4 Determination of the ejection parameters

The asymmetry in both brightness and position in the microquasar ejecta of Figs. 5 and 6 is normally interpreted in the context of special relativity effects. The jet component brighter and more distant from the core is the one approaching towards the observer, while the fainter and closer one is receding away from him/her. Figure 6 is also very reminiscent of the main figure in Mirabel & Rodríguez (1994) showing relativistic expansion in the superluminal microquasar GRS 1915+105, a source where such effects are stronger than in Cygnus X-3. The special relativity formulae as given by Mirabel & Rodríguez (1999) will be used throughout this discussion for quantitative estimates of the ejection parameters. The proper motion of the approaching ($\mu_a$, sign -) and receding jet ($\mu_r$, sign +) is then given by:

$$\mu_{\rm a,r} = {v \sin{\theta} \over D ( 1 \mp \beta \cos{\theta})},$$ (1)

where v is the true jet velocity, $\theta$ the inclination angle with the line of sight, $\beta=v/c$ and D the distance to the source.

In our case, the proper motion of the ejecta can be inferred from the third epoch map by using the offsets $\Delta \alpha \cos{\delta}$ and $\Delta \delta$ listed in Table 3 and assuming an age of 66.0 d since the ejection. This age is assumed because it seems natural that the most external ejecta should be also the oldest. For the brighter northern approaching component, the proper motion is $\mu_{\rm a}=9.3\pm0.3$ mas$\:$day^-1. Similarly, for the fainter southern receding component we find $\mu_{\rm r}=7.0\pm0.2$ mas$\:$day^-1. A more direct and independent estimate of $\mu_{\rm a}$ can be also obtained from the observed shift in the position of the northern jet component between the second and third epochs. Using the data in Table 3, the shift amounted to $176 \pm 24$ mas in a time interval of 15.05 days between the two epochs. The derived proper motion is therefore $\mu_{\rm a} = 12 \pm 2$ mas$\:$day^-1. An extrapolation of this independent proper motion backward in time gives JD $2451817\pm7$ as the ejection date of the northern component. This epoch is close to one of the strongest outburst in the series of events that activated our ToO proposal, but differs in two weeks from the assumed date of creation for the oldest and most external ejecta (JD 2451802.5). Considering the important uncertainties in this extrapolation, we prefer to keep this assumption as a very reasonable initial epoch for proper motion estimates.

Having obtained values for the proper motion of both the jet and the counterjet, the ejection parameter $\beta \cos{\theta}$ can be easily obtained from Eq. (1):

$$\beta \cos{\theta} = {\mu_{\rm a} - \mu_{\rm r} \over \mu_{\rm a} + \mu_{\rm r}}\cdot$$ (2)

It is remarkable that $\beta \cos{\theta}$ can be found without any knowledge of the distance D. Using the more accurate proper motion values derived from the third epoch ($9.3\pm0.3$ and $7.0\pm0.2$ mas day^-1), we obtain $\beta \cos{\theta} = 0.14 \pm 0.03$. This implies a true jet velocity $v \geq (0.14\pm0.03)c$ and an ejection angle $\theta \leq 82\pm 2^{\circ}$. We can also provide a relativistic upper limit for the Cygnus X-3 distance D given by

$$D \leq {c \over \sqrt{\mu_{\rm a} \mu_{\rm r}}}\cdot$$ (3)

The resulting value $D \leq 21 \pm 1$ kpc is, however, not very constraining when compared to modern distance estimates to Cygnus X-3. For example, based on X-ray data Predehl et al. (2000) recently obtained a distance value of 9 kpc.

The inferred proper motions imply apparent subluminal velocities of $(0.54 \pm 0.02)c[D/10~{\rm kpc}]$ and $(0.40\pm0.01)c[D/10~{\rm kpc}]$ for the northern and southern jet components, respectively. Our direct (not-inferred) proper motion measurement for the northern jet component translates into a consistent apparent velocity of $(0.69\pm0.12)c[D/10~{\rm kpc}]$. Accepting that the distance value to Cygnus X-3 is D=10 kpc, it is possible to derive both the true jet velocity and the inclination angle with the line of sight instead of the limits given above. Solving from Eq. (1) we get $\beta \sin{\theta}= 0.46 \pm 0.03$, which implies $v = (0.48 \pm 0.04)c$ and $\theta=73\pm 4^{\circ}$.

The flux density ratio between the approaching and receding jet components, $R \equiv S_{\rm a}/S_{\rm r}$, is related to the ejection parameters according to the Doppler boosting formula:

$$R = \left( {1+\beta \cos{\theta} \over 1 - \beta \cos{\theta}} \right)^{\!k-\alpha},$$ (4)

where $\alpha$ is the spectral index ( $S_\nu \propto \nu^{\alpha}$) and k=2, 3 for a continuous jet or a discrete jet condensation. This formula is strictly valid for equal angular distances from the core of the approaching and receding components. This fact is relevant in our case because of the noticeable jet asymmetry. Therefore, the flux densities whose ratio is considered are not simultaneous in time according to the observer. By interpolation between the values in Table 3, the flux density of the northern component at 460 mas from the core is $1.66\pm0.03$ mJy. This angular distance is the one at which we detect the southern component in the third epoch. Consequently, the flux density ratio is $R=1.87 \pm 0.08$. The Mioduszewski et al. (2001) maps suggest a continuous jet flow, so we will adopt k=2. Moreover, it seems natural to assume that the long-lived ejecta of Cygnus X-3 emits optically thin synchrotron radiation with a typical spectral index $\alpha =-0.6$. Under such circumstances and using Eq. (4), the $\beta \cos{\theta}$ parameter can be expressed as:

$$\beta \cos{\theta} = {R^{1/(k-\alpha)} - 1 \over R^{1/(k-\alpha)} +1},$$ (5)

that gives $\beta \cos{\theta} = 0.12\pm0.01$. The error quoted does not include any uncertainty in the assumed spectral index. The resulting value is in excellent agreement with the independent estimate $\beta \cos{\theta} = 0.14 \pm 0.03$ based on proper motion measurements.