4 Apparatus and Principal Noise Sources
The detector consists of the earth and a spacecraft as separated test masses, electromagneticallytracked using a precision Doppler system. The ground stations for the Doppler system are the antennas of the NASA/JPL Deep Space Network (DSN). Figure 3* shows DSS 25, the highprecision tracking station used in the Cassini gravitationalwave observations and other Cassini radio science investigations.Figure 4* shows an example of the other part of the Doppler system. This is the Cassini spacecraft during ground tests. (Reference [37*] gives a popular discussion of the Cassini mission, the spacecraft, and its instrumentation.) The Doppler system is shown functionally in Figure 5*: A precision frequency standard from the Frequency and Timing Subsystem (FTS) provides the frequency reference to both the transmitter and receiver chains. On the transmitter side, the socalled exciter produces a nearmonochromatic signal, referenced to the FTS signal but at the desired transmit frequency. This is amplified by the transmitter (with a closedloop feedback system around the power amplifier to ensure frequency stability is not degraded) and routed via waveguide to the transmitter feedhorn in the basement of the antenna. (To correct for aberration the Kaband transmit feed horn is on a table which is articulated in the horizontal plane. This allows the Kaband transmitted beam to be pointed correctly relative to the received beam. The Xband feed is common to both the transmit and receive chains.) In a beam waveguide antenna the transmitted beam is reflected off of six mirrors within the antenna up to the subreflector (near the prime focus), then back to the main dish and out to the spacecraft (passing first through the troposphere, ionosphere, and solar wind). When the signal is received at the spacecraft it is amplified and phasecoherently retransmitted to the earth. The received beam bounces off the main reflector to the subreflector and then, via mirrors and dichroic plates, to the receiver feed horn in the antenna basement. The received signal is downconverted to an intermediate frequency where it is digitized. The digital samples are processed to tune out the (very predictable) gross Doppler shift, and reduce the bandwidth of the samples. For GW operations, the bandwidth of the predetection data is typically reduced to 1 kHz, and those data are recorded to disk along with the tuning information. The phase of the signal is detected in software and, using the tuning information, the received sky frequency is reconstructed. This and the known frequency of the transmitted signal are used to compute the Doppler time series. Removal of the orbital signature and correction for charged particle and tropospheric scintillation gives Doppler residuals, which are used in subsequent processing steps to search for GWs (or for other radio science objectives [29*, 131*, 78*, 22*]).
Of course this cannot be done without introducing noise. The following Sections 4.1 – 4.11 summarize the principal noises, their spectra or Allan deviations^{3}, and their transfer functions to the twoway Doppler time series.
4.1 Frequency standard noise
In twoway Doppler coherence is maintained by the frequency standard to which the up and downlinks are referenced. Thus noise introduced by the frequency standard is of particular importance. Figure 6* shows fractional frequency stability as a function of integration time for several frequency standard technologies. In Cassiniera observations noise in the frequency and timing system (FTS) contributed less than 10^{–15} at 1000 s and, although fundamental, is not the leading noise source at the current level of sensitivity. (FTS stability required for future Doppler experiments is discussed in Section 7.)
FTS noise enters the twoway Doppler time series via the transfer function [52*, 46*, 125*] . The transfer functions of this and other principal noises are illustrated schematically in Figure 7*. (An example of the FTS transfer function using real data is shown in Figure 8*. Although the stability of the ground frequency standard is excellent, for a few days at the start of the first Cassini Gravitational Wave Experiment there was an intermittent problem with an FTS distribution amplifier at the Goldstone complex. The effect was to introduce isolated, fairly large, and very short glitches into the frequency reference for both the transmitter and the receiver. This produced characteristic anticorrelated glitches, separated by a twoway light time, in both the X and Kaband twoway Doppler time series; see Figure 8*)
4.2 Plasma scintillation noise
The radio waves of the Doppler system pass through three irregular media: the troposphere, the ionosphere, and the solar wind^{4}. Irregularities in the solar wind and ionospheric plasmas cause irregularities in the refractive index. The refractive index fluctuations for a cold unmagnetized plasma are and the phase perturbation is , where is the wavelength, is the classical electron radius, and is the electron density fluctuation along the line of sight . These phase perturbations mimic timevarying distance changes (thus velocity errors) and so are a noise source in precision Doppler experiments. The transfer function of plasma phase scintillation to twoway Doppler is shown schematically in Figure 7*. A solar wind plasma blob at a distance x from the earth (producing a oneway fractional frequency fluctuation time series y^{sw}) and an ionospheric plasma blob at negligible light time from the ground station (with oneway time series y^{ion}) produce twoway time series and , respectively.
Update* Plasma scintillation is mostly a statistical contribution to variability in the twoway Doppler time series. As such it can be seen in the autocorrelation function (acf) of the Doppler time series. Examples of Sband correlation functions which peak at (presumably ionospheric scintillation) and (localized solar wind scintillation^{5}) are shown in [9*]. Occasionally, however, large timelocalized plasma events can be seen in the raw time series. Figure 9* shows an example in Cassini data taken at DSS 25 on 2003 DOY 324. The top panel shows the time series of the twoway Xband, with two discrete events observed near 10:20 and 10:40 ground received time, echoed with positive correlation at about the two way light time. The middle panel is the time series of X(880/3344) Ka1Update*, which isolates the downlink plasma (and cancels nondispersive processes such as FTS noise, tropospheric noise, antenna mechanical noise, and gravitational waves; see Section 4.6). This indicates that the large events observed in the upper panel are due to plasma scintillation. The lower panel shows the acf of the twoway Doppler time series, . The arrow marks the twoway light time. The acf peaks slightly earlier than T_{2}, indicating that the features observed in the other panels are caused by nearearth plasma.Figure 10* summarizes the magnitude of the effect of plasma scintillation, tropospheric scintillation, and antenna mechanical noise (the last two discussed below) on the stability of a Doppler tracking system [46*, 9*, 10*, 11*, 12, 22*]. Shown in red are data and model curves for plasma phase scintillation: Circles are Sband (frequency 2.3 GHz) observations taken in the ecliptic using the Viking orbiters spacecraft taken over a wide range of sunearthspacecraft (SEP) angles [131, 21*]; crosses are Xband (frequency 8.4 GHz) taken near the antisolar direction using the Cassini spacecraft [19*, 22*]. Clearly plasma scintillation minimizes for observations near the antisolar direction. The model curves drawn through the data are described in [21*]. (Ionospheric phase scintillation is, of course, included in the data presented in Figure 10*. Based on very limited multiplestation observations [21*] and on transfer function studies [9*], highelevationangle plasma noise appears dominated by solar wind rather than ionospheric phase scintillation. In any case, the effect of any plasma scintillation effect can be made small by observing at high enough radio frequencies [128, 46*, 21] or by using multilink observations [65*, 41, 27, 122*, 121*, 29*] to solvefor and remove the plasma scintillation effect.)
4.3 Tropospheric scintillation noise
Phase fluctuations also arise from propagation through the neutral atmosphere. Here the socalled dry component of the troposphere is large but fairly steady with the wet component (water vapor fluctuations) being smaller but much more variable [20*, 95*, 94*, 96*, 76*]. Unlike plasma phase scintillation, propagation in the troposphere is effectively nondispersive at microwave frequencies [71]. Figure 10* shows the magnitude of the effect: The blue crosshatched region is the approximate level of uncalibrated tropospheric scintillation at NASA’s Goldstone Deep Space Communications Complex [75*]. Roughly, tropospheric scintillation is worse in the summer daytime and better on winter nights. Usually its raw magnitude is large compared with, e.g., antenna mechanical noise (discussed below.)
Experiments by George Resch and colleagues [95, 94*, 96*] were influential in showing that suitably boresighted water vapor radiometer measurements could calibrate and remove much of the tropospheric scintillation noise in both radioastronomical and precision spacecraft Doppler tracking observations. A watervaporradiometerbased Advanced Media Calibration (AMC) system (Figure 11*) was developed and installed near DSS 25 to provide tropospheric corrections for Cassini radio science observations. The AMC system [94*, 96*, 76*] consists of two identical units placed close enough to each other and to DSS 25 that the coherence of the tropospheric signal on the time scales of interest was high in all three time series (see [10*, 11*] for examples of the squaredcoherence as a function of Fourier frequency). The AMC calibrations were used successfully in both the Cassini gravitationalwave observations [19*] and in relativity and plasma experiments taken near solar conjunction [29*, 122*, 121*, 22*]. The transfer function of tropospheric scintillation to the twoway Doppler is . Examples of the cross correlation function of Doppler and the AMCestimated tropospheric scintillation are shown in [10, 11*].
4.4 Antenna mechanical noise
Figure 7* shows schematically how mechanical noise in the antenna enters the Doppler. If, for example, the antenna’s phase center suddenly moves toward the spacecraft, the received signal is blue shifted, causing an immediate effect in the Doppler. The motion also causes the transmitted signal to be blue shifted; this signal is echoed in the time series a twoway light time later. Early tests by Otoshi and colleagues [86, 87] indicated that antenna mechanical stability would contribute 10^{–15} for 1000 s integrations on a 34mclass antenna.^{6}
Examples of the temporal autocorrelation of a typical Cassini DSS 25 Kaband up and downlink tracks taken during the first Cassini GWE campaign in 2001 are shown in [19*, 11*]. Positive correlation at the twoway light time is characteristic of lowlevel residual antenna mechanical noise and is observed (with varying level of correlation at = T_{2}) in all the Cassini DSS 25 GW tracks. Antenna mechanical noise in this band ( 10^{–4} – 10^{–1} Hz) is thought to be caused by highspatialfrequency irregularities in the azimuth ring on which the antenna rolls, wind loading of the main dish, and uncorrected dish sag as the elevation angle changes. In addition to this lowlevel statistical antenna mechanical noise, discrete events positively correlated at the twoway light time and large enough to be visible by eye in the time series are (rarely) observed in operational tracks [11*]. Figure 12* shows an example (Cassini tracked by DSS 25 on 2001 DOY 330). The upper panel shows twoway Kaband Doppler residuals with approximately 10 s time resolution. The middle panel shows the time series of X(880/3344) Ka1, i.e., essentially the Xband plasma on the downlink, indicating the low level of plasma noise on this day. The AMC data (not plotted here) similarly show low tropospheric noise. The event at about 07:30 UT is echoed about a twoway light time later, and may be due to gusting wind on this day (another candidate pair is at about 09:45 UT and a twoway light time later). The lower panel shows the autocorrelation of the twoway Kaband data, peaking at T_{2}.
At lower Fourier frequencies (less than about 10^{–4} Hz) the apparatus operates in the LWL and the signature of antenna mechanical noise is lost [19*]. At these low frequencies aggregate antenna mechanical noise is probably composed both of approximately random processes (e.g., atmospheric pressure loading of the station [81, 123, 38], differential thermal expansion of the structure [100*]) and of lowlevel quasideterministic processes (e.g., lowspatialfrequency imperfections in the antenna’s azimuth track, systematic errors in subreflector focusing, etc.). Thermal processes (e.g., response of the structure to 10 K temperature variations during a track) can plausibly produce only several millimeters of radio path length variation. The subreflector is continuously repositioned to approximately compensate for elevationangle dependent antenna distortions; systematic errors in this focusing at the several millimeter level over the course of a track are not unreasonable. Additionally, there are systematic low and highspatialfrequency height variations, 6 mm peaktopeak, in the azimuth track which will cause pathlength variability. Independently determined VLBI error budgets (omitting components due to radio source structure, uncalibrated troposphere, and charged particle scintillation which are not common with Cassiniclass Doppler tracking observations) are believed dominated by station position and slowlyvarying antenna mechanical noises. These account for 1.3 cm rms path delay [104], occur on time scales 10super5 – 10^{6} s, and correspond to fractional frequency fluctuations 10^{–15} or smaller.
4.5 Ground electronics noise
The DSN ground electronics have been carefully designed to minimize phase/frequency noise and produce only a small contribution to the overall error budget. (Here I distinguish this noise from the white phase noise due to finite signaltonoise ratio; see Section 4.7). In a controlled test at DSS 25 (antenna stationary, FTS common to the transmit and received chains and thus cancelled in this zero twoway light time test) the sum of the noises from the exciter, transmitter, downconversion electronics, and receiver was measured (Figure 5*). The power spectrum of those test data is shown in [1*]; the corresponding Allan deviation at = 1000 s is 2 × 10^{–16}.
4.6 Spacecraft transponder noise
Transponders accept an input carrier signal and produce an output signal at a different frequency. The process is phasecoherent; that is, for every N integer cycles of the input there are M integer cycles of the output (with M/N being the transponding ratio). When this condition is achieved, the transponder is operating normally and produces an output that is “locked” to the input signal. The Cassini spacecraft has two transponders^{7}. The standard flight transponder (“KEX”) accepts the Xband uplink and produces two phasecoherent outputs, one at f_{X} 880/749 (= Xband downlink frequency) and another at f_{X} 3344/749 (= Ka1 downlink frequency), where f_{X} is the frequency of the Xband uplink signal observed at the spacecraft. These signals are amplified, routed to the spacecraft highgain antenna, and transmitted to the earth. Another flight unit, the Kaband Translator (“KaT”), accepts a Kaband uplink signal and produces a phase coherent signal with frequency f_{k} 14/15 (= Ka2 downlink frequency), where f_{k} is the Kaband signal frequency observed at the spacecraft.
Prelaunch test data of transponders similar in design to Cassini’s KEX showed negligible (i.e., < 10^{–15}) frequency noise. Prelaunch tests of the KaT similarly showed negligible frequency noise ( 10^{–16} at = 1000 s), provided the received Kaband uplink signal was relatively strong (greater than about –127 dBm, see [1*, 19*, 78*, 22*]) at the input to the KaT.
Appropriate linear combinations of the frequency time series of the three downlinks can be used to estimate and remove (at the Fourier frequencies of interest) downlink and roundtrip plasma noise [68, 65, 69*, 70*] in GW observations. For example, the downlink plasma noise time series can be determined by forming f_{X}(880/3344) f_{Ka1}, which is independent of FTS noise, antenna mechanical noise, spacecraft buffeting, and GWs (since these are all nondispersive.) These plasma corrections were also used with good success by Bertotti, Iess, and Tortora [29*] in a precision test of relativistic gravity involving Cassini tracking very close to the sun.
4.7 Thermal noise in the ground and spacecraft receivers
Finite signaltonoise ratios in the up and downlinks cause white phase noise. For observations to date, the signaltonoise ratio (SNR) of the downlink dominates. The onesided spectral density of phase fluctuations due to finite SNR, , is 1 / (SNR in 1 Hz band) rad^{2} Hz^{–1}. The associated Allan deviation [23*] is , where is the bandwidth of the detection system and assuming . For Cassini gravitational wave observations, B is typically 1 Hz and the typical X or Kaband SNR in a 1 Hz bandwidth can be 45 dB or more. At = 1000 s finite link SNR contributes negligibly to the overall noise budget in current generation Doppler experiments (see Table 2).
4.8 Spacecraft unmodeled motion
Unmodeled motion of the spacecraft enters directly into the twoway Doppler time series (see Figure 7*). The lack of a timedomain signature makes it difficult to isolate spacecraft motion using the Doppler data only. Such unmodeled motion can arise in principle from a variety of causes: fluctuations in the solar wind hitting the spacecraft, fluctuations in solar radiation pressure, physical articulation of spacecraft parts, leaking thrusters, sloshing of fuel in the spacecraft’s tanks, etc. Solar wind and solar radiation pressure fluctuations are computed to be far too small to affect observations at current levels of sensitivity [100*]. Articulation of instruments is restricted during quiet spacecraft periods where good Doppler sensitivity is required. Leaking thrusters and fuel sloshing are also thought to be small effects at current generation Doppler sensitivity [100].
In one case the asflown spacecraft motion noise was independently determined. Using telemetry from Cassini’s reaction wheel assembly, Won, Hanover, Belenky, and Lee [129*] inferred the time series of antenna phase center motion projected onto the earthspacecraft line (i.e., the sensitive axis for the Doppler system). This test was done when Cassini was in semiquiet cruise (thrusters off but with physical articulation of elements of one science instrument, the Cassini Plasma Spectrometer, at 0.0025 Hz) for 40 hours during 2001 DOY 152153. The resulting Allan deviation for unmodeled spacecraft motion at = 1000 s was computed to be 2.3 × 10^{–16}. Figure 13* shows the spectrum of velocity noise observed in that test. Unmodeled motion of the spacecraft – at least the Cassini spacecraft – is thus negligible compared with other noises at the sensitivity of currentgeneration Doppler experiments (see Figure 10* and Table 2).
4.9 Numerical noise in orbit removal
Update* Removal of the systematic variation in the Doppler time series due to the known motion of the earth and spacecraft can be done using the JPL/NASA Orbit Determination Program (ODP; [85]) or its successor the Missionanalysis, Operations, and Navigation Tool Environment (MONTE). Subtraction of the computed systematic Doppler frequency from the observed time series then gives residuals which can be searched for gravitational waves. The ODP computes Doppler by differencing the computed range to the spacecraft at two separated times and dividing by the time difference. Because of finite computer word length neither range nor time is expressed with perfect accuracy. These, coupled with finiteaccuracy computer arithmetic, give rise to numerical noise which propagates to the residuals. Because of the granularity in time and distance the magnitude of numerical noise depends on time past 12:00 UT January 1, 2000 (the ODP measures time from J2000), distance and relative radial velocity of the spacecraft [133*] and is small compared with other noises for the early 2000s Cassini gravitationalwave observations (thus not included in Table 2). However, it is clear in Cassini Doppler data taken later in the Saturn tour [133, 67] and likely relevant for future, e.g., Juno and BepiColombo, observations.4.10 Aggregate spectrum
Figure 14* shows the twosided power spectrum of twoway fractional Doppler frequency, S_{y2}(f), computed from data taken at DSS 25 during the 2001 – 2002 solar opposition (from [19*]). It is derived after using the multilink plasma corrections and the AMC tropospheric calibrations. The intrinsic frequency resolution of the spectrum is about 3 × 10^{–7} Hz. The spectrum in Figure 14* is smoothed to a resolution bandwidth of 3 × 10^{–6} Hz to reduce estimation error. Approximate 95% confidence limits for the logarithm of an individual smoothed spectral estimate are indicated [73*, 89*].
The lowfrequency part of the spectrum in Figure 14* consists of a continuum plus spectral lines between 10^{–5} – 10^{–4} Hz. The lines in the unsmoothed spectrum are near the resolution limit of the 40 day observation; their apparent width in Figure 14* is due to the spectral smoothing used to reduced estimation error. The lowest frequency line is near one cycle/day; the other lines are near harmonics of one cycle/day. Because of the multilink plasma correction, all random processes contributing to this spectrum are nondispersive. At frequencies greater than about 1/T_{2} there is clear, approximately cosinusoidal, modulation. This is characteristic of positive correlation in the time series at lag = T_{2}, i.e., either antenna mechanical noise or residual tropospheric noise. The level is too large to be dominated by residual tropospheric scintillation, however, and so is interpreted as mechanical noise. Many minima of the mechanical noise transfer function – at odd multiples of 1/(2 T_{2}) – are easily visible in Figure 14*. The spectrum appears to continue to be dominated by mechanical noise up to 0.01 Hz, with the signature of the transfer function being, however, difficult to see on this log plot (and also blurred at high frequencies since T_{2} changed with time by about 3% over the course of the 40 day observation.)
4.11 Summary of noise levels and transfer functions
To summarize the noise model: The principal noises are frequency and timing noise (FTS), plasma scintillation (solar wind and ionosphere), spacecraft electronics, unmodeled spacecraft motion, unmodeled ground antenna motion, tropospheric scintillation, ground electronics noise, thermal noise in the receiver, and systematic effects. The magnitudes of these noises in Cassiniera (2001 – 2008) observations are given in Table 2. Before any corrections, these noises enter the twoway Doppler as
where x is the effective distance of the solar wind perturbation from the earth. After multilink plasma calibration, phase scintillation due to charged particles is effectively removed. Watervaporradiometerbased tropospheric calibration removes 90% of the lowfrequency fluctuations due to the neutral atmosphere, so that the calibrated time series y_{2} is approximately where is all the nonGW (noise plus systematics) contributions to the twoway Doppler variability.Table 2 summarizes the noise model and the associated Allan deviation at = 1000 s for the principal noises (models of the spectra of the individual noises are given in [111*]). Figure 5* shows, highly schematically, the signal flow. This sketch, the GW transfer function (see Eq. (1*)), and the noise transfer functions (see, e.g., Figure 7*) are used in discussions of sensitivity, signal processing, and for qualifying/disqualifying candidate GW events.
Noise source 
[1000 s] 
Comment 
Frequency standard 
8 × 10^{–16} 

Antenna mechanical 
2 × 10^{–15} 

Ground electronics 
2 × 10^{–16} 

Plasma phase scintillation 
< 10^{–15}, for Kaband and SEP > 150° 
depends on SEP; dispersive; scales as square of radio wavelength; see Figure 10* 
Stochastic spacecraft motion 
2 × 10^{–16} 

Receiver thermal noise 
few × 10^{–16} 
depends on link SNR and detector bandwidth [23] 
Spacecraft transponder noise 
10^{–16} 

Tropospheric
scintillation

< 3 × 10^{–15} to 30 × 10^{–15} 

Tropospheric
scintillation

< 1.5 × 10^{–15} to 3 × 10^{–15} 
under favorable conditions [19*]; median conditions in connectedelement interferometry tests [96*] 