synchrotron radiation 47
%
Y F
$
5
2EVHUYHU
FIGURE 3.4
SR and remote observer.
Looking at this figure we conclude that the emitted radiation is observed during particle travel along the 2R/γ arc.
Let’s now keep in mind that the radiation travels at speed
c, while particles travel at v. At point B shown in Fig. 3.4 the
separation between radiation and the particles is given by
2R
v
ds ≈
1 −
(3.17)
γ
c
Therefore, the observer will see radiation during the following time interval:
ds 2R
R
Δt ≈
≈
(1 − β) ≈
(3.18)
c
c γ
c γ 3
We can thus proceed to estimate the characteristic frequency of emitted photons as the inverse of the time duration
of the flash, as seen by the observer:
1
c γ 3
ω c ≈
≈
(3.19)
Δt
R
Comparing the above with the exact formula that we reproduce here without derivation
3 c γ 3
ω c =
(3.20)
2 R
we can again see that our back-of-the-envelope estimations
give pretty accurate results while not hiding the physics of
the phenomena behind heavy math.
3.1.5 SR — number of photons
Having estimated the characteristic energy of the SR photons,
we can now estimate the number of photons emitted per unit
length by a single electron.