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3 Doppler velocity accuracy

Taking all major instrumental noise sources into account (spacecraft radio subsystem and ground station), the Doppler velocity error $\sigma_{v0}$ for the Rosetta transponder system at S-band and X-band was estimated (Table 2) for a one second integration time. For longer integrations, the accuracy $\sigma_v(\Delta t)$ scales with the square root of the integration time $\Delta t$:

 \begin{displaymath}\sigma_v(\Delta t) = \sigma_{v0} \frac{1}{\sqrt{\Delta t}}
\end{displaymath} (8)

and is shown in Fig. 4.


 

 
Table 2: Doppler velocity error budget
  Doppler velocity error $\sigma_{\rm v}$
  S-band X-band
Phase error 1.0 mm/s 0.3 mm/s
(thermal and ground    
station contribution    
Transponder quantisation 0.4 mm/s 0.1 mm/s
error in frequency    
Transponder quantisation 0.01 mm/s 0.004 mm/s
error in phase    
Total error (coherent mode) 1.06 mm/s 0.32 mm/s



  \begin{figure}
\par\resizebox{10.2cm}{!}{\includegraphics{h2690_f4.eps}}\hfill\parbox[b]{55mm}
{}
\end{figure} Figure 4: Estimated instrumental Doppler noise (from the radio subsystem and the ground station) at S-band (upper curve) and X-band (lower curve) as a function of integration time. Integration times of 600 s are planned to be used for the asteroid flybys

Another uncertainty which adds to the Doppler noise is the propagation of the radio signals through the turbulent solar wind (Bird 1982; Pätzold et al. 2000). Turbulent solar wind density variations along the radio ray path will induce dispersive carrier phase shifts on the radio carrier, increasing the phase noise. Using a dual-frequency downlink at S-band and X-band, it is possible to limit the solar wind noise by extracting the dispersive propagation contributions from the calculation of the differential Doppler and by correcting the X-band two-way radio link by this result.

At the time of the Siwa flyby (24th July 2008), the Siwa-Sun-Earth angle is $100^\circ$ (Fig. 5). The impact parameter of the radio ray path along the Earth-Siwa radius is about 191 solar radii still close enough for contributions from the coronal plasma (solar wind) to the radio link.

We estimate an enhancement of the Doppler noise by a factor of two with respect to the instrumental noise.


  \begin{figure}
\par\resizebox{10.5cm}{!}{\includegraphics{h2690_f5.eps}}\hfill\parbox[b]{55mm}
{}
\end{figure} Figure 5: Orbit positions of asteroid 140 Siwa and the Earth at the time of the Rosetta flyby (24th July 2008) projected onto the ecliptic plane. The angle between the heliocentric distance vectors toward Siwa and the Earth form an angle of 100$^\circ $. The impact parameter of the radio ray path (closest approach of the radio ray to the Sun) is about 191 solar radii

The integration time which will be used during the flybys are 600 s, yielding a theoretical $13~ \mu$m/s rms noise figure for the Doppler velocity from (8). The post-encounter Doppler velocity value for the 10 000 km distance in Fig. 2 is $100 ~\mu$m/s, approximately four $\sigma$ above Doppler noise level and therefore clearly detectable assuming the increase in noise by a factor of two with respect to (8) because of the propagation in the solar wind. Anderson et al. (1992) estimated the fractional GM error as

 \begin{displaymath}\frac{\sigma_{GM}}{GM} = \frac{v_0 d}{GM} \sigma_{\rm v}
\end{displaymath} (9)

where $\sigma_{\rm v}$ is the Doppler velocity noise. Applied to the simulation in Fig. 3, the fractional GM error $\sigma_{GM}$ for the Siwa flyby is estimated to be in the order of 1%. If the Sun proves quiet at the time of the flyby, this error might even be smaller. The direction to the Sun is located in the flyby plane and forms an angle of $167.61^\circ$ with $\vec{v}_0$. The radiation pressure acts in the anti-solar direction and forms an angle of $12.39^\circ$ leading to acceleration components along the track parallel to and across the track perpendicular to $\vec{v}_0$ (parallel to $\Delta \vec{v}_{\rm across}$). During a ten hour tracking pass, the radiation pressure will induce a velocity component along the line-of-sight of 0.7 mm/s (red shift), caused by a dominant velocity component parallel to $\vec{v}_0$. This is compared to the change in velocity by the gravitational attraction of the asteroid of -0.3 mm/s (Fig. 3) at the nominal flyby distance (blue shift). The radiation pressure contribution, depending on spacecraft mass, cross-sectional area exposed to solar radiation, absorption and reflection capabilities of the spacecraft's surfaces and the heliocentric distance, can be predicted to a large extent and removed from the observed Doppler data. Errors in the prediction will appear as a linear trend in the observed Doppler residuals (observed minus predicted Doppler).

While the determination of the mass of Siwa from the RSI experiment is likely to have an accuracy of 1%, the estimate of the volume and hence the density will be much more inaccurate. The OSIRIS camera system on board Rosetta has avery high resolution of 60 m/px at closest approach. Thus, in two dimensions, OSIRIS will determine the cross-sectional area of the target to about 0.5%. However, the third dimension is unseen by the camera because it is unilluminated. Hence, this dimension can only be constrained from remote observations taken several hours before and using the asteroid's rotation. Even in this case, the rotation is unlikely to be orthogonal to the Sun-asteroid line (i.e. there is a constantly unilluminated pole). Thus, estimates, based on the asteroid having a reasonably continuous shape, have to be made. We estimate the resulting error to be of the order of 20% in both volume and density.


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