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A fixed distant observer receives appreciable radiation while the velocity points within an angle of order $1/\gamma$ of the line of sight. The corresponding emission interval is $\Delta t_{\mathrm{em}}\sim R/(\gamma v)$. The source moves toward the detector during that interval, shortening the light-travel distance. In the far-field limit,where $\varphi$ is the angle between velocity and the detector direction. Near the tangent, $1-\beta \cos \varphi \sim {\gamma }^{-2}$, soThe emission interval itself is of order $3.2\times 10^{-10}$ s. Omitting the arrival-time compression would therefore miss many orders of magnitude.

For a definite angular window $-1/\gamma \le \varphi \le 1/\gamma$, integrating the arrival-time relation givesThe order-one coefficient depends on how the beam window or pulse width is defined. These are intrinsic single-electron radiation-flash scales. For bunch-length broadening of synchrotron radiation, a finite bunch moving through the same tangent point contributes an arrival-time spread of order its length divided by $v$; the longitudinal bunch profile and instrumental response must be convolved with the intrinsic flash. “Short bunch” alone does not specify an experimental bunch pulse duration at the attosecond scale, because no bunch length has been supplied.^[ref]

For an ultrarelativistic particle, relativistic beaming concentrates the synchrotron radiation within an angle of order $1/\gamma$ around its instantaneous velocity. A fixed viewing direction in the orbital plane receives an intense signal only when the tangent sweeps through that direction. One electron, or one sufficiently short bunch, therefore produces one pulse per revolution, separated by $\mathcal{T}\simeq 10.48\mu \mathrm{s}$. Several bunches produce the corresponding additional pulses per revolution.

The time during which the velocity points within that cone is of order $r/(\gamma c)$, but this is the emission interval. If $t_a$ is the arrival time at a distant detector,Near alignment, $1-\beta \cos \theta \simeq ({\gamma }^{-2}+{\theta }^2)/2=O({\gamma }^{-2})$. Thus synchrotron pulse compression reduces the observed duration by a further factor of order ${\gamma }^2$:A more explicit geometric estimate chooses the cone edges $\theta =\pm 1/\gamma$. With $\omega =u/r$, $x=r\sin \theta$ along the line of sight, the difference in arrival-time compression for a moving source isThe order-one coefficient depends on how pulse width is defined; the robust scale is $r/(c{\gamma }^3)$. The usual central-cone estimate $r/(2c{\gamma }^3)$ gives $4.1\times 10^{-18}\mathrm{s}$. These estimates agree on the attosecond scale and must not be confused with $r/(\gamma c)\simeq 2.84\times 10^{-10}\mathrm{s}$.^[ref]

Bunch qualification. The source says only that the bunches are short, without their longitudinal length $L_b$. Different electrons pass the same emitting tangent at times separated by up to about $L_b/u$, with essentially the same path length from that tangent to the detector. The bunch-length broadening of synchrotron radiation means that the measured bunch envelope is the single-electron pulse convolved with the bunch's passage-time profile; its scale is generally at least of order $\max \{L_b/u,r/(c{\gamma }^3)\}$, with additional detector bandwidth effects. For example, a $1\mathrm{mm}$ bunch, very short compared with this ring, gives an approximately $3.3\mathrm{ps}$ envelope, much longer than the single-electron duration. No unique numerical bunch-envelope width follows from the supplied data. The pulse argument also presumes that the accepted $+x$ direction is tangent to the orbit; a direction far outside the orbital plane receives no comparable intense forward pulse.