There have classically been two different approaches for modeling the emission of PWNe. On
one hand, Magnetohydrodynamic (MHD) simulations succeed on explaining the morphological properties of PWNe. On the other hand, spherically symmetric one-dimensional PWNe spectral models do not take into account the energy-dependent morphology of the PWN, but successfully explain the spectrum. We will briefly describe both approaches.
5.4.2.1 MHD models
Kennel & Coroniti (1984b) were the first presenting a model that reproduced the morphological and spectral properties of PWNe through MHD simulations. The model consisted on solving the analytical equations for the pulsar wind in the simplified case of a symmetrically spherical MHD flow. The solution depends on the magnetization parameter σ, the spin-down power and the radius of the termination shock. The injection spectrum considered is a power-law with a
cut-off that fits the observed spectrum. To calculate the synchrotron emission electron adiabatic
and synchrotron losses are taken into account. The photon fields used as targets to calculate IC emission are synchrotron photons and dust IR emission. Several variations of the model using the same approach were later proposed (de Jager & Harding 1992; Atoyan & Aharonian 1996;
de Jager et al. 1996; Hillas et al. 1998; Meyer et al. 2010) to account for different phenomena
inside the nebula.
5.4.2.2 One-dimensional spectral models
There are also several models that make a one-dimensional approach without taking into ac- count any energy dependence in the PWN morphology. Aharonian et al. (1997a) applied the
diffusion-loss equation (solved for the first time by Syrovatskii (1959)) to study the IC emission
from PWNe. To correctly account for the evolution of the lepton population inside the neb- ula, one has to introduce a time parameter in the equation. Some time-dependent models apply
different approximates such as neglecting the escape term (Tanaka & Takahara 2010), or substi-
tuting the energy losses by the particle’s escape time (Zhang et al. 2008), while others make no approximations (Mart´ın et al. 2012). The photon contributions considered to calculate the broad- band spectrum are also diverse, although Mart´ın et al. (2012) considers all of them: synchrotron emission, synchrotron self-Compton, IC, and bremsstrahlung. A more detail explanation about several of the time-dependent models is given in Section §7.3.1.
6
The Crab Nebula: a gamma-ray factory in our
backyard
6.1
Introduction
Figure 6.1: Lord Rosse’s drawing of the Crab Nebula.
The Crab Nebula is the remnant of the SN explosion in 1054 AD reported by Japanese and Chinese astrologers (Ho & Ho Ping-Y¨u 1962). The “new star” was visible during daytime for several weeks, and in the night sky for 22 months (Clark & Stephenson 1977). The remnant was discovered by the English astronomer John Bevis in 1731 and became the first object in Charles Messier’s catalog of nebulae and star clusters. The name of the nebula was given by William Parson, third Earl of Rosse in 1850, who found the similarity with the crustacean the first time he looked at the nebula as it can be seen in the drawing he made in Figure 6.1. Lundmark (1921) and Hubble (1928) proposed that the Crab Nebula was asso- ciated with the SN explosion in 1054, but it was not until 1941 when the Crab was unambiguously established as
the remnant of the SN 1054 (Duyvendak 1942; Mayall & Oort 1942) (an optical image of the Crab can be seen in Figure 6.2). The Crab is placed at a distance of ∼2 kpc (Trimble 1973)
and its composition can be divided in several observables. Let us go in detail over its different
Figure 6.2: Optical image of the Crab Nebula by the Hubble telescope. Image from Hester (2008).
The Crab pulsar
The Crab pulsar (also known as PSR J0534+2200), located at the center of the SNR, is the
engine that powers the nebula. It was discovered in radio by the Arecibo telescope (Staelin
& Reifenstein 1968; Comella et al. 1969). It has a period P=33 ms and its first derivative
is ˙P =4.21 ×10−13 s s−1. If we assume that the radius of the neutron star is ∼10 km and
its mass 1.4 M , the pulsar’s spin-down power is Lspin-down = 4π2I
˙ P
P3 = 5 × 1038erg s−1.
Lyne et al. (1988) assumed a breaking index n=2.51 and calculated that the initial period
of the pulsar was ∼19 ms, meaning that it has lost 3.6 ×1049erg since its origin, which is
less than 10% of the energy of the SN explosion that gave birth to the pulsar (assuming
the fiducial value of 1051 erg). One has to mention that most of the rotational energy lost
by the neutron star is not emitted in the form of pulsed emission, but it is carried away by a highly magnetized plasma. The Crab pulsar is detected at wavelengths ranging from radio to VHE gamma rays. It is characterized by a two-peaked light-curve with peaks varying across the electromagnetic spectrum in relative height but not in phase. After the aforementioned radio detection, pulsed emission was discovered from the Crab in optical (Cocke et al. 1969), X-ray (Fritz et al. 1969; Floyd et al. 1969) and γ-ray (Browning
et al. 1971; Kurfess 1971; Albats et al. 1972) wavelengths (see Figure 6.3 for a broadband spectrum). At VHE gamma rays the detection of pulsed emission from the Crab turned
out to be more difficult, and it was not until 2008 when MAGIC discovered it (Aliu et al.
2008). The detection of pulsed emission at such energies ruled out pulsed emission models as the polar-cap where the production of the pulsed radiation is too close to the pulsar to achieve the energies measured. The VHE spectrum was first extended by VERITAS and MAGIC up to 400 GeV (Aliu et al. 2011; Aleksi´c et al. 2012b) and very recently, up to TeV energies by MAGIC (Zanin 2014). To achieve such high energies on the pulsed emission constrains the emission region to be at a larger distance than the predicted by most of the models. The MAGIC collaboration has also reported VHE γ-ray emission from the so- called bridge region, or region between the two peaks of the light-curve (Aleksi´c et al. 2014b).
The Crab Nebula
The PWN confined by the thermal ejecta from the SN is known as the Crab Nebula. It will be described in more detail in Section §6.2.
The filaments and the shell
A third component of the SNR are the filaments that are visible in the outer ring of the nebula of Figure 6.2. They form a cage where the PWN is confined. The last component consists of freely expanding ejecta beyond the visible edge of the nebula. It has been a crucial part of all theoretical models for several years, but it has only recently been observed (Sollerman et al. 2000).