The constellation and the gravitational antennas
TianQin research cooperation, started in 2015, is led by prof. Jun Luo of the Sun-Yat Sen University, Guangzhou, Guangdong, China. The aim is to study and progressively realize a gravitational wave (GW) space observatory . The baseline is an equilateral constellation of three spacecraft (S/C) orbiting around the Earth at 100,000 km altitude and pointing to a specific GW source. Figure 1 sketches the constellation triangle orbiting around the Earth and the inertial orbital plane pointing toward a GW source. The constellation triangle is left to periodically deform due to orbit eccentricity and Lunar gravitation. The variation of the aperture angle between two arms may be referred to as the ‘breathing angle’. The baseline constellation plane is close to be orthogonal to the ecliptic plane. TianQin can be referred to as a LISA-like mission, its conception being in debt with the European LISA (Laser Interferometry Space Antenna, born as a US-Europe mission) under study since the 80’s of the past century . The main differences lie in the Earth orbit of the TianQin constellation compared to the LISA orbit around the Sun, and the observation of a single source compared to LISA scan of the whole celestial sphere. A solar orbit allows longer constellation arms (the present LISA design is km) and therefore better sensitivity to gravitational waves.
The three constellation arms (the triangle edges) should work as extremely sensible gravitational antennas capable of revealing picometer functuations of the 170,000 km spacecraft-to-spacecraft distance , in the frequency band from fractions of mHz to fractions of Hz (the science band). A fluctuation/distance ratio around corresponds to the expected strain of the four-dimensional space-time fabric induced by the candidate GW sources (the peak strain of the signal detected by the US LIGO ground observatories in September 2015 was close to 10-21 m/m ). Space observatories are expected to detect long-period waves from few seconds to few hours (from fractions of mHz to fractions of Hz in the Fourier frequency domain, the so-called science band), and thus a larger population of celestial phenomena, if compared to those observable from Earth ground which are restricted to fraction-of-second periods: the duration of the first signal detected by LIGO observatories was less than 0.2 s. Space-borne and ground observatories are expected to complement each other due to contiguous frequency bands and overlapping strain ranges. Due to the finite frequency band of observation, the expected observable strain should be better defined by one-sided (or unilateral) spectral density with unit .
Picometer fluctuations over distances larger than 100,000 km can be detected by measuring the phase variation of frequency-ultrastable electromagnetic waves walking at the speed of light from spacecraft to spacecraft along each constellation arm. The end wavefronts are reflected back to appropriate detectors by the mirror surfaces of two purely free falling test masses (TM), which are left moving along the electromagnetic wave direction under the sole action of initial conditions and local gravity field. The recent European technological mission LISA Pathfinder (2015-2017) has proved a ‘free falling purity’ better than in the Fourier frequency band , which lies within the science band . Like for the target strain, the purity index must be expressed as the spectral density of the residual TM acceleration since spectral density is not uniform but increases toward lower and higher frequencies. This purity index can been taken as a driving target for the TianQin mission. Optical phase variations due to picometer displacements can be appreciated from short-wavelength electromagnetic waves: near-infrared coherent light, generated by laser sources, with a wavelength of about 1 μm is the candidate, also due to on ground mature technology. The recent US-German gravity mission GRACE Follow-on, launched in 2018, has made the first test of scientific inter-satellite laser ranging along a 220 km distance . Interference of the variable-phase laser beam, from TM to TM, with a fixed-phase beam tuned on the same ultrastable frequency is the classical way to reveal, on a photodetector, known as the phasemeter, the variable phase and the proportional optical-path length variation. The technique is known as laser interferometry. The proportionality factor, being equal to 4π/λ, is inversely proportional to the wavelength λ as expected. Actually, since each test mass is shielded by a cage, the total optical path should be conveniently split, as in LISA design , into three segments: (i) TM to optical bench, inside the master spacecraft, about 0.5 m long, (ii) optical bench to optical bench, 170,000 km long, and (iii) optical bench to TM, inside a slave spacecraft, about 0.5 m long.
In the LISA design , each constellation S/C can act as the master instrument: the optical signal generated by an ultrastable oscillator is transmitted to the remote companions (the slave instruments), which in turn amplify and transmit the received optical wave back to the master. In this way, each S/C can measure the two-way length variation of both the arms departing from it. In TianQin, the received signals and noise will be delayed by more than 1 s from the oscillator signal and noise (the delay is time varying because of the constellation deformation), which imply that the master interference signal may be obscured by the delayed noise difference. Since, under equal arm length, the same delayed noise affects both received signals, taking the difference of their interference signals is a way of eliminating the common noise. The principle has been extended to unequal arms and delays – the actual condition for LISA and TianQin- by the off-line Time Delay Interferometry (TDI) . The method, tested on ground, is essential for allowing LISA and LISA-like missions to fix the on-board oscillator frequency stability to be of the order of – a standard stability index – in front of a target GW strain of about . Here [cps] (cycles per second) will be used as the unit of optical frequencies, and [Hz] as the unit of Fourier frequencies. Actually, the above gap can be reduced by two orders of magnitude, if an intermediate noise reduction is adopted as it has been envisaged by LISA design. The method, known as ‘arm locking’, is conceived to transfer the short-term stability of the arm length to laser frequency .
Laser interferometry allows the measurement of the picometer fluctuations of the optical-path length between the pair of test masses at the end of each arm, but an accurate control system is essential to leave the test masses free falling and to align the test mass sensitive axis to the laser beam itself. The sensitive axis is defined by the laser interferometer of the short segment. Since the main control target is to ensure free falling purity, the control system is given, for historical reasons, the nickname of ‘drag-free’, where drag-free means that the test mass motion is free of any environmental force like the atmospheric drag, which is the dominant perturbation, leading to satellite de-orbiting, roughly up to 1000 km altitude. The ‘drag-free satellite’ was conceived in the 60’s of the past century, during the early design stages of the US Gravity Probe B mission, later launched in 2004 [8, 9]. A small spherical mass (TM) was caged and left free in the cage by tiny S/C displacements driven by TM-to-cage distance measurements and actuated by gas thrusters. The first drag-free satellites belonged to the TRIAD and NOVA series, members of the US navigation TRANSIT constellation, now superseded by the GPS NAVSTAR . Drag-free control was employed to compensate the de-orbiting drag, thus increasing mission life. Of course, no appreciable atmospheric drag exists at 100,000 km altitude above the Earth ground (TianQin) and even more so along a solar orbit (LISA), the dominant perturbations being just solar pressure and wind.
The peculiar issue of TianQin and LISA drag-free control – and of the past LISA Pathfinder – is that the extreme free falling purity demands the cancellation of the internal stray acceleration due to attractive coupling between TM and cage: no mechanical TM-cage contact exists and no force should be nominally applied along the sensitive axis. If the natural angular frequency of the coupling happens to lie well below 1 mrad/s, the free falling purity is guaranteed by a residual TM-cage fluctuation below within the science band: an achievable requirement. The major contribution to residual fluctuations (jitter) comes from the S/C actuator noise, which should be compensated at least up to 0.1 Hz. Low-noise micro-thrusters either electric or cold gas need to be employed. Cold-gas technology was adopted on-board LISA Pathfinder, which tested also a kind of electric micro-propulsion . The solar pressure contribution turns out to be much smaller than the actuator noise. On each S/C, a pair of TM is mounted, each one at the extremes of two arms nominally separated by sixty degrees, but actually aligned to the fluctuating laser beams launched by the remote companion satellites (let us recall the already cited ‘breathing angle’ of the constellation). The micro-movements of the S/C (and of the shielding cages) should leave each TM free falling along its own sensitive axis and not along other directions. Along orthogonal directions (lateral and vertical) each TM is suspended to the cage by electrostatic actuators, whose main task is to keep the sensitive axis aligned to the laser beam direction. To exploit the three S/C degrees-of-freedom, each S/C can be made drag-free along the normal direction to the plane of the TM sensitive axes.
TM and telescope alignment to laser beam
The second control task is the TM alignment to the laser beam direction. A telescope must catch and focus the weak incoming laser spot, re-direct it to the long-segment photodetector and to the TM mirror. TM alignment must be accompanied by the alignment of the telescope optical axis with the incoming laser beam (ILB). In total, two pairs of axes (TM and telescope) have be aligned to each of two ILB directions, which sums up to eight degrees of freedom (DoF). By adding them to the drag-free degrees, we obtain the intermediate sum of eleven degrees to be continuously kept under control action. On the other hand, by subtracting the drag-free degrees from the remaining TM DoF (twelve minus four alignment degrees), we get the five degrees, which are responsible for centering and aligning TM and cage: they consist of the TM displacements orthogonal to the sensitive axes and of the rotation (roll) around the same axis.
As a first conclusion, the DoF number to be independently controlled amounts to sixteen. Actually, since only a single degree can be made drag-free along the normal of the sensitive axis plane (let us call it ‘the vertical direction’), the vertical displacement of one TM must be exactly opposite to other one.
In summary, the DoF cardinality to be controlled raises to seventeen, but a pair of them must be exactly opposite to each other.
To be completed
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