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For an ultrarelativistic particle, $\epsilon =pc$. Including the two spin states of an electron, counting wavevectors in a sphere gives the density of statesThe grand canonical partition function of the ideal Fermi gas is thereforeSince the grand potential satisfies $pV=k_{B}T\log \mathcal{Z}$, integration by parts givesThusThis is the equation of state of an ideal ultrarelativistic gas.

For an ultrarelativistic particle, $E\simeq |\mathbf{p}|$. Setting $\mu =0$ and writing the three-dimensional momentum integral radially givesThe standard Bose integrals yieldFor fermions, the corresponding integrals differ by the familiar factorsThus the number density scales as $T^3$, while the energy density and pressure scale as $T^4$.

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.