2 The Rosetta flyby at Siwa

RSI will use the two-way radio carrier signals at X-band uplink (7168 MHz) and S-band (2300 MHz) and X-band (8422 MHz) simultaneous downlink, provided by the spacecraft radio subsystem, in order to measure slight changes in the spacecraft velocity induced by the gravity attraction of the asteroid via the classical Doppler effect. If the spacecraft's velocity is changed by forces acting on the spacecraft, a frequency shift of the radio carrier with respect to a predicted Doppler shift (assuming no perturbing body present) is induced by

$$\Delta f = - f_0 \frac{\Delta v_r}{c}$$ (1)

where $\Delta v_r$ is the change of the spacecraft velocity along the line-of-sight (LOS) caused by non-accounted forces acting on the spacecraft, f₀ is the radio carrier frequency at X-band and c is the speed of light. If the two-way radio link is used as described above, the Doppler shift is twice the value from (1).

Table 1 gives the parameters for both flybys and Fig. 1 sketches the flyby geometry for the Siwa case.

 

 

Table 1: Flyby parameters of the Rosetta asteroids

	Otawara	Siwa
Estimated diameter	4 km (S type)a	110 kmb
	2.8 km (V type)a
Taxanomic type	S or V	C
Density	$\geq$ 1900 kg/m3	2000 kg/m3
		(estimate, used for simulation)
Nominal flyby distance	2183 km	3500 km
Nominal flyby velocity	10.63 km s-1	17.04 km s-1
Angle between the flyby trajectory and	169.96$^\circ$	174.04$^\circ$
the projected line-of-sight to Earth
Angle between the flyby plane and the	178.85$^\circ$	3.35$^\circ$
direction to Earth
Angle between the flyby trajectory and	149.14$^\circ$	167.61$^\circ$
the direction to the Sun

$\parbox{12.7cm}{ $^a$\space Doressoundiram et~al. (\cite{Doressoundiram99})\\ $^b$\space M. Fulchignoni (priv. comm.)}$

Figure 1: Rosetta flyby geometry at asteroid 140 Siwa on 24th July 2008. The flyby plane contains the flyby trajectory ($\vec{v}_0$) and the position vectors from spacecraft to asteroid. The projected direction to the Earth forms an angle of 174.04$^\circ$ with the flyby trajectory. The direction to Earth is located 3.35$^\circ$ below the plane. The change in velocity $\Delta v(t-t_0)$ can be split in two components along the track $\Delta v_{\rm along} (t-t_0)$ and across the track $\Delta v_{\rm across} (t-t_0)$. No net contribution to the total change in velocity is gained from the $\Delta v_{\rm along} (t-t_0)$ component. Observed via the classical Doppler effect is $\Delta v_r(t-t_0)$, the sum of $\Delta v_{\rm along} (t-t_0)$ and $\Delta v_{\rm across} (t-t_0)$ projected onto the line-of-sight. The direction to the Sun lies in the flyby plane and forms an angle of 167.61$^\circ$ with $\vec{v}_0$

The flyby plane is defined as the plane containing the relative flyby velocity vector ($\vec{v}_0$) and the direction vectors relative to the asteroid. The direction to the Sun lies also in this plane. During the flyby, the Rosetta spacecraft will point the instrument panel toward the asteroid. The spacecraft body will rotate during flyby in order to keep the remote sensing instruments pointed towards the asteroid. The fully steerable HGA will remain pointed toward the Earth which leads to a rotation of the HGA during flyby. A HGA rotation end point will be reached about five minutes before encounter resulting in the loss of radio signal. However, the instruments will remain pointed by the on-going rotation of the spacecraft body. The spacecraft is reoriented about 20 min after closest approach and spacecraft radio tracking will proceed after reestablishing the two-way radio link. The gravitational attraction from the asteroid acts onto the spacecraft at all times along the flyby trajectory (Fig. 1) leading to changes in the flyby velocity and trajectory. The deflection angle $\Psi$ is given by

$$\mbox{cosec} \frac{\Psi}{2} = 1 + \frac{r_0 v_0^2}{GM}$$ (2)

(Danby 1992), where M = mass of the asteroid and G = gravitational constant. For the nominal Siwa flyby at a closest approach distance of r₀=3500 km and relative flyby velocity v₀=17.04 km s^-1, this angle is $10^{-5 \circ}$ and the trajectory deflection can be neglected. Therefore, we assume a straight flyby trajectory in the following with the radial distance vector (spacecraft to Siwa) of

$$\vec{r} = \vec{r}_0 + \vec{v}_0 (t - t_0).$$ (3)

The currently agreed nominal flyby distance of 3500 km is still subject to discussion among the Rosetta instrument teams and may change in the future. The smallest feasible flyby distance is 1750 km constrained by the maximum slew rate of the spacecraft. It is not expected that a flyby distance larger than 3500 km will finally be decided on.

Defining $\vec{g}(t)$ as the gravity attraction vector directed toward the asteroid at time t

$$\vec{g}(t - t_0) = \frac{GM}{r^2} \vec{e}_{\rm r}$$ (4)

where t₀ = time of closest approach. The vector can be resolved into two components along and across the trajectory direction. In the existing literature (Anderson 1971; Anderson & Giampieri 1999; Rappaport et al. 2000) this is accomplished by integration over time with initial conditions at closest approach. Here, we choose to integrate with initial conditions at $-\infty$, which makes a small difference, equal to one-half of the total deflection angle (2), in the direction of the two components. The changes in velocity for both components because of the gravitational attraction are:

 

$$\Delta \vec{v}_{\rm along} (t - t_0) \frac{GM}{r_0 v_0} \frac{1}{\sqrt{1 + \frac{v_0^2}{r_0^2} [t - t_0]^2}} \vec{e}_{\rm y}$$ =

$$\Delta \vec{v}_{\rm across} (t - t_0) -\frac{GM}{r_0 v_0} \left[ 1 + \frac{\frac{v_0}{r_0} [ t - t_0 ]}{\sqrt{1 + \frac{v_0^2}{r_0^2} [t - t_0]^2}} \right] \vec{e}_{\rm x}.$$ = (5)

Only the LOS components of these velocity changes can be observed via the classical Doppler effect. Relevant is the angle between the along-track direction and the projected LOS ( $180^\circ - \alpha$) and the angle between the across-track direction and the projected LOS ( $90^\circ - \alpha$). Furthermore the Earth is located $\epsilon = 3.35^\circ$ below the flyby plane. The total change in the Doppler velocity along LOS due to gravity attraction during the flyby is then

 

$$\Delta \vec{v}_{\rm r} -\Delta \vec{v}_{\rm along} \cos(180^\circ - \alpha) \cos \epsilon$$ =

$$+ \Delta \vec{v}_{\rm across} \cos(90^\circ - \alpha) \cos \epsilon$$

$$\Delta \vec{v}_{\rm along} \cos \alpha \cos \epsilon + \Delta \vec{v}_{\rm across} \sin \alpha \cos \epsilon.$$ = (6)

The RSI flyby strategy is twofold. First, by comparing the predicted "force-free'' pre-encounter Doppler frequency, gained from a precise orbit determination after the last manoeuvre some days before the encounter, with the actual observed Doppler frequency long after the closest approach but before the first manoeuvre after the flyby. This procedure for detecting velocity changes is similar to the one used for the Giotto flybys at comets P/Halley and P/Grigg-Skjellerup (Pätzold et al. 1991a, 1991b, 1993). The difference in Doppler frequency is translated into Doppler velocity along the LOS via (1), taking into account that the two-way radio link contributes twice the Doppler frequency value of (1), and, from the encounter geometry (Fig. 1), the asteroid's mass can be determined from the absolute values

 

$$\Delta v_{\rm along}(\infty)$$ = 0

$$\Delta v_{\rm across}(\infty) 2 \frac{GM}{r_0 v_0}$$ =

$$\Delta v_{\rm r}(\infty) - \Delta v_{\rm accross} \sin \alpha \cos \epsilon$$ =

$$- 2 \frac{GM}{r_0 v_0} \sin \alpha \cos \epsilon.$$ = (7)

Second, the real-time Doppler frequency is analysed during the flyby. Again,the Doppler velocity along the line-of-sight can be computed from (1) at any time (t-t₀). Figure 2 shows the theoretical Doppler velocity signature for three different closest approach distances of 1750 km (the minimum feasible flyby distance), 3500 km (the nominal closest approach distance) and 10 000 km for the Rosetta/Siwa flyby.

Figure 2: Expected changes in velocity along the line-of-sight $\pm 2$ hours about the time of closest approach to the asteroid Siwa for three different flyby distances, 1750 km, 3500 km and 10 000 km

Figure 3 shows simulated noisy Doppler velocity measurements at 600 seconds integration time (solid circles) for the nominal flyby distance of 3500 km.

Also included in the simulation is a gap of 25 min about closest approach (grey area) lasting from 5 min before closest approach to 20 min after closest approach due to the loss of the radio link. A non-linear least squares fit according to Eqs. (3) and (4) was applied to the simulated data points in order to retrieve an estimate for GM of the asteroid.

Figure 3: Simulated noisy Doppler velocity values for an integration time of 600 s (solid points) according to Eqs. (5) and (6). A random Gaussian distributed noise of $\sigma_{\rm v} = 15 ~\mu$m/s was added to each simulated data point plus an grossly overestimated contribution of the same order of magnitude from the turbulent solar wind. The grey area marks 20 min of expected loss of radio signal. The solid line is an on-linear least squares fit to the simulated data points that yields GM as a result to an accuracy of 1%. The same accuracy was estimated from Anderson's (Anderson et al. 1992) rule of thumb (9)