3 Gravitational Wave Signal Response

The theory of spacecraft Doppler tracking as a GW detector was developed by Estabrook and Wahlquist [52*]. Briefly, consider the earth and a spacecraft as separated test masses, at rest with respect to one another and separated by distance L = cT2 / 2, where T2 is the two-way light time (light time from the earth to the spacecraft and back). A ground station continuously transmits a nearly monochromatic microwave signal (center frequency ν0) to the spacecraft. This signal is coherently transponded by the distant spacecraft and sent back to the earth. The ground station compares the frequency of the signal which it is transmitting with the frequency of the signal it is receiving. The two-way fractional frequency fluctuation is y2(t) = [ν(t − T2) − ν(t)]∕ν0, where ν (t) is the frequency of the actual transmitted signal. In this way the Doppler tracking system measures the relative dimensionless velocity of the earth and spacecraft: 2Δv ∕c = Δ ν∕ν0. In an idealized system (no noise, no systematic effects, no gravitational radiation), this time series would be zero.

A GW incident on this system causes perturbations in the Doppler frequency time series. The gravitational-wave response yg2w(t) of a two-way Doppler system excited by a transverse, traceless plane gravitational wave [84] having unit wavevector ˆk is [52*]

( ) ygw(t) = μ-−-1Ψ¯(t) − μΨ¯ t − 1-+-μ-T + 1-+-μ-¯Ψ (t − T ), (1 ) 2 2 2 2 2 2
where μ = ˆ k ⋅ ˆn, ˆn is a unit vector from the earth to the spacecraft, ¯ ˆ 2 Ψ (t) = (ˆn ⋅ h (t) ⋅ ˆn)∕(1 − (k ⋅ ˆn) ), and h(t) is the first order metric perturbation at the earth. (Here ¯ Ψ is distinguished from the Ψ used to analyze the LISA detector [16*, 51*, 119, 120]: Ψ = (1∕2)Ψ¯.) The GW amplitude at the earth is h (t) = [h+(t)e+ + h×(t)e×], where the 3-tensors e+ and e× are transverse to ˆk and, with respect to an orthonormal (ˆi,ˆj,ˆk) propagation frame, have components
( 1 0 0 ) ( 0 1 0) | | | | e+ = ( 0 − 1 0 ) , e× = ( 1 0 0) . (2 ) 0 0 0 0 0 0
(If general relativity and a transverse traceless perturbation are not assumed, the amplitude of the three-pulse response for a general tensor metric perturbation is given in [57].)

The Doppler responds to a projection of the time-dependent wave metric, in general producing a “three-pulse” response to a pulse of incident gravitational radiation: one event due to buffeting of the earth by the GW, one event due to buffeting of the spacecraft by the GW, and a third event in which the original earth buffeting is transponded a two-way light time later. The amplitudes and locations of the pulses depend on the arrival direction of the GW with respect to the earth-spacecraft line, the two-way light time, and the wave’s polarization state. From Eq. (1*) the sum of the three pulses is zero. Since the detector response depends both on the spacecraft-earth-GW geometry (T2, μ) and the wave properties (Fourier frequency content, polarization state) its distinctive three-pulse signature plays an important role in distinguishing candidate signals from competing noises. Figure 1* shows this three pulse response in schematic form.

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Figure 1: Schematic response of two-way Doppler tracking to a GW. The Doppler exhibits three pulses having amplitudes and relative locations which depend on the GW arrival direction, the two-way light time, and the wave’s strain amplitude and polarization state. The sum of the three pulses is zero, so the pulses overlap and partially cancel when the characteristic time of the GW pulse is comparable to or larger than the light time between the earth and spacecraft.

In the special case of the long-wavelength limit (LWL, where the Fourier frequencies of the GW signal are ≪ 1/T2), the gravitational wave can be expanded in terms of spatial derivatives. Equation (1*) then gives the LWL response for two-way Doppler tracking:

ygw,LWL (t) → T2(μ2 − 1)Ψ¯′(t). (3 ) 2 2
In this limit the three-pulses overlap in the tracking record causing partial cancellation, loss of signature, and loss of signal response. In the opposite limit, wave periods ≪ T2, the full three pulse character [Eq. (1*)] is expressed in the Doppler time series. Figure 2* shows the spectral response of a Doppler tracking system to sinusoidal GW signals from two specific directions and the average response from sources distributed isotropically on the celestial sphere. Because Figure 2* plots the transfer function to the spectral power, the dependence at low-frequency is ∝ f2. The algorithm used to average over GW polarization states in Figure 2* is described in [18*, 16*]. A related discussion (how to infer GW amplitudes, h – or limits to h – from measurements of y2 when the signal direction and polarization state are unknown) is in [18*]. The LWL of the three-pulse GW response has been used to analyze the GW response of ground-based Michelson gravitational wave interferometers [47]. The three-pulse response can also be constructed using the formalism of time-delay interferometry, the method LISA will use to cancel laser phase noise in an unequal-arm spaceborne GW detector (see Section 8). The formalism has also been applied to analysis of other spaceborne detector geometries, for example the candidate linear array, SyZyGy [49].
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Figure 2: Polarization-averaged power response of the Doppler system to a gravitational wave signal as a function of Fourier frequency. Earth-spacecraft geometry for Cassini’s gravity wave observations in 2000 – 2001 has been used. Blue: response from the direction of Virgo (μ ≃ 0.104). Black: response to randomly polarized sources distributed isotropically on the celestial sphere. Red: response from the direction of the galactic center (μ ≃ 0.9932). Note that in the case of waves from the Galactic center, the three-pulse response function ([52*] and Eq. (1*)) for the Cassini geometry is dominated by two pulses separated by only about 0.00034 T2 ≃ 20 s. This gives rise to the strong low-frequency suppression and the approximate sin2 modulation for the Cassini GWE1 geometry and GWs from the galactic center. Update*

In a practical GW observation spanning 20 – 40 days, the earth-spacecraft distance and the orientation of the earth-spacecraft vector on the celestial sphere change (typically slowly) with time. This modifies the idealized GW response (it is not strictly time-shift invariant) and has practical consequences in searches for long-lived signals (see Section 5.7).

To summarize the Doppler signal response:

  1. GW signals are observed in the Doppler tracking time series through the three pulse response [Eq. (1*)].
  2. The response depends on the two-way light time T2, the cosine of the angle between the GW wavevector and unit vector from the earth to the spacecraft, and GW properties (Fourier content and polarization state) [Eq. (1*)] and the expression for ¯ Ψ(t)).
  3. The GW response is not in general time-shift invariant if T2 or μ change during the time of observation.
  4. The GW response is a high-pass filter: In the long-wavelength limit (frequencies ≪ 1/T/sub2), the response is attenuated due to pulse overlap and cancellation (see Figure 2*).

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