Master thesis

On the possibility of nding exoplanets using

gravitational lensing of radio backgrounds.

Alexander Bartilsson

November 19, 2018

Abstract

We propose a new method for detecting exoplanets using gravita-tional lensing. The hypothesis is that the lensing caused by an exo-planet could distort the structures of a radio background, for instancea H II region. It cannot be done with todays telescope but with thesecond incarnation of the Square Kilometer Array (SKA), currently inits design phase we deem it possible. In order to test this hypothesis weconstructed a simulation that creates a simple background structurewith variable scale and then produces simulated images such as theones achievable with the SKA. We produce positive result with a clearsignature of the planet with certain background scales and suggestfurther investigation into this method of detecting exoplanets.

Institution of AstronomyMaster degree 45 HE credits2017-2018Supervisor: Markus Jansson

Contents

1 Introduction 3

2 Dierent stars and planets 5

2.1 Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Life on other planets 8

4 Methods for detecting exoplanets 9

4.1 Radial velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Transit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Microlensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4 Direct imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 134.5 Other methods . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6 Combining methods . . . . . . . . . . . . . . . . . . . . . . . 13

5 Gravitational lensing 14

5.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 How it works . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.2.1 Magnication . . . . . . . . . . . . . . . . . . . . . . . 175.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 The hypothesis 19

6.1 How to test it? . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7 The background 21

7.1 H II regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.2 Synchrotron radiation from supernova remnants . . . . . . . . 227.3 Combined radiation at GHz frequencies . . . . . . . . . . . . 22

8 Radio astronomy 23

8.1 Radiative transfer . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3 Aperture synthesis . . . . . . . . . . . . . . . . . . . . . . . . 258.4 Radio Interferometic Arrays around the world . . . . . . . . . 25

8.4.1 The Very Large Array . . . . . . . . . . . . . . . . . . 258.4.2 The European Very Large Baseline Interferometry Net-

work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9 The Square Kilometer Array 27

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10 The simulation 29

10.1 Approximations and idealizations . . . . . . . . . . . . . . . . 2910.1.1 Geometrical approximation . . . . . . . . . . . . . . . 2910.1.2 Sample points instead of extended source . . . . . . . 2910.1.3 Point lens . . . . . . . . . . . . . . . . . . . . . . . . . 2910.1.4 The gravitational inuence of intermediate mass . . . 3010.1.5 The antenna function . . . . . . . . . . . . . . . . . . 30

10.2 Step 1 - The background . . . . . . . . . . . . . . . . . . . . . 3110.3 Step 2 - Lensing . . . . . . . . . . . . . . . . . . . . . . . . . 33

10.3.1 The Einstein Ring . . . . . . . . . . . . . . . . . . . . 3510.4 Step 3 - The Telescope . . . . . . . . . . . . . . . . . . . . . . 36

10.4.1 Step 3a - Noise . . . . . . . . . . . . . . . . . . . . . . 3610.4.2 Step 3b - The PSF . . . . . . . . . . . . . . . . . . . . 38

10.5 Step 5 - Removing the background . . . . . . . . . . . . . . . 4010.6 Step 6 - The nal images . . . . . . . . . . . . . . . . . . . . . 41

11 Results 44

12 Future development 47

13 Usability of our method 48

13.1 Target planets . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.2 Number of possible stars within our target distance . . . . . . 48

14 Conclusions 49

References 51

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1 Introduction

A profound question, probably as old as humanity itself, is that of our placein this world. On this earth, around this Sun, in this Universe. Thoughwe might not have come much closer to the answer, scientic advances andthe evolution of the human intellect has broadened the question in a wayinapprehensible by the ones who rst conceived it. Our desire is not only tocomprehend our place in the world but also to determine the constancy ofour solitude. If the advent of life on this planet is just a coincident so unlikelythat is has not happened anywhere else in the vastness of the universe. Orif it is a natural consequence given the right conditions and time.

For the rst time in human history we could be on the brink of answeringthe question of our place in the world. The last quarter century has seenthe advent of a new observational eld in astronomy with the discovery ofplanets around other stars than our own. Now we nally might have thetools to shed some light on that big question.

The search for planets beyond our solar system is one of the most startlingdomains in science today. With new technical achievements taking placeevery year, what only a few decades ago was considered science ction isnow everyday science. Currently over 3800 planets have been detected andconrmed (Nasa Exoplanet Archive; The Extrasolar Planets Encyclopaedia)orbiting around almost 2800 stars with new added every week. It is nowevident that planetary systems is not something our of the ordinary. It mightinstead be more common with stars with orbiting planet(s) then without.

The rst conrmed discovery of a planet orbiting another star than ourown Sun was that of 51 Peg B in 1995 (Mayor et al. 1995). It was discoveredusing the radial velocity method which is one of four of the most commonmethods for detecting exoplanets1. The other three are direct imaging, thetransit method and microlensing.

In this thesis we will introduce a potential new method of exoplanet de-tection. Like the microlensing method it utilizes the physics of gravitationallensing but in a dierent way.

We will start giving a very short introduction to stars and planets inchapter 2, just to make the reader familiar with some terminology. Wediscuss the habitable zone and the possibility of life on other planets inchapter 3. In chapter 4 we present the most common ways of detectingexoplanets today. After this introductory part of the thesis we start toapproach the main topic. We try to explain the theory behind gravitationallensing and the hypothesis we wanted to test in chapters 5 and 6. We thencontinue with some of the background needed to understand our frameworkin chapters 7 through 9. We give an eloborate descripton of the simulationprocess (chapter 10), present the results of our simulation runs (chapter 11)

1An exoplanet is a planet outside our solar system.

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and present some ideas for further investigations (chapter 12) together witha description of the usability of our method (chapter 13). Finally we endwith some conclusive words in chapter 14.

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2 Dierent stars and planets

2.1 Stars

Stars are commonly classied based on their spectral characteristics. Moststars are classied under the Morgan-Keenan system from the hottest to thecoldest types using the letters O, B, A, F, G, K and M (Morgan et al. 1973).Each type is then subdivided into 10 subtypes using a single digit rangingfrom 0 to 9 with increasing temperature. A third symbol has also been addedto distinguish giant stars from dwarfs. This symbol is a Roman numeral andis called luminosity class. Luminosity class 0 or Ia+ stands for hypergiants,I for supergiants, II for bright giants, III for regular giants, IV for sub-giants,V for main-sequence2 stars and nally sd and D stands for sub-dwarf anddwarf respectively. The sun is a G5V star, indicating that it is on the mainsequence with a temperature of around 5800 K.

2.2 Planets

The rst exoplanet that was discovered was the planet around the pulsarPSR 1257+12 (Wolszczan et al. 1992) and the rst exoplanet discoveredaround an ordinary star was 51 Pegasi B, a gas giant with around half themass of Jupiter (Mayor et al. 1995; Birkby et al. 2017).

Prior to these discoveries the only planets we knew of were the ones inour own solar system. With the vast amount of detected exoplanets to dateit seems that the planets in our solar system are not typical or perhaps evencommon results of planet formation. A great deal of variety is found whenlooking closer at the exoplanet collection. A wide range of planets without aunied nomenclature has produced a sprawling terminology of planetary ob-jects with concept names such as "earthlike", "super-earths", "hot Jupiters","super-Jupiters", "mini-Neptunes", "dwarf-planets" etc. This has certainlynot made the exoplanetary science community any good and voices havebeen risen for the need of a unied vocabulary (Oppenheimer 2016). Thelast word is more than likely not said when it comes to this but Chen etal. (2016) did a probabilistic analysis of roughly 300 objects with well con-strained mass and radius. They ended up with three main groups of planets:Terran, Neptunian and Jovian worlds.

2A star spends most of its life on the main sequence. Its starts with the burning ofhydrogen which ignites the star and ends when the hydrogen in the core is depleted andthe Helium-burning begins which is the beginning of the path which ends with the "death"of the star.

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Figure 1: Figure from Chen et al. (2016) showing their classication of plan-ets and also the boundary between the dierent classes and the the boundarybetween planet and star. Notable is that brown dwarf are here classied asJovian worlds. Note that Chen uses the symbol ∼ (instead of ∝ which weuse) to denote "proportional to".

The main distinguisher is the way mass relates to radius. This of coursedepends on the density structure which depends on what the planet is madeof. We can describe the Mass-Radius relation as R ∝ Mα where α diers withclass as described below. Terran worlds are rocky planets with α ≈ 0.28.Mercury, Venus, Earth and Mars falls into this category. Around 2.0 M⊕

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is the maximum size for Terran worlds and the Neptunian region starts.Neptunian worlds are larger in size and contains more gas but with a solidcore making the radius more dependent on mass than for Terran worlds. Hereα ≈ 0.59. Neptune and Uranus are typical such planets with Saturn beingvery close to the boundary to Jovian worlds at M ≈ 0.41 MJ

4. To the rightof this limit are Jovian planets with a Mass-Radius-relation a ≈ −0.04 whichmeans a very limited mass dependency of the radius. At M ≈ 0.08 M

5 isthe boundary between planets and stars, consistent with current theory ofthe smallest mass needed to ignite hydrogen burning (e.g. Burrows et al.1993, 1997; Barae et al. 2003). Brown dwarfs are star-like objects with

3M⊕ denotes the mass of the Earth4MJ denotes the mass of Jupiter.5M denotes the mass of the sun.

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too little mass to sustain hydrogen burning but enough to fuse deuterium.These objects are by this denition placed in the Jovian worlds category.There has been debate regarding the existence of a distinct cut-o in massbetween giant planets and brown dwarfs. It is deemed likely that such acut-o does not exist and that the proposed boundary at 13 MJ more likelyis a diuse divider where the process in which the object has been created ismore important (born in interstellar medium in a star-like process or bornin a dierent process in protoplanetary disks) (Burrows et al. 2001).

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3 Life on other planets

Entangled in the probing of planets beyond our solar system is of coursethe possibility of nding life native to another world. Until such a daywhen actual expeditions to other planets are possible we have to search forsigns of life. At the center of the theory of possible life on exoplanets isthe concept of the habitable zone. The habitable zone is used to identifypotentially habitable planets and to act as selection factor for planets whichcould possibly harbor life. It was coined by Huang (1959) and even thoughthe term is widely used it should be pointed out that it by no means excludesall planets outside the habitable zone as possible for harboring life forms.Instead, it is a crude classication of the circular region around a star (ormultiple stars) in which liquid water could exist on an rocky planet's surface(Kaltenegger 2017, and references therin). The several thousand exoplanetsdetected to date give the rst statistical insights into the diversity of otherworlds (e.g. reviewed in Udry et al. 2007; Winn et al. 2015). Around ourclosest neighboring star, Proxima Centauri, a cool M5V dwarf only 1.3 pc6

from the Sun, a planet with minimum mass of 1.3M⊕ orbits in the habitablezone (Anglada-Escudé et al. 2016). The planetary system TRAPPIST-1orbiting a M9V star at a distance of 12 pc from us shows promise that threeor four of the seven planets are earth-sized and in the habitable zone (Gillonet al. 2017).

If both the radius and mass of a planet is known, the mean density can becalculated, which gives a strong clue to the composition of the planet throughcomparison with planets in our solar system. However, for most of thedetected exoplanets we only know either the mass or the radius, dependingon the detection method (see chapter 4). To ease the classication of planetsand also to facilitate follow up studies a rough classication scheme is oftenused. Planets with (minimum) masses below 10 M⊕ are considered rockyand planets with (minimum) masses above 10 M⊕ are considered gas planets.If only the radius is known, planets with radii below 2 R⊕ are commonlyconsidered rocky. It should be mentioned that there is no consistent limit forthe mass or radius divider between the terms mini-Neptune and super-Earth(Kaltenegger 2017).

The concept of the habitable zone and basically all our search for signsof life beyond the earth is based on the assumption that life elsewhere wouldshare fundamental characteristics with life on Earth such as being carbonbased and requiring liquid water. Life based on a dierent chemistry is notconsidered since how, or if, it would function and which signatures it wouldproduce in the atmosphere or on the surface of the planet is yet unknown(Kaltenegger 2017).

6The parsec (pc) is an astronomic distance unit equal to 3.086× 1016 m

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4 Methods for detecting exoplanets

There are several methods for detecting planets around other stars thanthe Sun. The radial velocity and transit methods are responsible for mostdetections to date as can be seen in gure 2. These two methods togetherwith the microlensing and direct imaging will be described briey belowfocusing on which kind of planets they are able to detect and also theiradvantages and disadvantages. Some other methods will then be mentioned,followed by some words on using multiple methods in combination.

Figure 2: Validated exoplanets plotted as mass versus orbital period (size oforbit) as of 2018-04-26 indicating detection method. Image courtesy of theNASA Exoplanet Archive (exoplanetarchive.ipac.caltech.edu)

4.1 Radial velocity

The radial velocity method is responsible for the second largest number ofexoplanet discoveries to date, only surpassed by the transit method. Over1000 detections have been made so far. It is also the method that startedthe planet search discipline. The rst hint of discoveries came in the late80s and early 90s (Campbell et al. 1988; Latham et al. 1989; Hatzes et al.1993) and culminated with the rst conrmed detection of a planet aroundanother another star, 51 Peg b, in 1995 (Mayor et al. 1995).

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The principle of the radial velocity method is that one measures the line-of-sight-velocity-component of a star as it circles around the center of massof the star-planet system. The velocity is measured by examining the shiftin wavelengths of the spectral lines which is due to the Doppler eect7 dueto the (radial) motion of the star. The radial-velocity method favors thediscovery of massive planets in close orbits around their host star.

Without knowing the inclination of the planet's orbit, the mass derivedwith the radial velocity method only represents the minimum mass of theplanet.

4.2 Transit

If an object orbits a star and the plane of the orbit is in the line of sight tothe observer this object is said to transit the disk of the host star. Thesetransits will cause the ux from the star to decrease in a periodic manner andthus enabling the detection of the orbiting object. This method of detectingexoplanets is called the transit method (or transit photometric method).

Figure 3: Schematic view of the transit method from Cameron (2016) show-ing the regions of the celestial sphere from which transits are visible. Thedark shadow shows full transits and the light shadow shows grazing transits.

7The Doppler eect (or Doppler shift) is a wave eect that occurs when an observer ismoving in relation to the wave source. The frequency (or wavelength) is then shifted inproportion to the relative velocity.

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The probability of a transit occuring in a way that is possible to detectfor an observer is:

Prtransit ' 0.0046

(R∗R

)(1 AU

a

), (1)

where R∗ is the radius of the star, a is the semi-major axis (Cameron 2016).This shows that the transit method favors the discovery of hot planets inclose orbits. The probability of detecting the Earth with the transit methodis 0.46% whereas for Jupiter, orbiting at 5.2 AU8 from the Sun, the prob-ability is only 0.09%. Even though the likelihood of a transit is small, thismethod is very suitable for survey missions; the largest one yet is the Ke-pler space observatory (Borucki et al. 2010) which was launched in 2009 andhas just retired after 9 years of collecting data. Kepler has found over 2300conrmed exoplanet with around the same amount of exoplanet candidates(Nasa Exoplanet Archive).

Figure 4: The light curve showing the detection of a planet using the transitmethod. Transit marks the position of when the planet is in front of the starand occultation mean that the planet is behind the star. From Cameron(2016).

If a radial velocity observation is made on a transiting planet, the realmass can be measured because then the inclination is known. Even withoutradial velocity observations, the mass can be estimated in a multiple planet

8AU is another astronomical distance unit based on the (mean) distance between theSun an the Earth (1 AU) which is ≈ 150× 109 m.

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system through variations in the transit timing, when the gravitational pullof one planet inuences the transit period of another planet. The mass canalso be estimated through orbital constraints on the system (e.g. Winn et al.2015).

4.3 Microlensing

Proposed by Mao et al. (1991) and Gould et al. (1992), microlensing is atechnique that is based on the physics that light bends when subjected togravitational force such as from the mass of a star. If two stars are exactlyaligned, the light from the background star can be magnied. If the fore-ground star has an orbiting planet, this planet can cause a deviation in thelensing and thus aect the magnication by a small fraction when perfectlypositioned. This eect can be seen in gure 5. The rst microlensing dis-covery of a planet was annonced in 2003 with the a 2.6 MJ planet orbitinga 0.6 M K star with a 4.3 AU wide orbit (Bond et al. 2004).

Figure 5: The discovery of exoplanet OGLE-2005-BLG-390Lb from a mi-crolensing event. The smaller peak to the right of the main peak shows theextra magnifying eect of the planet. Plot from Beaulieu et al. (2006).

Since there are many factors that need to be exactly right, discoverieswith this method are understandably scarce. Fewer than a percent of all con-rmed exoplanets and exoplanet candidates are found with the microlensingmethod (Gould 2016). Nevertheless, this method is still important since itcan discover planets that cannot be found with other methods, e.g. planetsat great distance from their host stars (or free-oating planets). As an ex-ample this methods has paved the way for the insights that so called cold

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Neptunes, which are planets with the mass roughly the same as Neptunequite far from the star (which is the reason that they are cold) (Suzuki et al.2016), are very common.

4.4 Direct imaging

Direct imaging might seem like the most straight forward method of detect-ing something with telescopes but it remains a very challenging prospect.This is of course since planets are so much dimmer than stars. And also thatthe angular separation from their host stars is so small on the distances in-volved that resolving them is very hard with current telescopes. For exampleJupiter is 109 times fainter than the Sun (in visible light) (Claudi 2016). Acouple of successful observation using direct imaging have been accomplishedthough, for example a planet orbiting the star β Pictoris (Lagrange et al.2009; Lagrange et al. 2010).

4.5 Other methods

A neutron star is a remnant of a star that has exploded in a supernova ex-plosion. Pulsars are rotating neutron stars and they emit electromagneticwaves in an extremely regular manner. So regular, that if a planet orbits theneutron star, the extremely small movement of the neutron star due to thegravitational pull from the planet will display anomalies in the timing of theradio pulses from the star. Originally not designed for planet discoveries,this method is so sensitive that it is capable of discovering smaller planetsthan any other current method. The rst conrmed planet outside the so-lar system was found using this method, around the pulsar PSR 1257+12(Wolszczan et al. 1992).

High precision astrometry could also be used for exoplanet detection. As-trometry means measurements of the position and movement of stars (andother celestial bodies) and a body orbiting a star will cause a (small) displace-ment of the host star due to their mutual orbit around the center of massof the system. The GAIA satellite launched in 2013 is hoped to contributewith new exoplanet detections with its planned over 1 billion measurementsof stars and other bodies (Lindegren et al. 2007; Perryman et al. 2014).

4.6 Combining methods

Since no method single-handedly is capable of measuring both mass andradius of an exoplanet, multi-method discoveries are highly desirable in orderto achieve better constraints on the properties of the planet.

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5 Gravitational lensing

5.1 History

About two centuries ago, when light was considered to be a particle phe-nomenon, several physicists and astronomers speculated that light rays maybe inuenced by a gravitational eld just as ordinary particles are. In thelate eighteenth and the rst years of the nineteenth century John Mitchelland Johann von Soldner calculated an expression for the deection of lightpropagating in a eld of a spherical mass. But it was not until after theformulation of general relativity by Albert Einstein in 1915 that the behav-ior of light under the inuence of a gravitational eld could be expressedand studied on solid physical ground (Schneider et al. 2006). Gravitationallensing was actually one of the rst real tests of Albert Einstein's theory ofrelativity which explains gravity as a geometric property of space and time.According to this theory, rays of light would bend under the inuence ofgravity (Schneider et al. 1992).

In 1919, the deection of light from stars by the Sun was studied dur-ing a total solar eclipse and meant a tremendous success for Einstein's newtheory of gravity (e.g. Schneider et al. 2006). Around the same time theterm 'lens' was used in context of gravitational light deection and in 1937Fritz Zwicky published two groundbreaking papers (Zwicky 1937a,b). Heused a somewhat dierent approach when he instead of stars in our Galaxyconsidered galaxies (then called "extragalactic nebulae") and estimated thetypical image separation caused by such objects on a background source (hegot it a magnitude too high though) and noted that should such an eect beobserved it would serve as another proof of general relativity. After Zwickyit was not until the 1960s that the scientic discipline of gravitational lensingstarted to prosper with milestone papers by Klimov (1963), Liebes (1964)and Refsdal et al. (1964). Their work was pivotal to lensing research. Therst widely accepted measurement of gravitational lensing in radio was doneby Walsh et al. (1979). Their measurements showed two occurrences of whatlater was conrmed to be the same quasar.

5.2 How it works

In the following we use the concept described in e.g. Mao (2012) and Schnei-der et al. (1992). For an exhaustive and exact description of gravitationallensing we need to use the tools of general relativity. But a much simplerapproach which is sucient for most astronomically relevant uses is an ap-proximative description of light rays which is called gravitational lens theory(Schneider et al. 2006). The basics of gravitational lensing physics says thata light ray that is aected by a gravitational eld from some sort of massis deected. The deection of course depends on the nature of the eld and

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thus the mass creating the eld. From calculations using general relativitywhere G is the gravitational constant, M is the mass of the lens, c is thespeed of light and ξ is the minimum distance at which the light ray passesthe lens, the deection angle a is:

a =4GM

c2ξ=

2RSξ, (2)

provided that the impact parameter ξ is much larger than the Schwarzschildradius of the mass:

Rs ≡2GM

c2(3)

This is valid for a spherically symmetric mass and that approximation issucient for our purposes (see further details in section 10.1.3).

We start by considering an arrangement, seen in gure 6. We consider apoint mass as our lens at a distance Dd from the observer. We let η denotethe two-dimensional position of the source on the source plane at a distanceDs, measured with respect to the intersection point of the optical axis (thestraight line from the observer through the lens, the dashed line in gure 6)with the source plane as can be seen in gure 6. β is the angular separationof the source as would be seen by the observer in the absence of a lens. From

Figure 6: Sketch of a simple gravitational lensing case. From Schneider et al.(2006).

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geometry we have that:

βDs =Ds

Ddξ − RS

ξDds, (4)

were Dds is the distance between the lens line and the source plane. If thesource is very far away, a cosmological model has to be used and the conceptof distance is ambiguous and need to be treated with care. In our case, fordistances within the Galaxy, we can use a Euclidean metric. We introduceθ as the angular separation between the lens and the deected ray as:

θ =ξ

Dd(5)

From (4) and (5) and expanding RS we get:

βDs =Ds

Ddξ − 2GM

c2

1

ξDds = Dsθ −

2GM

c2

1

θDdDds. (6)

If we then solve for θ we end up with:

θ2 − βθ − 2GM

c2

Dds

θDdDs= 0, (7)

which gives us:

θ1,2 =β

2± 1

2

√(4GM

c2

Dds

θDdDs

)2

+ 4β (8)

β, θ and ξ are two-dimensional vectors. There is a special case to considerwhen calculating gravitational lensing and that is the situation in which thesource, observer and lens are collinear, i.e. β = 0. In this case the lightpropagates in a rotationally symmetric conguration about the line-of-sightto the lens. If β = 0 the solution to (8) becomes:

θ1,2 = ±√

4GM

c2

Dds

DdDs≡ θE (9)

Because of this, due to symmetry, the whole ring of angular radius θ0 = θE(Einstein radius) is a solution to the lens equation. A point source exactlybehind the lens would thus appear as a circle around the lens. This is anidealized case which requires axially symmetric matter distribution of thelens, but extended sources can have ring-shaped images even if the lens isnot perfectly symmetric. Such images are called "Einstein rings".

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5.2.1 Magnication

One of the powerful implications of gravitational lensing is that the resultingimage not only is a deected version of the unlensed image but could also bemagnied, sometimes by a large magnitude. This is because the number ofrays hitting the observer could increase from the unlensed situation, and sincethe specic intensity is constant along a ray, this introduces an increased uxat the observer. The surface brightness for a lensed image is the same asfor the unlensed image. But when rays are deected by a lens the solidangle over which the source subtends the sky increases and since the surfacebrightness is identical, the total ux is greater than in the absence of thelens. This is illustrated in gure 7.

Figure 7: Illustration of the magnication by a gravitational lens. Sincethe surface brightness of the source is unchanged by the light deection, theapparent brightness of the source is magnied in proportion to the distortionof the solid angle. From Schneider et al. (2006).

The magnication, indicated by µ, is the ratio of the ux of the lensedimage to the unlensed source. If a point source is lensed by an arbitrary mat-ter distribution there is always at least one image as bright as the unlensedsource, i.e. µ ≥ 1 (Schneider 1984; Blandford et al. 1986). This can seemcontradictory and to go against the conservation of energy. If one imagines asource surrounded by equidistant observers and lenses are placed in betweenthe source and observers, each observer appears to receive more energy perunit time than before, although the luminosity of the source has not beenchanged. The solution to this and why it does not contradict the laws of

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physics has to do with general relativity and the fact that the masses ofthe lenses changes the space-time of the sphere spanned by the source andobservers. For a full derivation of this we refer the reader to (e.g. Schneideret al. 1992, 2006). We settle with presenting the result of calculating themagnication ratio and scaling β to the aforementioned Einstein radius θE :

β =β

θE(10)

µ± = ±1

4

β√β2 + 4

+

√β2 + 4

β± 2

(11)

where µ− will have negative and µ+ will have positive magnication. Theformer is said to have negative and the latter positive parity9. We see thatwhen β → 0, which is the case when the image of the source is a ring, weexpect the magnication to become large. It is also worth noting that if thesource is much further away than the lens, Ds Dd the distance to thesource cancels out:

Ds Dd ⇒ Dds Dd ⇒ θ2E =

4GM

c2

Ds

DdDds→ 4GM

c2Dd(12)

5.3 Applications

Apart from microlensing, one application of gravitational lensing is in cos-mology. By analyzing the lensing of far away galaxies caused by the inter-vening matter along the line of sight one can probe the content of ordinaryand dark matter on dierent redshifts.

The lensing by large scale structure (called "cosmic shear") and how itevolves with redshift can help constrain the physical models of the observedaccelerated expansion of the Universe (e.g. Weinberg et al. 2013) which isone of the most important tasks in cosmology today.

Until now the majority of cosmological applications of gravitational lens-ing has been made in the visual waveband (Bonaldi et al. 2016, and referencestherein) but measuring weak lensing10 in the radio band could be a promisingcomplement (Chang et al. 2004).

9Negative parity means the resulting image will be mirrored in respect to the source.10When the gravitational lensing eect is strong enough to create rings, arc and other

clearly visible alteration of the images it is called "strong gravitational lensing". "Weakgravitational lensing" on the other hand is when the eect is much smaller so that it cannotbe distinguished by a single background source but instead is measured as a statisticaleect aecting a (large) number of objects.

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6 The hypothesis

In microlensing and other applications using gravitational lensing, the lens-ing concerns a point source lensed by a point mass. Liebes (1964), Refsdalet al. (1964) and Bontz (1979) on the other hand introduce the idea of apoint mass lens and an extended background. Saslaw et al. (1985) takesthat idea further and suggest a number of areas in which gravitational lens-ing of extended sources could be a useful observational tool when used withVery Long Baseline Arrays of sucient resolution. Examples include deter-mination of pulsar masses, detection of black holes and other dark compactobjects, accurate mass determination of nearby stars, distance measuringand detection of subresolution structure in background clouds.

The hypothesis for this project came from Markus Janson, at the de-partment of Astronomy at Stockholm University and stems from the ideasof Saslaw et al. (1985) but suggest yet another area of utilization, namely thedetection of exoplanets (Markus Janson, private communication 2017). Theidea concerns the interstellar medium in the Galaxy and its radio emission.If observed at high enough resolution, small scale structure should becomeevident. These structures should not change over short time scales (years).If a planet would be in the line of sight it would act as a gravitational lensaltering the image of the background source. These changes would of coursebe very small but with a telescope with high enough resolving power andsensitivity it would be possible to detect these changes. In the remainderof this thesis we will investigate the plausibility of this using the upcomingsecond incarnation of the Square Kilometer Array (SKA) telescope currentlyin its planning phase.

6.1 How to test it?

To test this hypothesis, we devised a numerical simulation, consisting of threemain modules: the background, the lensing and the telescope modules. Thebackground module creates a synthethic background with a simple structure.It takes the scale of the structure as an input parameter. The lensing moduleputs a point mass in between the observer and the background at a chosendistance and then uses the physics (and geometry) described in section 5.2to calculate the deection of the light rays from the background source. Thelast module, the telescope module, simulates how the telescope, in our casethe SKA, aects the image. Mainly this is done by adding noise and byintroducing a point spread function to the incoming signal.

Apart from free oating planets the deecting exoplanet will have a hoststar. This host star will of course be of greater mass than the planet andthus the gravitational eect on the background signal will be far greaterthan that from the planet. However, if the Einstein radius of the star isa few times smaller than the angular separation between the star and the

20

planet it should be of little problem to distinguish the deection caused bythe planet from that caused by the star. To be certain of this we can use (9)and calculate the Einstein-radius, θE for a solar-mass star at a distance of 50pc. For this conguration θE ≈ 0.6 AU. This implies that if the planet is afew AUs from the star the separation of their gravitational lensing signatureshould be easily accomplished.

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7 The background

The radio background at ∼ 1 − 10 GHz is dominated by thermal emissionfrom H II regions and synchrotron radiation originating from cosmic-rayelectrons accelerated by supernova remnants such as the Cassipeia A andthe Crab Nebula.

7.1 H II regions

H II regions are clouds of hydrogen ionized by young bright stars. Thesestars, often of spectral type O or B, excite the hydrogen with its immenseemission of ultraviolet light. The H II regions in turn emit free-free emissionat radio frequencies. Free-free, or Brehmstrahlung, emission is produced byfree electrons scattering o ions. The electrons are free both before and afterthe interaction; hence the name free-free emission. For a more comprehen-sive description on free-free emission we direct the reader to e.g. Rybickiet al. (1985).

According to Paladini et al. (2003) and Anderson et al. (2014), who bothcollected a large number of studies of the galactic H II region population, thehighest frequency occurs in the galactic disk plane with almost a Gaussiandistribution centered at zero latitude. Although H II regions can be foundat all longitudes, approximately 76% of all the sources in Anderson et al.(2014) are within 60 deg of the Galactic center.

From Paladini et al. (2003) we calculate an average ux density from all1442 sources. Since the sources are presented as measured at 2.7 GHz, weextrapolate these values from 2.7 GHz with the theoretical spectral index α =−0.1 (S ∝ να), which is typical of thermal bremsstrahlung emission in a thinplasma (e.g. Rybicki et al. 1985; Paladini et al. 2003). The average brightnessof all the sources, at 2.7 GHz and extrapolated to the four frequencies weare using, can be found in table 1.

Table 1: Average brightness calculated from Paladini et al. (2003) and ex-trapolated to higher frequencies.

ν [GHz]Brightness

[mJy/arcsec2]Comment

2.7 0.52 From Paladini et al. (2003)5 0.49 Extrapolated14 0.44 Extrapolated20 0.43 Extrapolated24 0.42 Extrapolated

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7.2 Synchrotron radiation from supernova remnants

When charged particles are accelerated by a magnetic eld they emit elec-tromagnetic radiation. If the particles are relativistic, this emission is calledsynchrotron radiation and it can be detected in radio frequencies which iswhy this is interesting for our discussion. Synchrotron radiation plays a partin the total continuum background radiation which we will use in our lensingexperiment. For a more detailed explaination of synchrotron radiation seee.g. Rybicki et al. (1985).

7.3 Combined radiation at GHz frequencies

All current radio surveys of GHz frequencies have been conducted at muchlower resolution levels than that of SKA2. The combined brightness of allemission processes at frequencies 10− 30 GHz indicate that the brightnessis in the same magnitudes as derived from H II region surveys (e.g. Reichet al. 1986; Altenho et al. 1979; Pauls et al. 1976; Handa et al. 1987;Oliveira-Costa et al. 2008). However, what really is of interest to us, isnot the low resolution brightness though, but the high resolution brightnesscontrast. The brightness measured in low resolution will be divided intohigher resolution structures, which at the resolutions we are interested inhopefully will lead to high contrast structures with "dark" areas with lowbrightness and "light" areas with brightnesses much higher than the onespresented above. The exact constitution of the structures and its brightnesscontrast will only be known once high resolution observations are possible.

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8 Radio astronomy

The radio part of the spectrum is very broad and spans roughly betweenν ∼ 10 MHz and ν ∼ 1 THz. In wavelengths this corresponds to λ ∼30 m and λ ∼ 0.3 mm. More or less everything emits radio waves to someextent, through a wide variety of emission mechanisms. Since radio wavescan penetrate interstellar dust clouds and Compton-thick11 layers of neutralgas, few astronomical sources are obscured (Condon et al. 2016). Whenastronomers rst turned to observing in radio, unexpected new objects suchas radio galaxies, quasars, pulsars, cold molecular clouds and the whisperfrom the big bang itself in form of the cosmic microwave background wererevealed. Radio astronomy is a rewarding discipline. Since radio wavelengthsare much longer than atmospheric dust grains radio waves are not subject toso called Rayleigh scattering. And the Sun is not overwhelmingly bright inradio, which makes most observations possible day as well as night. Also, theatmosphere is nearly transparent to electromagnetic radiation in the radiowavelengths which is a condition only shared with (parts) of the visibleband as can be seen in gure 8. The boundaries of this "radio window" arenot absolute and variations in both altitude, geographical position and timeexists.

Figure 8: The only electromagnetic radiation to which the atmosphere isnearly transparent is in the visible and radio part of the spectrum. Theseare called the visible and radio atmospheric windows. Image from Condonet al. (2016).

11"Compton-thick" means that for certain wavelengths the light will be absorbed orscattered due to the Compton-eect in which photons are scattered by electrons.

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8.1 Radiative transfer

Here we avoid giving a detailed walk through of the basics of radiative trans-fer. For such we refer the reader to a vast number of sources including (e.g.Rybicki et al. 1985; Wilson et al. 2013). Instead we will just give a very briefintroduction to some of the concepts used later to aid the reader with littleknowledge in the subject. The reader already familiar with this subject mayskip this section.

Even though electromagnetic radiation in radio frequencies is a wave phe-nomenon, when the scale of the system is much larger than the wavelengthwe can consider the radiation to travel in straight rays. The innitesimalpower intercepted by an innitesimal surface can then be written as:

dP = Iν cos θ dΩ dσ dν, (13)

wheredP is the power in watts, WIν is the brightness in W m−2 Hz−1 sr−1

θ is the angle between the normal to dσ and the direction to dΩdΩ is the innitesimal solid angle subtended by the signaldσ is the innitesimal surface area, in m2

dν is the innitesimal bandwidth in Hz.

The total ux density of a source is calculated by taking the integral of(13) over the total solid angle Ωs subtended by the source.∫

ΩS

Iν(θ, φ) cosθ dΩ (14)

This ux density is measured in units of Wm−2 Hz−1. Flux densities ofradio sources are often very small and therefore a special radio astronom-ical ux density, the Jansky (abbreviated Jy) has been introduced (1 Jy= 10−26 Wm−2Hz−1). For extended sources Jansky per solid angle or perbeam12 are used. This density, the spectral power density, can be measuredby radio receivers. A such receiver can for instance be a radio telescope, oractually a device inside the radio telescope since the whole telescope consistsof several separate devices (antenna, amplier, A/D converter etc.) process-ing the electromagnetic signal.

8.2 Interferometry

Using a single radio telescope dish with the diameter D the resolution islimited to the diraction limit at θ ≈ λ/D. Even the largest single dish radiotelescopes (at least precision telescopes where the RMS13 reector surface

12The beam of a radio telescope is the opening of the smallest resolvable angular unit.13Root Mean Square.

25

error is small) are limited to θ >> 1 arcsec (Condon et al. 2016). In orderto (vastly) improve the resolution of radio observations one could combineseveral telescopes into an interferometer array. The array of antennas actas a single telescope giving the resolution equivalent of a telescope withthe diameter the size of the largest baseline14 of the array. By placing theindividual telescopes in certain patterns, great sensitivity and resolution canbe achieved. One of the most prominent advantages with interferometricsetups is that the resolution is a function of largest baseline while the eldof view is dependent of the disc size of the individual antennas, giving thepossibility of creating a telescope system with both high resolution and largeeld of view.

8.3 Aperture synthesis

Based on fourier transforms, a signal processing technique called aperturesynthesis combines the separate signals into one image. When using inter-ferometry, the radiation from the source reaches each individual antenna atdierent times, which introduces a phase shift. To combine all these signalswith dierent phases, the signals are processed and algorithms construct animage of the radiation from the source. The construction of the image alsohas to deal with the fact that in contrary to one big telescope, several smallerantennas means that there are gaps where no signal is received. These gapsneed to be accounted for and special processing algorithms need to be usedin order to compensate for the gaps. The more of the telescope area that arenon-receiving gaps, the harder it is to construct an image of the combinedsignals. The ability to combine dierent signals to one single image dependsof many factors such as the conguration in which the individual receptorsare places on the ground, minimal and maximum distance between antennapairs etc. (Levanda et al. 2010). Each baseline contributes one Fourier com-ponent so the more antennas the more details in the combined image. Themaximum number of baselines to be used for a system of antennas are thenN(N − 1)/2 were N is the number of antennas. More antennas also increasethe sensitivity of the system. For a more thorough review of the subject werefer the reader to (e.g. Levanda et al. 2010; Condon et al. 2016; Thompsonet al. 2017).

8.4 Radio Interferometic Arrays around the world

8.4.1 The Very Large Array

The Very Large Array (VLA) in the USA consists of 27 telescope dishes eachwith a diameter of 25 m (see Figure 9). The telescope operates between the

14The baseline is the distance between two dishes, antennas or telescopes.

26

Figure 9: The Very Large Array. Image courtesy of NRAO/AUI.

frequencies of 73 MHz and 50 GHz. The highest achievable angular resolutionis 40 mas15 (at 50 GHz) (Napier et al. 1983).

8.4.2 The European Very Large Baseline Interferometry Network

The European VLBI Network consists of around 20 telescopes around theworld (for example OSO16 in Sweden) giving it a very large maximum base-line. It operates on the frequencies between ν ≈ 1.6 GHz and ν ≈ 22GHz.It is the most sensitive VLBI array in the world and is mainly focused oncosmic radio sources.

15Milliarcsecond (mas).16Onsala Space Observatory in Onsala south of Gothenburg.

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9 The Square Kilometer Array

The Square Kilometer Array (SKA) will, when in operation, be the largesttelescope in the world. It is a telescope operating in the radio regime of theelectromagnetic spectrum and it will have an eective aperture of up to amillion square meters (Dewdney et al. 2009).

The project, a joint venture between 19 countries, began back in the earlynineties and the current estimate for completion is 2023. When completed,the SKA will be able to produce observe and produce data on wavelengthsfrom ∼ 6 m (50 MHz) down to ∼ 1 cm (30 GHZ). The rst incarnation of thearray, SKA1, will consist of two sub-arrays: SKA1-LOW, built in WesternAustralia, will be an aperture array operating at low radio frequencies from50 MHz to 350 MHz, whilst SKA1-MID will be a dish array with up tove observational frequency bands spanning the range 350 MHz to 15 GHzlocated in South Africa. Its resolution will far exceed current large radiotelescope arrays such as the Very Large Array (VLA) described in section8.4.1.

Figure 10: Artist rendition of the 15 m wide SKA-mid dishes in South Africa.Image credit: The SKA organization.

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Among other things the SKA is expected to shed new light on the fol-lowing areas of research (Braun et al. 2015):

• The period when the rst stars were born, some 100 million years intothe life of the Universe.

• The epoch of formation of the rst galaxies.

• Understanding the current epoch of accelerated expansion of the Uni-verse.

After the completion of the rst phase the mid-range of the telescope willreach 15 GHz (Braun 2017). This is only the rst phase of the SKA. Thereare plans for an upgrade of the SKA array which is designated SKA2. Thetimeline and exact details for this phase is yet to be worked out. As de-scribed in Dewdney et al. (2013), the full SKA (SKA2) will be a signicantexpansion of SKA1, with the current plan for SKA-MID increasing the num-ber of dishes from ∼ 200 to ∼ 2000 and spreading long baselines over SouthAfrica, undergoing construction between 2023 and 2030. As the sensitivityapproximately scales with the total collecting area, for SKA2 we assume aten times increase in sensitivity of the instrument and a twenty time increasein maximum angular resolution. The detailed design of the SKA2 is plannedto take place between 2018 and 2021.

For our purpose it is the second phase version of the SKA-MID whichis of interest. This instrument will provide unprecedented levels of bothinstrument sensitivity and spatial resolution. An increase in resolution for aradio telescope means the beam size decreases. The tradeo with a smallerbeam size is of course that the signal is divided amongst more beams whichmeans a fainter signal per beam. Therefore we need to nd a suitable beamsize for which we can nd a balance between the requirement of spatialresolution and signal to noise ratio. As we saw in section 5.2 the scale of thegravitational lensing deection is the Einstein ring radius which means thatour beam size need to be in the order of that or smaller for the eect to benoticeable.

Since the preparatory work of the second phase of SKA is in its very initialstage there exist no mission statements or science performance documentsso we have only those for the SKA1 to avail. From these we can assert thatin the frequency span of ∼ 10 GHz the anticipated galactic science to bedone are H2O, protoplanetary discs, magnetized plasma and high sensitivyobservations of continuum emitting objects (Dewdney et al. 2013).

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10 The simulation

Here we describe the simulation process step by step. We start by synthet-ically creating a background source and end up with a nal image of thegravitational lensing as seen through the SKA2 telescope. We also discusswhich idealizations and approximations we have used throughout the simu-lation.

10.1 Approximations and idealizations

10.1.1 Geometrical approximation

In our attempts at describing the propagation of gravitationally deectedrays in chapter 5.2 we fully utilize the approximation of geometrical opticsand ignore the wave properties of light. It is shown that this is an excel-lent approximation in nearly all astronomical cases (Schneider et al. 1992).However we need to show this explicitly for our systems in order to establishthat we use the correct approximation in our model. What will decide if theapproximation suces is the size of our lens in relation to the frequency ofthe light according to the following relation (Cremonese et al. 2018):

ML ≥ 105 M

( ν

Hz

)−1, (15)

where ML is the mass of the lens and ν is the frequency of the radio waves.Our interest lies in the frequency regime around ν ' 1 GHZ and puttingthat into (15) yields:

ML ≥ 10−4 M ≈ 33 M⊕ ≈ 0.1 MJ , (16)

which means we can use this approximation for planets with Jupiter-massesand above but not for Earth-mass planets.

10.1.2 Sample points instead of extended source

When calculating the resulting image of the source (both with and withoutlensing) the background source is treated as innitesimal points instead of anextended surface. The reason for this is that it makes the calculations simplerand the computations signicantly faster. Since the analytical treatment ofan extended source is just an integral over the surface (Schneider et al.1992) using sampling points with a small enough distance between them, isa commonly used method in numerical approximations, and the dierencebetween the two methods should be negligible.

10.1.3 Point lens

The planet in our simulation is approximated as a point lens instead of asan extended object. This greatly simplies our task and should be sucient

30

for our purposes. Of course a real planet is not a point but an extendedobject and also the mass of planet does not necessarily have to be sphericallysymmetrically distributed. Since we are interested in exoplanets the mass isat maximum in the order of that of Super-Jupiters (see above) with a massof M ∼ 0.01 M. That would give a Schwarzschild radius of:

RS =2GM

c2=

2G · 0.01Mc2

= 29.5 m, (17)

which at the distances we are simulating (∼ 1 pc) would be ∼ 10−10 arcsec.The requirement of using a point lens approximation is that the impactparameter ξ RS which is well satised for our purpose (Schneider et al.1992).

Also worth noting is that the point lens approximation provides relationsbetween e.g. lens mass, distances and angular separation between images ofthe lensed source which are of the same order-of-magnitude as those frommore realistic lenses (Schneider et al. 1992).

10.1.4 The gravitational inuence of intermediate mass

A great simplication that we have made is that the only mass in our sys-tem the we consider is that of the lens. We have not taken into account anyintermediate mass that of course would aect the path of the rays and alsonot other possible planets orbiting the same star. This simplication wasnecessary in order to create a manageable simulation to test the hypothe-sis within the scope of this work. Further investigation into the impact ofthe mass distribution of the intermediate space would be part of a furtherinvestigations concerning this method of detecting exoplanets.

10.1.5 The antenna function

The image resulting from the telescope is highly simplied. Observing usinginterferometry, especially interferometry with long baselines, is a complextask that requires the observer to take a number of dierent factors intoaccount. The image is aected by the so called antenna function which con-verts the signals from the telescopes via Fourier transform into a spatiallyresolved image. Apart from what we described in sections 8.2 and 8.3, wewill not go further into the specics of interferometric radio astronomy butwe direct the interested reader to e.g. Thompson et al. (2017) for a deeperunderstanding. We are fully aware of the fact that our simulation is a sim-plied version of reality but we deem it sucient for the purpose of testingthe viability of the hypothesis. In our idealized version we use a GaussianPSF instead of the actual antenna function. This simplication should beeligible due to the high number of antennas of the SKA2 (Woody 2001).

31

10.2 Step 1 - The background

For the background source we use a method similar to that in Saslaw et al.(1985). They constructed simulated background sources by placing straight-line isophotes17 with their brightness varying linearly with the x distancefrom the deector. We instead construct the background source plane usinga simple cosine function in one direction. The isophotes in this case are wavesin the x direction with all waves having the same amplitude. By altering thefrequency of the cosine wave the width of the waves changes and dierentscales of the structure of the background radio emission can be tested. Forthe creation of the waves there were some properties that needed to be met,namely that there be only positive values of the brightness and that thebackground be symmetrical. Such waves can be constructed in many ways,we chose the following:

I(x) = A

[cos

(2πf

fsx+ πf

)+ 1

], (18)

where I(x) is the brightness, fs is the sample rate and f is the frequency ofthe waves (how many waves one length unit contains). A is the amplitude ofthe wave. Integrating over an arbitrary area shows that the integrated valueof our background is exactly half that of a uniform background. So thereforeA = 2 and,

I(x) = 2

[cos

(2πf

fsx+ πf

)+ 1

], (19)

Some examples of this can be seen in gure 11 where we show 5 backgroundswith varying structural scales. These backgrounds are the ones used in thelater steps of the simulation.

The background plane is a quadratic grid with adjustable sample size(number of sample points). The sample size was determined by a combina-tion of computing time and resolution. The nal sample size was tested sothat further increase in size did not have a signicant eect on the result.

Since, as already stated, the SKA telescope will oer unprecedented res-olution we do not have past observational results to compare with in order toestimate common scales of the radio background at our frequencies. There-fore the size of the structure is a parameter that can be varied in order totest dierent structural scales and examine how these aect the lensing andsubsequently the possibility of planet detection.

17An isophote is a curve connecting points of equal brightness.

32

(a) Wave width = 1 mas (b) Wave width = 2 mas

(c) Wave width = 4 mas

(d) Wave width = 10 mas (e) Wave width = 20 mas

Figure 11: The background source as simulated by our code with the thewidth of the isophotes varied. The frequency is ν = 24 GHz. Note that theux density scale is linear.

33

10.3 Step 2 - Lensing

Using the physics from section 5.2 and a method close to the one that Schnei-der et al. (1992) calls "ray shooting", we constructed a code that takes lightrays from the background source and, depending on the properties of thelens i.e. its mass and position (distance from the background and observer),deects them accordingly.

As described above, the background source is divided into a uniform gridof points. Each point will be associated with a position on the source planeand a brightness calculated from the background structure described above.The lens plane is divided into a corresponding uniform grid.

We assume that each sample point of the background emits isotropically.All points except the point which is on the optical axis contributes with twoimage points. The sample on the optical axis contributes with an Einsteinring according to equation (9). As could be seen in section 5.2 each sourcepoint will be deected into two points in the lens plane with dierent mag-nication of the brightness. The properties of the lensing conguration canbe seen in table 2 and gure 12 shows lensed versions of the backgroundssources shown in gure 11. What we should direct our interest towards, is

Table 2: Properties used in the modeling.

Input parameters

Property Value

Distance to lens 50 pcDistance to source plane 800 pcMass of lensing planet 0.01 MFrequency 5, 14, 20 & 24 GHzBeam size varying (with frequency)Observation time 10 hours

Output parameters

Einstein radius 1.24 mas

the alteration of the source wave pattern when deected by the lens. As isevident, the smaller structure (smaller width of the waves) show the mostprominent changes and for the largest structures the eect is virtually non-existent. This is due to the fact that the structure of the lensing source needto be smaller than the Einstein radius in order for the deection to havea signicant eect. Inside the Einstein ring the waves are bent into "ears"starting and ending in origo and outside the Einstein ring, the waves aremerely deected around the ring. This is even more evident when we zoomin on one of the images as is shown in gure 13.

34

(a) Wave width = 1 mas (b) Wave width = 2 mas

(c) Wave width = 4 mas

(d) Wave width = 10 mas (e) Wave width = 20 mas

Figure 12: Lensing of parallel isophotes consisting of sinus waves. In allcases ML = 20 MJ the distance to the source is 800 pc and the distance tothe lens is 50 pc. The frequency is ν = 24 GHz. The ux density scale islogarithmic.

35

Figure 13: A zoom view of the lensed image in gure 12.

10.3.1 The Einstein Ring

As we learned in section 5.2 when the impact parameter ξ → 0, the raysdeect into an Einstein ring and the magnication µ→∞. This is unphys-ical and would violate the law of conservation of energy. But this requirea point source and a point lens, none of which is realistic. In order not tooverestimate the magnication for points on, or close to the Einstein ringin our simulation we create a thin annulus at the Einstein ring position andchange all pixel values on the annulus to an average value from the pixelsjust inside and outside the annulus. The result of this procedure can be seenin gure 14.

36

(a) Without Einstein ring correction (b) With Einstein ring correction

Figure 14: Zoom of lensed image without (a) and with (b) correction foroverestimated Einstein ring.

10.4 Step 3 - The Telescope

The telescope mainly aects two aspects of the images: the system noise,and the antenna function, which (as explained above) we approximate witha Gaussian point spread function.

10.4.1 Step 3a - Noise

An advantage of observing in the radio band is that, in contrast to ground-based optical observations, turbulence on arcsecond scales in the Earth'satmosphere is not expected to signicantly distort the images and decreasethe accuracy. This eect, called "seeing", is a major limiting systematicfor ground-based optical weak lensing experiments. However, at lower radiofrequencies, turbulence in the ionosphere and troposphere means this seeingeect returns. Fortunately this eect has a strong negative (typically ∼ν−2) scaling with frequency, which means it may be avoided by observing athigher frequencies. For the frequencies interesting to us, the eect should benegligible (Bonaldi et al. 2016).

The noise of the SKA1-mid telescope is calculated according to Braun(2017) as:

σSKA1−mid =SDSEFD

ηS√npol∆ν∆τ

, (20)

where (all from Braun 2017)

• SD is a "degradation factor relative to the natural array sensitivity forthe specic target Gaussian FWHM resolution of the image" which isapproximately SD ≈ 2

• SEFD is the System Equivalent Flux Density given by SEFD =2kbTsys/Aeff where kb is the Boltzmann constant and Tsys/Aeff is the

37

system temperature divided by the eective collecting area, a valuethat depends on frequency and that is tabulated in (Braun 2017)

• ηS is a system eciency that takes account of the nite correlatoreciency and other forms of incoherence and is assumed to be ηS = 0.9

• npol is the number of contributing polarizations assumed to be npol = 2

• ∆ν is the bandwidth. Here we used the same as used as example forcontinuum observations in Braun (2017): ∆ν/νc ≈ 0.3 where νc is thecentral wavelength of the observation.

• ∆τ is the integration time

All this is for the SKA1-mid telescope and not the later incarnation SKA2.Since the exact properties of the SKA2 telescope is far from decided on wehave used the anticipated improvement is comparison to SKA1 as statedon the SKA science website18 where it says that the SKA2 "is likely toinclude...", "...10 x SKA1 sensitivity in the frequency range of 350 MHz 24GHz." Accordingly, when calculating the error for the SKA2 we have used:

σSKA2 =σSKA1−mid

10(21)

The noise from (20) is the RMS noise which means it is the standard devia-tion of the noise. Here it refers to the noise per beam. So in order to simulatethe SKA noise we calculate the RMS per beam according to (20) and (21)and then create an array with the same shape as the image grid. Each arrayposition then corresponds to a point on the image grid. For every such arrayposition the noise level is calculated simply by randomizing a value from nor-mal distribution with zero as mean and σSKA2 as the standard deviation, seegure 15. In this way the total image noise will have a Gaussian distributionas intended. Since radio systems, unlike optical systems, are characterizedby Gaussian noise characteristics (optical detectors are limited by Poissonstatistics) the noise is simple added to the signal (Thompson et al. 2017).So therefore the noise array is then added to the image.

18https://astronomers.skatelescope.org/ska2/, retrieved 2018-05-20.

38

Figure 15: An example of the synthetically created noise which as added tothe signal. The ux density scale is logarithmic.

10.4.2 Step 3b - The PSF

Since the SKA2 is still in its early planning stage we do not currently knowthe specications of the telescope. In this work we have used numbers basedon the current target estimations of the SKA219, numbers from Dewdneyet al. (2009) and extrapolated properties from SKA1 performance Braun(2017). As stated before, this implies a ten times increase in sensitivity, atwenty times increase in maximum angular resolution and a maximum base-line of 3000 km. To simulate the interferometric aperture synthesis describedabove, we approximate each beam with a Gaussian function and convolvethe incoming rays with the beam functions. Each beam is simulated as aGaussian with a full width half maximum (FWHM) which is calculated from

19As stated on the SKA website (May 2018) https://astronomers.skatelescope.org/ska2/.

39

the angular diraction limit (e.g. Condon et al. 2016):

θD =λ

B, (22)

where λ is the wavelength in which we observe (λ = c/ν) and B is thebaseline of the interferometry array. If we compare the diraction limit(22) with the FWHMs of SKA1 presented in (Braun 2017) we see that forSKA1 the FWHM associated with the highest sensitivity grade is betweenθB ∼ 1.6 × θD and θB ∼ 200 × θD where θB is the beam FWHM. Usingthe same factor for the SKA2 but with the the baseline B of SKA2 (3000km) the FWHM of the PSF for an observation at 24 GHz at the highestsensitivity level should be θB ∼ 1.37 − 171 mas. See table 3 for the beamwidths of all the four frequencies used in the simulation.The simulation of the eect on the images by the telescope PSF is done

Table 3: Beam FWHM ranges for highest sensitivity level

ν [GHz] θmin [mas] θmax [mas]

5 6.60 82514 2.35 20620 1.64 19424 1.37 171

by convolving the image with a Gaussian PSF with the help of Pythonpackage Astropy (Astropy Collaboration et al. 2013; Price-Whelan et al.2018). An example of a lensed image with synthetically created noise addedand convolved with the PSF is presented in gure 16.

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Figure 16: An example of a lensed image after the noise is added and con-volved with the PSF. The simulation parameters are ν = 24 GHz, beamsize=1.37 mas. The background source structure width is 4 mas and thebackground is 100 times brighter than calculated in section 7. The uxdensity scale is linear.

10.5 Step 5 - Removing the background

Before the result is nalized, we remove the background in order to be leftwith only the changes due to the gravitational lensing by the interveningplanet. This is a straightforward task that involves creating an image ofthe unlensed background, adding noise and nally the PSF eect. Thisimage is then subtracted from the image created in the previous steps. Asa last substep the absolute value of all points are taken in order to treat alldeviations from the unlensed image equally20

20For example if a sample point has a higher value in the unlensed image than in thelensed image the resulting background subtracted image gets a negative value in thatpoint.

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10.6 Step 6 - The nal images

We now have a nal image of the simulated radio observation with the SKA2telescope. The nal images of the dierent setups described in the previoussteps are presented in gure 17 and in table 2 the properties used in thesimulation are presented.

As is evident by visual inspection, in gure 17, and also conrmed numer-ically, the eect on the lensed background is not signicant in the resultingimages. We therefore tested the simulations with a stronger source signal. Abackground source two magnitudes stronger gave a positive match for someof the widths. This is presented in gure 18. The simulation was tested withdierent telescope settings. For comparison the SKA1 settings were testedbut since the highest frequency of that conguration is ν ≈ 14 GHz and thesmallest beam size (with the highest sensitivity) is ≈ 60 mas the eect of adeecting planet is much to small. The highest frequency we test is ν = 24GHz. Since the conguration of the second phase of the SKA2 is still to bedecided on we want to be modest in our anticipation of the highest limits andregarding the frequency, reading about the SKA2 in the SKA1 documentsthis seems as a reasonable level. We also tested with the SKA2 congurationwith lower frequencies.

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(a) Wave width = 1 mas (b) Wave width = 2 mas

(c) Wave width = 4 mas

(d) Wave width = 10 mas (e) Wave width = 20 mas

Figure 17: The nal result of the simulations for the dierent widths of thebackground structure after the unlensed images are removed. The frequencyis ν = 24 GHz and the beam size is θB = 1.37 mas. The brightness scale islinear.

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(a) Wave width = 1 mas (b) Wave width = 2 mas

(c) Wave width = 4 mas

(d) Wave width = 10 mas (e) Wave width = 20 mas

Figure 18: The nal result of the simulations with a stronger background.The frequency is ν = 24 GHz and the beam size is θB = 1.37 mas. Thebrightness scale is linear.

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11 Results

There are many ways possible to detect a signature from an exoplanet usingour method. In this work we use a very simple one which is very likely notthe most accurate. We estimate the signal to noise ratio by making 5 circularapertures in the resulting simulated background subtracted image. One ofthose apertures is around the planet lensed signal (fluxsignal) and the fourothers are spread out in the signal-free part of the image (fluxnoise,i, withi=1-4), see gure 19. We calculate the signal to noise ratio in the followingway:

SNR =|fluxsignal− < fluxnoise >|

σnoise, (23)

where σnoise is the standard deviation of the four noise aperture values. Ifthe SNR is ' 20 we consider it a positive match.

Figure 19: An example showing the apertures with which we calculate thesignal to noise ratio. The one in the center is the signal and from the fourothers we calculate the noise with equation (23).

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The results for the dierent background source structure widths anddierent frequencies are presented in gure 20, where we present the meanof a Monte Carlo test run with 10 simulations at each frequency (excludingthe highest and lowest values for each width).

When simulating a telescope with the characteristics described above wend that the ux from only H II regions is too low using the simple waveisophotes we describe above. The signal to noise ratio (SNR) can be improvedin a number of ways. If the contrast of the background structure were higherthan that of the isophotes of our simulation that would increase the SNR.Of course the structure of the background is very unlikely to show suchordered structure as simple one-directional waves. The brightness we usein our simulation comes from extrapolations from observations with muchlower resolution than what is possible with the SKA so there could verywell exist smaller scale structures showing a much higher contrast than wetested for in our simulations. Also, increasing the observation time couldincrease the sensitivity and thus possibly the SNR. And since the secondincarnation of the SKA is still on the design table it might have an evenhigher sensitivity than our crude extrapolation from the SKA1 propertiesshowed. Therefore we consider this method for nding exoplanets plausiblefor future instruments and we suggest that further studies of its use shouldbe initiated.

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Figure 20: A plot showing the mean signal to noise ratio from a Monte Carlorun of 10 simulations per source structure width. We repeated the tests forfour dierent frequencies.

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12 Future development

Since the completion of the telescope is at least a decade away this leavesgood time for further testing and for preparing the method. More complexsimulations would be welcomed and also algorithms responsible for locatingthe positives signals in survey data should be developed. He we present afew topics that would be interesting to devote time on in order to analyzethe versatility of this exoplanet detection method.

• A bayesian calculation of the possibility of nding positive exoplanetmatches would be benecial where the adequate priors would give fur-ther insights in the plausibility of this method.

• Finding the best way to detect a positive signal. The way we did it withve apertures to measure the signal and the noise is a very rudimen-tary method. A more elaborate way to detect positive matches wouldincrease the usability of this method and decrease the false matches.

• Since the wavelength of our interest are quite long an examination ofthe possibility of detecting signal matches through interference wouldbe interesting.

• An investigation of best frequency region. We tested four frequenciesand although positive matches (when we increased the backgroundsignal) was made will all four there is likely a lower and upper frequencylimit between where the detection is most easily made in respect tobeam size, sensitivity and interfering radio noise.

• Testing our model with more irregular background structures to en-sure that the positive result does not depend on the regularity of thestructures.

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13 Usability of our method

13.1 Target planets

The planet signature in our method is proportional to the mass of the planetso the limit here is set by the resolution of the telescope and the brightness ofthe background. As can be seen in gure 2 there is a clear bias in detectionso far towards planets with shorter orbits P < 104 days which shows a clearvalue for a detection method like our.

13.2 Number of possible stars within our target distance

The number of stars within a distance of 10 pc is around 1,700 and within adistance of 50 pc the number is in the vicinity of 70,000 (Prusti et al. 2016;Brown et al. 2018) so the number of possible target should be enough for alarge number of possible detections.

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14 Conclusions

Since the basis for this method is a telescope that is around a decade forwardin time, the nal properties it will have are far from certain. The aim for thiswork has therefore not been to achieve a ready-to-use manual for searchingfor exoplanets with the SKA telescope but to make a rst, rough, try-out ifthis method could be plausible at all and to point out further investigationsthat need to be done before achieving a real world astronomical tool. Withthat said, we have concluded that using the second phase of the SKA as atool for searching for exoplanets using gravitational lensing shows promise.The biggest uncertainty of the possibility of this method is the nature ofthe background source on these scales. Given that the SKA would oer aresolution by far unprecedented by earlier telescopes the structure of thebackground is not known and therefore poses the biggest uncertainty tothe possible functionality of this method. The brightness contrast of thestructural element and its scale seems to be the biggest obstacle to overcome.But since there are no previous observations at the resolution of interest, theanswer is something that lays further down the road. If it would work itwould be a complement to other exoplanet search methods already in use.One of the advantages with this method is that it does not need to allocatetelescope time since it can use already collected data from other observations.Since the eld of view of the SKA (in its second phase) is so large (almost3 mas) and the survey speed is very high, large portions of the sky could besurveyed at the same time. In addition our method would also be perfectfor oine analysis which means archive datasets could very well be used.

Our simulations indicate that this method could be of interest for thepurpose of nding exoplanets. When testing with several parameters we ndthat a positive signal is found for certain input parameters. The gravitationallensing created by a planet is, to our knowledge, smaller than any that hasbeen observed this far. Our simulations show that for certain values of thebackground structure size, the sensitivity of the observations and the beamsize the gravitational lensing by a Super-Jupiter sized planet will show apositive match with our method. The ndings of our work suggests thatfurther modeling, simulations and investigations would be worthwhile. Ifthis method would prove useful, this would suggest a new tool to add to thetoolbox for detecting exoplanets. And just maybe, it could be an componentin the quest for an answer to the question of our place in the Universe.

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Acknowledgements

I want to thank my supervisor Markus Janson for giving me such an inter-esting topic to work with and for all his help during this master thesis.

I would also like to thank Edvard Mörtsell at the department of Physicsfor a number of discussions regarding the physics of gravitational lensing;Poonam Chandra at the department of Astronomi for helping me under-stand the mysterious ways of radio telescopes and Tobia Carozzi at OnsalaSpace Observatory at the Department of Earth and Space Sciences, ChalmersUniversity for helping me with noise calculations.

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