The broadband electromagnetic radiation emitted by a pulsar travels through the in- terstellar medium (ISM) on its way to the Earth. The ISM is not empty; it is mostly a cold, ionised plasma, in many cases inhomogeneous and turbulent. The pulsar ra- diation interacts with the charged particles of the ISM as it propagates through it, in particular with the free electrons10. As a result, the signal gets dispersed, can be
scattered and can suer of intensity modulations. Furthermore, if the pulsar signal is polarized (as it is commonly the case), and there is a magnetic eld along the line of sight, the polarization properties of the pulsar signal are also aected. While these propagation eects generally complicate the detection and study of pulsar signals, at the same time provide a means to obtain valuable information about the ISM. The next sections explains the basics of the relevant propagation eects for the work presented in this thesis: dispersion, scattering, and scintillation.
1.3.1 Dispersion
In vacuum, an electromagnetic (EM) wave of any frequency will travel at exactly the speed of light. This occurs because the refractive index of vacuum is one, and the velocity of the EM wave is
vg =c n, (1.13)
where vg is its group velocity, c is the speed of light, and n is the refractive index of
the medium where the EM travels. However, since the ISM is not empty its refractive index is not equal to 1. The refractive index of the ISM (as a cold, ionised plasma) depends on the so-called plasma frequency (see e.gWilson et al.,2013),
n= s 1−ν 2 p ν2, (1.14)
where νp is the plasma frequency, and ν is the frequency of the EM wave. Combining equations 1.13 and1.14, vg=c s 1−ν 2 p ν2. (1.15)
The plasma frequency, νp, depends on the density of free electrons in the plasma (see e.g. Lorimer & Kramer,2005),
νp= s Nee2 π me '8.978 Ne cm−3 0.5 [kHz], (1.16)
10The radiation also interacts with the ions, but we neglect here this eect because the ion's high mass with respect to the electrons (mion'2000me) decrease their contribution by this same factor.
where Ne is the number of free electrons, and e and me are the electric charge and mass of the electron, respectively.
These simple relations show that the group velocity of an EM wave travelling in a medium will depend on the free electron content and the specic frequency of the EM wave. Since the pulsar radiation is broadband, Equation1.15implies that the dierent frequencies will propagate at dierent velocities through the ISM. As a result, the low frequencies arrive to the observer delayed in time with respect to the high frequencies. This frequency-dependent time delay is known as dispersion.
Expanding in series Equation 1.15 and using Equation 1.16, we can calculate the time delay between two frequencies,
∆t= e 2 2πcme 1 ν12 − 1 ν22 Z L 0 Ne(l)dl, (1.17) The rst factor is a constant usually referred as the dispersion constant
D ≡ e
2 2πcme
'4.1488056·103MHz2pc−1cm3s. (1.18) The integral of the number of free electrons along the line of sight is called dispersion measure, DM,
DM=
Z L
0
Ne(l)dl. (1.19)
From equations1.17,1.18, and 1.19we can write ∆t≈4.15·106 1 ν12 − 1 ν22 DM [ms]. (1.20)
which corresponds to the time delay in ms between the frequenciesν1 andν2 expressed in MHz.
One remark from this analysis is the fact that if we can measure the distance to a pulsar with precision (for example through astrometric measurements, precise timing, or spectroscopy), we can obtain the average electron density in the line of sight to that pulsar. Similarly, knowing the average electron content in the line of sight we can estimate the distance to a pulsar. Some of the most complete models of the electron content in the Milky Way are based on pulsar DM and distance measurements (Taylor & Cordes,1993;Cordes & Lazio,2002), and their precision can be continuously improved as new pulsars are discovered and more accurate measurements of DM values and distances become available.
1.3.2 Scattering
As was already mentioned, the ISM is not homogeneous. Dierences in the electron densities between the observer and a pulsar, as well as relative motion between the observer, the pulsar, and the inhomogeneous ISM regions will cause distortions of the originally coherent EM wave front (see Figure 1.5). The multi-path propagation and
Coherent Wave Front Time Delays / Distorted Phases vPSR vobs θ θd Pulsar vscreen Diffraction Pattern
Figure 1.5: Sketch showing the eects of an inhomogeneous ISM (represented by a thin screen model) on pulsar radiation. The initially coherent wave front gets distorted as a result of multi-path propagation and dierences in the refractive index of the medium (from variations of the electron density, see Equation 1.14 and 1.16). The dierences in electron density can be inherent to the medium (e.g. if crossing an HII region or supernova shock front), to turbulence, or due to relative motion between the pulsar, the inhomogeneous medium, and the observer. The multi-path propagations translates into time delays of the radiation reaching the observer (scattering) and into phase dierences that can lead to constructive or destructive interference (scintillation). Figure adapted from Lorimer & Kramer(2005).
dierences in the refractive index of dierent paths result in dierent time delays of the signal arriving at the observer. Such eect produces scattering of the signal
To describe the eect, a simple model is usually adopted in which a thin scatter- ing screen, representing the averaged eects of all the inhomogeneities in the line of sight, is placed middle way between the pulsar an the observer (Scheuer, 1968). The calculations depend on the turbulence spectrum of the ISM, typically assumed to be a Kolmogorov spectrum. Under these assumptions and using simple geometrical rela- tions, the observer of a scattered point-like source will see a diuse disk with angular radius (see e.g. Lorimer & Kramer,2005)
θd = θ
2, (1.21)
whereθis the bending angle of the wavefront at the scattering screen (see Figure1.5), and an exponentially decaying angular intensity distribution,
I(θ)dθ∝exp(−θ2/θd2) 2πθdθ. (1.22) Since the light rays observed after being bended an angle θ will arrive with a
geometrical time delay, ∆t(θ) =θ2d/c, with respect to the rays taking the direct path travelling a distance dfrom the pulsar to the Earth, we can derive that the observed intensity as a function of time,
I(t)∝exp(−c∆t/θd2d)≡exp(−∆t/τs) (1.23) where τs≡θd2 d c ∝∆N 2 e d2f −4 (1.24)
is referred as the scattering time scale, i.e. a one-side exponential function that rep- resents the distortion of the pulsed emission due to the scattering. This expression gives us important information about the scattering eect in pulsar emission. First, the temporal scattering of the emission is proportional to the square of the variations in electron density and the square of the distance to the pulsar. Secondly, the scat- tering decreases rapidly with frequency, as the fourth power. Consequently, a way to decrease the deleterious eects of scattering is to use high frequency observations. This is particularly useful where the scattering is expected to be strong like, for instance, when observing pulsars in the Galactic Centre (Cordes & Lazio,1997, see Chapter 3). 1.3.3 Scintillation
The same geometrical time delays that causes scattering results in waves arriving at the observer plane with dierent phases,∆φ∼2πτs. The constructive and destructive interference pattern will produce a modulated variation of the detected intensity of the source. To produce interference, the phases of the waves must dier less than 1 radian. This means that there is a limiting bandwidth (also called scintillation bandwidth or decorrelation bandwidth) in which waves show the intensity variations. The condition for interference is (see e.g. Lorimer & Kramer,2005),
2π∆ντs ∼1, (1.25)
implying that ∆ν∝1/τs∝ν4.
The eects of scintillation can dier depending on the dierent scales of the electron density inhomogeneities in the ISM and the distance to the pulsar. Following this dependency, scintillation can be classied in weak and strong scintillation, and the strong scintillation is subcategorised in diractive and refractive. The regime can be determined calculating the eld coherence scale, s0, (i.e. the size of a circular region on the scattering screen centred at the source with phase dierences less than 1 rad), and comparing it to the rst Fresnel zone, lF. Following Lorimer & Kramer (2005), (see also Cordes & Lazio,1991)
s0 = 1 k θd
, (1.26)
lF =
r
d
k. (1.27)
Ifs0 lF the distortions are caused only by small phase perturbations within the Fresnel zone, producing weak scintillation. Otherwise, the signal can suer of larger phase perturbations, which lead to strong scintillation.
Using the velocity of the relative motion of the pulsar, VISS, we can estimate the observable eects of the scintillation on the intensity of the signal. Most pulsar ob- servations occur in the strong regime (and this is also the case for the observations presented in this thesis), and we focus here on this case. Typically, the estimated quantity is the standard deviation, σS, of the mean ux density, which can be used to
obtain the so-called modulation index
m= σS
hSi. (1.28)
The modulation index of strong diractive scintillation depends on the number of scintles within the observation, both in time (Nt) and frequency (Nf),
mDISS = r 1 NtNf , (1.29) where Nt≈1 +κ Tobs ∆tDISS , (1.30) Nν ≈1 +κ ∆ν ∆νDISS , (1.31) ∆tDISS = s0 VISS , (1.32) and ∆νDISS ∼11 ν GHz 4.4 d kpc −2.2 [MHz], (1.33)
being Tobs the length of the observation, ∆tDISS the scintillation time scale, ∆ν the observation bandwidth, ∆νDISS is the scintillation bandwidth, and κ is a parameter empirically estimated in the range 0.1−0.2. Furthermore, if the refractive scale lR = lF2/s0 lF, strong refractive scintillation must also be taken into account. Strong refractive scintillation produces intensity modulations with an index
mRISS= lF s0 −1/3 , (1.34)
in longer time scales, of the order of
∆tRISS = lR VISS