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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

, p(t)

2(t)

3(t))()()2()()(

)()()2()()(

3333

2222

tntpLtptht

tntpLtptht

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

)()(:

)()(][)()(

)()(][)()(

33333

22222

iiLttDwhere

tntpIDDtht

tntpIDDtht

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

)(][)(][)(

33222233

33222233

322233

tpIDDIDDIDDIDDtnthIDDtnthIDD

tIDDtIDDtX

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

itwo-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 offer 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

2n

1n

3n

o

1*

3

2*

• One can actually regard X as given by the interference of two beams that propagate within the two arms of 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.

L3L2

1

2

3

)]()([)]()([)(32222333tDDttDDttX

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

)()(13,133,322112,122,2331

)()()(

)()(

1,123,11,123,1

1,213,233,132,122,321,31

noisesSecondaryGWpppp

..

.L1

L2

L3

3

2

1

--

? ?


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, 2003N. 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/Hz1/2

the 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 ().

• @ 10-3 Hz the laser frequency fluctuations remaining in () would be about 30 times larger than the secondary noise sources.

kmLLLLLL 14||4|)()(| 3'

2'

1'

321

..

.

L1

L’2

L’1L’3

L2

L3

3

2

1

..

.L1

L2

L3

3

2

1

“Flexy”

32

1


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

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

''''

'''

jiijjiji

ijijiji

LVLVLLttDDtDDLtLtLttDD

Systematic Approach (Cont.))(),(),(),(

31132112tttt

1'222;1331

13'3'3;1221

][][pIDDpIDD

1'223'313'3'22

2;13313'3'3;1221'22

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

pIDDIDDpIDDIDDIDDIDDX

.

..

L2

L3

3

2

1L’2

L’3

= 0

1'223'3'33;2;1331'3;1221

13'3'222'2;'3;12212;1331

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

0])()][([])()][(['33;2;1331'3;12213'3'22

2'2;'3;12212;1331'223'31

IDDDDIDDDDX

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[)(~)(~)22()()(

)22(2

1

321

32LLifefXfX

LLtXtXtX

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