Download Ay121 Problem Set 3

Ay121 Problem Set 3
due Tuesday, October 18, 2016, 1:00 pm
1. Stokes Parameters (3 points) Read Rybicki & Lightman 2.4 before you answer this question. The
electric field component of an electromagnetic wave can be described in complex notation as
~ =E
~ 0 e−iωt
E
The vector can be decomposed into its x and y components as follows:
~
E
E1
=
=
x̂E1 + ŷE2
ǫ1 eiφ1
E2
=
ǫ2 eiφ2
(a) An electromagnetic wave has the following magnitudes and polarization angles:
ǫ1
=
0.7 V m−1
ǫ2
φ1
=
=
3.2 V m−1
20◦
φ2
=
37◦
(Volts per meter is the SI unit of electric field.) Calculate the Stokes parameters I, Q, U , and V .
(b) For a different electromagnetic wave, ǫ1 , ǫ2 , φ1 , and φ2 are unknown. Instead, its Stokes parameters are known to be
I
Q
=
=
513.6 V2 m−2
117.2 V2 m−2
U
V
=
=
227.8 V2 m−2
445.3 V2 m−2
Calculate ǫ1 , ǫ2 , φ1 , and φ2 .
(c) For both cases, calculate Π, the degree of polarization.
2. Radiation Spectrum (4 points) Consider a medium containing a large number of radiating particles.
Each particle emits a pulse of radiation with an electric field E0 (t) as a function of time. An observer
will detect a series of such pulses, all with the same shape but with random arrival times t1 , t2 , t3 , ...,
tN . The measured electric field will be
E(t) =
N
X
i=1
1
E0 (t − ti )
(a) Show that the Fourier transform of E(t) is
Ê(ω) = Ê0 (ω)
N
X
eiωti
i=1
where Ê0 (ω) is the Fourier transform of E0 (t).
(b) Argue that
2
N
X
iωti e =N
i=1
when averaged over random arrival times.
(c) Thus, show that the measured spectrum is simply N times the spectrum of an individual pulse.
(Note that this result still holds if the the pulses overlap.)
(d) By contrast, show that if all the particles are in a region much smaller than a wavelength and they
emit their pulses simultaneously, then the measured spectrum will be N 2 times the spectrum of
an individual pulse. These particles are now emitting coherently, which is a concept that we will
soon explore in depth.
3. Pulsars (9 points) Rybicki & Lightman problem 3.1. A pulsar is conventionally believed to be a
rotating neutron star. Such a star is likely to have a strong magnetic field, B0 , because it traps lines
of force during its collapse. If the magnetic axis of the neutron star does not line up with the rotation
axis, there will be magnetic dipole radiation from the time-changing magnetic dipole, m(t).
~
Assume
that the mass and radius of the neutron star are M and R; that the angle between the magnetic and
rotation axes is α; and that the rotational angular velocity is ω.
(a) Find an expression for the radiated power P in terms of ω, R, B0 , and α.
(b) Assuming that the rotational energy of the pulsar is the ultimate source of radiated power, find
an expression for the slow-down time scale, τ , of the pulsar.
(c) For M = 1 M⊙ , R = 106 cm, B0 = 1012 G, and α = 90◦ , find P and τ for ω = 104 s−1 , 103 s−1 ,
and 102 s−1 . (Newly formed pulsars are believed to have ω = 104 s−1 .)
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