Les gens ont des étoiles qui ne sont pas les mêmes. Pour les uns qui voyagent, elles sont des guides.

Transcription

1 Les gens ont des étoiles qui ne sont pas les mêmes. Pour les uns qui voyagent, elles sont des guides. Pour les autres, elles ne sont rien que des petites lumières. Pour d autres, qui sont savants, elles sont des problèmes. Le Petit Prine (Antoine de St Exupéry ry) 5/9/9 1 Cours Ing., J. Surdej 1

2 . Gravitational lenses 4. PHYSICAL BASIS OF GRAVITATIONAL LENSES: 4.1. General remarks: Geometrial quantities haraterizing the defletion of a light ray by a point mass lens (M). 5/9/9. Gravitational Lenses: 4. PHYSICAL BASIS OF GRAVITATIONAL LENSES; 4.1. General remarks: The physial basis of gravitational lensing essentially onsists in the defletion of light, and eletromagneti waves in general, in gravitational fields as predited by Einstein's theory of General Relativity. In the regime of small defletion angles and of weak gravitational fields, whih are of pratial interest to us here, the so-alled Einstein defletion of a light ray passing near a ompat mass at a distane is 4GM α( ) = = R s << 1, (4.1) where G and stand for the onstant of gravitation and the veloity of light, respetively, and where R s represents the Shwarzshild radius assoiated with the mass M (see the above figure). For an extended mass, it is easy to alulate the defletion angle by just summing up (integrating) the individual defletions due to all the mass elements onstituting the lens. Sine there is usually just one mass onentration whih ats as a lens, and whih has a small size relative to the distanes involved, the thin lens approximation is usually justified. We therefore introdue a lens plane (ζ = (, η)) through the enter of mass of the lens and perpendiular to the line defletor-observer (see Fig.). All the mass an then be onsidered to be loated in the lens plane and the defletion to take plae where the ray rosses the lens plane. The defletion an therefore be expressed as a two-dimensional angle vetor, ' ' ) ( η α ) )( ( 4 ζ d d = G Σ, (4.) Cours Ing., J. Surdej ζ ζ ' ζ ' ζ ζ '

3 . Gravitational lenses 4. PHYSICAL BASIS OF GRAVITATIONAL LENSES: 4.1. General remarks: R 4GM α( ) = = α ( ζ 4G ) = x α s ( ζ ')( ζ ζ ' Σ ζ ζ ' << 1, (4.1) ) d ' d η i O b '. (4.) r n = 1 U/ Fig.: Gravitational light defletion by a massive objet 5/9/9 3. Gravitational Lenses: 4. PHYSICAL BASIS OF GRAVITATIONAL LENSES; 4.1. General remarks: where Σ(ζ ) represents the surfae mass density of the lens at the loation ζ (assuming that Σ(ζ ) = M δ(ζ ), where δ(ζ ) is the Dira funtion, the defletion law (4.1) is easily retrieved). In Newtonian terms, the Einstein defletion also follows if one assumes a refrative index n whih depends on the Newtonian gravitational potential U of the lens via the relation n = 1 - U/. We see here one more some analogy with atmospheri lensing. For those students who already have taken a ourse in General Relativity, the above results may be determined as follows. From the general expression of the spae-time metris ds = -(1+U/ ) dt + (1-U/ )(dx + dy + dz ), applied to the ase of a photon (ds =) travelling along the x spae axis, we find to the first order approximation that (dx/dt) = v ~ (1+U/ ) and therefore n = /v ~ 1-U/. As for the ase of atmospheri lensing (Desarte s law: n os(i) = Cte), the expression of the defletion angle takes the form (sine di/dx = 1/n dn/db where b orresponds to the impat parameter -see the above figure- and using the relation tg(i) = db/dx) α = - (di/dx) dx = - 1/n b n dx = ( / ) b U dx. For the ase of the Newtonian potential orresponding to a point-mass lens, we find that U = -GM/r = -GM/(b +x ) 1/ and b U = GM b /(b +x ) 3/ and finally, α = ( / ) GM b /(b +x ) 3/ dx = (4GM/ ) b / b.(to set y = x/b and y = tg(µ) in the last equation). Cours Ing., J. Surdej 3

4 . Gravitational lenses 4. PHYSICAL BASIS OF GRAVITATIONAL LENSES: 4.. The lens equation: θ - θ s = α(θ) = -(D ds / D os ) α(θ). (4.3) or, equivalently, ζ -ζ s = D od α(θ) = -(D od D ds / D os ) α(θ). 5/9/9 4. Gravitational Lenses: 4. PHYSICAL BASIS OF GRAVITATIONAL LENSES; 4.1. General remarks: Sine the Einstein defletion is independent of wavelength, gravitational lenses are ahromati (indiret hromati effets may however be indued by miro-lensing, see setion 9.4.). Furthermore, geometrial optis an be used sine physial optial effets are negligible in realisti situations. 4.. The lens equation: Let now the true position of the soure S on the sky be defined by the angle θ s and the image position(s) by θ i (i=1,,...). These orrespond of ourse to the solutions of the lens equation (f. the previous diagram), θ - θ s = α(θ) = -(D ds / D os ) α(θ), (4.3) where D ds and D os represent respetively the "defletor- soure" and "observer-soure" angular size distanes and where α is the displaement angle, α = -(D ds / D os )α (by appliation of the sine rule in the previous diagram from whih sin(α)/d ds = sin(α )/D os and assuming that sin(α ) ~ α and sin(α ) ~ α, α and α being infinitesimal angles). We note that a given image position always orresponds to a speifi soure position whereas a given soure position may sometimes orrespond to several distint image positions. Suh ases of multiply imaged soures onstitute of ourse the most spetaular and interesting aspets of gravitational lensing. Cours Ing., J. Surdej 4

5 . Gravitational lenses 4. PHYSICAL BASIS OF GRAVITATIONAL LENSES: 4.3. Magnifiation and amplifiation: (a) Soure Image (b) dθ s θ s θ i dθ i µ i = θ i dθ i / θ s dθ s (a) Image of a lensed soure as seen projeted on the sky. (b) Idem, axially symmetri defletor at enter! µ i = dω i / dω s = det( θ s / θ i ) -1. (4.4) 5/9/9 5. Gravitational Lenses: 4. PHYSICAL BASIS OF GRAVITATIONAL LENSES; 4.. The lens equation: A typial lens situation is shown on the next figure, where soure and image positions (one image in this ase) are seen projeted on the plane of the sky. We see again that the image position is shifted by α relative to the soure position; note however that α is usually not onstant over the soure and this results in possible (de-)magnifiation and deformation of extended soures Magnifiation and amplifiation: Sine gravitational lensing preserves the surfae brightness of a soure, the ratio (magnifiation) between the solid angle dω i overed by the lensed image and that of the unlensed soure dω s immediately gives the flux amplifiation µ i due to lensing. More formally, this is given for an infinitesimal soure by the inverse jaobian of the transformation matrix between the soure and the image(s): µ i = dω i / dω s = det( θ s / θ i ) -1. (4.4) If there are several images of a given soure, the total magnifiation (amplifiation) is of ourse given by the sum of all individual image magnifiations (amplifiations). We shall hereafter use the term 'magnifiation' whenever the lensed images (f. luminous ars, radio rings, et.) are resolved by the observer, thus emphasizing the hange in angular size, and the term 'amplifiation' otherwise (f. when referring to miro-lensing effets or to the integrated flux of unresolved marolensed images). Cours Ing., J. Surdej 5

6 . Gravitational lenses 4. PHYSICAL BASIS OF GRAVITATIONAL LENSES: 4.4. Axially symmetri lenses: α ) = ( 4G M ( ), (4.5) µ i = θ i θ s d d θ i θ s. (4.6) 5/9/9 6. Gravitational Lenses: 4. PHYSICAL BASIS OF GRAVITATIONAL LENSES; 4.4. Axially symmetri lenses: Considering a thin gravitational lens that is axially symmetri with respet to the line-of-sight, we an, in virtue of Gauss's law applied to the two-dimensional ase, rewrite the defletion given in Eq.(4.) as the salar angle α ) = ( 4G M ( ). (4.5) It is as if only the mass M() loated inside the ylinder defined by the impat parameter was ontributing to the light defletion and may be thought of as ating like a single point mass loated at the enter (ompare with Eq.(4.1)). The defletion aused by the matter distributed outside this ylinder exatly anels out. All this reminds us of ourse of the situation for the ase of gravitational (or Coulomb) fores aused by spherially symmetri mass (or eletri harge) distributions. Sine light defletion by an axially symmetri lens redues to a one-dimensional problem (beause light rays are defleted in a plane), it is also straightforward to simplify expression (4.4) for the magnifiation µ i of the lensed images as follows (see the upper image at right on the previous page) µ i = θ i θ s d d θ i θ s. (4.6) Cours Ing., J. Surdej 6

7 (A): Gravitational lensing (B): Optial lens experiment O O O 1 O 1 Defletor plane Observer plane Lens plane Observer plane 5/9/9 7 Cours Ing., J. Surdej 7

8 (A): Gravitational lensing (B): Optial lens experiment α O O O 1 O 1 Defletor plane Observer plane Lens plane Observer plane 5/9/9 8 Cours Ing., J. Surdej 8

9 (A): Gravitational lensing (B): Optial lens experiment α O O ε O 1 O 1 Defletor plane Observer plane Lens plane Observer plane 5/9/9 9 Cours Ing., J. Surdej 9

10 n i r ε() 5/9/9 1 Cours Ing., J. Surdej 1

11 . Gravitational lenses 5. THE OPTICAL GRAVITATIONAL LENS (GL) EXPERIMENT: 5.1. Shapes of axially symmetri optial lenses: Defletion of a light ray passing through an axially symmetri optial lens. 5/9/9 11. Gravitational Lenses; 5. THE OPTICAL GRAVITATIONAL LENS EXPERIMENT: For didatial purposes (see the appliations in setion 6), it is very useful to onstrut and use optial lenses that mimi the defletion of light rays as derived in Eq.(4.5) for the ase of axially symmetri gravitational lenses. Suh optial lenses should of ourse be rotationally symmetri, flat on one side (for simpliity) and have, on the other side, a surfae determined in suh a way that rays haraterized by an impat parameter gets defleted by the angle ε() = α() (see Eq.(4.5) and the above figure) 5.1. Shapes of axially symmetri optial lenses: Applying Desartes's law (f. Eq.(3.)) to the ray depited in the above figure and assuming that the angles (r and i) between the normal n to the optial surfae and the inident and refrated rays are very small, we may write the relation sin n = sin ( i) i ( r) r, (5.1) where n represents here the refrative index of the lens with respet to the air. Furthermore, sine we have i = ( ) + r ε 4GM ( ) = + r, (5.) and that the tangent to the optial surfae at the point (, ) is merely given by (see the above figure, stands for the thikness of the lens) Cours Ing., J. Surdej 11

12 . Gravitational lenses 5. THE OPTICAL GL EXPERIMENT: 5.1. Shapes of axially symmetri optial lenses: sin n = sin i = ( i) i ( r) r ( ) + r,, 4GM ( ) = ε d = r d + r (5.1), (5.) (5.3) d 4GM ( ) = d ( n 1) (, Rs ) = ( ) + ln( ) n 1 5/9/9 1 (5.4). (5.5). Gravitational Lenses: 5. THE OPTICAL GRAVITATIONAL LENS EXPERIMENT: 5.1. Shapes of axially symmetri optial lenses: d = r d, (5.3) it is straigtforward to derive the shape of a lens by means of the following differential equation d = d 4GM ( ) ( n 1) The optial point mass lens:. (5.4) By definition, the mass M of a point lens model is onentrated in one point suh that we have M() = M. It is then simple to solve Eq.(5.4) and derive the thikness () of the orresponding optial lens as a funtion of the impat parameter. We find that, (5.5) Rs ( ) = ( ) + ln( ) n 1 where R s represents the Shwarzshild radius of the ompat lens (f. Eq. (4.1)). In pratie, the point (, ( )) is hosen in order to speify a given thikness (e.g. ( ) = 1 m) for the optial lens at a seleted radius (e.g. = 15 m). The resulting shape of suh an optial 'point mass' lens is illustrated on the next figures (see Fig. a on left and the left lens on the right photograph). Cours Ing., J. Surdej 1

13 . Gravitational lenses 5. THE OPTICAL GL EXPERIMENT: 5.1. Shapes of axially symmetri optial lenses: Below: several examples of axially symmetri optial lenses simulating the light defletion properties due to a point mass (a), a SIS galaxy (b), a spiral galaxy (), a uniform disk (d) and a trunated uniform disk of matter (e). Right: examples of (upper left) a 'point mass' lens (8 m in diameter) manufatured at the Hamburg Observatory and of (lower right) a 'spiral galaxy' optial lens (3 m in 5/9/9 diameter) produed by the authors at the European Southern 13 Observatory (Garhing bei Münhen).. Gravitational Lenses: 5. THE OPTICAL GRAVITATIONAL LENS EXPERIMENT: 5.1. Shapes of axially symmetri optial lenses; The optial point mass lens: It looks very muh like the foot of some glasses of wine whih, therefore, have been ommonly used in the past by well known astronomers to simulate lensing effets. A realisti 'point mass' lens, made of plexiglas-like material (refrative index n = 1.49 and a diameter of 8 m), has been manufatured at the Hamburg Observatory for the partiular value of R s =.3 m. This orresponds in fat to the Shwarzshild radius of one third of the Earth mass. We have used suh lenses (see the above models on the photograph and the simulations presented during the letures), made of plexiglas-like material (n = 1.49), to simulate the formation of multiple images of a distant soure (see setion 6). Our optial gravitational lens experiment is desribed in setion 5.. Very reently (Otober 6), we have begun a mass prodution of point mass lenses in plexiglas with the following haraterization : diameter of 15 m, n = 1.49 and an equivalent Shwarzshild radius of R s =.6 m, orresponding to a blak hole with a mass of /3 that of the Earth The SIS optial lens: For the ase of a singular isothermal sphere (hereafter SIS) lens model, it is well known that the mass of suh a galaxy inreases linearly with the impat parameter, i.e. M(). We may thus rewrite Eq. (5.4) in the form d / d = -K, (5.6) where K represents a positive onstant. Integration of the above equation leads to the solution () = ( ) + K ( - ). (5.7) Cours Ing., J. Surdej 13

14 . Gravitational lenses 5. THE OPTICAL GL EXPERIMENT: 5.1. Shapes of axially symmetri optial lenses: d / d = -K, (5.6) () = ( ) + K ( - ). (5.7) Σ() = Σ exp(-/ ), (5.8) M ( ) π Σ ( ') ' d ' =, (5.9) M ( ) = π Σ 1 exp( / )(( / ) + 1), (5.1) 5/9/9 14. Gravitational Lenses: 5. THE OPTICAL GRAVITATIONAL LENS EXPERIMENT: The shape of the resulting SIS lens is thus merely an axially symmetri one as illustrated in Fig. b (what would be the shape of suh a refleting lens?) Shapes of axially symmetri optial lenses; The 'spiral galaxy' optial lens: Given the exponential surfae mass density (f. the similar surfae brightness distribution) Σ() = Σ exp(-/ ), (5.8) whih desribes reasonably well the mass distribution of a spiral galaxy disk having a harateristi size, we may derive the mass distribution M() of suh a defletor by means of the relation = M ( ) π Σ ( ') ' d ', (5.9) Integration of this last expression leads immediately to the result M ( ) = π Σ 1 exp( / )(( / ) + 1). (5.1) Inserting this result into Eq. (5.4) and performing the integration, we find that n 1) ( z /. (5.11) = ( exp( ) exp( dz ) ln( Σ z / ) ( ) ) exp( ) π 8 G The general shape of a 'spiral galaxy' optial lens is illustrated in the previous Fig.. A 3 m diameter 'spiral galaxy' lens, produed by the authors at the European Southern Observatory (Garhing bei Münhen), is shown on the previous photograph (right figure, at bottom right). Cours Ing., J. Surdej 14

15 . Gravitational lenses 5. THE OPTICAL GL EXPERIMENT: 5.1. Shapes of axially symmetri optial lenses: ( ) = ( Σ 8πG ) + ( n 1) ln( ) exp( ) + exp( / exp( z) ) + dz, (5.11) / z M() = π Σ, if, (5.1a) M() = π Σ, if >. (5.1b) 5/9/9 15. Gravitational Lenses; 5. THE OPTICAL GRAVITATIONAL LENS EXPERIMENT: 5.1. Shapes of axially symmetri optial lenses; The 'spiral galaxy' optial lens: This manufatured lens is haraterized by the following physial parameters: an equivalent Shwarzshild radius of one third of the Earth mass, i.e. R s = GM( ) / = 4π G Σ / =.3 m (see Eq. (5.1)), =. m, = 15 m and ( ) =.7 m. We should also like to mention that Vanderriest (1985) has manufatured a similar type of lens (~ m in diameter) diretly from a piee of glass, at the Meudon Observatory The uniform disk optial lens: As we shall see in setion , a uniform irular disk of matter leads to a perfet onvergene of all inoming light rays from a distant soure into a single 'foal' point, assuming that the line-ofsight is perpendiular to the disk. We onlude that a lassial onverging optial lens onstitutes the natural ounterpart of suh a defletor (f. Fig. d) The trunated uniform disk optial lens: For the ase of a trunated uniform disk of matter, let us assume that the surfae mass density Σ() is onstant (= Σ o ) for values of the impat parameter in the range [, ] and that it is zero outside. We then get M() = π Σ, if, and M() = π Σ, if >. (5.1a) (5.1b) Inserting these results into Eq. (5.4), a simple integration leads to the solution for the thikness () of the optial lens as a funtion of the impat parameter Cours Ing., J. Surdej 15

16 . Gravitational lenses 5. THE OPTICAL GL EXPERIMENT: 5.1. Shapes of axially symmetri optial lenses: ( ) = ( 4 G ) + ( n 1) Σ ln π, if, (5.13a) ( ) = ( 4π ) + ( n 1) Σ ln G (5.13b) ( ) = ( Σ πg ) + ( ) ( n 1), if. (5.13) 5/9/9 16. Gravitational Lenses: 5. THE OPTICAL GRAVITATIONAL LENS EXPERIMENT: 5.1. Shapes of axially symmetri optial lenses; The trunated uniform disk optial lens: 4πG, if, (5.13a) = + Σ ( ) ( ) ln ( n 1) from whih we may derive ( and ) = ( 4π ) + ( n 1) Σ ln G (5.13b) πg, if. (5.13) ( ) = ( ) + Σ ( ) ( n 1) We have illustrated in a previous figure (see Fig. e) the shape of suh an optial lens that simulates the gravitational lensing effets due to a trunated uniform disk of matter. Cours Ing., J. Surdej 16

17 GL mirage simulator for the ase of grazing inidene light refletion (point-like mass lens) r i α x α y α = 4GM / ( x) dy/dx = -tg(r)) with i + r + α = π, i = r, and thus r = π/ - α/ and finally, 5/9/9 17 y = y + K (x x ), with K = 1/( s ) ), with K = 1/( R s Cours Ing., J. Surdej 17

18 GL mirage simulator for the ase of grazing inidene light refletion (point-like mass lens) 5/9/9 18 Cours Ing., J. Surdej 18

19 GL mirage simulator for the ase of grazing inidene light refletion (point-like mass lens) 5/9/9 19 Cours Ing., J. Surdej 19

20 GL mirage simulator for the ase of normal inidene light refletion (point- like mass lens) α x y α = 4GM / ( x) dy/dx = -r r with i + r = α, i = r, and thus r = α/ and finally, y = y + K ln(x / x) with K = Rs) 5/9/9 Cours Ing., J. Surdej

21 GL mirage simulator for the ase of normal inidene light refletion (point- like mass lens) 5/9/9 1 Cours Ing., J. Surdej 1

22 GL mirage simulator for the ase of normal inidene light refletion (point- like mass lens) 5/9/9 Cours Ing., J. Surdej

23 GL mirage simulator for the ase of grazing inidene light refletion (uniform disk lens) y = y + K ln(x / x), with K = /(GπΣ ) 5/9/9 3 Cours Ing., J. Surdej 3

24 GL mirage simulator for the ase of normal inidene light refletion (uniform disk lens) y = y + K (x x ) with K = (4G/ )πσ 5/9/9 4 Cours Ing., J. Surdej 4

25 The Optial GL Experiment (light reletion) 5/9/9 5 Cours Ing., J. Surdej 5

26 THE OPTICAL GL EXPERIMENT: 5/9/9 6. Gravitational Lenses: 5. THE OPTICAL GRAVITATIONAL LENS EXPERIMENT: Lens by refletion! Kamehameha Floral Parade in Waikiki on 1 June 6! Cours Ing., J. Surdej 6

27 THE OPTICAL GL EXPERIMENT: 5/9/9 7. Gravitational Lenses: 5. THE OPTICAL GRAVITATIONAL LENS EXPERIMENT: Lens by refletion! Kamehameha Floral Parade in Waikiki on 1 June 6! Cours Ing., J. Surdej 7

28 THE OPTICAL GL EXPERIMENT: 5/9/9 8. Gravitational Lenses: 5. THE OPTICAL GRAVITATIONAL LENS EXPERIMENT: Lens by refletion! Kamehameha Floral Parade in Waikiki on 1 June 6! Cours Ing., J. Surdej 8

29 5/9/9 9 Cours Ing., J. Surdej 9

30 Gravitational lensing inversion 5/9/9 3 Cours Ing., J. Surdej 3

31 Gravitational lensing inversion Formation of a 3-lensed image mirage by gravitational lensing. 5/9/9 31 Cours Ing., J. Surdej 31

32 Gravitational lensing inversion Superposition of the ray traing diagram shown in the previous figure and a opy of it, rotated by 18. In this way, we learly see that the light rays passing through the pinhole sreen (PS) get similarly defleted, but in an opposite diretion, to those oming from the distant soure (S). The three light rays passing the seond, inverted, defleting galaxy (D) thus ontinue their travel as a beam of three parallel light rays. 5/9/9 3 Cours Ing., J. Surdej 3

33 Gravitational lensing inversion Same as previously but for the ase of a defletor (D) at left being loated at a distane three times loser to the pinhole sreen (PS) and being also three times less massive than the original defletor (D1) 5/9/9 33 Cours Ing., J. Surdej 33

34 Gravitational lensing inversion A two lensed image mirage produed by a osmi point mass lens (P1) at right. The two inoming rays pass through the pinhole sreen (PS) and get defleted by a point-mass lens objet (P) at left, at a distane 3 times loser from the pinhole and whih mass is also 3 times smaller than that of the original osmi point mass lens (P1). Alike for the ase disussed previously, the outgoing light rays are again parallel. 5/9/9 34 Cours Ing., J. Surdej 34

35 Gravitational lensing inversion Shape (left) and photograph (right) of an optial lens simulator orresponding to the ase of a point mass lens. Suh a lens is made of Plexiglas and its shape, similar to the foot of a wine glass, is essentially determined by the mass of the point mass defletor (Refsdal and Surdej 1994). 5/9/9 35 Cours Ing., J. Surdej 35

36 Gravitational lensing inversion Same as beore but the small size osmi lens at left (P) has been replaed by an optial point mass lens simulator (S) orresponding to the same mass. Sine the separation between the two outgoing light rays is now redued to several tens of entimetres, or even smaller, it is easy to plae at left a lassial onverging lens (CL) so that a perfet, lens inverted image of the distant soure at right (S) is formed in its foal plane. 5/9/9 36 Cours Ing., J. Surdej 36

37 Gravitational lensing inversion The first point mass gravitational lens simulator (S1) at right produes a doubly imaged soure as seen from the pinhole (PS) while the seond lens simulator (S) inverts the mirage into two parallel light rays whih are then foused at left by a lassial onverging lens (CL). 5/9/9 37 Cours Ing., J. Surdej 37

38 Gravitational lensing inversion Optial benh in the laboratory showing from right to left the laser point soure (LS) obtained with a spatial filter (mirosope objetive ombined with a pinhole sreen), followed by a lens (C) that ollimates the light rays into a parallel beam whih enters the first point mass gravitational lens simulator (S1). Some of the defleted rays then enounter the pinhole sreen (PS) and enter the seond optial lens simulator (S). The outgoing light rays are then foused by means of a onverging lens (CL) on the white sreen (WS) set at the extreme left. 5/9/9 38 Cours Ing., J. Surdej 38

39 Gravitational lensing inversion Another view of the laboratory optial benh. In this ase, the first point mass gravitational lens simulator (S1) has been somewhat tilted with respet to the optial axis. By plaing a white sreen (WS) just behind the pinhole sreen (PS), one sees the formation of a quadruply imaged soure, in aordane with preditions made for the ase of a point mass lens in presene of an external shear (see Refsdal and Surdej 1994). 5/9/9 39 Cours Ing., J. Surdej 39

40 Gravitational lensing inversion (a) (b) () In ase of perfet alignment between the soure (LS), the pinhole (PS) and the optial lens (S1), set perpendiularly with respet to the axis of the optial benh, there results the formation of an Einstein ring (a) as seen on the white sreen (WS) set between the pinhole (PS) and the S lens. If we slightly translate along a diretion transverse to the optial axis the S1 lens with respet to the pinhole, the Einstein ring breaks in two lensed images (b). If instead, we slightly tilt the S1 lens with respet to the axis of the optial benh, the Einstein ring breaks into four lensed images () (see Refsdal and Surdej 1994). 5/9/9 4 Cours Ing., J. Surdej 4

41 Gravitational lensing inversion (a) (b) Reonstrution of the inverted lensed images of the laser soure on the white sreen (WS) plaed at the extreme left, for the ase of a single (a) point-like soure image S and that of a double one (b). The double soure was obtained by means of a beam splitter. The diameter of the spot(s) seen in (a) and (b) is approximately mm. In these two ases, the S1 and S lenses were set perpendiularly to the optial benh axis, slightly translated in opposite transverse diretions and symmetrially plaed with respet to the pinhole (PS). 5/9/9 41 Cours Ing., J. Surdej 41

42 Gravitational lensing inversion 3-D ray traing simulations performed with the Matlab software for the ase of the two optial point mass lens simulators (S1 and S) being slightly tilted ( ) and translated along the vertial diretion with respet to the pinhole (PS). Note that one of the lenses (S) is twie less massive than the other one (S1) and plaed at appropriate distanes with respet to the pinhole in order to provide an overall optimal gravitational lens refrator. 5/9/9 4 Cours Ing., J. Surdej 4

43 Gravitational lensing inversion Another example of 3-D ray traing simulations performed with the Matlab software for the ase of the two equal optial point mass lens simulators (S1 and S) slightly tilted (1 ). 5/9/9 43 Cours Ing., J. Surdej 43

44 Gravitational lensing inversion A simple unfolded model of the optimal gravitational lens telesope (OGLT). It is here assumed that the astronomial telesope pointed towards a gravitational lens system produes in its foal plane at right two lensed images (A & B) of a distant point-like soure as well as the image of the defletor (D). The latter has been entred on the phase mask (PM) of the oronagraphi devie shown in this figure. The lens C ollimates the light forming an exit pupil where a Lyot stop is plaed. The optial point mass lens simulator S then inverts the observed gravitational lens mirage into two parallel beams of light rays whih are then foused at left by a lassial 5/9/9 onverging lens (CL). A single image of the original point-like soure, observed 44 as a multiply imaged objet, has thus been restored in the foal plane of CL. Cours Ing., J. Surdej 44

45 . Gravitational lenses 5. THE OPTICAL GL EXPERIMENT: 5.. Setup of the optial gravitational lens experiment: Setup of the optial gravitational lens experiment. 5/9/9 45. Gravitational Lenses: 5. THE OPTICAL GRAVITATIONAL LENS EXPERIMENT: 5.. Setup of the optial gravitational lens experiment: In order to simulate the formation of lensed images by a given mass distribution (point mass, et.), we have used the optial setup that is shown on the next figure. The optial gravitational lens experiment is omposed of a light soure (on the left on the piture), the optial lens followed by a white sreen with a small hole at the enter (pinhole) and further behind, a large sreen on whih is (are) projeted the lensed image(s) of the soure as it would be seen if our eye was loated at the position of the pinhole (the pinhole orresponds to the observer s eye). In the example illustrated here, the pinhole is set very preisely on the optial axis of the gravitational lens so that the soure, the lens and the pinhole (observer) are perfetly aligned. In the present ase, the resultant image seen on the sreen is an Einstein ring. Considering other relative positions between the soure, the lens and the observer, and also for the additional ase of an asymmetri lens, we shall illustrate in setion 6 the resulting lensed images as a funtion of the pinhole position in the observer plane. Note that the bright regions seen on the lens in the figure shown above are aused by sattered light. Cours Ing., J. Surdej 45

46 . Gravitational lenses 6. GRAVITATIONAL LENS MODELS: 6.1. Axially symmetri lens models: X α α S. D i (=θ). O On the ondition for an observer O to see a light ray from a distant soure S, deviated by a defletor D so that more than one image an be seen. Note that O, D and S are oaligned and that axial symmetry is assumed. No sale is respeted in this and all subsequent diagrams. 5/9/9 46. Gravitational Lenses: 6. GRAVITATIONAL LENS MODELS: 6.1. Axially symmetri lens models; Generalities: In the ase of perfet alignment between a soure (S), an axially symmetri defletor (D) and an observer (O) (see the above figure), we easily see that, with the exeption of the diret ray propagating from the soure to the observer, the ondition for any other light ray to reah the observer is θ / D ds ~ α / D os, (6.1) as obtained from the diret appliation of the sine rule to the triangle SXO and assuming that the angles θ and α remain very small. Of ourse, this will also be true if the following ondition (*) is fulfilled: α α sine it will then be always possible to find a light ray with a larger impat parameter suh that Eq. (6.1) is fulfilled. Expressing the angle θ between the diret ray and the inoming defleted ray as θ ~ / D od, making use of Eqs. (4.5) and (6.1), we may thus rewrite ondition (*) as follows Σ( <) Σ, average surfae mass density of the lens Σ( <) = M() / (π ), (6.4) (6.) and (6.3) i.e. the evaluated within the impat parameter, must simply exeed the ritial surfae mass density Σ, defined by, (6.5) Σ = 4πG D Cours Ing., J. Surdej 46

47 . Gravitational lenses 6. GRAVITATIONAL LENS MODELS: 6.1. Axially symmetri lens models: θ / D ds ~ α / D os, (6.1) θ ~ / D od, (6.) Σ( <) Σ, (6.3) Σ( <) = M() / (π ), (6.4) Σ = 4πG D D = D od D ds / D os,, (6.5) (6.6) 5/9/9 47. Gravitational Lenses: 6. GRAVITATIONAL LENS MODELS: 6.1. Axially symmetri lens models; Generalities: where D = D od D ds / D os. (6.6) Let us note that the latter quantity is essentially determined by the distanes between the soure, the defletor and the observer. As we shall see in setion , a irular disk with uniform surfae mass density Σ, ats as a perfet onverging lens, and we have Σ = Σ for the partiular ase when the observer is preisely loated at its foal point. Although the above reasoning essentially applies to a stati Eulidian spae, Refsdal (1966b) has shown that it also remains valid for Friedmann, Lemaître, Robertson, Walker (FLRW) expanding universe models, provided that D os, D od and D ds represent angular size distanes. Adopting typial osmologial distanes for the defletor (redshift z d ~.5) and the soure (z s ~ ), we find that Σ ~ 1 g m -. Substituting M() and in Eq. (6.4) with a typial mass M and a radius R for the defletor, we have listed in the appended Table values for the ratio Σ(<R) / Σ onsidering a star, a galaxy and a luster of galaxies loated at various distanes. We see that only stars and very ompat, massive galaxies and galaxy lusters, for whih Σ(<R) / Σ 1, onstitute promising 'multiple imaging' defletors. More preisely, we may write, (6.7) z ( ) d D = L( z ) 1 d H ( ) zs Cours Ing., J. Surdej 47

48 . Gravitational lenses 6. GRAVITATIONAL LENS MODELS: 6.1. Axially symmetri lens models: D = H L( z ( ) 1 ( z d d zs ) ). (6.7) The quantities L(z) and (z) as a funtion of the redshift z for various osmologial models 5/9/9 48. Gravitational Lenses: 6. GRAVITATIONAL LENS MODELS: 6.1. Axially symmetri lens models, Generalities: where L(z d ) is a funtion presenting a well defined extremum around z d ~.7, almost independently of the adopted osmologial model (see the next figures). A natural onlusion is that foreground objets loated near z d ~.7 should prove to be very effiient lenses. In the ase of axial symmetry, it is lear that in the presene of an effiient defletor, an observer loated on the symmetry axis will atually see a ring (the so-alled 'Einstein ring', f. the previous photograph) of light from a distant soure. Combining Eqs. (4.5)-(6.1) and (6.), the angular radius of this ring may be onveniently expressed as θ E = R' Dod Dod D 4GM ( θ ) E os D. (6.8) ds Cours Ing., J. Surdej 48

49 . Gravitational lenses 6. GRAVITATIONAL LENS MODELS: 6.1. Axially symmetri lens models: 4GM ( R' Dodθ ) E Dds θ = E Dod Dos. (6.8) Table: Ratio of the average Σ(R < R) and ritial Σ surfae mass densities, angular (θ E ) and linear ( E ) radii of the Einstein ring for different values of the mass M, distane D od and radius R of the defletor, assuming that D os = D od (1parse = 1 p = 3.6 lightyears = m) Defletor M D od R Σ(R <R)/Σ θ E E = θ E D od (p) Star 1 M 1 4 p 1-8 p " p Star 1 M 1 9 p 1-8 p " 1 - p Galaxy ore 1 1 M 1 9 p p 4 " 1 4 p Cluster ore 1 14 M 1 9 p 1 5 p 1 " 1 5 p 5/9/9 49. Gravitational Lenses: 6. GRAVITATIONAL LENS MODELS: 6.1. Axially symmetri lens models: Generalities: We have also listed in the previous Table typial values of θ E for different types of defletors loated at various distanes. As we shall see in the next setions, the value of θ E derived above is very important beause it an usually be used to estimate the angular separation between multiply imaged soures in more general ases where the ondition of a perfet alignment between the soure, defletor and observer is not fulfilled or even for lens mass distributions whih signifiantly depart from the axial symmetry. Observed image separations (~ θ E ) an therefore lead to the value of M / D od, or to the value of M H, if the redshifts z d and z s are known. This is the simplest and most diret astrophysial appliation of gravitational lensing. We see from the previous Table that for a soure and a lens loated at osmologial distanes (z d ~.5 and z s ~ ), the angle θ E an vary from miro-arse (stellar defletion) to arse (galaxy lensing), and up to some tens of arse in the ase of luster lenses. Let us also finally note that the ondition (6.3) for a defletor to produe multiple images of a lensed soure turns out to be usually appliable, even when there is no axial symmetry. We shall now desribe in more detail some of the best known lens models. Cours Ing., J. Surdej 49