San Diego S a e Uni e si y
REVERBERATION TESTS WITH REAL DATA
The new LCCF was compu ed and compa ed agains he s anda d CCF on wo Sey e 1
sys ems, NGC 5548 (Fausnaugh, e al. 2016) and IC 4329A (Ben z, e al. 2023). A andom-
walk was used o pa ch da a gaps and c ea e equally-sampled ligh cu es. In gene al, he
same lags a e eco e ed, bu he LCCF p oduces mo e sha ply peaked co ela ion unc ions.
This is pa icula ly ue o he ACF case (uppe igh panels).
DISCUSSION
These and o he es s show ha he esonance-based Cauchy-Lo en z co ela ion
unc ion p o ides highe esolu ion han he s anda d CCF. The highe esolu ion should
ansla e o highe p ecision and a smalle unce ain y in he lag. No e ha he undesi able
“blu ing” in a s anda d CCF comes om he con olu ion inhe en in i s cons uc ion and
canno be educed. Thus he esonance LCCF hold p omise as a quick and easy
al e na i e me hod o measu ing lags and shi s; i is ce ainly no wo se.
Fo ue AGN e e be a ion mapping cases whe e he lag is o en a sizeable ac ion o
he du a ion o he obse a ions, he usual ouble holds. We ind ha while he LCCF
bea s he s anda d CCF, i does no ha e a signi ican ad an age o e he “local” CCF
used on de ended da a o measu e he lag. The ad an age is in he highe p ecision,
no in he accu acy.
An analy ic exp ession can be de i ed o he unce ain y in he lag, and he esul is
analogous o he educed chi-squa e s a is ic. Howe e , he es ima e seems oo small in
some cases. This is cu en ly being in es iga ed.
INTRODUCTION
The ime lag be ween AGN con inuum and b oad emission-line ligh cu es
gi es us an es ima e o he ligh - a el ime ac oss he B oad Line Region
(BLR), and hus i s physical size. The c oss co ela ion unc ion (CCF) is he
s anda d way o es ima e he ime-delay lags. In his wo k, we p esen a new
unc ional o m o measu ing he lag ha o e s se e al ad an ages o e he
ene able CCF. This echnique is based on he concep o a physical
esonance o es ima e he lag - i is no an imp o ed nume ical me hod o
compu e he s anda d CCF no a new way o handle da a gaps. I s main
s eng h is ha i o e s highe esolu ion han he s anda d CCF. I can also
p o ide unce ain ies in he lag, and i is e y simple o compu e. While i is a
gene al echnique ha can be used o measu e he shi be ween any wo ime
se ies, he me hod wo ks well o he e e be a ion mapping case whe e he
AGN con inuum ligh cu es con ain signi ican ed-noise and hus he CCF is
e y b oad. The highe esolu ion yields sha pe 2-d eloci y-delay maps.
A VELOCITY-RESOLVED TEST
Since he CCF and he new LCCF usually yield he same lags, he eal
bene i o he LCCF lies in i s highe esolu ion. This is made clea in a
eloci y- esol ed e e be a ion example shown below. Simula ions in ended
o mimic he p oposed K onos Mission da a (Ho ne, e al. 2004; Pe e son, e
al. 2004) we e used. The 2-d image o ime-lag e sus eloci y is shown in
he uppe panel; he lag a each eloci y in he middle panel; and he
co ela ion coe icien in he bo om panel (scaled o compa ison). While he
LCCF image is in e io o he ac ual ans e unc ion ( ,) and canno
cap u e he spi al a ms in his simula ion, he double-peaked s uc u e ha
indica es a Keple ian acc e ion disk is seen. And he LCCF co ela ion
s eng h clea ly shows a much s onge double-peaked line p o ile han he
s anda d CCF.
A TRANSFER FUNCTION TEST CASE
A simula ed emission-line ligh cu e was gene a ed by con ol ing a andom-
walk con inuum ligh cu e wi h he ans e unc ion show below (in blue).
The LCCF ( ed) has subs an ially highe esolu ion han he adi ional CCF
(black). Fu he mo e, he LCCF emains posi i e jus as does.
We p esen a new me hod o measu ing ime lags ha has ad an ages o e
he s anda d c oss co ela ion unc ion (CCF). This me hod uses he concep
o a esonance and is based on he Cauchy/Lo en z unc ion. I p o ides
highe esolu ion han he ene able CCF and is e y simple o compu e. The
me hod wo ks especially well o AGN 2-d eloci y- esol ed e ebe a ion. In
gene al, his new Lo en z-Cauchy co ela ion unc ion (LCCF) holds p omise
o any case whe e a lag o shi needs o be measu ed (e.g. adial eloci y
Dopple shi s, image egis a ion, e c).
ABSTRACT
CAUCHY-LORENTZ CORRELATION
We p opose a new o m o he co ela ion unc ion, based on he in ui i e
concep o a physical esonance. A spike in he co ela ion s eng h will occu
when he wo da a se s a e mos simila as de ined by a Cauchy / Lo en z
unc ion. Thus we ha e he esonance Cauchy-Lo en z Co ela ion Func ion:
LCCF( ):
To simpli y, we ha e sub ac ed o he means and no malized by he s anda d
de ia ions. The 𝛾is he hal -wid h o he Lo en zian and he peak ampli ude is
1/(𝝿𝛾). The alue o 𝛾is a bi a y, bu a good choice is he a e age noise
alue (i.e. e o ba , o he RMS i unce ain ies a e no known). This gi es a
no maliza ion ha is easy o in e p e : a peak o 1 o pe ec co ela ion, a
alue o 0.5 o a s a is ically good co ela ion, and alue close o ze o o a
e y low co ela ion. No e ha unlike he s anda d CCF, he Lo en z-Cauchy
CF is always > 0.
We hank John Hood, J o his suppo o ou esea ch
