Background for gravitational wave signal at LISA from refractive index of solar wind plasma
BBackground for gravitational wave signal at LISA from refractive index of solar windplasma
Adam Smetana,
Institute of Experimental and Applied Physics, Czech Technical University in Prague, Prague, Czech Republic
Abstract:
A strong indication is presented that the space-based gravitational antennas, in par-ticular the LISA concept introduced in 2017 in response to the ESA call for L3 mission concepts,are going to be sensitive to a strong background signal interfering with the prospected signal ofgravitational waves. The false signal is due to variations in the electron number density of the solarwind, causing variations in the refractive index of plasma flowing through interplanetary space. Ascountermeasures, two solutions are proposed. The first solution is to deploy enough solar wind de-tectors to the LISA mission to allow for reliable knowledge of the solar wind background. The secondsolution is to equip the LISA interferometer with a second laser beam with a distinct wavelength toallow cancelling of the background solar wind signal from the interferometric data.
PACS numbers:
I. INTRODUCTION
Interferometers are highly appreciated in physics for their excellent sensitivity to the signal of interest. Examples oftheir common and old usages are the Rayleigh interferometer for measurement of refractive index of a transparentgas, or the Michelson interferometer designed to detect the Earth’s motion through the supposed luminiferous aether.At the cutting edge of Michelson laser interferometry are the ground-based gravitational wave detectors LIGO [2]and VIRGO [3], which reached unprecedented sensitivity for displacement of their four-kilometers-long arm ends atthe current upgraded level of ∼ − m in the gravitational wave frequency range ∼ − . × km allowing for sensitivity at the level of ∼ − m in the low frequencyrange 10 − − − Hz, opening the window for the heaviest and most diverse gravitational-wave sources which areinaccessible from the ground.Placing the gravitational wave observatory into space is motivated by dropping the necessity to construct high volumevacuum tubes, by lack of limitations on the length of the arms of the interferometer, and by avoiding the jitter noise ofseismic origin. On the other hand, the ground-based facilities allow for controlling the vacuum impurity and its timestability. Mainly the latter is important to avoid spurious signals as pointed out in this work. In this work the effectof vacuum impurity of the interplanetary space environment of the LISA caused by time-variable solar activity hasbeen studied. Namely, under the assumption of validity of the Sellmeier equation for the refractive index of plasma,extrapolated to extremely small electron densities, it is shown that the solar wind plasma density at the 1 AU solarorbit provides small deviation of electromagnetic wave phase velocity from its pure vacuum value. The time variationsof the solar wind electron density produces an accountable signal at gravitational-wave antennas of the sensitivityproposed for LISA. The resulting large background for significant portion of the prospected gravitational wave signalsof interest would jeopardize the scientific mission of the LISA. Possible countermeasures are proposed.
II. REFRACTIVE INDEX OF SOLAR WIND PLASMA
The solar wind composition is dominated by electrons and protons. Smaller component of solar wind is given by α -particles and heavier ions. The quasi-neutrality of the solar wind is guaranteed by comparable number densitiesof electrons and protons. The solar wind at the distance of 1 AU from the Sun has the average particle density ∼
10 cm − . At the scales of the optical wavelength range, the solar wind is practically a collision-less ideal plasma.Therefore, for the purpose of this work, its electron component can be treated as a gas of free electrons characterizedby the plasma frequency ω p = n e e (cid:15) m e , (1) a r X i v : . [ a s t r o - ph . I M ] A ug where n e is the electron number density, m e is the electron mass, e is the value of the elementary electric charge and (cid:15) is the permittivity of vacuum. This situation is analogous to the ionospheric plasma, where the index of refractionis given by the well-known Sellmeier equation [6]. Due to the inverse-power dependence on the mass of the plasmaparticles, contributions from protons and ions can be neglected. Extrapolating its validity to the range of near-opticalwavelengths and into the extremely low electron densities of solar wind, the formula for refractive index n of the solarwind plasma can be used n = (cid:114) − ω p ω . = 1 − ω p ω + . . . , (2)where ω is the angular frequency of the electromagnetic wave passing through the plasma. The assumption of validityof the formula Eq. 2 is justified by negligible values of gyrofrequencies of the solar wind plasma in comparison with ω p and ω p (cid:28) ω . For the values of the electron density ∼
10 cm − and for ω in the optical range, the deviation of n from unity is truly negligible, at the level of ∼ − − − , which would require measurement by an instrument ofcomparable sensitivity. III. LISA PARAMETERS
The space-based gravitational wave antenna LISA is proposed to be based on a system of free-falling test masses insolar orbit, which serve as both geodetic reference and mirrors for laser beams at the ends of the interferometer arms.The test masses are shielded against the space weather and against the solar wind by bodies of drag-free spacecraft,which absorb a major portion of momentum transfer from the environment by constantly adjusting their trajectoriesto the trajectories of the test masses. The solar activity and solar wind as possible sources of noise for the LISAhas been analysed in [9]. Due to enormous effort in suppressing LISA’s noise level [4], the displacement noise linearspectral density, √ S IFO , of the interferometric test-mass-to-test-mass distance has been proposed to be (cid:112) S IFO ≤ − m √ Hz (cid:115) (cid:18) f (cid:19) for 10 − Hz ≤ f ≤ − Hz , (3)where f is the frequency parameter spanning the gravitational wave frequency range of interest. Such a noise levelhas been required in order to be smaller than the strain level of the signal of interest from the gravitational waves andnot to cover it. That was successfully demonstrated and even improved by the the LISA Pathfinder single-test-massmission [5].The proposed interferometric configuration of the LISA observatory is based on three arms with six active laser linksoperating on the wavelength λ = 1064 nm , (4)between three identical spacecraft in an equilateral triangular formation, each separated by a mean distance of L = 2 . × km , (5)and carrying two identical test masses each. The LISA reference orbit is a stable Earth-trailing heliocentric orbitabout 50 million km from Earth.The amplitudes of typical gravitational wave signals for the LISA with the proposed parameters can be found in theLISA proposal [4], or in more detail in [7]. IV. SOLAR WIND DATA
For the present study, the relevant solar wind data are that of the electron number density n e with sufficient samplingfrequency (cid:38) . . ∼
10 periods. The selection has been done to include some of the calm space conditions data as wellas some of the solar storm events data. In the Tab. I and Tab. II the data samples with their main characteristics arelisted. It shows the mean, n e , maximum, n max e , and minimum, n min e , values of each data set. The data samples usedin Tab. I are of 24h duration, chosen arbitrarily to be from the 11th day of each month. The data from the Tab. IIcover periods pre-selected by the WIND collaboration to cover certain significant solar events. Figure 1: The upper panels show two typical calm-condition ( n e < .
0) 24-hours data samples of the solar wind electrondensity. The lower panels show two typical data samples of the solar wind electron density during the solar storm events oractive conditions ( n e > . V. ESTIMATION OF THE BACKGROUND FOR LISA FROM THE SOLAR WIND PLASMA EFFECT
Based on the Eq. 2 and Eq. 1, the deviation of refractive index depends linearly on the electron number density, whichis the only source of its time variation. The time variation of the refractive index can be described in terms of timevarying effective displacement h SW ( t ) = 12 n e ( t ) e (cid:15) m e ω L , (6)mimicking the signal of displacement due to the gravitational wave distortion of the space fabric.
Table I: List of data samples arbitrarily chosen from the period Feb 1997 - Nov 1998. Initial time is always 00:00:00.no. initial date length n e n max e n min e [cm − ]1. 1997/07/11 24 h 5 . . .
02. 1997/08/11 24 h 6 . . .
53. 1997/09/11 24 h 4 . . .
74. 1997/10/11 24 h 7 . . .
05. 1997/12/11 24 h 4 . . .
86. 1998/01/11 24 h 7 . . .
68. 1998/03/11 24 h 3 . . .
69. 1998/04/11 24 h 5 . . . . . . . . . . . . . . . . . . n e n max e n min e [cm − ]A. ISTP-SEC 1997/11/04 24 h 6 . . . . . . . . . . . . . . . . . . In order to compare the solar wind signal with the required strain sensitivity of LISA, √ S IFO given in Eq. 3, the linearspectral density √ S SW ( f ) for the effective displacement h SW ( t ) is calculated. The result is shown in the Fig. 2 plottingthe linear spectral densities √ S SW for all selected data samples together (in colors) with the required sensitivity √ S IFO (in black). In the frequency range10 − . Hz ≤ f ≤ − . Hz for calm conditions (7)10 − . Hz ≤ f ≤ − . Hz for active conditions (8)the solar wind signal in the analysis has amplitude greater than the required noise level, √ S IFO , by factor up to ∼ − . Hz, or by factor up to ∼
10 (active conditions) around the frequency10 − . Hz. Out of all used data sets only a single one (no. 8.) provides signal, which does not exceed the requirednoise level, √ S IFO , and three (no. 3.,10.,14.) only very little.The result of the analysis indicates that the solar wind causes a significant background for the expected gravitationalwave signal √ S h , noted in the LISA proposal [4]. This result indicates a severe constraint of the scientific performanceof the proposed LISA mission. VI. DISCUSSION OF THE RESULT AND PROPOSAL OF COUNTERMEASURES TO SAFE THE LISA
The danger of the signals from solar wind plasma effects observable by LISA are applicable to any space-basedinterferometric observatory sensitive to the sub-Hz frequency range used here. It shows the weak point of replacingthe laboratory vacuum by the vacuum of the interplanetary open space, which exhibits uncontrollable time variationsof the solar wind plasma density caused by solar activity. The characteristic frequencies of the solar activity span alarge range covering the frequency range of LISA’s interest. The present analysis, based on the formulas Eq. 1 andEq. 2, indicates that the solar wind plasma effect on the refractive index of the LISA’s space environment for thewavelength of its laser is strong enough to interfere with expected gravitational wave signals.The result presented in this work is just an estimate of the effect. The main uncertainty of the result lies in ourignorance about the space variation of the solar wind density. The available data carries the information only aboutthe time variation being collected in a single spot - the WIND spacecraft position - at a given time. If the spacevariation of the solar wind is dominated by a characteristic wavelength λ SW λ SW (cid:28) L , (9)
Figure 2: In black (thick) the proposed required sensitivity performance of the LISA observatory, √ S IFO , given in Eq. 3, isshown. The noise level should not exceed the black noise level. In color (thin) the linear spectral densities of the effectivedisplacement, h SW ( t ) introduced in the Eq. 6, are shown, representing the solar wind plasma signal due to the variations of thesolar wind electron number densities. In the left column the comparison of the displacement linear spectral density relative tothe arm length L is shown. In the right column the ratio of the solar wind effective displacement linear spectral density andthe required sensitivity performance is shown. In the upper row the signals of the calm-condition data samples (1.–12.,14.)from Tab. I are shown, and in the lower row the signals of the solar storm events (A.–F.) from the Tab. II and active-conditiondata samples (13.) from Tab. I are shown. then the solar wind electron density variations would be averaged out along the laser trajectory of the interferometricarm, effectively suppressing the solar wind signal.Similar effective suppression of the solar wind signal is expected for the arm in the case it appears parallel with thesolar wind velocity vector at some instances on the orbit. Taking the typical value of the solar wind velocity to be v SW ∼
500 km / s, the integration over the time t = L/v SW ∼ λ (cid:48) = rλ , (10)using the same optical system including the same test-mass mirrors, where λ is the wavelength of the originallyproposed laser given in Eq. 4 and the factor r is their smartly chosen ratio different from unity. Such an improvementto LISA would allow the subtraction of the two data sets, h ( t ) and h (cid:48) ( t ), corresponding to the laser wavelengths λ and λ (cid:48) , respectively, in the following way ∆ h ( t ) = r h ( t ) − h (cid:48) ( t ) . (11)The resulting data set ∆ h ( t ) is effectively free of the solar wind signal. Under the assumption that the gravitationalwave signal scales with the laser wavelength only very weakly (or at least differently than the solar wind signal), thedata set ∆ h ( t ) still carries desired information. Of course, by such manipulation with data, the noise level and thesystematic uncertainties increase. VII. CONCLUSIONS
This brief letter reports about the indication for significant sensitivity of space-based gravitational antennas, inparticular of the LISA concept introduced in 2017, to the time-variable solar wind background, with signal comparableto that prospected for gravitational waves. Namely, it has been shown that the solar wind plasma density at the 1 AUsolar orbit provides small time variations of the refractive index for the laser electromagnetic wave from its purevacuum value, which would be effectively measured as a displacement of the test masses. It has been shown thatthe signal from the solar wind may provide a measurable background for a significant portion of the prospectedgravitational wave signal of interest. This result severely constrains the scientific performance of the proposed LISAmission.Two possible solutions have been proposed. The first solution is to deploy enough solar wind detectors to the LISAmission allowing for reliable knowledge of the background. The second solution is to equip the LISA interferometer witha second laser beam at a different wavelength to allow the cancellation of the solar wind signal from the combinedmeasurement taken at each wavelength. The improved LISA mission would gain an added value of providing anexcellent tool for research of solar wind plasma and would generate a valuable and unique data for the plasma theoryin extreme conditions.The purpose of this letter is to bring attention to this phenomenon and encourage the community to consider it whendesigning the LISA mission. More detailed analysis is needed, including the reliable 4D simulations of the spaceweather environment of the LISA and including theoretical studies and evaluations of the full variability of the solarwind plasma effects. It may turn out, after all, that the effect of the solar wind plasma will not be so harmful asindicated here, e.g., thanks to the averaging the solar wind plasma density variations out of the frequency range ofthe LISA’s interest.
Acknowledgements
I acknowledge the support of the Institute of Experimental and Applied physics, Czech Technical University in Prague,for many years of support. The work is dedicated to my Teacher.
Data availability
The data underlying this article are available in Goddard Space Flight Center - Space Physics Data Facility(https://cdaweb.gsfc.nasa.gov/istp public/), at https://dx.doi.org/10.1007/BF00751326. [1] Abbott B. P., et al., 2016, Phys. Rev. D, 93, 122004[2] Abramovici A., et al., 1992, Science, 256, 325 [3] Acernese F., et al., 2002, Classical and Quantum Gravity, 19, 1421[4] Amaro-Seoane P., et al., 2017, Laser Interferometer Space Antenna ( arXiv:1702.00786 )[5] Armano M., et al., 2019, LISA Pathfinder. pp 185–204, doi:10.1142/9789811207402˙0013,