List Of Footnotes

List of Footnotes

1		Gravitational wave bands conventionally divide based on detector technology [102 , 33 ]: Future extremely-low-frequency ( $∼$ 10^–18 to $∼$ 10^–15 Hz) search programs will be based on mapping the intensity and polarization of the cosmic microwave background; very-low frequency observations ( $∼$ 10^–9 to $∼$ 10^–6 Hz) mostly use pulsar timing observations; low-frequency ( $∼$ 10^–6 to $∼$ 10^–1 Hz) observations currently use Doppler tracking of spacecraft (in the 2020’s a laser interferometer in space); high-frequency ( $∼$ 1 to $∼$ 10⁴ Hz) observations involve ground-based laser interferometers or resonant bar detectors. For general reviews see [85 , 102 , 34 ].
2		Although conceptually important – and used with excellent success as part of the hydrogen-maser-based suborbital GP-A experiment [119 ] – one-way Doppler presents a practical problem for precision tracking of deep space probes: Flight-qualified frequency standards for deep space are substantially less stable than ground-based standards. The quality of one-way spacecraft Doppler GW measurements is severely limited by noise in the flight frequency generator. Deep space tracking systems circumvent this by measuring two-way Doppler. In the two-way mode the ground station transmits a radio signal referenced to a high-quality frequency standard. The spacecraft receives this signal and phase-coherently retransmits it to the earth. The transponding process adds noise, but at negligible levels in current observations (see Table 2), and does not require a good oscillator on the spacecraft. The ground station then measures the two-way Doppler shift by comparing the frequency of the received signal against the frequency of a local reference derived from the ground frequency standard.
3		Noises are characterized in the time domain by Allan deviation, $σy(τ)$ , or in the frequency domain by the power spectra of fractional frequency fluctuations, $Sy(f)$ . These are related by $∫ ∞ σ2(τ) = 4 Sy(f)sin4(πτf)df, y 0 (πτf)2$ where $Sy (f)$ is the two-sided spectrum [23 ].
4		There is a large literature on wave propagation through random media. Excellent general references for radiowave propagation observations include [100 , 75 , 37 , 92 , 65 ].
5		The mechanical stability of the DSN’s 70-m antennas has not been systematically studied. A few observations done with Cassini in 2003 suggest mechanical noise of the 70-m antennas is substantially larger than for the 34-m beam-waveguide antennas.
6		In principle, Cassini has one transponder and one translator. The distinction is that a transponder performs functions in addition to phase-coherent generation of the downlink signal from the uplink signal.
7		The bispectrum is the Fourier transform of the third-order lagged product $⟨x(t)x(t+ τ1)x(t+ τ2)⟩$ ; it gives the contribution to the third moment from the product of three Fourier components having frequencies which add to zero. It has been used in many fields, notably geophysics, to study weak nonlinearities (see, e.g., [53 , 76 ]).
8		An exception was with Mars Observer [60 , 13 ] where spacecraft engineering telemetry was crucial for correcting the Doppler for the (slow) spacecraft spin.
9		This was suggested by Estabrook as giving “noncommittal” directions on the celestial sphere and have separations which are reasonably matched to the effective angular response of a typical Doppler tracking observation.
10		Signal processing procedures which exploit differences in the signal and noise transfer functions can give improved sensitivity at selected frequencies (see, e.g.,[104 , 9 , 11 ]).
11		One-way tracking emphasizes the symmetry of the LISA array and simplifies the analysis of the apparatus; for technical reasons the actual implementation of LISA may involve some of the links being two-way [95 ].
12		TDI developed in increasing sophistication to account for unequal armlengths, differences between the (unequal) armlengths on given up- and down-links due to aberration, and time-dependences of the unequal, aberrated armlengths. For a discussion of this development see [109 ] and references therein. TDI also allows LISA’s laser noises to be canceled in many ways [17 , 48 , 110 ]. In particular, one laser-noise-free combination is insensitive to GWs, but responds to the instrumental noises; this combination will be used to discriminate a stochastic GW background due to galactic binary stars from instrumental noises [108 , 42 ]. Because multiple laser-noise-free combinations can be simultaneously constructed, the optimum sensitivity of the LISA array can be achieved by appropriately linear combinations of the TDI data streams [86 ].