Browsing by Author "Vinet, J-Y."

Now showing 1 - 4 of 4

Algebraic approach to time-delay data analysis for LISA
(2001-12-20) Dhurandhar, Sanjeev; Nayak, K. R.; Vinet, J-Y.
Cancellation of laser frequency noise in interferometers is crucial for attaining the requisite sensitivity of the triangular 3-spacecraft LISA conﬁguration. Raw laser noise is several orders of magnitude above the other noises and thus it is essential to bring it down to the level of other noises such as shot, acceleration, etc. Since it is impossible to maintain equal distances between spacecrafts, laser noise cancellation must be achieved by appropriately combining the six beams with appropriate time-delays. It has been shown in several recent papers that such combinations are possible. In this paper, we present a rigorous and systematic formalism based on algebraic geometrical methods involving computational commutative algebra, which generates in principle all the data combinations cancelling the laser frequency noise. The relevant data combinations form the ﬁrst module of syzygies, as it is called in the literature of algebraic geometry. The module is over a polynomial ring in three variables, the three variables corresponding to the three time-delays around the LISA triangle. Speciﬁcally, we list several sets of generators for the module whose linear combinations with polynomial coeﬃcients generate the entire module. We ﬁnd that this formalism can also be extended in a straight forward way to cancel Doppler shifts due to optical bench motions. The two modules are infact isomorphic. We use our formalism to obtain the transfer functions for the six beams and for the generators. We speciﬁcally investigate monochromatic gravitational wave sources in the LISA band and carry out the maximisiation over linear combinations of the generators of the signal-to-noise ratios with the frequency and source direction angles as parameters.
General relativistic treatment of LISA optical links
(2008-05) Dhurandhar, Sanjeev; Vinet, J-Y.; Nayak, K. R.
LISA is a joint spae mission of the NASA and the ESA for deteting low frequeny gravitational waves in the band 10−5 − 1 Hz. In order to attain the requisite sensitivity for LISA, the laser frequeny noise must be suppressed below the other seondary noises suh as the optial path noise, a eleration noise et. This is ahieved by ombining time-delayed data for whih preise knowledge of time-delays is required. The gravitational eld, mainly that of the Sun and the motion of LISA a e t the time-delays and the opti al links. Further, the e e t of the gravitational eld of the Earth on the orbits of spaeraft is inluded. This leads to additional exing over and above that of the Sun. We have written anumerial ode whih omputes the optial links, that is, the time-delays with great auray∼ 10−2 metres - more than what is required for time delay interferometry (TDI) - for most of the orbit and with su ient auray within ∼ 10 metres for an integrated time window of about six days, when one of the arms tends to be tangent to the orbit. Our analysis of the optial links is fully general relativisti and the numerial ode takes into aount e ets suh as the Sagna, Shapiro delay, et. We show that with the deemed parameters in the design of LISA, there are symmetries inherent in the on guration of LISA and in the physi s, whih may be used e etively to suppress the residual laser noise in the modi ed rst generation TDI. We demonstrate our results for some important TDI variables.
Optimising LISA orbits: The projectile solution
(2008-09-11) Dhurandhar, Sanjeev; Nayak, K. R.; Vinet, J-Y.
LISA is a joint spae mission of the NASA and the ESA for deteting low frequeny gravitational waves (GW) in the band 10−5 − 0.1 Hz. The proposed mission will use oherent laser beams whih will be exhanged between three idential spae raft forming a giant (almost) equilateral triangle of side 5×106 kilometres. The plane of the triangle will makean angle of ∼ 60◦ with the plane of the elipti. The spaeraft onstituting LISA will be freely oating in the ambient gravitational eld of the Sun and other elestial bodies. To ahieve the requisite sensitivity, the spaeraft formation should remain stable, one requirement being, the distanes between spaeraft should remain as onstant as possible - that is the exing of the arms should be minimal. In this paper we present a solution - the projetile solution - whih onstrains the exing of the arms to below 5.5 metres/se in a three year mission period. This solution is obtained in the eld of the Sun and Earth only, whih prinipally a et the motion of the spaeaft, espeially the exing of LISA's arms.
Time delay interferometry for LISA with one arm dysfunctional
(2010-01-27) Dhurandhar, Sanjeev; Nayak, K. R.; Vinet, J-Y.
In order to attain the requisite sensitivity for LISA - a joint space mission of the ESA and NASA- the laser frequency noise must be suppressed below the secondary noises such as the optical path noise, acceleration noise etc. By combining six appropriately time-delayed data streams containing fractional Doppler shifts - a technique called time delay interferometry (TDI) - the laser frequency noise may be adequately suppressed. We consider the general model of LISA where the armlengths vary with time, so that second generation TDI are relevant. However, we must envisage the possibility, that not all the optical links of LISA will be operating at all times, and therefore, we here consider the case of LISA operating with two arms only. As shown earlier in the literature, obtaining even approximate solutions of TDI to the general problem is very difficult. Since here only four optical links are relevant, the algebraic problem simpliﬁes considerably. We are then able to exhibit a large number of solutions (from mathematical point of view an inﬁnite number) and further present an algorithm to generate these solutions.