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Laser Interferometer Space Antenna (LISA)

Time-Delay Interferometry with Moving Spacecraft Arrays

Massimo Tinto

Jet Propulsion Laboratory, California Institute of Technology 8th GWDAW, Dec. 17-20, 2003, Milwaukee, Wisconsin

W.M. Folkner et al., C.Q.G., 14, 1543, (1997)

Unequal-arm Interferometers

P.D

Laser

P.D

L2 L3 ν0, p(t)

φ2(t) φ3(t)

) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) (

3 3 3 3 2 2 2 2

t n t p L t p t h t t n t p L t p t h t + − − + = + − − + = φ φ

• M. Tinto, & J.W. Armstrong, Phys. Rev. D, 59, 102003 (1999).

) ( ) ( : ) ( ) ( ] [ ) ( ) ( ) ( ) ( ] [ ) ( ) (

3 3 3 3 3 2 2 2 2 2 i i

L t t D where t n t p I D D t h t t n t p I D D t h t − Ψ ≡ Ψ + − + = + − + = φ φ ) ( )] )( ( ) )( [( )] ( ) ( ][ [ )] ( ) ( ][ [ ) ( ] [ ) ( ] [ ) (

3 3 2 2 2 2 3 3 3 3 2 2 2 2 3 3 3 2 2 2 3 3

t p I D D I D D I D D I D D t n t h I D D t n t h I D D t I D D t I D D t X − − − − − + + − − + − = − − − ≡ φ φ

S.V. Dhurandhar, K.R. Nayak, and J-Y. Vinet, Phys. Rev. D, 65, 102002 (2002).

φi = two-way phase measurements

Time-delay Interferometry

• It is best to think of LISA as a

closed array of six one-way delay lines between the test masses.

• This

approach allows us to reconstruct the unequal-arm Michelson interferometer, as well as new interferometric combinations, which

• ffer

advantages in hardware design, in robustness to failures of single links, and in redundancy of data.

• M. Tinto: Phys. Rev. D, 53, 5354 (1996); Phys. Rev. D, 58, 102001 (1998)

J.W. Armstrong, F.B. Estabrook, and M. Tinto: Ap. J., 527, 814 (1999) F.B. Estabrook, M. Tinto, & J.W. Armstrong, Phys. Rev. D, 62, 042002 (2000)

• M. Tinto, D.A. Shaddock, J. Sylvestre, & J.W. Armstrong: Phys. Rev. D 67, 122003 (2003)

1 3* 2

L2 L1 L3

2

n r

1

n r

3

n r

• l

l l

1* 3 2*

• One can actually regard X as given

by the interference of two beams that propagate within the two arms

• f LISA, each experiencing a delay

equal to (2L2 + 2L3) .

• X is actually a zero-area Sagnac

Interferometer, synthesized by properly combining measurements from each arm.

L3 L2

1 2 3

)] ( ) ( [ )] ( ) ( [ ) (

3 2 2 2 2 3 3 3

t D D t t D D t t X φ φ φ φ + − + =

Unequal-arm Interferometers (Cont.)

D.A. Shaddock, M. Tinto, F.B. Estabrook & J.W. Armstrong, Phys. Rev. D, 68, 061303 (R) (2003).

Six-Pulse Data Combinations

α, β, γ, ζ

) ( ) (

13 , 13 3 , 32 21 12 , 12 2 , 23 31

η η η η η η α + + − + + =

) ( ) ( ) ( ) ( ) (

1 , 1 23 , 1 1 , 1 23 , 1 1 , 21 3 , 23 3 , 13 2 , 12 2 , 32 1 , 31

noises Secondary GW p p p p + + − − − = + − − + − = η η η η η η ζ

. . .

L1 L2 L3 3 2 1

• ?

?

D.A. Shaddock, M. Tinto, F.B. Estabrook & J.W. Armstrong, Phys. Rev. D, 68, 061303 (R) (2003).

Eight-Pulse Data Combinations

. . .

L2 L3 3 2 1

. . .

L1 L2 L3 3 2 1

. . .

L1 L2 L3 3 2 1

. . .

L1 L2 L3 3 2 1

(X, Y, Z) (P, Q, R) (U, V, W) (E, F, G) Monitor Relay Beacon Unequal-arm Michelson

Moving spacecraft Arrays and Clocks Synchronization

• The analysis above assumed the clocks onboard the LISA S/Cs to

be synchronized to each other in the frame attached to the LISA array.

• In a rotating reference frame, the Sagnac effect prevents the

implementation of the Einstein’s Synchronization Procedure, i.e. synchronization by transmission of electromagnetic signals (GPS is a good example of this problem!)

• To account for the Sagnac effect, one introduces an hypothetical

inertial reference frame, and time in this frame is the one adopted by the spacecraft clocks!

• In other words, the onboard receivers have to convert time

information received from Earth to time in this inertial reference frame (SSB).

• M. Tinto, F.B. Estabrook, & J.W. Armstrong, gr-qc/0310017, October 6, 2003
• N. Asbby, “The Sagnac effect in the GPS System”, http://digilander.libero.it/solciclos/

Moving spacecraft Arrays and Clocks Synchronization (Cont.)

• In the SSB frame, the differences between back-forth

delay times are very much larger than has been previously recognized.

• The reason is in the aberration due to motion and

changes of orientation in the SSB frame.

• With a velocity V=30 km/s, the light-transit times of

light signals in opposing directions (Li, and L’i) will differ by as much as 2VL (a few thousands km)

• They will also change in time due to rotation (0.1

m/s); this however is significantly smaller than the spacecraft relative velocity (10 m/s).

TDI with Moving spacecraft Arrays

• The “first-generation” TDI expressions do not

account for:

– The Sagnac Effect – Time-dependence (velocity) of the arm lengths in the TDI expressions (the “Flex-effect”)

• Both effects prevent the perfect cancellation of the

laser frequency fluctuations in the “first-generation” TDI combinations.

• With a laser frequency stability of 30 Hz/Hz

1/2the

remaining laser frequency fluctuations could be as much as 30 times larger than the secondary noise sources.

D.A. Shaddock, Phys. Rev. D: to appear; gr-qc/0306125 Cornish & Hellings, Class. Quantum Grav. 20 No 22 (21 November 2003) 4851-4860 D.A. Shaddock, M. Tinto, F.B. Estabrook & J.W. Armstrong, Phys. Rev. D, 68, 061303 (R) (2003).

The Sagnac Effect and the Sagnac Combinations

• In presence of rotation, the amount of time spent

by a beam to propagate clockwise is different by the time it spends to propagate counterclockwise along the same arm => (L1, L2, L3, L’1, L’2, L’3).

• The Sagnac effect prevents the perfect

cancellation of the laser frequency fluctuations in the existing expressions of the Sagnac combinations (α, β, γ, ζ).

• @ 10-3 Hz the laser frequency fluctuations

remaining in (α, β, γ, ζ) would be about 30 times larger than the secondary noise sources.

(α, β, γ)

km L L L L L L 14 | | 4 | ) ( ) ( |

3 ' 2 ' 1 ' 3 2 1

≅ Α ⋅ Ω = + + − + + r r

. . .

α1, α2, α3

L1 L’2 L’1 L’3 L2 L3

α, β, γ

3 2 1

. . .

L1 L2 L3 3 2 1

“Flexy”

3 2 1

D.A. Shaddock, M. Tinto, F.B. Estabrook & J.W. Armstrong, Phys. Rev. D, 68, 061303 (R) (2003).

Systematic Approach

• Is there a general procedure for deriving the

“2nd generation” TDI combinations?

• YES!
• M. Tinto, F.B. Estabrook, & J.W. Armstrong, gr-qc/0310017, October 6, 2003

) )( ( ) ( ] , [ ) ( )) ( ) ( ( ) (

' ' ' ' ' ' ' j i i j j i j i i j i j i j i

L V L V L L t t D D t D D L t L t L t t D D − − − Ψ ≅ Ψ Ψ ≠ − − − Ψ = Ψ

Systematic Approach (Cont.)

) ( ), ( ), ( ), (

31 13 21 12

t t t t η η η η

1 ' 2 2 2 ; 13 31 1 3 ' 3 ' 3 ; 12 21

] [ ] [ p I D D p I D D − = + − = + η η η η

1 ' 2 2 3 ' 3 1 3 ' 3 ' 2 2 2 ; 13 31 3 ' 3 ' 3 ; 12 21 ' 2 2

] ][ [ ] ][ [ ) ]( [ ) ]( [ p I D D I D D p I D D I D D I D D I D D X − − − − − = + − − + − = η η η η

. . .

L2 L3 3 2 1 L’2 L’3

= 0

1 ' 2 2 3 ' 3 ' 33 ; 2 ; 13 31 ' 3 ; 12 21 1 3 ' 3 ' 2 2 2 ' 2 ; ' 3 ; 12 21 2 ; 13 31

] [ ) ( ) ( ] [ ) ( ) ( p I D D D D p I D D D D − = + + + − = + + + η η η η η η η η

] ) ( ) ][( [ ] ) ( ) ][( [

' 33 ; 2 ; 13 31 ' 3 ; 12 21 3 ' 3 ' 2 2 2 ' 2 ; ' 3 ; 12 21 2 ; 13 31 ' 2 2 3 ' 3 1

≅ + + + − − + + + − = η η η η η η η η I D D D D I D D D D X

How Does the LISA Sensitivity Change?

• Once the laser frequency fluctuations are removed, the

corrections to the signal and the secondary noises (optical path, proof-mass, etc.), introduced by the extra delays due to the Sagnac (14 km) and flexy (~ 300 m) effects, are many orders of magnitude below the signals and secondary noises determined by the “1st generation” TDI expressions:

] 1 [ ) ( ~ ) ( ~ ) 2 2 ( ) ( ) (

) 2 2 ( 2 1 3 2 1

3 2

L L if

e f X f X L L t X t X t X

+

− = − − − =

π