Turning Quasar Microlensing From A
Curiosity Into a Tool
•C.S. Kochanek, X. Dai, N. Morgan, C. Morgan, S. Poindexter (OSU)
•G. Chartas (PSU)
•Introduction to gravitational lenses
•The nature of the data
• Results
•Physical, computational, statistical and
observational challenges
The Gravitational Lens RXJ1131-1231
Quasar image B
Quasar image A
Quasar image C
Quasar image D
“Einstein Ring”
image of quasar
host galaxy
lens galaxy
Monitor Quasar Image Brightness
Two sources of time variability
•Intrinsic quasar variations, which appear with a “time” delay between each image
•Uncorrelated variations due to “microlensing” by the stars in the lens galaxy
(For non-astronomers, magnitudes are –2.5log(flux)+constant)
First Determine Time Delays
•For RXJ1131-1231 they are +12, +10 and –87 days for images B, C
and D relative to A (B leads, D trails)
•The delays can be used to study the mass distribution of the lens or
to estimate the Hubble constant, but this is not our present topic
What’s left is the “microlensing”: Variations in the flux ratios
of the images after correcting for the time delays
What Determines an Image’s Flux?
•The local magnification is determined by the local derivatives of the
potential, 
sin 2
 , xx , xy  1    cos2





M 1 


sin 2
1    cos2 
 , xy , yy  
   /  c  convergenc e  surface density
1
  (tidal) shear
total magnificat ion  det( M )  ( 1     ) 1( 1     ) 1
•What contributes to these derivatives?
•Overall smooth potential – the “macro” model
•Satellites/CDM substructure – millilensing
•Stars – microlensing
•Finite source sizes  smoothing of the small scale structures in the
magnifications, in particular, smoothing of the caustics on which the
magnification diverges
Length Scales
Set by the mass of the lenses and the distances
 4GM DLS 
Einstein radius  E   2

c
D
D
OL
OS 

1/ 2
1/ 2
M 
 8
 as
M 
1/ 2
M 
 10 h 
 cm on the source plane
M 
17
1
•Lens galaxy
M~1010M• E~1arcsec
•Satellite galaxy M~106M• E~10 milliarcsec
•Star
M~M•
E~10 microarcsec
The stars produce complex magnification patterns with different
structures near each image
Source plane scale=40<E>
2h-1<M/M•>1/2pc=335<M/M•>1/2as
B
C
D
A
For a 109M• black hole
RBH=0.0001pc
=0.01as
=0.l<M/M•>-1/2pixels
What can we study using microlensing?
•Quasar Structure – microlensing “resolves” quasar accretion disks,
allowing us to measure their structure as a function of wavelength
•Dark matter – microlensing depends on the fraction of the mass
near the lensed images comprised of stars
•Stellar populations – microlensing can estimate the mean stellar
mass the halos of cosmologically distant galaxies
Quasar Accretion Disks Have A Very Similar Size Scale
Gravitatio nal Radius
 M

rg  1.5  1014  9 BH  cm
 10 M  
Thin disk size


  
R  9.7  1015 


m



Should be able to study structure
of quasar accretion disks because
variability amplitude  source
size

4/3
 M BH 
 9

10
M
 

2/3
1/ 3
 M 


 
M
 Edd 
cm
Minima
Saddle Points
Depends on the
fraction of the surface
High
density in stars
optical
depth
Low
optical
depth
Or in the statistics
(Schechter & Wambsganss
2002)
Mean Stellar Mass
All “observable” properties of the microlensing are in “Einstein
units” of <M/M>1/2cm. Converting to just cm requires a prior on
•The mean microlens mass <M>
•The true physical velocities
•The true source size
We have good physical priors for the physical velocities, which
means we can sensibly estimate <M> in cosmological distant
galaxies
But how do you go from light curves to physics?
One of the two basic problems in using quasar
microlensing for astrophysics.
The other is the sociological difficulty of doing the
observations.
OGLE (Wozniak et al.
2000ab) microlensing light
curves of Q2237+0305
For Galactic microlensing events you just fit the light curves.
Binary microlensing event
MACHO 98-SMC-1
In this case solutions must
include binary orbital motion…..
Afonso et al. 2000
We will just do the same, using computer power and
the Reverend Bayes
•Given the local properties (, *, ) and the stellar mass function
•Generate random realizations of the magnification patterns
•Given a model for the quasar accretion disk
•Randomly choose disk parameters, convolve with patterns
•Given a random selection of a source velocity
•Randomly pick nuisance parameters (direction, starting
point)
•Fit the resulting light curve to data to estimate a 2 statistic
•Combine all the trials using Bayesian methods to estimate
probability distributions for the values of interesting parameters
We Obtain Statistically Acceptable Fits To The Data
Good fits mean fitting all 6 difference light curves between the 4 images
(the average light curve gives the intrinsic source variability)
Are Galaxies Composed of Stars?
RXJ1131-1231
Q2237+0305
Most lenses, like RXJ1131-1231, should only have a small fraction of the
surface density near the quasar images comprised of stars (/*~0.1 to 0.2),
but one lens, Q2237+0305, where we see the images through the bulge of a
low redshift spiral galaxy, should be almost all stars (/*~1)
The Microlensing Knows…..
(although for most lenses it has yet to converge
significantly)
RXJ1131-1231 should be
mostly dark matter
Q2237+030 should be
mostly stars
What is Mean Mass of the Microlenses?
All directly measured quantities are in “Einstein units” <M>1/2cm
Best physical priors are for the true velocities P(ve)
Roughly, M  v e2 /v̂ e , formally need convolutio n
2

P M    dv e P( v e ) P v̂e  v e M / M *
For any one lens, the uncertainties
will be large:
•Mass goes as (velocity)2
•Physical prior for any one less has
a velocity uncertainty of order a
factor of two from the unknown
peculiar velocity
Best single case Q2237+0305
<M> = 0.61 M
0.12 M  <M> 2.85 M
1 / 2

Q2237+030
Ensembles of Lens Should Provide An Accurate Estimate
•Dominant uncertainty in priors is a random variable (peculiar velocities) whose
dispersion is known, but not the value for a particular system
•Multiply P(<M>) for each system to get a joint estimate for ensemble of lenses
•Must hit a systematic floor at some point, but almost certainly not yet
Combining the 8 systems (mostly)
analyzed as of Saturday…
<M> = 0.09 M
0.04 M  <M>  0.19 M
Answer stable to dropping any one
lens, but it would be nice to see the
outliers shift towards the median as
we accumulate longer light curves.
Disk Scale Lengths Well-Determined
Why? Because the mass scale uncertainties affect the source size little
Measure the effective velocity and source size in Einstein units
 M 

v̂ e  v e 
M
  
1 / 2
 M 

and r̂s  rs 
M
  
1 / 2
with a " degeneracy direction" that r̂s  v̂e
 Means that the physical
source size is little affected by
the uncertainties in the mass,
which is very convenient!
Beginning to Test Accretion Disk Theory
Thin disk size
  
R  9.7  1015 

 m 


4/3
 M BH 
 9

 10 M  
2/3
1/ 3
 M 



 M Edd 
cm
Black hole masses estimated from
emission line-width/mass
correlations
Disk Structure Best Probed As Size Versus Wavelength
First tests, surprisingly, are optical versus X-ray sizes
CXO Spring 2004
Blackburne et al
CXO Spring 2006
OSU/PSU
•Optical and X-ray flux ratios very
different, and change differently with
time (now known for 4 lenses)
•Smaller sources will show greater
microlensing variability
Partial results for 3 systems now, additional
data being collected, but X-ray sources are
roughly 1/10 the size of the optical sources
As expected, size ratios are less uncertain than
absolute sizes (remember, the X-ray size is
being determined from 4 data points!)
Issues of Physics
• Mass function of microlenses – extensive prior theoretical studies showing that
this is very hard to probe and has little effect on results
• Disk structure at fixed wavelength – extensive prior theoretical studies show
that microlensing data measures typical size rather than details of the surface
brightness distribution – better to focus on size versus wavelength, black hole
mass etc..
• Only time variability or also observed flux ratios – observed flux ratios are also
affected by substructure (satellites) and extinction/absorption, but they are
powerful constraints on where you sit in the microlensing magnification pattern
• The stars move. Except in experiments, we have used fixed magnification
patterns, but in real life they change with time as the stars move. Theoretical
studies suggest that we are safe so far, but not forever.
Issues of Computation
• With some physically irrelevant fiddles, it is possible to make the image and source
regions periodic – allows use of FFTs to generate magnification patterns and leads to
periodic magnification patterns that simplify the Bayesian analysis method
• Dynamic range of magnification patterns – need to maintain a large enough outer scale to
get a fair sample of stars, and a small enough inner scale to deal with compact sources –
40962 maps marginally OK
• Computationally challenging to allow stars to move – we need ~3 GByte to analyze a
systems at 2 wavelengths with static patterns, but ~300 GBytes if we allow the stars to
move and need an animated sequence of patterns. Probably doable on shared memory
machines (and we have experimented with this), but could lead to catastrophic time
penalties on other parallel machines because of the need for random access to all the
patterns (awaits a really good computational student).
• Fiddles to speed execution. We have incorporated various fiddles to make the analysis
run FAST. The most serious of which is an inner loop that does a local maximum
likelihood search over nuisance variables that is not strictly in accordance with the
Bayesian outline of the calculations.
• Monte Carlo verification. We need to spend more effort on generating fake data sets and
verifying that the analysis performs as expected. So far, so good, but….
Issues of Statistics
All looks pretty good, except for:
•The interaction of priors, static magnification maps and the reality that the stars
move and we need to animate the maps – current approach does not do this
properly but is designed (hopefully!) to lead to overly broad uncertainty
estimates on the physically interesting quantities
•The inner, hidden, maximum likelihood loops where we allow a local
optimization of the nuisance variables over restricted ranges. Needs more study
to better approximate the true Bayesian integrals
•Stratified/likelihood sampling of physical variables needs to be
understood/implemented to minimize wasted time on low likelihood regions of
parameter space
•Testing with simulated data needs to be massively expanded
Issues of Observation
Obtaining the necessary observations remains our biggest problem
• We monitor ~20 lenses well at at one wavelength (R band), with lesser coverage at J, I and B
• The Babylonian observer problem for ground based optical/near-IR remains a major
problem. For example, all our results are for lenses visible from the queue-scheduled
SMARTS telescope at CTIO – we cannot do similar analyses for Northern lenses.
• To study how the region near the last stable orbit differs from regions further from the last
stable orbit, we need UV observations with HST – success depends on obtaining long
(years) time baselines before HST fails. No luck so far ….
• To study the X-ray emitting region at “low resolution” we need continued support for
short (<10ksec) CXO observations – measurement accuracy largely depends on having
reasonable sampling over long temporal baselines. Good luck so far….
• To study the X-ray emitting region at “high resolution,” meaning the relative sizes of the
hard, soft and X-ray line emitting regions, will require a major effort (~100 ksec exposure
times). Hopefully, we prove our method with the current observations, propose and get
shot down next year, propose and succeed the following…..
On the plus side, this is much better than we achieved in the proceeding 20 years, during which
we completely wasted the opportunity to do this physics because of the sociological barriers!
Summary
•We are achieving physical interesting results that no one expected from this
approach. We can estimate
•The surface density of stars relative to dark matter
•The average mass of the stars
•The structure of the accretion disk as a function of wavelength
•All of which are new and unique probes of great astrophysical relevance
•We are primarily data limited at the present time  given the ability to collect the
necessary data, we can dramatically improve over our already completely
unexpected results
•There are challenging computational and statistical issues if we can get that data