[tel-00726959, v1] Caractériser le milieu interstellaire ... - HAL - INRIA

A&A 517, A12 (2010)tel-00726959, version 1 - 31 Aug 2012A.5. Gridding in both planesStarting from the definition of SV G (Eq. (41)), we Fouriertransformit along the sky dimension at constant u p .Usingthatthe gridding along the u p dimension can be factored out of theFourier transform, we derive∫SV G u p(u s ) = g ( )u p − u ′ γp SV u (u ′ ps) du ′ p.(A.36)u ′ pUsing Eq. (A.27), we now replace SV γ u (u ′ ps) in the previous equationto get∫SV G u p(u s ) = γ (u s ) g ( u p − u p) ′ SVu ′ p(u s ) du ′ p, (A.37)oru ′ pSV G u p(u s ) = γ (u s ) SV g u p(u s ) .From this relation, it is easy to deduce thatΣ G ( u p , u s , u ′) = γ (u s ) Σ g ( u p , u s , u ′) .(A.38)(A.39)Using the convolution theorem when taking the inverse Fouriertransform of Σ G along the u s dimension and replacing Σ g (u p , α ′ s,u ′ ) with Eq. (A.24), we finally deriveΣ G ( u p ,α s , u ′) ≡∫∫g ( u p − u p) ′ ( (γ αs − α ′ s) Σ u′p ,α ′ s , u′ p − u′) du ′ p dα′ s . (A.40)u ′ pα ′ sA.6. Wide-field vs. single-field dirty beamsThe notation (59) yields W(u ′ , u ′′ ) = Ω(u ′ , u ′′ ). Using this inEq. (35)givesD ( u ′ , u ′′) =∫Ω ( ) (u p , u ′ + u ′′ − u p S up , u ′′) B ( u p − u ′) du p . (A.41)u pTaking the inverse Fourier transform along the u ′′ axis ofEq. (A.41) and reordering the integral to factor out the termindependent of u ′′ , we can writeD ( u ′ ,α ′′) ∫= B ( u p − u ′) FT 1 (u p , u ′ ,α ′′ )du p , (A.42)u pwhereFT 1 (u p , u ′ ,α ′′ ) ≡∫Ω ( ) (u p , u ′ + u ′′ − u p S up , u ′′) e +i2πu′′ α ′′ du ′′ . (A.43)u ′′We now introduce the following definitionS ( u p , u ′′) ∫≡ S ( )u p ,α s e−i2πα s u ′′ dα s ,(A.44)α sto deriveFT 1 (u p , u ′ ,α ′′ ) =∫α sS ( u p ,α s) [∫ u ′′ Ω ( u p , u ′ + u ′′ − u p)e+i2πu ′′ (α ′′ −α s ) du ′′ ]dα s .Page 18 of 21Using the following change of variables v ≡ u ′′ + u ′ − u p , u ′′ =v − u ′ + u p and dv = du ′′ on the innermost integral, we getFT 1 (u p , u ′ ,α ′′ ) =∫S ( ) ( )u p ,α s Ω up ,α ′′ − α s e+i2π(u p −u ′ )(α ′′ −α s ) dα s .α sSubstituting this result into Eq. (A.42) and taking the inverseFourier transform along the u ′ axis, we can writeD ( α ′ ,α ′′) =∫∫Ω ( ) ( )u p ,α ′′ −α s S up ,α s FT2 (u p ,α s ,α ′ ,α ′′ )du p dα s , (A.45)u p α swhereFT 2 (u p ,α s ,α ′ ,α ′′ ) ≡∫B(u p − u ′ )e +i2πu p(α ′′ −α s ) e +i2πu′ (α ′ −α ′′ +α s ) du ′ .u ′Using the following change of variables v ≡ u p − u ′ , u ′ = u p − vand dv = du ′ ,wegetFT 2 (u p ,α s ,α ′ ,α ′′ ) = B ( α ′′ − α ′ − α s) e+i2πu p α ′ .(A.46)Substituting this result into Eq. (A.45) and re-ordering the terms,we can writeD ( α ′ ,α ′′) ∫= B ( α ′′ − α ′ )− α s FT3 (α s ,α ′ ,α ′′ )dα s , (A.47)α swhere∫FT 3 (α s ,α ′ ,α ′′ ) ≡ Ω ( ) ( )u p ,α ′′ − α s S up ,α s e+i2πu p α ′ du p .u pA simple application of the convolution theorem gives∫FT 3 (α s ,α ′ ,α ′′ ) ≡ Ω ( ) ( )α ′ − α p ,α ′′ − α s Δ αp ,α s dαp ,α pwhereΔ ( α p ,α s) α p⊃u pS ( u p ,α s). (A.48)Substituting this result into Eq. (A.47), we finally derive the desiredexpression, i.e., Eq. (57).Appendix B: From the celestial sphereonto a single tangent planeEquation (1) neglects projection effects, known as non-coplanarbaselines. Any method which deals with interferometric widefieldimaging must take this problem into account. After a shortintroduction to the problem, we show how wide-field synthesisis compatible with at least one method, namely the uvwunfacetingof Saultetal.(1996b). This method tries to builda final wide-field uv plane from different pieces, just as ourwide-field synthesis approach does. Another promising methodis the w-projection, based on original ideas of Frater & Docherty(1980) and first successfully implemented by Cornwell et al.(2008). We did not look yet at its compatibility with wide-fieldsynthesis.