unusual LISA Jean-Yves Vinet A.R.T.E.M.I.S. Observatoire de la Cte - PowerPoint PPT Presentation

unusual LISA Jean-Yves Vinet A.R.T.E.M.I.S. Observatoire de la Côte d’Azur NICE (France)

Contents 1) Gravitational coronography 2) Signals from asteroids GGI/FLORENCE 28-30 J-Y. Vinet 2 Sept 2006

Gravitational coronography Tinto & Larson CQG 22/10 S531 (2005) Nayak, Dhurandhar, Pai & Vinet PRD 68 122001 (2003) Resolved Source of GW θ ϕ Data combination giving a zero response For direction ( , ) θ ϕ GGI/FLORENCE 28-30 J-Y. Vinet 3 Sept 2006

Benefits (conjecture) • Occultation of a strong source for better analysis of its angular neighborhood • Improving angular resolution GGI/FLORENCE 28-30 J-Y. Vinet 4 Sept 2006

LISA 2’ 2 U V 2 2 δν ฀ , U V ν r L r L 1 n 3 n 3 1 U V 3 1 r 3’ 1 n 2 3 1’ L V U 2 3 1 GGI/FLORENCE 28-30 J-Y. Vinet 5 Sept 2006

Recall • 6 main data channels (1 per phasemeter) • There exist families of combinations of the 6 flows with properly chosen time delays that cancel dominant instrumental noises � TDI (Time Delay Interferometry) (Tinto, Armstrong, Estabrook, 99) • They form a module and have generating parts (sets of generators). (Dhurandhar,Nayak,Vinet, 02) • One may combine these generators for special purposes, keeping the noise cancelling property GGI/FLORENCE 28-30 J-Y. Vinet 6 Sept 2006

Recall • The generators have the form: = ( , , , , , ) g p p p q q q 1 2 3 1 2 3 • Where are formal polynomials ( , ) p q i i in the 3 delay operators D a ≡ − = ( )( ) ( ) ( 1,2,3) D f t f t L a a a GGI/FLORENCE 28-30 J-Y. Vinet 7 Sept 2006

Recall 6-uple of data: = U( ) [ ( ), ( ), ( ), ( ), ( ), ( )] t V t V t V t U t U t U t 1 2 3 1 2 3 Generic noise-cancelling combination g : 3 ∑ = + | U ( ) [ ( ) ( )] g t p V t q U t i i i i = 1 i = 0 when U,V represent laser phase fluctuations GGI/FLORENCE 28-30 J-Y. Vinet 8 Sept 2006

Notation r w θ ϕ Source oriented unit vector : ( , ) r r r ∂ ∂ r r 1 w w θ = ϕ = 3 orthonormal vectors : , , w ∂ θ θ ∂ ϕ sin r a = Unit vector along arm #a : ( 1,2,3) n a Directional functions (spin 2 harmonics): r r r r ξ + = θ ⋅ − ϕ ⋅ 2 2 ( ) ( ) n n a a a r r r r ξ × = θ ⋅ ϕ ⋅ 2( )( ) n n a a a r r r µ = ⋅ Location of node #a : r notation w r a a a GGI/FLORENCE 28-30 J-Y. Vinet 9 Sept 2006

Notation , : the 2 polarization components of the GW h h + × Data flow at node # 1 : − µ − − µ − ( ) ( ) h t h t L + × + × = − ξ , 1 , 3 2 ( ) U t r r + × + × + ⋅ 1 , 2 , 2(1 ) w n 2 − µ − − µ − ( ) ( ) h t h t L + × + × = ξ , 1 , 2 3 ( ) V t r r + × + × − ⋅ 1 , 3 , 2(1 ) w n 3 Others are obtained by circular permutation of indices GGI/FLORENCE 28-30 J-Y. Vinet 10 Sept 2006

Notation Fourier space : transfer functions: % % % % = + = + , U F h F h V F h F h + + × × + + × × a U U a V V a a a a ω µ + ω µ + ωµ ωµ − − ( ) ( ) i i L i i L e e e e 2 3 3 2 1 1 = ξ = − ξ , F F r r r r + × + × − ⋅ + ⋅ 3 , 2 , V U + × + × 2(1 ) 2(1 ) 1 , w n 1 , w n 3 2 + circular permutations 6-uple transfer: = F ( , , , , , ) F F F F F F × + × + × + × + × + × + × +, , , , , , , V V V U U U 1 2 3 1 2 3 GGI/FLORENCE 28-30 J-Y. Vinet 11 Sept 2006

Notation • Example of generators (Tinto et al.) α = (1, , ,1, , ) D D D D D D 3 1 3 1 2 2 β = ( ,1, , ,1, ) D D D D D D 1 2 1 3 2 3 γ = ( , ,1, , ,1) D D D D D D 2 2 3 1 3 1 ς and α ⎡ ⎤ r ⎢ ⎥ = ⎢ ⎥ β • Vector generator: Y ⎢ ⎥ γ ⎣ ⎦ GGI/FLORENCE 28-30 J-Y. Vinet 12 Sept 2006

Algebraic solution Generic combination C : r r = α + β + γ = ⋅ C Y C C C C 1 2 3 Transfer function: % % % = + = ( ) | F ( ) | F ( ) h f C h f C h f + × × + C r r r r % % ⋅ + ⋅ C Y|F ( ) C Y|F ( ) h f h f + × × + GGI/FLORENCE 28-30 J-Y. Vinet 13 Sept 2006

Algebraic formal solution For cancelling any signal, we must have simultaneously: r r r r ⋅ = ⋅ = C Y|F 0, C Y|F 0 × + Thus: r r r = × C Y|F Y|F × + GGI/FLORENCE 28-30 J-Y. Vinet 14 Sept 2006

Explicit solution = α 1) The transfer functions | F F α + × + × , , May be considered as scalar products with ξ The directional functions : r ξ = ξ ξ ξ ( , , ) + × + × + × + × , , 1 2 , 3 , r r r r r r r ∃ α β γ = α β γ ⋅ ξ then , , : ( , , ) F α β γ + × + × , , , , r r r r ⎡ ⎤ β × γ ⋅ ξ × ξ ( ) ( ) + × ⎢ ⎥ r r r And r r = γ × α ⋅ ξ × ξ ⎢ ⎥ C ( ) ( ) + × r r r ⎢ ⎥ r α × β ⋅ ξ × ξ ( ) ( ) ⎢ ⎥ GGI/FLORENCE 28-30 J-Y. Vinet ⎣ ⎦ + × 15 Sept 2006

Explicit solution Recall: ωµ ω µ + ωµ ω µ + − − ( ) ( ) i i L i i L e e e e 1 2 3 1 3 2 = ξ = − ξ , F F r r r r + × + × − ⋅ + ⋅ 3 , 2 , V U + × + × 2(1 ) 2(1 ) 1 , w n 1 , w n 3 2 ωµ ω = = Notation : i i L , g e e e a a a a r r r r = − ⋅ = + ⋅ 1 , 1 v n w u n w a a a a − − g g e g g e = = 1 2 3 1 3 2 , G H 3 2 v u 3 2 GGI/FLORENCE 28-30 J-Y. Vinet 16 Sept 2006

Explicit Solution then : − − − ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ e G e H G e e H e e G H 3 1 2 1 1 2 3 1 2 3 1 1 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ r r 1 1 1 α = − β = − γ = − , , e e G H e G e H G e e H ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 3 2 2 1 2 3 2 2 1 3 2 2 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − − − ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ G e e H e e G H e G e H 3 1 2 3 1 2 3 3 2 3 1 3 Invariance under simultaneous circular permutation of: - Vectors - Components - Indices GGI/FLORENCE 28-30 J-Y. Vinet 17 Sept 2006

Explicit Solution r The direction of the source is constant w B In the barycentric frame There exists a linear mapping to the LISA frame : r r = ⋅ ( ) R( ) w t t w o o B Orbital time parameter, very slowly varying with respect to The « signal time » GGI/FLORENCE 28-30 J-Y. Vinet 18 Sept 2006

Explicit Solution r All functions may be expressed in terms of w ⎧ ⎪ = µ = m ( , ) 1 , / 3 u v w Lw 1 2 1 1 ⎪ ⎪ 1 = ± − µ = − + ⎨ ( , ) 1 ( 3 ), ( 3 ) / 2 3 u v w w L w w 2 2 1 2 1 2 2 ⎪ ⎪ 1 = ± + µ = − − ( , ) 1 ( 3 ), ( 3 ) / 2 3 u v w w L w w ⎪ ⎩ 3 2 1 3 1 2 2 ⎡ ⎤ + − + 2 2 2 1 2 2 w w w 3 1 2 ⎢ ⎥ r r r 3 η ≡ ξ × ξ = + + − − 2 2 2 ⎢ ⎥ 1 2 3 w w w w w w + × 3 3 1 2 1 2 4 ⎢ ⎥ + + − + 2 2 2 ⎢ 1 2 3 ⎥ w w w w w ⎣ ⎦ 3 1 2 1 2 GGI/FLORENCE 28-30 J-Y. Vinet 19 Sept 2006

Explicit Solution η = − ⎡ − − + − − − + ⎤ 2 2 2 1 (1 ) ( ) ( ) C e u v g e g e g e e g u v e e g e g e g e e g ⎣ ⎦ 1 1 2 3 1 3 2 2 3 2 3 1 3 2 2 3 1 2 2 3 3 2 3 1 4 u v u v 2 2 3 3 η { ⎡ + − − − + 2 2 (1 ) ( ) e e e v v g e g e g e e g ⎣ 1 2 3 1 3 1 3 2 1 2 1 3 3 4 u v u v 1 1 3 3 ⎤ − − − + 2 ( ) u u e g e g e g e e g ⎦ 1 3 2 3 1 2 3 2 1 3 1 ⎡ + − − − + 2 ( ) ( ) e e e u v g e g e g e e g ⎣ 1 2 3 1 3 2 3 3 1 1 1 3 2 } ⎤ − − − + 2 ( ) u v e g e g e g e e g ⎦ 3 1 2 2 1 3 3 1 1 3 2 η { + − ⎡ − − + 2 3 (1 ) ( ) e e e u u g e g e g e e g ⎣ 1 2 3 1 2 1 3 3 1 3 1 2 2 4 u v u v 1 1 2 2 ⎤ − − − + 2 ( ) v v e g e g e g e e g ⎦ 1 2 3 2 1 3 2 3 1 2 1 ⎡ + − − − + 2 ( ) ( ) e e e v u g e g e g e e g ⎣ 1 2 3 1 2 3 2 2 1 1 1 2 3 } ⎤ − − − + 2 ( ) 33 different delays u v e g e g e g e e g ⎦ 1 2 2 3 2 1 1 2 1 2 3 GGI/FLORENCE 28-30 J-Y. Vinet 20 Sept 2006

Explicit Solution For retrieving the time domain, simply replace the Phase factors ω ωµ = = i L i , e e g e a a a a By delay operators = − Γ = − µ ( )( ) ( ), ( )( ) ( ) D f t f t L f t f t a a a a µ the delays et are slowly varying L a a Due to the orbital deformation of the triangle (flexing) L a µ Due to the apparent motion of the source viewed from LISA a GGI/FLORENCE 28-30 J-Y. Vinet 21 Sept 2006