1 Introduction to microlensing

XV International Conference on

[1ex] Gravitational Microlensing

[4ex] Conference Book

January 20–22, 2011

[2ex] University of Salerno, Italy


V. Bozza, S. Calchi Novati, L. Mancini, G. Scarpetta

(Local Organizing Committee)

Organized by the Astrophysics Group of the

Physics Department of the University of Salerno


Sponsored by IIASS (International Institute for Advanced Scientific Studies),

INFN (Istituto Nazionale di Fisica Nucelare), International Ph.D. in Astrophysics, and University of Salerno




This volume collects the abstracts in extended format of the Microlensing Conference held in the University of Salerno on January 20-22, 2011. The Conference has gathered 68 scientists from 17 countries confirming microlensing as a mature and established tool of research over a broad range of astrophysical issues, from dark matter searches to the detection of new extrasolar planets of very low mass, down to Earth-size or below. The 3-days Conference has been preceeded by a 2-days school on “Modelling Planetary Microlensing Events” focused on this last issue. The abstracts collected here offer an updated snapshot of the current researches in this field. The topics include: the status of current surveys, planetary events, dark matter searches, cosmological microlensing, theoretical investigations and an outlook towards the future, with in particular a discussion on the possible role to be played by microlensing searches for exoplanets in the forecoming space missions, WFIRST and EUCLID. Finally, the conference has been enriched by a series of topical speeches on related issues from a non-microlensing point of view: the physics of giant planet accretion and evolution, “new physics” and dark matter in the LHC era and an update on the GAIA mission and its potential to characterize planetary systems with high-precision astrometry.

Scientific Rationale

25 years after the seminal intuition by Bohdan Paczynski, microlensing has rapidly evolved into a new promise for modern astrophysics. Indeed, the detection of celestial bodies by their gravitational effects on the light of background sources has proved to be a very powerful tool for the study of many aspects of our Galaxy and beyond.

The original proposal of using microlensing to estimate the amount of baryonic dark matter in the form of compact substellar bodies is still topical. The observation of microlensing events toward nearby galaxies still represents a hard challenge for present observational facilities. Very strong efforts are under way for upgrading the current strategies in particular toward the galaxy M31.

Hundreds of microlensing events are discovered toward the Galactic Bulge every year. This considerable amount of statistics can be used to characterize the stellar populations of the bulge and the disc of our Galaxy. After several years of observations, the significance of the microlensing sample in the characterization of our Galaxy is growing more and more.

Yet, the most intriguing perspective offered by microlensing is represented by its power to discover new extrasolar planets of very low mass, down to Earth-size or below. Several planetary events have already been detected and studied, while many more are expected as soon as future dedicated telescopes become operational. Interestingly, microlensing is already being used to estimate the abundance of planets around stars in the disc of our Galaxy and to characterize their distributions in distance and mass. As the planetary anomalies typically last only a few hours, the cooperation among all observing groups is mandatory, in order to characterize the events properly and maximize the scientific achievements. Even amateur astronomers are now giving their fundamental contribution. In this respect, microlensing stands as a perfect example of how science can unite the whole mankind in a common path toward pure knowledge.

Salerno Microlensing Conference 2011 will gather all people active in this field, providing the state of the art of microlensing searches and the perspectives opened by new methodologies and new observational and computational facilities. Colloquia on dark matter searches and planet formation theories are also foreseen as a central part of the conference. The three-days conference will be preceded by a school dedicated to the delicate issue of efficient modelling of planetary microlensing events, which requires major efforts and new ideas from new talents in order to get access to the precious physical information hidden in microlensing light curves. For one week in January 2011, Salerno will thus be the place in which the present and the future of microlensing will be unveiled.

Valerio Bozza
Local Organizing Committee

Welcome address

On behalf of the Rector of Salerno University, professor Raimondo Pasquino, the Dean of the Science Faculty, professor Mariella Transirico, the Decan of the Department of Physics, professor Ferdinando Mancini and the Organizing Committee, I welcome all the participants to this exciting Conference.

This University has the peculiar status to be at the same time one of the youngest Universities in Italy and the oldest one.

The re-establishment of Salerno University in the Modern Era dates back to about forty years ago, and as you can see, in a very short time it has grown in the Irno valley as a university campus, with about forty thousand students, more than one thousand professors and researchers, distributed in ten faculties.

Five years ago the Campus was completed with the Medical Science Faculty, realizing the ideal link with one of the oldest Medical Schools of the West, the so called “Scuola Medica Salernitana”.

Salerno was not the seat of the first Medieval University, but the Medical School was the first to organize studies and diffuse culture on international scale during the medieval period.

According to the legend, the “Scuola Medica Salernitana” traces its origins to four erudite: the Greek Pontus, the Jew Helinus, the Arab Abdela, and the Latin Salernus. Thanks to the encounter of these four cultures the Medical School established its knowledge divulgating the Greek, Jewish, Arab and Latin medical knowledge. The divulgation in the West of the Islamic and Greek Medical Science is certainly due to Constantino the African (XI century), thanks to his translations into Latin of the most important Arabic and Greek medical treatises.

I like to read some words extracted from the nobel lecture of Abdus Salam (1979)

“Scientific thought and its creation is the common and shared heritage of mankind. In this respect, the history of science, like the history of all civilization, has gone through cycles. Perhaps I can illustrate this with an actual example.

Seven hundred and sixty years ago, a young Scotsman left his native glens to travel south to Toledo in Spain. His name was Michael, […] Michael reached Toledo in 1217 AD. […] From Toledo, Michael travelled to Sicily, to the Court of Emperor Frederick II. Visiting the medical school at Salerno, chartered by Frederick in 1231, Michael met the Danish physician, Henrik Harpestraeng […] Henrik had come to Salerno to compose his treatise on blood-letting and surgery. Henrik’s sources were the medical canons of the great clinicians of Islam, Al-Razi and Avicenna, which only Michael the Scot could translate for him. Toledo’s and Salerno’s schools, representing as they did the finest synthesis of Arabic, Greek, Latin and Hebrew scholarship, were some of the most memorable of international assays in scientific collaboration”.

The “Scuola Medica Salernitana” was at the height of its fame in the XIV century and carried on its teaching for nine centuries, until Giocchino Murat’s decree, dated the 25th of January 1812, prescribed the closure.

After two centuries, we all, coming here in the Salerno University from east and west countries, renew in some sense the old tradition, establishing an ideal link with the cultural heritage of the “Scuola Medica Salernitana” .

I hope you will enjoy your stay in Salerno and I wish you good and useful work.

Gaetano Scarpetta
Chair of the Local Organizing Committee

Local Organizing Committee

V. Bozza
S. Calchi Novati
L. Mancini
G. Scarpetta (Chair)

Scientific Organizing Committee
J.-P. Beaulieu France
D. Bennett USA
I. Bond New Zealand
S. Calchi Novati (Chair) Italy
M. Dominik UK
A. Gould USA
C. Han South Korea
Ph. Jetzer Switzerland
E. Kerins UK
M. Moniez France
T. Sumi Japan
Y. Tsapras UK
A. Udalski Poland
\@fontswitchL. Wyrzykowski UK
Scientific Secretary
O. De Pasquale
V. Di Marino
T. Nappi
S. Russo


Foreword iii
Scientific Rationale v
welcome vi
Organization ix
Participants xvi
School Program xix
Conference Program xx

School on Modelling Planetary Microlensing Events 1

Introduction to microlensing 2
     Philippe Jetzer

From microlensing observations to science 3
     Martin Dominik

The theory and phenomenology of planetary microlensing 4
     Scott Gaudi

From raw images to lightcurves: how to make sense of your data 5
     Yiannis Tsapras

The efficient modeling of planetary microlensing events 6
     David Bennett

Contour integration and downhill fitting 7
     Valerio Bozza

Microlensing modelling and high performance computing 8
     Ian Bond

Conference topical speeches 9

Giant planet accretion and dynamical evolution: considerations
on systems around small-mass stars
     Alessandro Morbidelli

The dark matter –- LHC endeavour to unveil TeV new physics 14
     Antonio Masiero

Characterization of planetary systems with high-precision
astrometry: the Gaia potential
     Alessandro Sozzetti

Status of current surveys 18

Status of the OGLE-IV survey 19
     Andrzej Udalski

MOA-II observation in 2010 season 20
     Takahiro Sumi

The RoboNet 2010 season 22
     Yiannis Tsapras

Cosmological Microlensing 25

Dark matter determinations from Chandra observations of
quadruply lensed quasars
     David Pooley

Cosmic equation of state from strong gravitational lensing systems 27
     Marek Biesiada & Beata Malec

Galactic Microlensing: the dark matter search 31

PAndromeda - the Pan-STARRS M31 survey for dark matter 32
     Arno Riffeser, Stella Seitz, Ralf Bender, C.-H. Lee, Johannes Koppenhoefer

Final OGLE-II and OGLE-III results on microlensing towards
the LMC and SMC
     \@fontswitchLukasz Wyrzykowski

Analysis of microlensing events towards the LMC 37
     Luigi Mancini & Sebastiano Calchi Novati

Simulation of short time scale pixel lensing towards the Virgo cluster 39
     Sedighe Sajadian & Sohrab Rahvar

M31 pixel lensing and the PLAN project 40
     Sebastiano Calchi Novati

Planetary events 44

MOA-2009-BLG-266LB: the first cold Neptune with a measured
     David Bennett

Increasing the detection rate of low-mass planets in high-magnification
events and MOA-2006-BLG-130
     Julie Baudry & Philip Yock

The complete orbital solution for OGLE-2008-BLG-513 49
     Jennifer Yee

Planetary microlensing event MOA-2010-BLG-328 51
     Kei Furusawa

Binary microlensing event OGLE-2009-BLG-020 gives orbit predictions
verifiable by follow-up observations
     Jan Skowron

Theoretical investigations 59

Microlensing and planet populations - What do we know,
and how could we learn more
     Martin Dominik

The Frequency of extrasolar planet detections with microlensing
     Rieul Gendron & Shude Mao

A semi-analytical model for gravitational microlensing events 66
     Denis Sullivan, Paul Chote, Michael Miller

GPU-assisted contouring for modeling binary microlensing events 70
     Markus Hundertmark, Frederic V. Hessman, Stefan Dreizler

Red noise effect in space-based microlensing observations 74
     Achille Nucita, Daniele Vetrugno, Francesco De Paolis, Gabriele Ingrosso,
Berlinda M. T. Maiolo, Stefania Carpano

Light curve errors introduced by limb-darkening models 76
     David Heyrovsky

Isolated, stellar-mass black holes through microlensing 77
     Kailash Sahu, Howard E. Bond, Jay Anderson, Martin Dominik,
Andrzej Udalski, Philip Yock

The observability of isolated compact remnants with microlensing 80
     Nicola Sartore & Aldo Treves

Gravitational microlensing by the Ellis wormhole 82
     Fumio Abe

The deflection of light ray in strong field: a material medium
     Asoke Kumar Sen

Rapidly rotating lenses - repeating orbital motion features in
close binary microlensing
     Matthew Penny, Eamonn Kerins, Shude Mao

Towards the future: new facilities/instrumentation/procedures 107

Microlensing with the SONG global network 108
     Uffe G. Jørgensen, Kennet B. W. Harpsøe, Per K. Rasmussen, Michael I. Andersen, Anton N. Sørensen, Jørgen Christensen-Dalsgaard, Søren Frandsen, Frank Grundahl, Hans Kjeldsen

Next-generation microlensing pilot planet search and the frequency of planetary systems 112
     Dan Maoz & Yossi Shvartzvald

Kohyama Astronomical Observatory: current status 116
     Atsunori Yonehara, Mizuki Isogai, Akira Arai, Hiroki Tohyama

Optimal imaging for gravitational microlensing 118
     Kennet B.W. Harpsøe, Uffe G. Jørgensen, Per K. Rasmussen, Michael I. Andersen, Anton N. Sørensen, Jørgen Christensen-Dalsgaard, Søren Frandsen, Frank Grundahl, Hans Kjeldsen

IPAC’s role as the science center for NASA’s WFIRST mission 121
     Kaspar von Braun

EUCLID microlensing planet hunt 122
     Jean-Philippe Beaulieu & Matthew Penny

Space-based microlensing exoplanet survey: WFIRST and/or Euclid 124
     David Bennett

Microlensing with Gaia satellite 125
     \@fontswitchLukasz Wyrzykowski

Poster session 127

Critical curve topology in special triple lens configurations 128
     Kamil Danek

PAndromeda - a dedicated deep survey of M31 with
     Chien-Hsiu Lee, Arno Riffeser, Stella Seitz, Ralf Bender, Johannes Koppenhoefer

SMC2011 Participants

Abe, Fumio Nagoya University, Japan
Baudry, Julie University of Orsay, France
Bachelet, Etienne University of Toulouse, France
Beaulieu, Jean-Philippe Institut d’Astrophysique de Paris, France
Bennett, David University of Notre Dame, USA
Bond, Ian Massey University, New Zealand
Bonino, Donata INAF - Turin Astronomical Observatory, Italy
Bozza, Valerio University of Salerno, Italy
Browne, Paul University of St Andrews, UK
Calchi Novati, Sebastiano University of Salerno, Italy
Danek, Kamil Charles University, Czech Republic
De Paolis, Francesco University of Salento, Italy
Dominik, Martin University of St Andrews, UK
Dominis Prester, Dijana University of Rijeka, Croatia
Fouqu, Pascal University of Toulouse, France
Furusawa, Kei Nagoya University, Japan
Gardiol, Daniele INAF - Turin Astronomical Observatory, Italy
Gaudi, Scott Ohio State University, USA
Gendron, Riel University of Manchester, UK
Gould, Andrew Ohio State University, USA
Harpsøe, Kennet Niels Bohr Institute, Denmark
Harris, Pauline Victoria University of Wellington, New Zealand
Henderson, Calen Ohio State University, USA
Heyrovsky, David Charles University, Czech Republic
Horne, Keith University of St Andrews, UK
Hundertmark, Markus University of Göttingen, Germany
Jetzer, Philippe University of Zurich, Switzerland
Jørgensen, Uffe G. Niels Bohr Institute, Denmark
Lambiase, Gaetano University of Salerno, Italy
Lee, Chien-Hsiu University Observatory Munich
Liebig, Christine University of St Andrews
Lubini, Mario University of Zurich, Switzerland
Malec, Beata Copernicus Center for Interdisciplinary Studies, Poland
Mancini, Luigi University of Salerno, Italy
Maoz, Dan Tel-Aviv University, Israel
Masiero, Antonio University of Padova, Italy
Mirzoyan, Sergey University of Salerno, Italy
Miyake, Noriyuki Nagoya University, Japan
Morbidelli, Alessandro Observatoire de la Cote d’Azur, France
Muraki, Yasushi Konan University, Japan
Nucita, Achille University of Salento, Italy
Orio, Marina INAF - Padova Astronomical Observatory, Italy
Paulin-Henriksson, Stphane CEA - Paris, France
Payandeh, Farrin Payame Noor University Tabriz, Iran
Penny, Matthew University of Manchester, UK
Pooley, David Eureka Scientific, USA
Retana Montenegro, Edwin F. Universidad de Costa Rica, Costa Rica
Riffeser, Arno Max Planck Institute for Extraterrestrial Physics, Germany
Sahu, Kailash Space Telescope Science Institute, USA
Sajadian, Sedighe Sharif University of Technology, Iran
Sartore, Nicola INAF - IASF Milano, Italy
Scarpetta, Gaetano University of Salerno, Italy
Sen, Asoke Kumar Sen Assam University, India
Shvartzvald, Yossi Tel-Aviv University, Israel
Skowron, Jan Ohio State University, USA
Sozzetti, Alessandro INAF - Turin Astronomical Observatory, Italy
Sullivan, Denis Victoria University of Wellington, New Zealand
Sumi, Takahiro Nagoya University, Japan
Tortora, Crescenzo University of Zurich, Switzerland
Tsapras, Yiannis Queen Mary University, UK
Udalski, Andrzej Warsaw University Observatory, Poland
Vetrugno, Daniele University of Salento, Italy
Vilasi, Gaetano University of Salerno, Italy
von Braun, Kaspar California Institute of Technology, USA
Yee, Jennifer Ohio State University, USA
Yock, Philip University of Auckland, New Zealand
Yonehara, Atsunori Kyoto Sangyo University, Japan
Wyrzykowski, \@fontswitchLukasz University of Cambridge, UK

School on Modelling Planetary Microlensing Events


Tuesday, January 18, 2011
09:00 Introduction to microlensing                     Philippe Jetzer
10:50 Coffee Break
11:15 From microlensing observations to science                     Martin Dominik
13:00 Lunch Break
14:45 The theory and phenomenology of planetary                     Scott Gaudi
16:45 Coffee Break
17:15 From raw images to lightcurves: how to make                     Yiannis Tsapras
sense of your data
Wednesday, January 19, 2011
09:00 The efficient modeling of planetary microlensing                     David Bennett
10:50 Coffee Break
11:15 Contour integration and downhill fitting                     Valerio Bozza
13:00 Lunch Break
14:45 Microlensing modelling and high performance                     Ian Bond
16:45 Coffee Break

SMC2011 Program

Thursday, January 20, 2011
09:00 Registration
09:30 Greetings
09:40 Communications
Chair: Gaetano Scarpetta
Status of current Surveys I
09:50 Status of the OGLE-IV survey                     Andrzej Udalski
10:20 MOA-II observation in 2010 season                     Takahiro Sumi
10:50 Coffee Break
Topical Speech I
11:20 Formation and evolution of our                     Alessandro Morbidelli
solar system
Theoretical investigations I
12:10 Microlensing and planet populations:                     Martin Dominik
what do we know, and how could we
learn more?
12:40 The frequency of extrasolar planet                     Rieul Gendron
detections with microlensing simulations
13:00 Lunch Break
Chair: Philippe Jetzer
Dark Matter Search
14:40 PAndromeda - the Pan-STARRS M31                     Arno Riffeser
survey for dark matter
15:10 Final OGLE-II and OGLE-III results on                     \@fontswitchLukasz Wyrzykowski
microlensing towards the LMC and SMC
15:30 Analysis of microlensing events towards the LMC                     Luigi Mancini
15:50 Simulation of short time scale pixel lensing                     Sedighe Sajadian
towards the Virgo cluster
16:10 M31 pixel lensing and the PLAN project                     Sebastiano Calchi Novati
16:30 Coffee Break
Towards the future I
17:00 Microlensing with the SONG global network                     Uffe G. Jørgensen
17:20 Next-generation microlensing pilot planet search                     Dan Maoz
and the frequency of planetary systems
17:40 Kohyama Astronomical Observatory:                     Atsunori Yonehara
current status
18:00 The lucky imaging technique for microlensing                     Kennet Harpsøe
Friday, January 21, 2011
Chair: P. Yock
Status of current Surveys II
09:00 The 2010 MicroFUN season                     Jennifer Yee
09:30 The RoboNet 2010 season                     Yiannis Tsapras
Theoretical investigations II
10:00 A semi-analytical model for gravitational                     Denis Sullivan
microlensing events
10:20 GPU-assisted contouring for modeling                     Markus Hundertmark
binary microlensing events
10:40 Rapidly rotating lenses - repeating orbital motion                     Matthew Penny
features in close binary microlensing
11:00 Coffee Break
Topical Speech II
11:30 The dark matter –- LHC endeavour to unveil                     Antonio Masiero
TeV new physics
Cosmological microlensing
12:20 Dark matter determinations from Chandra                     David Pooley
observations of quadruply lensed quasars
12:40 Cosmic equation of state from strong                     Beata Malec
gravitational lensing systems
13:00 Lunch Break
Chair: Pascal Fouqu
Planetary events
14:30 MOA-2009-BLG-266LB: the first cold Neptune                     David Bennett
with a measured mass
14:50 Increasing the detection rate of low-mass                     Julie Baudry
planets in high-magnification events and
15:10 The complete orbital solution for                     Jennifer Yee
15:30 Planetary microlensing event MOA-2010-BLG-328                     Kei Furusawa
15:50 Binary microlensing event OGLE-2009-BLG-020                     Jan Skowron
gives orbit predictions verifiable by follow-up
16:10 Coffee Break
17:30 Tour of the old town of Salerno
20:00 Social dinner
Saturday, January 22, 2011
Chair: Francesco De Paolis
Towards the future II
09:00 IPAC’s role as the science center for                     Kaspar von Braun
NASA’s WFIRST mission
09:30 EUCLID microlensing planet hunt                     Jean-Philippe Beaulieu
09:50 Simulating the planet hunting capability of Euclid                     Matthew Penny
10:10 Space-based microlensing exoplanet survey:                     David Bennett
WFIRST and/or Euclid
10:40 Coffee Break
Topical Speech III
11:10 Characterization of planetary systems with                     Alessandro Sozzetti
high-precision astrometry: the Gaia Potential
12:00 Microlensing with Gaia satellite                     \@fontswitchLukasz Wyrzykowski
Theoretical investigations III
12:20 Red noise effect in space-based microlensing                     Achille Nucita
12:40 Light curve errors introduced by limb-darkening                     David Heyrovsky
13:00 Lunch Break
Chair: Scott Gaudi
14:40 Isolated, stellar-mass black holes through                     Kailash Sahu
15:00 The observability of isolated compact                     Nicola Sartore
remnants with microlensing
15:20 Gravitational microlensing by the Ellis wormhole                     Fumio Abe
15:40 The deflection of light ray in strong field:                     Asoke Kumar Sen
a material medium approach
16:00 How to stop a runaway (Monte Carlo Markov)                     Keith Horne
16:20 Coffee Break
16:50 Open session
Poster Session
Critical curve topology in special triple lens                     Kamil Danek
PAndromeda - a dedicated deep survey of M31                     Chien-Hsiu Lee
with Pan-STARRS 1

School on

Modelling Planetary Microlensing Events

Introduction to microlensing
Philippe Jetzer
From microlensing observations to science
Martin Dominik
The theory and phenomenology of planetary microlensing
Scott Gaudi
From raw images to lightcurves: how to make sense of your data
Yiannis Tsapras
The efficient modeling of planetary microlensing events
David Bennett
Contour integration and downhill fitting
Valerio Bozza
Microlensing modelling and high performance computing
Ian Bond

Chapter 1 Introduction to microlensing

Philippe Jetzer
Institute of Theoretical Physics, University of Zurich

In the lecture I will present the basic formalism of gravitational lensing such as the lens equation and the special case of the Schwarzschild lens with its application in microlensing. I will discuss the microlensing probability for different targets such as the galactic bulge, LMC, SMC and the Andromeda galaxy as well as give a short review of the present status of microlensing searches conducted by the various collaborations in particular with respect to the problem of the galactic dark matter content in form of MACHOs.

Chapter 2 From microlensing observations to science

Martin Dominik
SUPA, University of St Andrews, School of Physics & Astronomy
United Kingdom

An elaborate integrated strategy incorporating target selection and scheduling, data flow, assessment, and final analysis is required to ensure that the scientific goals that we aim for are achieved. Specific issues that need to be taken care of are dealing with uncertainties, ambiguities, and degeneracies, as well as having the capacity to keep track with data being acquired at ever increasing rate. I will present you with some challenges.

Chapter 3 The theory and Phenomenology of Planetary Microlensing

Scott Gaudi
Department of Astronomy, Ohio State University

I discuss the theory and phenomenology of planetary microlensing. I begin with a review of the basic theoretical formalism: starting with the time delay surface, and continuing with the lens equation, I describe how critical curves, caustics, and magnifications can be computed. I then explore the properties of the caustic curves of planetary microlenses. I illustrate the topology of caustic curves and how these change with the parameters of the planetary system. In addition, I review the generic, universal behavior of images near caustics. I then delve into the rich phenomenology and salient observable properties of planetary microlensing light curves, and discuss how these can be intuitively understood based on consideration of the microlensed images and shape of and magnification near the caustics. Finally, I demonstrate how all of these considerations can be used to roughly estimate the properties of a planetary system giving rise to an observed light curve based purely on visual inspection.

Chapter 4 From raw images to lightcurves: how to make sense of your data

Yiannis Tsapras
Queen Mary University

I will present an overview of astronomical image processing with special focus on the difference imaging technique in the context of microlensing observations towards the galactic bulge. Starting from how to calibrate the raw data, I will discuss the various steps involved in the analysis, possible pitfalls, and how to extract the photometric information from the images in order to construct a clean lightcurve. Examples using the RoboNet DIA software will be given.

Chapter 5 The Efficient Modeling of Planetary Microlensing Events

David P. Bennett
University of Notre Dame, Notre Dame, IN

I present a general method for the modeling of planetary microlensing events, with an emphasis on the most difficult events which involve more than two lens masses, microlensing parallax and/or orbital motion. Finite source calculations are done with the image centered ray-shooting method, which can be made both highly efficient and flexible enough to model any events.

Chapter 6 Contour integration and downhill fitting

Valerio Bozza
Department of Physics, University of Salerno

By Green’s theorem, the two-dimensional integration on the microlensed images is written as a line integral on the image boundaries. We will discuss the advantages and the shortcomings of this method, presenting several improvements of the basic idea: parabolic correction, error control, optimal sampling, limb darkening. We will also review some basic downhill fitting methods, which rapidly provide preliminary models for microlensing events starting from suitable initial conditions.

Chapter 7 Microlensing Modelling and High Performance Computing

Ian Bond
Massey University
New Zealand

I will go over the practical aspects of the modelling and analysis of microlensing events. I will discuss two programming environments for high performance computing: cluster computers and GPU (graphical processor units) platforms. I will describe, with examples, how to design and implement software to run on these platforms. In particular, GPUs are emerging as a powerful tool for scientific computation and their potential in microlensing modelling is promising. The use of GPUs will be particularly emphasized in this seminar.

Conference topical speeches

Giant planet accretion and dynamical evolution: considerations on systems around
small-mass stars
Alessandro Morbidelli
The dark matter -– LHC endeavour to unveil TeV new physics
Antonio Masiero
Characterization of planetary systems with high-precision astrometry: the Gaia
Alessandro Sozzetti

Chapter 8 Giant planet accretion and dynamical evolution: considerations on systems around small-mass stars

Alessandro Morbidelli
Observatoire de la Cote d’Azur, Nice

1 Introduction

The classical model of giant planet formation envisions that multi-Earth mass solid cores formed from the accretion of planetesimals and that these cores then captured by gravity a massive atmosphere from the gas in the proto-planetary disk [1]. A problem in this scenario is that the solid cores have to grow on a timescale shorter than the gas-dissipation timescale, which observations set to be a few My only [2]. This is not easy, particularly in disks with small densities, such as those around low-mass stars. For this reason it is expected that giant planets cannot form around low-mass stars [3] or they form very rarely [4]. However, micro-lensing and radial velocity detections show that giant planets do exist around stars down to at least 0.2. This conflict between models and observations suggests that the process of giant planet accretion needs to be revisited

2 The problem of core accretion

It has been recently pointed out that our ideas on the accretion of the cores of the giant planets were simplistic even in the case of disks around solar-mass stars.

In fact, N-body simulations show that, once the cores become sufficiently massive (about 1), they tend to scatter the planetesimals away, rather than accrete them. In doing this, they clear their neighboring region, which in turn limits their own growth, even if initially there are a lot of solids available in the system. If damping effects are included in the simulation, in the hope of limiting the ability of the cores to scatter the planetesimals away, the result is that the planetesimals end up in stable circular orbits that are dynamically separated from the cores (i.e. no close encounters with the cores are possible). Effectively, the cores open radial gaps in the planetesimal distribution [5]. One could think that core migration or planetesimal radial drift due to gas drag can help to break the isolation of the cores from the planetesimals. However the simulations show that, in these cases, most planetesimals end up locked in resonances with the cores; resonances prevent collisions, so radial migration is unlikely to boost accretion in a significant way [5].

3 Hints for a new model of formation of giant planets cores

New results on the migration of proto-planets in gas-disks show that the concept of inward migration of Earth-mass objects (the so-called Type I migration [6]) is valid only in idealized isothermal disks. In more realistic radiative disks, the migration of the proto-planets is outward in the inner part of the disk and inward in the outer part [7][8][9]. There are therefore one (or sometimes two) orbital radii where migration is cancelled. All proto-planets formed in the disk tend to migrate towards these equilibrium radii [10]. This can concentrate the proto-planets in a narrow region of space and favor further mutual accretion, thus leading to the formation of giant planets cores. A proof-of-concept simulation will be presented at the conference, leading to the formation of a 12 body from a system of 1 planetary embryos.

If this idea is correct, the formation of the cores of the giant planets is not due to the local runaway accretion of small planetesimals, but to the concentration of all planetary embryos that formed throughout the disk and their subsequent mutual collisions. This is a big change of emphasis. The concept of the local surface density is not relevant any more because the material is migrated to the site of growth of the core from a large range of distances. This gives hope that the formation of the cores is less sensitive to the surface density of the disk than previously expected, and that therefore giant planet formation is more likely also around low-mass stars.

4 Dynamical evolution of giant planets

Once giant planets are formed, they open gaps in the gas distribution in the proto-planetary disk. Once this happens, giant planets are expected to undergo Type II migration towards the central star [11].

Also Type-II migration, though, is an idealized concept. It is valid in the case where planets are much less massive than disks and they open extremely deep gaps that gas cannot pass through. In the other cases, the planets do not migrate at the Type-II rate; they can even move outward and, instead of gaps, they can open cavities in the inner part of the disk, removing most the gas between the star and the orbit of the planet [12].

In case of two (or more) planets, the migration pattern is even more complex. For two planets in resonance, migration is inward if the outer planet has a mass comparable to the inner one, or larger. If the outer planet has a mass that is a sizeable fraction of the inner one (as in the Jupiter-Saturn case), migration is outward. If the outer planet has a negligible mass (say Jupiter and Neptune), migration is inward again [13]. Thus, for our solar system we can envision that Jupiter migrated inwards while Saturn was growing and then, once Saturn reached a mass close to its current one, they migrated outward. This kind of evolution may be relevant for the system around the star OGLE-06-109L, discovered by microlensing, which looks like a Sun-Jupiter-Saturn system slightly scaled down in mass.


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Chapter 9 The Dark Matter - LHC Endeavour to Unveil TeV New Physics

Antonio Masiero
INFN - University of Padova

After more than four decades of relentless tests of the Standard Model (SM) of particle physics, one can safely state that it correctly describes the fundamental interactions all the way up to the energy scale of 100 GeV. Yet, the observational evidences that neutrinos are massive and that a large amount of non-baryonic dark matter exists corroborate the theoretical demand for the presence of new non-SM physics at the TeV scale. Interestingly enough, the main theoretical motivation for new physics (NP) at the electroweak scale (i.e., the presence of an ultraviolet SM completion to enforce the stability of the electroweak breaking scale) nicely joins the need for some form of cold matter: indeed, most theoretically dictated extensions of the SM (low-energy supersymmetry, extra-dimensions, etc.) entail the existence of some new stable particle which can play the role of dark matter candidate. I’ll discuss the interplay between the searches for TeV NP which are going on at the LHC and the searches for dark matter related to TeV NP both in direct and indirect DM probes. It is exciting that the coming decade has the potentiality to witness the simultaneous success of the high-energy (LHC) and astroparticle (DM) roads in our endeavour to unveil the presence of NP at the electroweak scale.

Chapter 10 Characterization of Planetary Systems with High-Precision Astrometry: The Gaia Potential

Alessandro Sozzetti
INAF - Osservatorio Astronomico di Torino

1 Introduction

In its all-sky survey, the ESA global astrometry mission Gaia, due to launch in late 2012, will perform high-precision astrometry and photometry for 1 billion stars down to V = 20 mag. The data collected in the Gaia catalogue, to be published by the end of the decade, will likely revolutionize our understanding of many aspects of stellar and Galactic astrophysics. One of the relevant areas in which the Gaia observations will have great impact is the astrophysics of planetary systems. There are many complex technical problems related to and challenges inherent in correctly modelling the signals of planetary systems present in measurements collected with an observatory poised to carry out precision astrometry at the micro-arcsecond (as) level. Provided these are well understood, Gaia as astrometry has great potential for important contributions to the astrophysics of planetary systems, particularly when seen in synergy with other indirect and direct methods for the detection and characterization of planetary systems.

2 Astrometric Modeling of Planetary Systems

The problem of the correct determination of the astrometric orbits of planetary systems using Gaia data (highly non-linear orbital fitting procedures, large numbers of model parameters) will present many difficulties. For example, it will be necessary to assess the relative robustness and reliability of different procedures for orbital fits, with a detailed understanding of the statistical properties of the uncertainties associated with the model parameters. For multiple systems, a trade-off will have to be found between accuracy in the determination of the mutual inclination angles between pairs of planetary orbits, single-measurement precision and redundancy in the number of observations with respect to the number of estimated model parameters. It will be challenging to correctly identify signals with amplitude close to the measurement uncertainties, particularly in the presence of larger signals induced by other companions and/or sources of astrophysical noise of comparable magnitude. Finally, for systems where dynamical interactions are important (a situation experienced already by Doppler surveys), fully dynamical fits involving an n-body code might have to be used to properly model the Gaia astrometric data and to ensure the dynamical stability of the solution (see Sozzetti 2005). All the above issues could significantly impact Gaia’s planet detection and characterization capabilities. For these reasons, a Development Unit (DU), within the pipeline of Coordination Unit 4 (object processing) of the Gaia Data Processing and Analysis Consortium, has been specifically devoted to the modelling of the astrometric signals produced by planetary systems. The DU is composed of several tasks, which implement multiple robust procedures for (single and multiple) astrometric orbit fitting (such as Markov Chain Monte Carlo algorithms) and the determination of the degree of dynamical stability of multi-planet systems.

(pc) (AU) () 0-50 1.0 - 4.0 1.0 - 13.0 50-100 1.0 - 4.0 1.5 - 13.0 100-150 1.5 - 3.8 2.0 - 13.0 150-200 1.4 - 3.4 3.0 - 13.0
Case N. of systems Detection Orbits and masses to accuracy Successful coplanarity tests (b)
Table 1: Left: Number of giant planets of given ranges of mass () and orbital separation () that could be detected () and measured () by Gaia, as a function of increasing distance () and stellar sample (). Right: Number of planetary systems that Gaia could potentially detect, measure, and for which coplanarity tests could be carried out successfully. See Casertano et al. (2008) for details.

2.1 The Gaia Legacy

Gaia’s main contribution to exoplanet science will be its unbiased census of thousands of planetary systems (see Table 1) orbiting hundreds of thousands nearby ( pc), relatively bright () stars across all spectral types, screened with constant astrometric sensitivity. As a result, the actual impact of Gaia measurements in exoplanets science is broad, and rather structured. The Gaia data have the potential to: a) significantly refine our understanding of the statistical properties of extrasolar planets; b) help crucially test theoretical models of gas giant planet formation and migration; c) achieve key improvements in our comprehension of important aspects of the formation and dynamical evolution of multiple-planet systems; d) aid in the understanding of direct detections of giant extrasolar planets; e) provide important supplementary data for the optimization of the target selection for future observatories aiming at the direct detection and spectral characterization of habitable terrestrial planets. Finally, ongoing studies are now focusing on the detailed understanding of the planet discovery potential of Gaia as far as low-mass stars (Sozzetti et al., in preparation) and post-main-sequence objects (Silvotti, Sozzetti, & Lattanzi 2011) and are concerned.

In conclusion, the Gaia mission is now set to establish the European leadership in high-precision astrometry for the next decade. The largest compilation of high-accuracy astrometric orbits of giant planets, unbiased across all spectral types up to pc, will allow Gaia to crucially contribute to several aspects of planetary systems astrophysics (formation theories, dynamical evolution), in combination with present-day and future extrasolar planet search programs.


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Status of current surveys

Status of the OGLE-IV Survey Andrzej Udalski
MOA-II observation in 2010 season Takahiro Sumi
Session Chair: Martin Perl
The deflection of light ray Asoke Kumar Sen
in strong field: a material medium approach

Chapter 11 Status of the OGLE-IV Survey

Andrzej Udalski
Warsaw University Observatory

On the night of March 4/5, 2010, after ten months long break, the OGLE survey resumed regular observations entering the OGLE-IV phase. With the new 32 chip mosaic camera covering 1.4 square degrees on the sky with the resolution of 0.26 arcsec/pixel and fast reading time of the entire array of 20 seconds the observing capabilities of the OGLE survey increased by an order of magnitude compared to the previous OGLE-III phase. With this observing set-up the OGLE project continues its 18 years long tradition of being one of the largest sky surveys worldwide with the current outcome of about 40 Terabytes of raw data per year.

During the 2010 Galactic bulge observing season OGLE-IV regularly monitored large fraction of the Galactic bulge conducting pilot observations for the second generation microlensing survey. About 4.5 square degrees of the Galactic bulge regions with the highest microlensing rate were observed with the cadence of 20 minutes while additional 8.5 square degrees with the cadence of 1 hour. Moreover, large area of the Galactic bulge was observed once or twice a day.

The automatic data pipeline of OGLE-IV images reductions has been recently finished. Its full implementation at the telescope, expected in early 2011, will allow to restart on-line (real time) reductions of the huge data-stream coming from the OGLE telescope and restart the data analysis systems like the OGLE Early Warning System.

Chapter 12 MOA-II observation in 2010 season

Takahiro Sumi
Solar-Terrestrial Environment Laboratory
Nagoya University

1 Introduction

Since the first discovery of exoplanets orbiting main-sequence stars in 1995 [17], more than 500 exoplanets have been discovered via the radial velocity method and more than 50 have been detected via their transits. Several planetary candidates have also been detected via direct imaging, and astrometry.

The gravitational microlensing was proposed as a unique method to search exoplanets [18, 19]. The planet’s gravity induces small caustics, which can generate small deviations in standard [82] single-lens microlensing light curves. Compared to other techniques, microlensing is sensitive to smaller planets, down to an Earth mass [21], and in wider orbits of 1-6 AU. Because microlensing observability does not depend on the light from the lens host star, it is sensitive to planets orbiting faint host stars like M-dwarfs and even brown dwarfs. Furthermore, it is sensitive to distant host stars at several kpc from the Sun, which allows the Galactic distribution of planetary systems to be studied.

In 2003, the gravitational microlensing method yielded its first definitive exoplanet discovery [141]. So far ten planetary systems with eleven planets have been found by this technique [23], which have very distinct properties from those detected by other techniques.

Although the radial velocity and transit discoveries are more numerous, microlensing is uniquely sensitive to the cold Neptunes outside of the snow-line, and the microlensing results to date indicate that this class of planets may be the most common type of exoplanet yet discovered [24].

2 Observations

The MOA-II carries out survey observations toward the Galactic Bulge (GB) to find exoplanets and toward the large and small Magellanic clouds (LMC and SMC) for searching MACHOs via the gravitational microlensing using a 1.8m telescope, equipped with a mosaic camera with 10 chips of 2kx4k-pixel CCD[25], at Mt. John Observatory in New Zealand. We observe our target fields very frequently (every 15-90 min) and analyze data in real-time to issue alerts. This high cadence is specifically designed to find the short timescale planetary signature. Our real-time anomaly alert system searches for planetary signatures in ongoing microlensing events in which new data points are available on the light curves within 5 min after exposure. In 2010 season, we continued this observational strategy.

3 Results

In 2010 season, we detected 607 microlensing events and issued the alerts toward the GB. Among these, 5 events show planetary or brown dwarf signatures in their light curves thanks to the data by OGLE-IV survey and intensive follow-up observations by FUN, PLANET, RoboNet, MiNDSTEp.

In 2010, we found one event toward the LMC that shows a clear asymmetry due to the parallax. This indicates that the lens object is likely a foreground disk star.

We are grateful to OGLE, FUN, PLANET, RoboNet, MiNDSTEP collaborations for close cooperation.


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Chapter 13 The RoboNet 2010 season

Yiannis Tsapras
Queen Mary University

1 The Las Cumbres Observatory Global Telescope Network

The Las Cumbres Observatory (LCOGT) [26] is a privately funded institute which owns and operates the 2m Faulkes Telescopes in Hawaii and Australia. LCOGT is in the process of developing a global network of 1m and 0.4m telescopes capable of robotic around-the-clock observations. These will be used for science as well as education. The 0.4m and 1m prototypes are currently being tested and deployment around the world will begin in 2011. The first ones will be deployed in Chile and South Africa, followed by Tenerife, Australia and China. The complete network will comprise of twelve to fifteen 1m and twenty-four 0.4m telescopes and is expected to be complete by the end of 2014. The final numbers depend on the relative costs. The SUPA-II Planet Hunter bid from the University of St Andrews will finance the construction of an extra three 1m telescopes with exactly the same specifications.

The 0.4m telescopes will be primarily used for educational purposes. Schools from around the world will have access to these telescopes through web interfaces and will be able to make observations during daylight hours in Europe by using the telescopes in Chile, Hawaii and Australia. Furthermore, they will have the option to get involved in certain science projects, like microlensing or transits, and contribute observations. The LCOGT education pages provide descriptions of possible projects available to the students. The 1m telescopes will be used for science observations. Each site will eventually host two to four telescopes which can be operated individually or in parallel, allowing simultaneous observations of targets in different pass-bands. Each telescope will be fitted with a fast readout CCD camera that has a field of view of about half a degree and filters in the UBVRI and Pan-STARRS Z and Y bands. Each cluster of telescopes will have an optical fiber-feed to a shared medium-resolution spectrograph.

The observation requests sent out to the telescopes will be scheduled by a highly dynamic scheduling algorithm that will constantly farm the database of TAC-approved science projects for the most appropriate observations to perform at any given time given the current conditions and the scientific priority.

2 RoboNet: A microlensing search for cold planets

In order to draw conclusions about planetary populations, it is not enough to just detect planets, it is necessary to understand the selection bias of the surveys. This requires the adoption of an observing strategy that is not based on human intervention, since this cannot be simulated. RoboNet’s unique advantage is that our entire system is completely automated, which makes it easier to understand the selection biases of our strategy. As the system is designed for a network with continuous Bulge coverage, we expect to address this question with the help of the full LCOGT network in future seasons.

The methodology used by the RoboNet team is described in [27]. Our pilot campaign (RoboNet-I) contributed to 5 of the first 6 microlensing planet discoveries and provided the impetus to develop our system further. We are continuing this development in preparation for the deployment of the LCOGT network of 1m and 0.4m telescopes.

RoboNet has pioneered the technique of adaptive scheduling of microlensing targets through the use of a prioritisation algorithm that continuously shuffles the observable microlensing targets, giving higher priority to the ones that are more likely to reveal a planetary signal [28]. It calculates the optimal frequency at which each ongoing microlensing event needs to be sampled at in order to maximize the planet detection probability. Once an observation is obtained for a specific event, it’s priority is adjusted upwards in the case that it deviates from the expected single-lens case, or downwards if it doesn’t. The relative priorities are reassessed every few minutes and active events that have not been observed recently will start to appear higher and higher in the list until an observation is obtained. At that point their relative priority will be readjusted according to the previous scheme. In order to be sensitive to ”cold Earths”, a sampling interval of  15-45 mins is required, which the extended robotic network will be able to deliver.

RoboNet makes use of difference image analysis software [29] to extract accurate brightness measurements from the incoming images which are used to construct the event lightcurves. One image is selected as a template and all other images are geometrically and photometrically aligned to and subtracted from it. Any stars that show variability are immediately obvious on the resulting subtracted images, while stars of constant brightness leave no residuals.

The resulting lightcurves are immediately shared with the extended microlensing community via automated rsync transfers but are also available on the project website for downloading. The lightcurves are continuously assessed by our SIGNALMEN software, which uses robust methods to automatically detect the onset of anomalous features. If such a feature is detected, an override request is sent to the network to obtain prompt observations that will confirm or disprove the anomaly. While most observations are performed in queued observing mode, we sometimes use our override capability in order to respond to anomaly alerts. Once more telescopes are included in our network, we will be able to operate in a fully automated mode throughout the microlensing season.


  • [26] www.lcogt.net
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Cosmological Microlensing

Dark matter determinations from Chandra observations of quadruply lensed quasars
David Pooley
Cosmic equation of state from strong gravitational lensing systems
Marek Biesiada & Beata Malec

Chapter 14 Dark Matter Determinations from Chandra Observations of Quadruply Lensed Quasars

David Pooley
Eureka Scientific

I present all publicly available Chandra observations of 14 X-ray bright quadruply lensed quasars. The X-ray data reveal flux ratio anomalies which are more extreme than those seen at optical wavelengths, confirming the microlensing origin of the anomalies originally seen in the optical data. The X-ray emitting regions are essentially point sources and therefore give a microlensing signal unencumbered by source size considerations. Building on our previous work, we have completed a thorough investigation of the best way to analyze the X-ray data, resulting in improved determinations of the X-ray flux ratios. We have constructed custom microlensing magnification maps for a range of stellar fractions for the four images of each quasar, and we have implemented a more sophisticated Bayesian analysis of the data to determine the most likely dark matter fraction in the lensing galaxies.

Chapter 15 Cosmic Equation of state from Strong Gravitational Lensing Systems

Marek Biesiada
Department of Astrophysics and Cosmology, Institute of Physics
University of Silesia Poland

Beata Malec
Copernicus Center for Interdisciplinary Studies
Cracow Poland

1 Introduction

Accelerating expansion of the Universe is a great challenge for both physics and cosmology. In light of lacking the convincing theoretical explanation, an effective description of this phenomenon in terms of cosmic equation of state turns out useful.

The strength of modern cosmology lies in consistency across independent, often unrelated pieces of evidence. Therefore, every alternative method of restricting cosmic equation of state is important. Strongly gravitationally lensed quasar-galaxy systems create such new opportunity by combining stellar kinematics (central velocity dispersion measurements) with lensing geometry (Einstein radius determination form position of images).

2 Idea and methods

Strong gravitationally lensed systems create opportunity to test cosmological models of dark energy in a way alternative to Hubble diagrams (from SNIa or GRBs), CMBR or LSS. The idea is that formula for the Einstein radius in a SIS lens (or its SIE equivalent)

depends on the cosmological model through the ratio: (angular-diameter) distance between lens and source to the distance between observer and lens. Provided one has reliable knowledge about lensing system: the Einstein radius (from image astrometry) and stellar velocity dispersion (central velocity dispersion inferred from spectroscopy) one can used such well studied systems to test background cosmology. This method is independent on the Hubble constant value ( gets cancelled in the distance ratio) and is not affected by dust absorption or source evolutionary effects. It depends, however, on the reliability of lens modelling (e.g. SIS or SIE assumption). The method was proposed by Biesiada [30] and also discussed in details by Grillo et al. (2008) [31].

Here we apply such method to a combined data sets from SLACS and LSD surveys of gravitational lenses [31]. In result we obtain the cosmic equation of state parameters, which generally agree with results already known in the literature. This demonstrates that the method can be further used on larger samples obtained in the future [32]. We have also performed joint analysis taking into accout standard rulers (combining lensing system data with CMB acoustic peak location and BAO data) and standard candles (SNIa data - we used Union08 compilation by Kowalski et al. (2008) [33] ). The observables we used had different parameter degeneracies and so different restrictive power in the parameter spaces of cosmological models. It can be best seen in figures.

Figure 1: Best fits (dots) and (68 %, 95%) confidence regions in plane for quintessence model. Confidence regions displayed separately for standard rulers, standard candles and joint analysis.

We considered five cosmological scenarios of dark energy, widely discussed in current literature. These are CDM, Quintessence, Chevalier-Polarski-Linder model, Chaplygin gas and Braneworld scenario. Then we performed fits to different cosmological scenarios on described samples. The probes were combined by calculating joint likelihoods.

Because standard rulers and standard candles probe distance measures based on different concepts (angular diameter distance and luminosity distance), one step before making a full joint fit we performed fits based on rulers and candles separately.

Figure 2: Best fits (dots) and (68 %, 95%) confidence regions in plane for Chevalier-Linder-Polarski model. Confidence regions displayed separately for standard rulers, standard candles and joint analysis.

3 Conclusions

The best fits we obtained for the model parameters in joint analysis turned out to prefer cases effectively equivalent to CDM model. They are also in agreement with other combined studies performed by other authors on different sets of diagnostic probes. As comparison between models in terms of chi-square values does not account for the relative structural complexity of the models we also made use of information theoretic methods. According to Akaike criterion (AIC) CDM it is only slightly preferred over the quintessence and both models with a dynamical equation of state (CPL parametrization) and Chaplygin gas scenario get considerably less support from the data. Odds against the brane-world scenario are so high that it can be considered as ruled out by the data. According to the Schwartz Bayesian Information criterion (BIC) CDM wins, the quintessence model is considerably less supported by data, and the other ones are ruled out.

This work was supported by the Polish Ministry of Science Grant no. N N203 390034.


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Galactic Microlensing:

the dark matter search

PAndromeda - the Pan-STARRS M31 survey for dark matter
                 Arno Riffeser, Stella Seitz, Ralf Bender, C.-H. Lee, Johannes Koppenhoefer
Final OGLE-II and OGLE-III results on microlensing towards the LMC and SMC
\@fontswitchLukasz Wyrzykowski
Analysis of microlensing events towards the LMC
Luigi Mancini & Sebastiano Calchi Novati
Simulation of short time scale pixel lensing towards the Virgo cluster
Sedighe Sajadian & Sohrab Rahvar
M31 pixel lensing and the PLAN project
Sebastiano Calchi Novati

Chapter 16 PAndromeda - A Dedicated Deep Survey of M31 with Pan-STARRS 1

Arno Riffeser, Stella Seitz, Ralf Bender, Chien-Hsiu Lee, Johannes Koppenhoefer
Max Planck Institute for Extraterrestrial Physics, Garching

Pan-STARRS 1 Science Consortium University of Hawaii, Pan-STARRS Project Office, Max Planck Institute for Astronomy, Max Planck Institute for Extraterrestrial Physics, Johns Hopkins University, University of Durham, University of Edinburgh, Queens University of Belfast, Harvard-Smithsonian Center for Astrophysics, Los Cumbres Observatory Global Telescope Network, National Central University of Taiwan

The 1.8 m Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) on Haleakala, Maui, began its regular survey in spring 2010. With its 7 deg field of view and the use of orthogonal transfer array CCDs [34] it represents the system with the highest data-flow today. In addition to the large area 3Pi Survey some fields are exposed deeper, and visited more frequently. One of these so called medium deep fields (MD) targets the Andromeda galaxy. PAndromeda monitors M31 for 2% of the overall PS1 time. This means that M31 is observed for 0.5 h per night during a period of 5 months per year. PAndromeda is designed to identify gravitational microlensing events, caused by bulge and disk stars (self-lensing) and by compact matter in the halos of M31 and the MW (halo lensing, or lensing by MACHOs). The main science goals of PAndromeda are measuring the masses and mass-fraction of compact objects in the M31 and MW halos, and constraining the M31 bulge mass function at the low mass end. As a side product PAndromeda is also able to search for microlensing events towards M32 and NGC 205.

The interpretation of the microlensing events (see [35]) requires i) understanding the mix of stellar ages and metalicities in the bulge, disk, and halo of M31 as obtained from resolved stellar populations (census of supergiants, OB-associations, analysis of CMD diagrams as a function of location) variability studies (from Cepheids to LPVs), and color gradients in the light profiles, ii) deriving improved 3 dimensional models for the density and velocity distributions of the bulge and disk, iii) including improved constraints on the extinction in M31 [36]. All these informations can directly extracted from the PAndromeda data itself.

The continuous monitoring in the and bands with PAndromeda allows to confirm the achromaticity of events; the main filter (detection filter) is the -band to optimize the number of source stars accessible for measurable lensing events, which depends on the brightness of the stars, the photon noise (mostly by M31), and the sensitivity of filters in each band. During the first season we focussed on the integration depth in the filters r’ and i’. We split the -band imaging into two integrations (separated by 4-6 hours) per night, for the whole season. This allows to measure light curves with shorter time scales than 1 day more accurately, since a large fraction of events is predicted (and found in previous surveys) with short time scales. PAndromeda monitored M31 from 07/23/2010 till 12/27/2010 on 91 nights (58%). In total 1782 images were exposed, 1179 in r’ (90 nights, 70740 sec) and 603 in i’ band (66 nights, 36180 sec). The total amount of reduced data is 14 TB. Depending on our observing strategy in 2011 the remaining filters and integration times for PAndromeda will be chosen such, that in turn these observations can be used to study many other aspects of M31 with the aim to improve the predictions of the microlensing event rates. These microlensing predictions depend on the mass function of stars, their luminosities and sizes as determined by their ages and metallicities, on their density and velocities distributions, and on the extinction.

The PS1 Gigapixel Camera (GPC1) produces images consisting of 3840 CCDs () with roughly pixels each. This results in pixels which are exposed and read-out. The GPC1 data are de-biased, flat-fielded, and astrometrically registered with the Pan-STARRS Image Processing Pipeline (by Magnier E.). A reduced GPC1 exposure requires a disk storage of 8.0 GB consisting of frame, weight and mask. After this basic reduction the data are further processed by our own image processing software (MUPIPE, [37]). For the PAndromeda difference imaging analysis we adapted the MUPIPE pipeline to the specifications of PS1 data and data flow and implemented it into the Astro-WISE system.

From the 2010 season we analyzed the central field of M31 (). This is to test the detection process in the field where we expect the highest lensing rate because of self lensing. So far we detected 3 high quality microlensing light-curves. The third one is very bright with 19 mag in r’. Note that high flux excess events are more difficult to reconcile with self-lensing than with halo-lensing [38]. The full data set is currently analyzed.


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Chapter 17 Final OGLE-II and OGLE-III results on microlensing towards the LMC and SMC

Lukasz Wyrzykowski
Institute of Astronomy, University of Cambridge

1 Introduction

For almost two decades compact halo objects (MACHOs) have been one of the best candidate for Dark Matter (DM). The main channel of detecting them is through the microlensing phenomenon, however the results from two microlensing surveys, MACHO and EROS, are in contradiction as to the abundance of MACHOs in the Galaxy.

We present an independent and the most comprehensive result so far from the OGLE microlensing survey of the Large and Small Magellanic Clouds running continuously for over 13 years. In both Clouds, we report the detection of the total of 8 events with 2 more plausible candidates. All but one of them are consistent with the expected signal from lensing by Clouds own stars (self- lensing), therefore there is no need of introducing dark microlenses.

2 LMC: 2+2 events and 2 candidates

There were 4 events found in the OGLE-II and OGLE-III data. Another 2 more were flagged as potential candidates in OGLE-III data. The optical depth derived for these events was and for OGLE-II and OGLE-III, respectively ([40], [42]).

Time-scales, locations and positions of these events on the colour-magnitude diagram indicate all of them are very likely to be solely caused by self-lensing. Source stars in all events seem to belong to the LMC and events occurred in the areas where the self-lensing contributes the most to the overall microlensing optical depth.

Small number of events seen by OGLE indicates the excess of events seen by MACHO group is probably not caused by microlensing. OGLE data ruled out as microlensing one of the MACHO events (#7), which showed another bump after many years.

3 SMC: 1+3 events

Previously there were 2 events found in the SMC by MACHO and EROS groups and both were confirmed to be due self-lensing.

In the OGLE data there were 4 events detected giving [41] and . Events 03 and 04 are good candidates for self-lensing due to their time-scales and locations. Event 01 is a rather weak candidate and could be also due to a variable star.

In the most likely microlensing model for the event OGLE-SMC-02 the lens is a 10 binary black hole from the Galactic halo [39]. Such events, however, are not seen towards the LMC, which calls for further observations and theoretical studies.

Figure 1: MACHOs contribution to the total halo mass as measured by MACHO group and upper limits from EROS and OGLE data.

4 Conclusions

Based on our detection efficiency, we derived new constrains on the MACHOs presence in the Galactic Halo of 6% for M=0.1-0.4 and below 4% for masses in range 0.01-0.1 (Fig. 1). For MACHOs with mass of 1 the upper limit is % and % for .

Our result indicates that baryonic DM in the form of relics of stars and very faint objects in the sub-solar-mass range are unlikely to inhabit the Milky Way’s dark matter halo in any significant numbers. Presence of a black-hole lens candidate towards the SMC agrees with expected no more than 2% contribution of black-holes to the mass of the Galactic halo.


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Chapter 18 Analysis of microlensing events towards the LMC

Luigi Mancini & Sebastiano Calchi Novati
Department of Physics, University of Salerno, Italy
IIASS, Vietri Sul Mare

The nature of the observed microlensing events towards the Magellanic Clouds is still an open issue. Indeed, in order to draw meaningful insights into the contribution of dark matter objects in form of MACHOs, it is essential first to estimate accurately all the possible contributions due to (luminous) lenses belonging to known populations located along the line of sight (to which we refer too as “self lensing”). A possibly non exhaustive list includes lenses belonging to the luminous components of the LMC (the disc and the bar) that can act as sources , the disc of the Milky Way and the somewhat elusive stellar halo for both the Milky Way and the LMC.

The results reported so far by the collaborations who carried out observational campaigns towards the LMC as for the MACHO contribution to the Galactic halo are in agreement to exclude MACHOs as a viable dark matter candidate for masses below . However, a relevant discrepancy still exists as for compact halo object in the mass range . The MACHO collaboration claimed for an mass halo fraction in form of MACHOs of about out of observations towards the LMC (Alcock et al., 2000), a result more recently confirmed by Bennett (2005). On the other hand, the EROS (Tisserand et al., 2007) and the OGLE (Wyrzykowski et al., 2009, 2010) collaborations, out of observations towards both the LMC and SMC, concluded that the expected self-lensing rate is sufficient to explain the observed rate also in this mass range. It is therefore important to address the issue of the nature of the observed events, either to be attributed to MACHO lensing or to self lensing.

In previous analyses we have considered the set of events reported by the MACHO collaboration and shown that, on the basis of both their number number and their characteristics (event duration and spatial distribution), they cannot all be attributed to self lensing (Mancini et al., 2004). In Calchi Novati et al. (2006) we have considered the possible role played by the LMC dark matter halo, in particular suggested that the halo fraction in form of MACHOs for the Milky Way and the LMC might not be equal. Finally, in Calchi Novati et al. (2009) we have discussed the results of the OGLE-II campaigns towards the LMC.

Here we report on a detailed analysis of the recent results of the OGLE-III campaiagn towards the LMC (Wyrzykowski et al, 2010). For all the possible lens populations (both luminous and dark), we present maps of the optical depth and a study of the expected characteristics (duration, spatial distribution and number), to be compared with the observed events. This is done through an evaluation of the microlensing rate towards all the observed fields. Finally, we evaluate the probability distribution for the mass halo fraction in form of MACHO via a likelihood analysis. Overall, we find the observed rate to be compatible with the expected lensing signal by known luminous populations.


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Chapter 19 Simulation of pixel lensing of M87 by HST

Sedighe Sajadian & Sohrab Rahvar
Department of Physics, Sharif University of Technology, Tehran

In this work we propose a new strategy of pixel lensing observation of the unresolved stars of M87 in the Virgo cluster by HST. We show that in contrast to the previous observational strategy in Baltz t al. (2004) with one observation per night for the duration of one month, a few days intensive observation with taking one image per one HST orbit, we can substantially increase the number of events more than one order of magnitude. In this observational strategy, high magnification microlensing events is predicted to be observed with the rate of event per day with a typical transit time scale of hours. We examine the possible detection of dark matter mini-halos with this observational strategy.


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Chapter 20 M31 pixel lensing and the PLAN project

Sebastiano Calchi Novati
Department of Physics, University of Salerno Italy
IIASS, Vietri Sul Mare

1 Introduction: M31 Pixel Lensing and the issue of MACHO versus self lensing

Pixel lensing is a unique tool for the study of dark matter objects in form of MACHOs up to distant galaxies as M31. More than 20 pixel lensing events towards M31 have been reported so far from several different campaigns, but no firm conclusions on the MACHO issue have been reached yet (Calchi Novati, 2010). A fundamental problem in the analysis is the difficulty to disentangle microlensing signal due to dark matter MACHOs from “self lensing”, microlensing signal due to lenses belonging to known luminous populations. This is complicated first by the additional degeneracy in the microlensing event parameter space characteristics of pixel lensing, where one studies flux variations of unresolved objects. Second, by the rather large expected self-lensing signal, as compared to the MACHO lensing one, at least in the inner M31 region, where in any case most of the events, either MACHO or self lensing, are expected. In this respect, as for many others, M31 pixel lensing is peculiar with respect to microlensing towards the Magellanic Clouds, LMC and SMC (Moniez, 2010). Still, besides the underlying technique, also the fundamental physical issue one addresses to, that of MACHOs, is the same. The recent analyses of OGLE-II/SMC (Wyrzykowski et al., 2010) and OGLE-III/LMC (Wyrzykowski et al., 2010) both indicates that the observed rate of events is compatible with the expected self-lensing signal. This is in agreement with the previous OGLE-II/LMC (Wyrzykowski et al., 2009) and the EROS results (Tisserand et al. 2007). On the other hand, the MACHO collaboration claimed for a MACHO signal of about for a mass halo fraction in form of MACHO (Alcock et al., 2000), a result more recently confirmed by the further analyses of Bennett (2005). To address the issue of MACHOs along a different line of sight is therefore important. Besides, looking towards M31 one can map its own full dark matter halo, which is not possible for the Galaxy one.

To explore the MACHO versus self-lensing issue, and more in general to address the problem of the nature of the observed events, the careful and through analysis of single events have shown to be a very efficient tool even if it can only give partial indications. Relevant results have been reported for the M31 microlensing event PA-N1 (Aurière et al., 2001), PA-S3/GL1 (Riffeser et al., 2008), and by the more recent analysis of OAB-N2 by the PLAN collaboration (Calchi Novati et al., 2010). Remarkably, though with different level of confidence, all these analyses conclude that MACHO lensing is to be preferred to self lensing for the events studied. This approach runs in parallel with the analysis of full data set completed by a careful comparison of the observed and the expected rate (an approach which still has to deal with the limitation given by the rather small statistics involved). The more relevant results have been reported by the POINT-AGAPE (Calchi Novati et al., 2005) and the MEGA (de Jong et al., 2006) collaborations, working on the same data set. POINT-AGAPE claimed for an evidence of a MACHO signal, in the same mass range indicated by the MACHO LMC analysis, whereas MEGA concluded that self lensing was sufficient to explain the detected rate. An extremely promising new project for M31 pixel lensing observation is that of PAndromeda (Riffeser et al, 2011).

2 The PLAN project: an update

Within this still open framework we discuss some of the results we have obtained with the PLAN collaboration which is carrying out a long term monitoring project of M31 (with observational campaigns at the 1.5m OAB telescope from 2006 to 2010 and a pilot campaign at the 2m HCT telescope in October 2010). In Calchi Novati et al. (2009) as a result of a fully automated pipeline we have reported 2 microlensing candidate events out of the 2007 campaign (OAB-N1 and OAB-N2). The observed rate turned out to be relatively large, still, compatible with the expected self-lensing one. The available statistics was however too small to allow us to draw stringent conclusions on the MACHO issue. Since then we have followed both the indicated approaches to address the MACHO versus self-lensing issue.

First, some additional data made available by the WeCAPP collaboration along the OAB-N2 light curve and a more through analysis of KPNO and HST archive images used to better constrain the possible event source characteristics, have allowed us a much more detailed analysis of OAB-N2 (Calchi Novati et al., 2010). In particular, through a study of the lens proper motion, we have shown that this event is more likely to be attributed to MACHO lensing than to self lensing.

As for our full data set, more statistics is badly needed. Here we report some preliminary results of the 2008 and 2009 campaigns. As an output, no news microlensing candidate events have been detected. In particular we discuss the crucial role played by the selected bump unicity analysis we carry out on archive POINT-AGAPE light curves. We compare the observed rate to the expected one evaluated through a Monte Carlo simulation completed by an analysis of the efficiency of the pipeline. The observed rate is compatible with the expected self-lensing one but still does not allow us to constrain MACHO lensing (Calchi Novati et al, 2011).


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Planetary events

OA-2009-BLG-266LB: the first cold Neptune with a measured mass
David Bennett
Increasing the detection rate of low-mass planets in high-magnification events and
Julie Baudry & Philip Yock
The complete orbital solution for OGLE-2008-BLG-513
Jennifer Yee
Planetary microlensing event MOA-2010-BLG-328
Kei Furusawa
Binary microlensing event OGLE-2009-BLG-020 gives orbit predictions verifiable by
follow-up observations
Jan Skowron

Chapter 21 MOA-2009-BLG-266LB: The First Cold Neptune with a Measured Mass

David P. Bennett
University of Notre Dame, Notre Dame, IN

for the MOA, FUN, RoboNet, PLANET, MiNDSTEp, and OGLE Collaborations

The MOA-II survey detected beginning of the planetary signal in microlensing event MOA-2009-BLG-266, so MOA’s prompt anomaly alert led to nearly complete sampling of the planetary anomaly. Good light curve coverage from MOA and a number of the follow-up telescopes after the anomaly enabled the measurement of a strong microlensing parallax signal that allows the mass to be measured.

MOA-2009-BLG-266 is also the first planetary microlensing event to be observed from a telescope in Heliocentric orbit, as it was observed for days by the High Resolution Instrument (HRI) of the Deep Impact (DI) spacecraft. The short duration and unfortunate timing of the space-based observations imply that these observations have only a weak constraint on the microlensing parallax measurement. But this data also indicates that future observations with the DI/HRI instrument in 2011-2014 will allow mass measurements of many microlens planets and their host stars, including the first mass measurements of planets in the Galactic bulge.

Chapter 22 Increasing the detection rate of low-mass planets and MOA–2006-BLG-130

Julie Baudry
University of Orsay, Paris Sud XI

Philip Yock
Department of Physics, University of Auckland
New Zealand

1 Introduction

Extra-solar planets detection is one of the applications of the gravitational microlensing phenomenon: a perturbation on the light curve can betray the presence of a planet orbiting the lens star [74, 75]. Most microlensing planets have been found in events of high magnification. Events of highest magnification occur when the lens star transits the source star, but these events provide relatively poor sensitivity to planets. Previous research concentrated on highest magnification events, but, on purely geometrical grounds, near-perfect alignement is relatively rare and there must be more events at low magnification than at high magnification. It will be presented in a first part the interest of monitoring events of lower magnification, in order to increase the detection rate of low-mass planets.

MOA-2006-BLG-130 (OGLE-2006-BLG-437) is a microlensing event that occurred in August 2006 which exhibited a slight perturbation on the light curve. In a second part, we will present the analysis of this event, in order to see whether or not it is a binary lens.

2 Planetary perturbation detectability and magnification

Events of lower magnification may be subdivided into two classes : those in which the planet lies closer to the Einstein ring than the length of either Einstein arc (type I), and those in which the planet lies further away (type II), Figure 1. Both type I and type II events enjoy good sensitivity to planets. In type II events, the planet is closer to the lens, and the planetary perturbation is greater when the magnification is higher. In type I events, the Einstein arc is affected by the proximity of the planet, and it will be shown that the sensitivity of these events to low-mass planets is almost independent of magnification. Using simulation programs of the MOA@UoA project [76], producing theoritical light curves, it will be demonstrated that the planetary perturbation is reasonably detectable in type I events, even at low magnification. It follows that monitoring further events of this type could increase the detection rate of low-mass planets.

Figure 1: Different types of events, depending on the geometrical construction of the lens.

3 Moa-2006-Blg-130

This event, for which the data light curve is given in Figure 2, was tested with the MOA@UoA simulation code. We used a marginalisation method, to test 27 binary lens, operation being repeated for each for 17 values of the source track angle. Results and analysis method will be presented. However, the coverage of this event was really sparse, so other phenomena were considered as possible cause of the perturbation.

Figure 2: Data light curve for MOA-2006-BLG-130, from the MOA microlensing alerts [77].


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Chapter 23 The Keplerian Orbit of OGLE-2008-BLG-513/MOA-2008-BLG-401

Jennifer Yee
Ohio State University

The light curve of OGLE-2008-BLG-513/MOA-2008-BLG-401 is characterized by two strong peaks due to companion to the lens with a mass ratio of . The timescale of this event is fairly long ( = 32 days) and the span of the 2-body perturbation is days because the caustic is resonant (the separation between the lens star and its companion in Einstein radii is ). Because of the timescales involved we were able to measure both the parallax effect and the orbital motion of the lens. Using information from the parallax and orbital motion effects, we find that the companion is a 7.8 giant planet orbiting an 0.3 host with a semi-major axis of 2.2 AU.

We initially fit the data with a static, 2-body lens model including parallax effects but were unable to find a model that satisfactorily data. We are able to fit the data successfully by allowing orbital motion in the 2-body lens. We use two different models: the first uses only the lens velocity projected onto the plane of the sky and the second uses a full Keplerian description for the lens motion. We find that the full Keplerian model is slightly preferred ( improvement of  10). This preference for full Keplerian motion indicates that we are measuring the curvature of the orbit during the event and allows us to place constraints on most of the orbital parameters.

The source star in this event is unusually blue compared to the other stars in the field: it is 0.95 mag bluer than the red clump. Given its blue color, one possibility for the source is that it is an A dwarf in the bulge or behind the the bulge in the far disk. An alternative hypothesis is that the source is well in the foreground, in front of most of the dust. In this case, it would be a low mass dwarf that appears blue because it experiences much less reddening than the other stars in the field. Various inconclusive arguments can be made supporting each hypothesis, drawing on the observed relative proper motion, astrometry, source limb-darkening, and arguments about the observed color and magnitude of the source. However, in our analysis of the light curve, we find that in order for the orbit of the lens planet to be bound, the source must be a nearby M dwarf. If the source star is a nearby, metal-poor thick disk star, this explanation accounts for almost all of the information we have on the source. In order to clarify the nature of the source, we have applied for and been granted time on the VLT to get a spectrum of the source. Given our orbital motion observations, we predict that the source will be a low-mass star.

Regardless of the nature of the source, the mass ratio between the companion and the lens star is robust. A mass ratio of 0.027 is very rare among binary/brown dwarf/planetary systems regardless of the specific nature of the companion. If the spectrum confirms our interpretation of the event, this will be yet another example of a massive planet orbiting an M dwarf. Such planets are extremely difficult to form using present theories of planet formation. The core accretion theory predicts that such planets should be extremely rare because M dwarfs have less massive disks and short orbital timescales at the snow line where giant planets should form. The gravitational instability theory is able to form giant planets around M dwarfs but only at much larger distances than we observe in this system. This planet, and others like it, increasingly form a challenge to the accepted theories of planet formation.

Chapter 24 Planetary microlensing event MOA-2010-BLG-328

Kei Furusawa
Nagoya University

1 Observation

Microlensing Observations in Astrophysics (MOA) monitors 40 degree of the Galactic bulge to find Microlensing events by 1.8m MOA-II telescope in New Zealand. Our real-time anomaly alert system checks new data points of each of microlensing events after five minutes of observation to find deviations due to planets. MOA has discovered 607 microlensing events and some anomaly events by the real-time anomaly alert system toward the Galactic bulge in 2010.

The microlensing event MOA-2010-BLG-328 was detected and alerted by MOA on 16 June 2010. Around UT 12:30 27 July, we found the deviations from single lens light curve of the event by anomaly alert system and issued the anomaly alert. Two days after the anomaly alert was issued, the deviations could be seen more clearly and David Bennett sent his preliminary planetary model. Following his e-mail, follow-up groups, FUN: CTIO (Chile), Farm Cove Observatory (New Zealand),Palomar Observatory (CA, USA), PLANET: Canopus (Australia), RoboNet: Faulkes Telescope North (Hawaii, USA), Faulkes Telescope South (Australia), Liverpool Telescope (Spain), MiNDSTEp: SAAO (South Africa), Danish (Chile) conducted intensive observation. Fortunately, the event occurred in OGLE-IV monitored fields so we got also OGLE data.

2 Analysis

The light curve is fitted by Markov Chain Monte Carlo (MCMC). Initial parameters for best fit model search are used over the range of the planet-star mass ratio () log and planet-star separation () log, the total number of initial parameters is 858.

We estimate the dereddened source color and magnitude from fitting and color magnitude diagram (CMD). Then the estimated source star is G type star and we adopt Limb Darkening parameter of G type star.

Currently, as the result of the detail analysis of the light curve, the mass ratio and the separation are yielded , respectively. It remains possible that the light curve also shows the microlensing parallax which is caused by the orbital motion of the Earth around the Sun during the event. The parallax effect enables us to estimate the physical parameters of the planet.

3 Conclusion

We analyzed planetary microlensing event MOA-2010-BLG-328. As the result of this analysis, the lens has a companion with planet mass ratio. This result is preliminary because the parallax signal is very sensitive to systematic errors of data. The data are gradually improving, so we continue evaluating parallax in detail

Chapter 25 Binary microlensing event OGLE-2009-BLG-020 gives orbit predictions verifiable by follow-up observations

Jan Skowron
Department of Astronomy, Ohio State University

1 Introduction

We present the first example of binary microlensing for which the parameter measurements can be verified (or contradicted) by future Doppler observations. This test is made possible by a confluence of two relatively unusual circumstances. First, the binary lens is bright enough () to permit Doppler measurements. Second, we measure not only the usual 7 binary-lens parameters, but also the “microlens parallax” (which yields the binary mass) and two components of the instantaneous orbital velocity. Thus we measure, effectively, 6 ’Kepler+1’ parameters (two instantaneous positions, two instantaneous velocities, the binary total mass, and the mass ratio). Since Doppler observations of the brighter binary component determine 5 Kepler parameters (period, velocity amplitude, eccentricity, phase, and position of periastron), while the same spectroscopy yields the mass of the primary, the combined Doppler + microlensing observations would be overconstrained by degrees of freedom. This makes possible an extremely strong test of the microlensing solution.

We also introduce the uniform microlensing notation for single and binary lenses, define conventions, summarize all known microlensing degeneracies and extend set of parameters to describe full Keplerian motion of the binary lenses (see Appendix A in the main paper Skowron et al. 2011, in prep.).

2 Observational data

On 15 February 2009, heliocentric Julian Date (HJD) 2454878, the Optical Gravitational Lensing Experiment (OGLE) team announced ongoing microlensing event OGLE-2009-BLG-020 detected by Early Warning System (EWS) and observed on the 1.3m Warsaw Telescope in Las Campanas Observatory in Chile. The event was also monitored by the Microlensing Observations in Astrophysics (MOA) 1.8m telescope on Mt. John in New Zealand. On HJD’ 4915 () it could be seen that light curve was deviating from the standard Paczyński model, and follow-up observations by other telescopes began. Intense monitoring of the event was performed until HJD’ 4920. During this period source crossed caustic twice and both crossings where observed.

In this work, beside the data obtained by survey telescopes, we use follow-up data from 8 additional observatories: the 2.0m Faulkes South (FTN), the 2.0m Faulkes North (FTN), the 36cm telescope at Kumeu Observatory, the 36cm telescope at Bronberg Observatory, the 1m telescope on Mt. Canopus (Utas), the 36cm telescope in Farm Cove Observatory (FCO), the SMARTS 1.3m Cerro Tololo Inter-American Observatory (CTIO), and the 40cm telescope in Campo Catino Austral Observatory (CAO). Light curve of the event is shown on Figure 1.

Figure 1: Light curve and best-fit model of OGLE-2009-BLG-020.

The whole light curve consists of 9 years of data with 121 days during the course of the visible magnification, of which 5 days constitute intense follow-up observations. The OGLE telescope performed observations in and bands, and CTIO telescope in , and bands, which permit measurements of the color of the magnified source star. Together we have 5333 data points in the light curve with 2247 during the course of the magnification event.

3 Microlensing fit with full Keplerian orbit

To find a microlensing model we use the method developed by S. Dong and described in Dong et al. (2006) and its modification in Dong et al. (2009, Section 3). Although we introduce a few changes in parametrization of the model. In order to allow comparison with the radial velocity (RV) measurements it is profitable to use the full Keplerian orbit parametrization. In addition to being more accurate, the additional advantage of this approach is to avoid all unbound orbital solutions (with eccentricity ) and to enable introduction of priors on the values of orbital parameters directly into MCMC calculations, if one decides to adopt that approach.

We describe the orbit of the secondary component relative to the primary by giving 3 cartesian positions and 3 velocities at one arbitrarily chosen time (). See Figure 2. Our extended microlensing model (with parallax parameters) carries information about the mass ratio, the total mass of the lens, and the physical scale in the lens plane, so together with the six instantaneous phase-space coordinates it comprises a complete set of system parameters (except systemic radial velocity). This, for example, allows us to calculate the relative RV at any given time. We can use this for comparison with the RV curve from follow-up observations.

Figure 2: Definition of the phase-space parameters at , describing motion of the secondary binary lens component () relative to the primary (). Vertical plane is the plane of the sky.

It is possible to calculate all properties in physical units as well as standard Kepler parameters of the orbit – i.e., eccentricity (), time of periastron (), semi-major axis () and 3 Euler angles: longitude of ascending node (), inclination () and argument of periapsis (). (See Figure 3 for illustration of the orbital parameters and they relation to the phase-space parameters specified at the fixed time .)

Figure 3: Figure shows an example of the relative binary orbit which is rotated by three Euler angles. The binary components at the time are marked with 2 dots. The axis points toward the observer; axes marked with symbols and define the plane that is parallel to the plane of the sky and crosses the primary component of the binary. The portions of the line that lie behind the plane are dashed. The base coordinate system is related to the microlensing event such that at the time the first axis coincides with the binary axis projected onto plane of the sky.

Likelihoods obtained from MCMC procedure suffer from degeneracy between one of the parallax parameters ( – perpendicular to the Earth acceleration) and angular velocity of the lens in the plane of the sky. We break this degeneracy by choosing subset of solutions which yields observed brightness of the lens consistent with the theoretical main-sequence isochrone (we assume that all blended light is coming from the lens, and, with mass ratio , we expect that the secondary component of the lens does not contribute to the light). This leads to estimations of the lens distance and the mass of its primary component:


The observed magnitudes of the blend and the source are:


4 Test with radial velocity

Every link in the MCMC chain consist a complete set of parameters of the binary lens. It yields not only the mass, distance and separation in physical units, but also all Keplerian parameters of the orbit. Thus we can calculate the RV curve for any period of the binary.

Assuming the observations of the RV curve of the primary are taken, we can assign to every link the likelihood that it is consistent with those data. We derive the radial velocity for any time the data were taken and, allowing for systemic velocity of the center of mass of the binary to vary, we calculate the likelihoods for every point. In this way we construct new set of links weighted by, both, the microlensing light curve and radial velocity curve. This new set of solutions yields new, most probable, values of all lens parameters, which may or may not coincide with the values derived from the microlensing only solution. This would be a test of microlensing solution.

If new values of parameters will lie outside 3- limits of microlensing solution we can say that our method failed. By contrast, if solutions, which are consistent with the RV curve, lie near the best-fit values we obtained from microlensing we can not only believe our solution, but we can also read all parameters of the binary, which are not given by one or the other method alone. For example, radial velocity curve will give us period and systemic radial velocity, which we cannot read from the microlensing light curve alone. However, microlensing will yield the inclination, orientation on the sky and the 2-d velocity of the binary projected on the plane of the sky.

4.1 Useful information for RV observations

The selective power of the RV curve holds when the observation are taken through at least one period of the binary. We propose a 2 year span of observations since the period we derive is between 200 and 700 days (2- limit).

The binary coordinates are (, , ). The finding chart can be found on the OGLE EWS webpage111http://ogle.astrouw.edu.pl/ogle3/ews/2009/blg-020.html.

The binary is blended with the “microlensing source” which, as its position on the CMD suggests, is a Galactic bulge giant. The binary has a mass ratio of , so assuming both components are the main-sequence stars, majority of the light is coming from the primary. The binary is magnitude brighter in and mag brighter in than the giant, however they are of similar brightness in .

Since the binary is located in the Galactic disk, we anticipate it will be clearly separated in the velocity space from the blended Bulge giant. The radial velocity amplitude of the primary is expected to be of order of a few .

5 Conclusions

The binary star that manifested itself in the microlensing event OGLE-2009-BLG-020 is the first case of a lens that is close enough and bright enough to allow ground-based spectroscopic follow-up observations. This makes it a unique tool to test the microlensing solution.

We derive lens parameters using the same method by which majority of planetary candidates discovered by microlensing are analyzed. We detect a signal from the orbital motion of the lens in the microlensing light curve. This signal, as well as our measurements of the orbital parameters of the binary lens, can be confirmed or contradicted by the future observations.

Combining the microlensing solution with the radial velocity curve will yield a complete set of system parameters including 3-d Galactic velocity of the binary and all Kelperian orbit elements.

This work undertakes an effort to establish a uniform microlensing notation, extending work of Gould (2000) by including full set of orbital elements of the binary lens (see Appendix A in the paper). We also summarize all known microlensing degeneracies.

The method of deriving orbital elements from the 6 phase-space coordinates, used to parametrize microlensing event, is described in Appendix B in the paper.


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Theoretical investigations

Microlensing and planet populations - What do we know, and how could we learn more
Martin Dominik
The frequency of extrasolar planet detections with microlensing simulations
Rieul Gendron & Shude Mao
A semi-analytical model for gravitational microlensing events
Denis Sullivan, Paul Chote, Michael Miller
GPU-assisted contouring for modeling binary microlensing events
Markus Hundertmark, Frederic V. Hessman, Stefan Dreizler
Red noise effect in space-based microlensing observations
         Achille Nucita, Daniele Vetrugno, Francesco De Paolis, Gabriele Ingrosso,
         Berlinda M. T. Maiolo, Stefania Carpano
Light curve errors introduced by limb-darkening models
David Heyrovsky
Isolated, stellar-mass black holes through microlensing
         Kailash Sahu, Howard E. Bond, Jay Anderson, Martin Dominik,
         Andrzej Udalski, Philip Yock
The observability of isolated compact remnants with microlensing
Nicola Sartore & Aldo Treves
Gravitational microlensing by the Ellis wormhole
Fumio Abe
The deflection of light ray in strong field: a material medium approach
Asoke Kumar Sen
Rapidly rotating lenses - repeating orbital motion features in close binary microlensing
Matthew Penny, Eamonn Kerins, Shude Mao

Chapter 26 Microlensing and planet populations - What do we know, and how could we learn more?

Martin Dominik
SUPA, University of St Andrews, School of Physics & Astronomy
United Kingdom

1 Gravitational microlensing campaigns

Following a concrete suggestion brought up by Bohdan Paczyński [82], the first dedicated gravitational microlensing surveys [83, 84] were designed to measure a potential clumpy dark matter content in the galactic halo by means of observations towards the Magellanic Clouds. Observations towards the Galactic bulge have then been proposed for robustly testing whether the experimental set-ups are able to detect signals [85, 86], given that a possible negative result towards the Magellanic Clouds would remain ambiguous. Wider-reaching astrophysical applications of gravitational microlensing simply became add-ons to existing opportunities, and have subsequently shaped the evolutionary process of experiments. Exploiting the discovery potential of planets orbiting stars other than the Sun [87] by means of follow-up observations on ongoing microlensing surveys [88] became possible with real-time processing and public dissemination of data [89, 90]. The explicit goal to detect planets by microlensing was in particular engraved in the acronym of the Probing Lensing Anomalies NETwork (PLANET) [91].

2 Efficiency and planet population statistics

Planetary abundance statistics cannot be extracted simply from counting claimed detections. We need to understand what the characteristics of the monitoring campaigns are, which might have changed over time. In fact, the observed sample is a probabilistic realisation of the product of the underlying planet population and the detection efficiency of the experiment(s). This however means that the considered sample and the detection efficiency need to refer to the same detection criterion and monitoring strategy. Not only does the detection efficiency need to be evaluated in a statistical meaningful way, but also do the ”detections” need to be counted accordingly [92].

Moreover, not suprisingly, with different approaches being followed by the various surveys and follow-up monitoring campaigns, their detection efficiencies are substantially different functions of the planet mass and orbital separation . For the most massive planets, there is not much gain with less than daily sampling, so that survey experiments with their larger number of observed events have the largest sensitivity [93, 94], whereas less massive planets with their shorter signals become undetectable by falling into the data gaps. With regular quasi-continuous high-precision round-the-clock monitoring, as carried out by PLANET [95], RoboNet [96, 97], or MiNDSTEp [98], the planet detection efficiency drops roughly as , until the finite size of the source stars at a few Earth masses causes a faster drop-off and subsequently undetectability [99, 100]. If, like FUN, one focuses on highly-magnified peaks, probing the central caustic close to the lens star, the detection efficiency is close to 100 % for a wide range of planetary masses, with just the range of orbital separations shrinking towards lower masses, and it then falls to zero rather suddenly [101].

The history of claimed microlensing planet detections seems to follow a pattern that matches the characteristics of the different campaigns: A first super-Jupiter from the OGLE and MOA surveys [102], the 5-Earth-mass planet OGLE-2005-BLG-390Lb from PLANET observations [103], and planets from high-magnification peaks monitored by FUN [104, 105]. Moreover, a larger number of detections in the region between Neptune and Saturn [106, 107] arose in the more recent years due to increased survey cadence.

However, rough planet abundance estimates give rather puzzling results. A well-defined sample of events with peak magnifications , densely covered mostly by FUN data [101] indicates an abundance for massive gas giants that is about 10 times larger than what one would estimate from PLANET observations and the corresponding detections [108]. Despite the fact that both estimates are statistically compatible within their large uncertainties, these findings are somewhat surprising given that the PLANET observations are more powerful in this mass range, but have provided less detections. In fact, the 4-year FUN sample contains just 13 events, which means that the adopted strategy is not powerful enough for providing a planetary mass function in foreseeable time; increased efforts of monitoring a larger number of events at more moderate magnifications, as follow-up efforts or high-cadence surveys, are required [108]. The detection of a planetary signal in event MOA-2009-BLG-266, resembling the path that led to OGLE-2005-BLG-390Lb, clearly demonstrates the power of the regular follow-up approach (which in fact is just copied by the MOA high-cadence monitoring).

With the current small number of planetary detections, microlensing observations are still in their very infancy on the path to determining planet abundances, and any kind of planetary mass function is a long way ahead. A particular complication arises from the fact that the planet abundances strongly depend on the properties of the parent star, and microlensing events come with a probabilistic mixture of host stars. Should more massive stars come with more or less planets of a certain kind, what planetary mass function are we talking about? Distinguishing between masses of planet host stars requires a much larger sample for drawing meaningful conclusions. Stellar metallicity is a further relevant parameter that is difficult to control, and the fact that microlensing probes both Galactic disk and bulge stars is a challenge as well as an opportunity.

While planetary microlensing naturally depends on the dimensionless separation parameter and the planet-to-star mass ratio , planet formation does not obey such a scaling law, regardless of whether planet formation theories give us good guidance or not. Moreover, the adoption of any specific functional form, just for its simplicity, that is not clearly supported by the data, may not be adequate to probe certain features. In particular, smooth monotonic functions are ill-suited to describe planetary deserts or cut-offs.

3 Moving forward

Maximizing the number of detections without adopting well-defined monitoring procedures might lead to further spectacular findings, but will be useless for studying planet populations. High-cadence surveys have already proven their extended sensitivity to planets with smaller masses. In order to obtain meaningful results on planet population statistics, deviations from a deterministic survey programme must be made in a deterministic way only as well. Otherwise, claimed detections cannot be used to gather statistical information. It has also been demonstrated that the automated detection of anomalies [109] increases the detection efficiency, and in particular allows for the realistic detection of planets of Earth mass and below, while keeping the deterministic process chain intact. The real-time identification of potential deviations from ordinary light curves is essential for the current ability to push further down in mass, but if we want to determine the abundance of such planets as well, this identification must not be left to human judgement. If planets of Earth mass and below are common, recent survey data give rise to speculating that opportunities have been missed for claiming such a detection. If we want to make an impact now, the real-time provision of photometric data within minutes (not within hours) is likely to make the crucial difference. Beyond that, and in the longer term, precision photometry on fainter stars would allow for far better statistics on low-mass planets.


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Chapter 27 The frequency of extrasolar planet detections with microlensing simulations

Rieul Gendron
Jodrell Bank Centre for Astrophysics, University of Manchester
United Kingdom

Shude Mao
Jodrell Bank Centre for Astrophysics, University of Manchester, UK
National Astronomical Observatories, Chinese Academy of Sciences

It was first proposed by [110] that gravitational microlensing could be used to detect extrasolar planets. To date, over 500 have been discovered, with microlensing planets, although relatively few in number, probing a region of the mass versus semi-major axis plane that is currently out of reach of the other methods, with the sensitivity peaking just beyond AU. Microlensing planets are therefore very useful for placing constraints on planet formation models.

We simulate light curves for 1000 extrasolar planet systems around host stars of mass drawn from the Ida & Lin core accretion models of planet formation [111]. The simulated data is first fitted with a single-lens model. If the fit is poor, combinations of the planets most likely to be causing perturbations are placed around the star and new light curves are generated to attempt to reproduce the original light curve.

We found that the majority of the light curves were well-fitted by single-lens models. However, 26 were not and hence were determined to contain pertubations due to at least one planet. Of the 26 interesting light curves, it was found that 16 could be explained by single planets. A greater number of planets were necessary to explain the remaining 10 cases. These results place an upper limit on the number of planets that would be deduced by a fitting procedure since it might be possible to produce the original light curve with a fewer number of planets.


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Chapter 28 A Semi-Analytical Model for Gravitational Microlensing Events

Denis Sullivan & Paul Chote, Michael Miller
School of Chemical & Physical Sciences, Victoria University of Wellington
New Zealand

1 Introduction

This paper describes the latest version of the software we have developed for modelling gravitational microlensing events that involve multiple lensing masses. Of particular interest is the modelling of planetary microlensing events requiring two or more lensing masses.

The basic thin lens equation relating a point source image position, , to the positions, , of the multiple images created by the lensing masses is simple enough,

The positions in this expression are in units of the Einstein ring radius corresponding to the total lens mass. Except for the single lens case, inverting the equation in order to determine each image position directly is not a trivial exercise. In fact, the inversion procedure is relatively complicated for even the two-mass lens and it rapidly becomes unmanageable when the number of lensing massses increases much beyond two.

2 Inverse Ray-tracing

The above inversion difficulty has led to wide-spread use of the “brute force” approach called inverse ray tracing: “all” potential positions in the image plane are checked in order to determine which ones are consistent with a given source position. Each determination is straight forward since it is clear that a simple substitution of an image position in the above lens equation yields a unique source position origin.

Due to the potentially large number of calculations required, access to significant computing power makes this technique a practical proposition, but it can also be improved by use of efficient image plane search techniques or better still recording all image plane, source plane correspondences in order to create a so-called magnification map. This map can be used to generate a microlensing light curve, since the source magnification at a given position can be directly estimated from the area density of points (inverse rays) in the source plane relative to the density in the image plane. A particular light curve can be rapidly computed by calculating the changing “ray” density ratio for any chosen source track. This method includes finite source size effects directly along with source limb darkening if required, by weighting ray densities in the source disk. Observer/source parallax effects can also be included via appropriately curved source tracks. However, internal lens configuration changes will alter the magnification map and accuracy depends directly on the density of rays covering the image plane, which of course has computational implications.

Modelling code using this methodology was developed as part of a PhD programme at VUW [112] and it has been used to model a range of planetary microlensing events. Recently, we decided to pursue the approach of directly inverting the lensing equation. One particular motivation for this work was to enable satisfactory modelling of lens motion effects, but the overall approach is also more efficient, at least for systems with no more than four lenses.

3 The Polynomial Method

The algebra is greatly simplified by representing the two dimensional plane positions using complex coordinates as first introduced by Witt [113]. The lens equation relating source position () to image positions () for lenses () then becomes:

Combining both forms of this lensing equation into one equation for image position for a two mass lens system produces a complex fifth order polynomial, whose five roots are potential image positions. Back substitution is required in order determine if a root corresponds to a real image position, as there can be three or five images depending on the geometry. In general, the complex fifth order polynomial can only be solved numerically but there are algorithms to accomplish this. We use the method of Jenkins and Traub [114] in combination with root polishing using the Laguerre method [115] to obtain accurate root values.

Given that the light bending in a gravitational field does not change the intrinsic source brightness, the point source magnification at each image position, , is given by the ratio of the infinitesimal area element in the image plane to the corresponding elemental area in the source plane at the position . This quantity is given by the determinant of the Jacobian for the transformation of the point in the image plane to the point in the source plane, as governed by the above lensing equation. But, it is the ‘Jacobian’ () of the inverse transformation from the source position to the image position that is readily determined and, given the geometrical nature of the quantity, the absolute value of the inverse provides the value of interest. We have:

The point source magnification of each image formed at the position is therefore and the combined magnification for all unresolved images is simply the sum over these quantities.

The critical curves (and the corresponding caustics) are the locii of points where the point source magnification formally diverges and are thus determined from the requirement that this inverse Jacobian is zero; this leads to a set of fourth order complex polynomials for the two mass lens system.

Assembling the expressions that give the coefficients of the complex polynomials is rather tedious [116] and a symbolic algebra computer package such as Maple is of considerable assistance. However, we have found that (substantial) further manual manipulation can lead to more compact and efficient expressions. This is particularly the case for lensing systems with three and even four masses. And, we have included both these cases in our code. The three-mass case yields a complex order polynomial [116], while the four-mass system leads to a order polynomial requiring the determination of no less than 18 complex coefficients. The critical curve polynomials for these systems are of and order respectively.

Finite source effects become important when the source disk is near a caustic curve and are important in planetary events. We have included these in the software package by using a combination of two methods: a hexadecapole expansion and a numerical integration procedure we call the polygon method. The hexadecapole expansion method [117] employs multiple point source magnification values to estimate the magnification resulting from a finite source, including limb darkening effects if required. The polygon method is based on the work of Gould and Gaucherel [118] and uses polygons to represent the source disk and image figures and thereby estimate the areas. Limb darkening effects are included in this method by using a network of “concentric” polygons.

A method promoted by Bennett [119] for finite source effects first employs the polynomial approach to locate the image centres and then adopts the inverse ray-tracing method in the vicinity of each image to estimate the limb-darkening integrals over the images. We are looking at implementing this approach as well.

4 Conclusion

In spite of the expression complexity in calculating the lensing polynomial coefficients, we have found our new code to be competitive speed-wise with the inverse ray-tracing approach. The rapidly rising level of complexity and the large numbers of terms in the coefficient expressions makes the polynomial method impractical for more than four lensing masses. For lenses the order of the polynomial is , so for lenses we have 27 coefficients to determine. Furthermore, we need to employ numerical methods to extract 26 complex root values from the order polynomial and determine which ones correspond to real image positions. It is reasonably clear that the inverse ray-tracing approach will be more efficient for these cases, at least for examples with no lens motion.

When lens motion is involved, the polynomial method is both more efficient and faster than inverse ray tracing. A three-mass lensing system consisting of a star and two planets has already been discovered and successfully modelled [120], and it could be the case that a system of a star and three planets is found and usefully modelled. But it is probably the case that endeavouring to model a microlensing light curve with five lensing masses or more (and perhaps even four) is very unlikely to produce useful physical information.


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Chapter 29 GPU-assisted contouring for modeling binary microlensing events

Markus Hundertmark, Frederic V. Hessman, Stefan Dreizler
Institut für Astrophysik, Georg-August-Universität Göttingen

1 Introduction

Due to the continuously improving computer power available in dedicated hardware, clusters and computing grids, real-time binary modeling is now being tested and used by different microlensing groups. In addition to CPU-based computing, one can exploit the possibility of speeding-up calculations using graphic cards which are available at reasonable prices and have a reduced carbon footprint. These Graphics Processing Units (GPUs) offer a increased capability concerning the number of floating point operations per second but at the cost of careful flow control management. Naturally, these GPUs are well suited for ray-tracing but in this work a pointwise approach for simulating lightcurves on the basis of contouring techniques is also possible, despite some system dependent drawbacks concerning the accuracy.

2 GPU-contouring

The use of GPUs for gravitational microlensing simulations seems natural as their design is driven by the needs of intensive ray-tracing computer games applications. The advantages and disadvantages when used for scientific ray-tracing have been shown, e.g. by [1],[2]. One bottle-neck for a fast fitting environment is the limited data transfer capacity between CPU and GPU. In order to ease the burden of transfer back and forth to the GPU a large fraction of the analysis must be executed on the GPU and needs to be optimized for these machines, since they offer the highest performance for standard single-precision floating point numbers.

As an alternative, we have implemented the contouring technique on GPUs aiming to preserve the pointwise calculation of lightcurves by parallelizing the root-finding and subsequent integration of the image contours as by [3].

Figure 1: Two-grided polar parameterization of the lens plane for root-finding.

Due to the technical limitations of a computing system with many processing units, the number of required computations and variables has to be kept low. This comes at the price of careful flow control management. For this purpose, the squared deviation between lens and source position as introduced by [4] has been used. A point-wise parallelisation of the root-finding was achieved by solving the lens equation on a polar grid (cf. [5]), where the radial search for potential roots was carried out as one thread, as illustrated in Fig. 1. Afterwards, the elements are integrated with a simple Gaussian quadrature. This approach is particularly efficient for planetary events with modest angular separations and source positions . A test implementation on a system with 240 streaming processors111NVIDIA® Tesla™ C1060 is able to simulate a lightcurve consisting of 1000 points in one second. Root-finding threads cannot communicate in this approach and thus it would be more difficult to implement a hierarchical approach as by [6].

In contrast to existing CPU-based solutions, the demanded accuracy cannot always be achieved, as most commonly used GPUs exclusively compute with single-precision222IEEE 754 single-precision (32 bits) floating numbers or require a substantial overhead for operating at double-precision. On single-precision machines the root-finding accuracy is limited by in the radial direction, which directly affects the maximal achievable accuracy and accessible sources star radii . Estimating the magnification as an integration of evenly sized squares in search directions per grid gives at zeroth order an uncertainty of


Models for ground-based microlensing lightcurves require a model accuracy of better than and thus the applicability of the model is limited to source star radii . Beyond that limit, only relative precision can be guaranteed. The effect of missing solutions at the edges is and thus the aforementioned number of elements is a compromise between runtime, single-precision round-off error, and numerical uncertainty of the integration. The benefits of refining the grids for keeping the requested accuracy is shown in Fig. 2. Comparing current simulations with magnification maps [7],[8], indicate that the numerical accuracy stays below at one sigma level as demanded.

Figure 2: Lightcurves simulated with GPU-assisted contouring are shown for different source star radii with a constant number of integration elements and for a number of integration elements adapted to a predefined accuracy limit.

3 Summary and further work

A first point-wise implementation of the contouring technique on GPUs has been achieved. The rapidly increasing number of streaming processors and the recently introduced double-precision capability333NVIDIA® Fermi™ supports the development of these systems as the existing software can be scaled to a new state-of-the-art equipment just by recompiling. In addition, this approach can be used with existing sophisticated fitting codes and the simulation can even be mixed with other pointwise contouring techniques which are more suitable for highly accurate simulations, such as the system introduced by [9].

M.H. would like to acknowledge the support from the DFG and the DFG Research Training Group GrK - 1351 “Extrasolar Planets and their host stars”.


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Chapter 30 Red noise effect in space-based microlensing observations

Achille A. Nucita, Daniele Vetrugno, Francesco De Paolis, Gabriele Ingrosso, Berlinda M. T. Maiolo
Department of Physics, University of Salento, Lecce, Italy
INFN, Sez. di Lecce, Lecce, Italy

Stefania Carpano
ESA, Research and Scientific Support Department, ESTEC, Noordwijk
The Netherlands

1 Introduction

Gravitational microlensing, first developed to investigate the distribution of Galactic dark matter [1], represents now a powerful and important tool to explore many other astrophysical contexts. For example, in the last years, the possibility to investigate the distribution of exo-planetary systems by means of the microlensing has been pointed out to the scientific community. The advantages for using it are huge and span from the high sensitivity to relatively low mass planets and large observer-lens distances to the unique possibility of detecting free-floating planets. Gravitational microlensing surveys may also give the possibility of a planetary census throughout the Galaxy [2]. Another technique used to search for exo-planets is that of detecting transit events by means of the differential photometry analysis of the sources.

Both transits and microlensing techniques are very close from the observational point of view, since both are photometric-based and the observer has to search for deviations from the baseline flux. In the transit case one seeks for a decrease in the signal flux while for microlensing events an amplification is present. In both cases, the analysis is to be made on light curves.

To this aim, we decided to study and characterize the noise (typically red or pink noise which could mimic a microlensing feature) of a sample of light curves from a space-based experiment. In this prospective we have considered a sample of light curves taken from the CoRoT111More detailed information are avaible at the web site http://smsc.cnes.fr/COROT/ satellite, a space-based experiment projected by the “Centre National d’Etudes Spatials” (CNES) in collaboration with the French laboratories and other international partners (Europe, Brasil) launched in December .

D.Vetrugno acknowledges support from the Faculty of the European Space Astronomy Centre (ESAC) and for the kind hospitality. The CoRoT space mission, launched on 2006 December 27, was developed and is operated by the CNES, with participation of the Science Programs of ESA, ESA’s RSSD, Austria, Belgium, Brazil, Germany and Spain.


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Chapter 31 Light curve errors introduced by limb-darkening models

David Heyrovský
Institute of Theoretical Physics, Charles University in Prague
Czech Republic

During the analysis of caustic-crossing microlensing events the source star is typically modeled by analytical limb-darkening laws - most often linear, in a few cases higher-order laws (square-root, quadratic). An alternative option is to use a non-analytical limb-darkening model based on principal component analysis of model atmospheres [121, 122], which gives a more accurate description of the source-star intensity profile [123]. To guide the choice of limb-darkening model during event analysis we need to know the photometric error introduced by assuming a particular model. In order to answer this question, we compute single-lens caustic-crossing light curves for stellar model atmospheres from Kurucz’s ATLAS9 grid in a representative set of photometric bands, and compare them with light curves of sources with different approximations of the underlying limb darkening. We demonstrate both the amplitude of the introduced photometric error and the corresponding residual pattern. We discuss the implications of our results for the analysis of single and two-point-mass caustic-crossing events.


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Chapter 32 Detecting Isolated Stellar-Mass Black Holes through Astrometric Microlensing using HST

Kailash C. Sahu, Howard E. Bond, Jay Anderson
Space Telescope Science Institute, Baltimore, USA

Martin Dominik
School of Physics & Astronomy, University of St. Andrews, St. Andrews, UK

Andrzej Udalski
Warsaw University Observatory, Warszawa, Poland

Philip Yock
Department of Physics, University of Auckland, Auckland, New Zealand

1 Introduction

Stars with masses greater than are thought to end their lives as stellar-mass black holes (BHs) (e.g., Fryer & Kalogera 2001). Yet, there has never been an unambiguous detection of an isolated black hole, since they emit no radiation. The only detected BHs to date are through their X-ray emissions in binary systems, where their masses have been measured through radial velocity measurements. Solitary BHs are detectable with current microlensing searches like OGLE and MOA. These campaigns have already detected several BH candidates. Bennett et al. (2002) have described two events: MACHO 96-BLG-5 and MACHO 98-BLG-6, with mass estimates of solar mass. Mao et al. (2002) concluded that MACHO 99-BLG-22 is a BH candidate with a minimum mass of . However, all these claims are statistical in nature, since determination of the lens mass from the microlensing light curve alone suffers from a degeneracy between the distance to the lens, the mass of the lens, and the source-lens relative proper motion.

2 Black Hole Detections through Microlensing

Microlensing survey programs typically detect microlensing events towards the Galactic bulge per year. A few of them have long durations, and these long-duration events ( days) could potentially be arising from lenses with masses larger than 10 . For most of them, the light from the lens itself is negligible. So it has been suggested that a majority of them are due to massive, non-luminous stellar remnants. However, a long-duration microlensing event can be also by a low-mass lens passing in front of the source with a small relative motion.

Figure 1: The lower panel shows a photometric microlensing light curve with minimum impact parameter u=0.05, and the upper panel shows the corresponding expected astrometric shifts. The astrometric shift peaks at u1.4, which is about 1.4 milliarcsec for a BH lens with a mass of 5 in the Sagittarius arm. The timing of our 5 proposed observations is shown by red dots in the upper panel. These 5 measurements will unambiguously separate the shifts caused by microlensing from the shifts caused by blending and proper motion of the source.

3 Astrometry Resolves the Degeneracy

Microlensing not only causes a photometric amplification, but it also causes a small astrometric shift in the position of the source (see Fig. 1 for illustrations). This astrometric shift is generally undetectable from ground-based observations. But, for lenses with masses of , the astrometric shift is expected to be mas, which is easily detectable with HST. This astrometric shift, coupled with the parallax effect generally observed for the long-duration events, allows a determination of the mass of the lens unambiguously, thus resolving the degeneracy between the low-mass and the high-mass lenses (Sahu & Dominik 2001).

4 Mass Determinations of the BH Candidates and Implications

We expect to achieve an astrometric precision of 0.2 mas at each epoch of HST observations. Thus a shift of 1.4 mas as expected from a 5 BH would be detected at 7 sigma. The 0.2 mas error translates to an error of about 0.75 in mass measurement. Thus we can determine masses with about precision at 5 or at 10 .

Thus our program has the potential of making the first unequivocal detections of isolated black holes, and the first direct mass measurements for isolated stellar-mass BHs through any technique. We expect to observe 5 promising long-duration events with WFC3/HST, possibly leading to detection of some confirmed BHs. Detection of these BHs will provide the very first clues on the frequency of isolated BHs in the Galaxy, which have strong implications on the slope of the IMF at high masses.

This project uses the NASA/ESA Hubble Space Telescope, operated by AURA for NASA.


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Chapter 33 The observability of isolated compact remnants with microlensing

Nicola Sartore
INAF - Istituto di Astrofica Spaziale e Fisica Cosmica, Milano

Aldo Treves
Dipartimento di Fisica e Matematica, Universitá dell’Insubria, Como

1 Introduction

The observability of old isolated compact remnants is a long standing problem. To date, blind searches for neutron stars and stellar-mass black holes (NSs and BHs herafter) likely accreting from the interstellar medium (ISM) have failed to reveal any promising candidates.

Dark compact objects may still manifest themselfs through the effect of their gravitational field on the light of background sources, i.e. as microlensing events, in particular towards the Galactic bulge, where the density sources is larger. The contribution of NSs and BHs to the microlensing rate towards the bulge has been estimated by e.g. [129], [130] and [131]. Here we present a quantitative analisys of this contribution, under the hypothesis that both NSs and BHs are born with large kick velocities (see e.g. [132]) and compare this contribution with that of “normal” stars, that is brown dwarfs, main sequence stars and white dwarfs.

2 Model and Results

We built a model of the distribution of normal stars in the bulge and disk of the Galaxy (see [133] for a detailed description of the model and procedure adopted) and followed the orbits of synthetic bulge-born and disk-born isolated compact objects in the Galactic potential. Following [129], the normalization of the phase space distribution of NSs and BHs was estimated from the stellar initial mass function assuming that all stars with initial mass greater than are in the remnant phase. In particular, stars with inital mass between 8 and 40 solar masses are now NSs with , while stars with mass are now BHs with .

Then, we calculated the optical depth and the distribution of event time scales for normal stars, NSs and BHs towards different lines of sight (l.o.s. hereafter). The optical depth obtained was lower than the case in which NSs and BHs are born without kicks. This can be easily explained since objects with large velocity have a larger scale height. Thus, the spatial density along the l.o.s. is lower. On the other hand, a fast moving object is expected to give a higher rate of events with respect to an object with lower velocity. In fact, we found that the fraction of microlensing events due to isolated NSs and BHs is larger than that expected in the case of non kicks. In particular, for events longer than 100 days, we estimated that about of the events are likely due to NSs and BHs.

3 Conclusions and Future Prospects

Cosidering the large number of microlensing events observed by several microlensing surveys, our results suggest that a non-negligible fraction of these events is likely related to NSs and BHs, in particular among long duration events. Indeed, an excess of long duration events has been reported in the literature. Thus, microlensing surveys are a suitable tool to probe the statistical properties of old isolated compact remnants. However, the nature of single lenses has to be assessed as further confirmation. We are performing a cross-check of archival X-ray data to find counterparts of long duration events. The detection of X-ray emission would be a strong indication that the lens is a compact object.


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Chapter 34 Gravitational microlensing by the Ellis wormhole

Fumio Abe
Solar-Terrestrial Environment Laboratory
Nagoya University

1 abstract

A method to calculate light curves of the gravitational microlensing of the Ellis wormhole is derived in the weak-field limit. In this limit, lensing by the wormhole produces one image outside the Einstein ring and one other image inside. The weak-field hypothesis is a good approximation in Galactic lensing if the throat radius is less than km. The light curves calculated have gutters of approximately 4% immediately outside the Einstein ring crossing times. The magnification of the Ellis wormhole lensing is generally less than that of Schwarzschild lensing. The optical depths and event rates are calculated for the Galactic bulge and Large Magellanic Cloud fields according to bound and unbound hypotheses. If the wormholes have throat radii between and km, are bound to the galaxy, and have a number density that is approximately that of ordinary stars, detection can be achieved by reanalyzing past data. If the wormholes are unbound, detection using past data is impossible.

2 Introduction

A solution of the Einstein equation that connects distant points of space–time was introduced by [149]. This “Einstein–Rosen bridge” was the first solution to later be referred to as a wormhole. Initially, this type of solution was just a trivial or teaching example of mathematical physics. However, [157] proved that some wormholes are “traversable”; i.e., space and time travel can be achieved by passing through the wormholes. They also showed that the existence of a wormhole requires exotic matter that violates the null energy condition. Although they are very exotic, the existence of wormholes has not been ruled out in theory. Inspired by the Morris–Thorne paper, there have been a number of theoretical works (see [174, 155] and references therein) on wormholes. The curious natures of wormholes, such as time travel, energy conditions, space–time foams, and growth of a wormhole in an accelerating universe have been studied. Although there have been enthusiastic theoretical studies, studies searching for real evidence of the existence of wormholes are scarce. Only a few attempts have been made to show the existence or nonexistence of wormholes.

A possible observational method that has been proposed to detect or exclude the existence of wormholes is the application of optical gravitational lensing. The gravitational lensing of wormholes was pioneered by [146], who inferred that some wormholes show “negative mass” lensing. They showed that the light curve of the negative-mass lensing event of a distant star has singular double peaks. Several authors subsequently conducted theoretical studies on detectability [165, 142]. Another gravitational lensing method employing gamma rays was proposed by [171], who postulated that the singular negative-mass lensing of distant active galactic nuclei causes a sharp spike of gamma rays and may be observed as double-peaked gamma-ray bursts. They analyzed BASTE data and set a limit for the density of the negative-mass objects.

There have been several recent works [167, 161, 159, 163, 148] on the gravitational lensing of wormholes as structures of space–time. Such studies are expected to unveil lensing properties directly from the space–time structure. One study [148] calculated the deflection angle of light due to the Ellis wormhole, whose asymptotic mass at infinity is zero. The massless wormhole is particularly interesting because it is expected to have unique gravitational lensing effects. The Ellis wormhole is expressed by the line element


where is the throat radius of the wormhole. This type of wormhole was first introduced by [151] as a massless scalar field. Later, [157] studied this wormhole and proved it to be traversable. The dynamical feature was studied by [168], who showed that Gaussian perturbation causes either explode to an inflationary universe or collapse to a black hole. [147] showed that the tchyon condensate can be a source for the Ellis geometry.

In this paper, we derive the light curve of lensing by the Ellis wormhole and discuss its detectability. In Section 2, we discuss gravitational lensing by the Ellis wormhole in the weak-field limit. The light curves of wormhole events are discussed in Section 3. The validity of the weak-field limit is discussed in Section 4. The optical depth and event rate are discussed in Section 5. The results are summarized in Section 6.

3 Gravitational lensing

Magnification of the apparent brightness of a distant star by the gravitational lensing effect of another star was predicted by [150]. This kind of lensing effect is called “microlensing” because the images produced by the gravitational lensing are very close to each other and are difficult for the observer to resolve. The observable effect is the changing apparent brightness of the source star only. This effect was discovered in 1993 [172, 134, 138] and has been used to detect astronomical objects that do not emit observable signals (such as visible light, radio waves, and X rays) or are too faint to observe. Microlensing has successfully been applied to detect extrasolar planets [141] and brown dwarfs [143, 152]. Microlensing is also used to search for unseen black holes [137, 140, 162] and massive compact halo objects [136, 170, 175], a candidate for dark matter.

The gravity of a star is well expressed by the Schwarzschild metric. The gravitational microlensing of the Schwarzschild metric [164, 154, 160] has been studied in the weak-field limit. In this section, we simply follow the method used for Schwarzschild lensing. Figure 1 shows the relation between the source star, the lens (wormhole), and the observer. The Ellis wormhole is known to be a massless wormhole, which means that the asymptotic mass at infinity is zero. However, this wormhole deflects light by gravitational lensing [145, 144, 159, 148] because of its curved space–time structure. The deflection angle of the Ellis wormhole was derived by [148] to be


where is the closest approach of the light. In the weak-field limit (), the deflection angle becomes


The angle between the lens (wormhole) and the source can then be written as


where , , , and are the distances from the observer to the lens, from the observer to the source, and from the lens to the source, and the impact parameter of the light, respectively. In the asymptotic limit, Schwarzschild lensing and massive Janis–Newman–Winnicour (JNW) wormhole lensing [148] have the same leading term of . Therefore, the lensing property of the JNW wormhole is approximately the same as that of Schwarzschild lensing and is difficult to distinguish. As shown in Equation (\theequation), the deflection angle of the Ellis wormhole does not have the term of and starts from . This is due to the massless nature of the Ellis wormhole and indicates the possibility of observational discrimination from the ordinary gravitational lensing effect. In the weak-field limit, is approximately equal to the closest approach . For the Ellis wormhole, . We thus obtain


The light passing through the other side of the lens may also form images. However, Equation (\theequation) represents deflection in the wrong direction at . Thus, we must change the sign of the deflection angle:


It would be useful to note that a single equation is suitable both for and images in the Schwarzschild lensing. However, such treatment is applicable only when the deflection angle is an odd function of .

If the source and lens are completely aligned along the line of sight, the image is expected to be circular (an Einstein ring). The Einstein radius , which is defined as the radius of the circular image on the lens plane, is obtained from Equation (\theequation) with as


The image positions can then be calculated from




where is the angle between the image and lens, and is the angular Einstein radius. Using reduced parameters and , Equations (\theequation) and (\theequation) become simple cubic formulas:




As the discriminant of Equation (\theequation) is , Equation (\theequation) has two conjugate complex solutions and a real solution:




The real positive solution corresponds to the physical image.

The discriminant of Equation (\theequation) is . Thus it has a real solution if :




with . This solution corresponds to a physical image inside the Einstein ring. For , Equation (\theequation) has three real solutions. However, two of them are not physical because they do not satisfy . Only the solution


corresponds to a physical image inside the Einstein ring.

Figure 2 shows the calculated images for source stars at various positions on a straight line (source trajectory). The motion of the images are similar to those of the Schwarzschild lensing. Table 1 shows the Einstein radii and angular Einstein radii for a bulge star ( kpc and kpc are assumed) and a star in the Large Magellanic Cloud (LMC, kps and kpc are assumed) for various throat radii. The detection of a lens for which the Einstein radius is smaller than the star radius ( km) is very difficult because most of the features of the gravitational lensing are smeared out by the finite-source effect. Thus, detecting a wormhole with a throat radius less than 1 km from the Galactic gravitational lensing of a star is very difficult.

4 Light curves

The light curve of Schwarzschild lensing was derived by [160]. The same method of derivation can be used for wormholes. The magnification of the brightness is


where and are magnification of the outer and inner images, and correspond to outer and inner images, respectively. The relation between the lens and source trajectory in the sky is shown in Figure 3. The time dependence of is


where is the impact parameter of the source trajectory and is the time of closest approach. is the Einstein radius crossing time given by


where is the transverse velocity of the lens relative to the source and observer. The light curves obtained from Equations (\theequation) and (\theequation) are shown as thick red lines in Figure 4. The light curves corresponding to Schwarzschild lensing are shown as thin green lines for comparison. The magnifications by the Ellis wormhole are generally less than those of Schwarzschild lensing. The light curve of the Ellis wormhole for shows characteristic gutters on both sides of the peak immediately outside the Einstein ring crossing times (). The depth of the gutters is about 4% from the baseline. Amazingly, the star becomes fainter than normal in terms of apparent brightness in the gutters. This means that the Ellis wormhole lensing has off-center divergence. In conventional gravitational lensing theory [166], the convergence of light is expressed by a convolution of the surface mass density. Thus, we need to introduce negative mass to describe divergent lensing by the Ellis wormhole. However, negative mass is not a physical enitity. As the lensing by the Ellis wormhole is convergent at the center, lensing at some other place must be divergent because the wormhole has zero asymptotic mass. For , the light curve of the wormhole has a basin at and no peak. Using these features, discrimination from Schwarzschild lensing can be achieved. Equations (\theequation) and (\theequation) indicate that physical parameters (, , and ) are degenerate in and cannot be derived by fitting the light-curve data. This situation is the same as that for Schwarzschild lensing. To obtain or constrain these values, observations of the finite-source effect [158] or parallax [135] are necessary.

The detectability of the magnification of the star brightness depends on the timescale. The Einstein radius crossing time depends on the transverse velocity . There is no reliable estimate of for wormholes. Here we assume that the velocity of the wormhole is approximately equal to the rotation velocity of stars ( km/s) if it is bound to the Galaxy. If the wormhole is not bound to our Galaxy, the transverse velocity would be much higher. We assume km/s [165] for the unbound wormhole. Table 2 shows the Einstein radius crossing times of the Ellis wormhole lensings for the Galactic bulge and LMC in both bound and unbound scenarios. As the frequencies of current microlensing observations are limited to once every few hours, an event for which the timescale is less than one day is difficult to detect. To find very long timescale events ( days), long-term monitoring of events is necessary. The realistic period of observation is years. Thus, the realistic size of the throat that we can search for is limited to km km both for the Galactic bulge and LMC if wormholes are bound to our Galaxy. If wormholes are unbound, the detection is limited to km km.

5 Validity of the weak-field hypothesis

First, we consider the outer image. In the previous section, we applied the weak-field approximation to the impact parameter and the deflection angle . As previously mentioned, the impact parameter is written as


The condition to neglect the second term is . As the image is always outside the Einstein ring,


From the deflection angle, we obtain a similar relation from Equation (\theequation):


The values of and in Table 1 show that the weak-field approximation is suitable for km in the Galactic microlensing. More generally, is derived from Equation (\theequation) for . This means that is much greater than if . Thus, the weak-field approximation is suitable if the throat radius is negligibly small compared with the source distance. For the inner image, the higher-order effect is expected to be greater than that for the outer image. However, the contribution of the inner image to the total brightness is small and decreases quickly with ( for and for ). On the other hand, the absolute value of the corresponding does not decrease as quickly ( for and for ). Thus the contribution of the higher order effect of the second image to the total brightness is expected to be small.

Another possibility of deviation from the weak-field approximation is the contribution of relativistic images. Recently, gravitational lensing in the strong-field limit [173] has been studied for lensing by black holes. In this limit, light rays are strongly bent and wound close to the photon sphere. As a result, a number of relativistic images appear around the photon sphere. However, it has been shown that there is no photon sphere [148] in Ellis wormhole lensing. Therefore, there is no contribution of relativistic images to the magnification in Ellis wormhole lensing. We thus conclude that the weak-field hypothesis is a good approximation unless the throat radius is comparable to the galactic distance.

6 Optical depth and event rate

The probability of a microlensing event to occur for a star is expressed by the optical depth :


where is the number density of wormholes as a function of the line of sight. Here we simply assume that is constant ():


The event rate expected for a source star is calculated as


There is no reliable prediction of the number density of wormholes. Several authors [153, 155] have speculated that wormholes are very common in the universe, at least as abundant as stars. Even if we accept such speculation, there are still large uncertainties in the value of because the distribution of wormholes is not specified. Here, we introduce two possibilities. One is that wormholes are bound to the Galaxy and the number density is approximately equal to the local stellar density. The other possibility is that wormholes are not bound to the Galaxy and are approximately uniformly distributed throughout the universe. For the bound hypothesis, we use pc, where is the local stellar density in the solar neighborhood, pc, and is the average mass of stars. We use , a typical mass of an M dwarf; i.e., the dominant stellar component in the Galaxy. For the unbound hypothesis, we assumed that the number density of the wormholes is the same as the average stellar density of the universe. The stellar density of the universe is estimated assuming that the fraction of baryonic matter accounted for by star is the same as that of the solar neighborhood. Then we obtain pc, where p is the critical density, is the baryon density of the universe divided by the critical density, and pc and pc are the local star and local baryon densities, respectively.

Using these values, we calculated the optical depths and event rates for bulge and LMC lensings. Table 3 presents the results for the bulge lensings. In an ordinary Schwarzschild microlensing survey, observations are made of more than 10 million stars. Thus, we can expect approximately events in a year. However, the situation is different in a wormhole search. As mentioned previously, the magnification of wormhole lensing is less than that of Schwarzschild lensing, and a remarkable feature of wormhole lensing is the decreasing brightness around the Einstein radius crossing times. Past microlensing surveys have mainly searched for stars that increase in brightness. The stars monitored are those with magnitudes down to the limiting magnitude or less. However, we need to find stars that decrease in brightness in the wormhole search. To do so, we need to watch brighter stars. Therefore, far fewer stars can be monitored than in an ordinary microlensing survey. Furthermore, the detection efficiency of the wormhole is thought to be less than that for Schwarzschild lensing because of the low magnification. Here we assume that the effective number of stars monitored to find a wormhole is . To expect more than one event in a survey of several years, must be greater than . The values in Table 3 indicate that the detection of wormholes with km is expected in the microlensing survey of the Galactic bulge in the case of the bound model. The results of the optical depths and the event rates for LMC lensing are presented in Table 4. On the basis of the same discussion for bulge lensing, we expect to find a wormhole. The event rates expected for LMC lensing are greater than those for bulge lensing. We expect detection of a wormhole event if km for the bound model. If no candidate is found, we can set upper limits of and/or as functions of . To convert these values to physical parameters ( and ) requires the distribution of . Right now, there is no reliable model of the distribution except for using the bound or unbound hypothesis. On the other hand, the event rates for the unbound model are too small for the events to be detected.

In past microlensing surveys [136, 170, 175, 169], large amounts of data have already been collected for both the bulge and LMC fields. Monitoring more than stars for about 10 years can be achieved by simply reanalyzing the past data. Thus, discovery of wormholes can be expected if their population density is as high as the local stellar density and km. Such wormholes of astronomical size are large enough for humans to pass through. Thus, they would be of interest to people discussing the possibility of space–time travel. If no candidate is found, the possibility of a rich population of large-throat wormholes bound to the Galaxy can be ruled out. Such a limit, however, may not affect existing wormhole theories because there is no prediction of the abundance. However, theoretical studies on wormholes are still in progress. The limit imposed by observation is expected to affect future wormhole theories. On the other hand, the discovery of unbound wormholes is very difficult even if their population density is comparable to that of ordinary stars. To discover such wormholes, the monitoring of a much larger number of stars in distant galaxies would be necessary. For example, and days for the M101 microlensing survey ( Mpc) if the throat radius is km. To carry out such a microlensing survey, observation from space is necessary because the resolving of a large number of stars in a distant galaxy is impossible through ground observations.

Only the Ellis wormhole has been discussed in this paper. There are several other types of wormholes [167, 159, 163] for which deflection angles have been derived. These wormholes are expected to have different light curves. To detect those wormholes, calculations of their light curves are necessary. The method used in this paper can be employed only when we know the analytic solutions of the image positions. If no analytic solution is found, the calculation must be made numerically.

7 Summary

The gravitational lensing of the Ellis wormhole is solved in the weak-field limit. The image positions are calculated as real solutions of simple cubic formulas. One image appears on the source star side and outside the Einstein ring. The other image appears on the other side and inside the Einstein ring. A simple estimation shows that the weak-field hypothesis is a good approximation for Galactic microlensing if the throat radius is less than km. The light curve derived has characteristic gutters immediately outside the Einstein ring crossing times. Optical depths and event rates for bulge and LMC lensings are calculated for simple bound and unbound hypotheses. The results show that the bound wormholes can be detected by reanalyzing past data if the throat radius is between and km and the number density is approximately equal to the local stellar density. If the wormholes are unbound and approximately uniformly distributed in the universe with average stellar density, detection of the wormholes is impossible using past microlensing data. To detect unbound wormholes, a microlensing survey of distant galaxies from space is necessary.

We would like to thank Professor Hideki Asada of Hirosaki University, Professor Matt Visser of Victoria University, and Professor Tomohiro Harada of Rikkyo University for discussions and their suggestions concerning this study. We thank Professor Philip Yock of Auckland University for polishing our manuscript. Finally, we would like to thank an anonymous referee who pointed out the existence of a second image.


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Figure 1: Sketch of the relation between the source star, lens (wormhole), and observer.
Figure 2: Source and image trajectories in the sky from the position of the observer.
Figure 3: Sketch of the relation between the source trajectory and the lens (wormhole) in the sky. All quantities are normalized by the angular Einstein radius .
Figure 4: Light curves for (top left), (top right), (bottom left), and (bottom right). Thick red lines are the light curves for wormholes. Thin green lines are corresponding light curves for Schwarzschild lenses.
Bulge LMC
(km) (km) (mas) (km) (mas)
1 0.001
10 0.003 0.001
0.013 0.004
0.061 0.018
0.283 0.083
1.31 0.387
6.10 1.80
28.3 8.35
131 38.7
610 180
2 832 835
13 143 3 874
Table 1: Einstein radii for bulge and LMC lensings
Bulge LMC
(km) (day) (day)
Bound Unbound Bound Unbound
1 0.019 0.001 0.035 0.002
10 0.089 0.004 0.164 0.007
0.413 0.018 0.761 0.033
1.92 0.084 3.53 0.155
8.90 0.392 16.4 0.721
41.3 1.82 76.1 3.35
192 8.44 353 15.5
890 39.2 1 639 72.1
4 130 182 7 608 335