Browsing by Author "Nayak, K. R."

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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.
Fundamentals of the LISA Stable Flight Formation
(2011-07-06) Dhurandhar, Sanjeev; Nayak, K. R.; Koshti, S.
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.
Improving the sensitivity of LISA
(2002-10-04) Nayak, K. R.; Pai, A.; Dhurandhar, Sanjeev
It has been shown in several recent papers that the six Doppler data streams obtained from a triangular LISA conﬁguration can be combined by appropriately delaying the data streams for cancelling the laser frequency noise. 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. A rigorous and systematic formalism using the powerful techniques of computational commutative algebra was developed which generates in principle all the data combinations cancelling the laser frequency noise. The relevant data combinations form a ﬁrst module of syzygies. In this paper we use this formalism to advantage for optimising the sensitivity of LISA by analysing the noise and signal covariance matrices. The signal covariance matrix is calculated for binaries whose frequency changes at most adiabatically and the signal is averaged over polarisations and directions. We then present the extremal SNR curves for all the data combinations in the module. They correspond to the eigenvectors of the noise and signal covariance matrices. A LISA ‘network’ SNR is also computed by combining the outputs of the eigenvectors. We show that substantial gains in sensitivity can be obtained by employing these strategies. The maximum SNR curve can yield an improvement upto 70 % over the Michelson, mainly at high frequencies, while the improvement using the network SNR ranges from 40 % to over 100 %. Finally, we describe a simple toy model, in which LISA rotates in a plane. In this analysis, we estimate the improvement in the LISA sensitivity, if one switches from one data combination to another as it rotates. Here the improvement in sensitivity, if one switches optimally over three cyclic data combinations of the eigenvector is about 55 % on an average over the LISA band-width. The corresponding SNR improvement increases to 60 %, if one maximises over the module
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.
Optimising the directional sensitivity of LISA
(2011-07-05) Nayak, K. R.; Dhurandhar, Sanjeev; Pai, A.
It was shown in a previous work that the data combinations canceling laser frequency noise constitute a module - the module of syzygies. The cancellation of laser frequency noise is crucial for obtaining the requisite sensitivity for LISA. In this work we show how the sensitivity of LISA can be optimised for a monochromatic source - a compact binary - whose direction is known, by using appropriate data combinations in the module. A stationary source in the barycentric frame appears to move in the LISA frame and our strategy consists of coherently tracking the source by appropriately switching the data combinations so that they remain optimal at all times. Assuming that the polarisation of the source is not known, we average the signal over the polarisations. We ﬁnd that the best statistic is the ‘network’ statistic, in which case LISA can be construed of as two independent detectors. We compare our results with the Michelson combination, which has been used for obtaining the standard sensitivity curve for LISA, and with the observable obtained by optimally switching the three Michelson combinations. We ﬁnd that for sources lying in the ecliptic plane the improvement in SNR increases from 34% at low frequencies to nearly 90% at around 20 mHz. Finally we present the signal-to-noise ratios for some known binaries in our galaxy. We also show that, if at low frequencies SNRs of both polarisations can be measured, the in
Reducing the flexing of the arms of LISA
(2005-07-01) Nayak, K. R.; Koshti, S.; Dhurandhar, Sanjeev; et al.
The joint NASA-ESA mission LISA relies crucially on the stability of the three spacecraft constellation. All three spacecraft are on heliocentric and weakly eccentric orbits forming a stable triangle. It has been shown that for certain spacecraft orbits, the arms keep constant distances to the ﬁrst order in the eccentricities. However, exact orbitography shows the so-called ‘breathing modes’ of the arms where the arms slowly change their lengths over the time-scale of a year. In this paper we analyse the breathing modes (the ﬂexing of the arms) with the help of the geodesic deviation equations to octupole order which are shown to be equivalent to higher order Clohessy-Wiltshire equations. We show that the ﬂexing of the arms of LISA as given by the ‘exact’ solution of Keplerian orbits, which gives constant armlengths to the ﬁrst order in eccentricity and whose maximum ﬂexing amplitude is ∼ 115, 000 km, can be improved, by tilting the plane of the LISA triangle slightly from the proposed orientation of 60◦ with the ecliptic to obtain a maximum ﬂexing amplitude of ∼ 48, 000 km, reducing it by a factor of ∼ 2.4. The reduction factor is even larger if we consider the corresponding Doppler shifts, for which the reduction factor reaches almost a factor of 6. We solve the second order equations and obtain the general solution. We then use the general solution to establish the optimality of the solutions that we have found.
Time delay interferometry and LISA optimal sensitivity
(2011-07-05) Pai, A.; Nayak, K. R.; Dhurandhar, Sanjeev; et al.
The sensitivity of LISA depends on the suppression of several noise sources; dominant one is laser frequency noise. It has been shown that the six Doppler data streams obtained from three space-crafts can be appropriately time delayed and optimally combined to cancel this laser frequency noise. We show that the optimal data combinations when operated in a network mode improves the sensitivity over Michelson ranging from 40% to 100%. In this article, we summarize these results. We further show that the residual laser noise in the optimal data combination due to typical arm-length inaccuracy of 200 m is much below the level of optical path and the proof mass noises.
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.