A GW incident on this system causes perturbations in the Doppler frequency time series. The gravitational-wave response of a two-way Doppler system excited by a transverse, traceless plane gravitational wave  having unit wavevector is [52*][16*, 51*, 119, 120]: .) The GW amplitude at the earth is , where the 3-tensors and are transverse to and, with respect to an orthonormal propagation frame, have components .)
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, ) 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.
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: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 . 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 .
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:
- GW signals are observed in the Doppler tracking time series through the three pulse response [Eq. (1*)].
- 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 ).
- The GW response is not in general time-shift invariant if T2 or change during the time of observation.
- 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*).