Motivation and goals

For many years, astronomers have struggled with the application of sophisticated statistical methodologies to analyze their rich datasets and address complex astrophysical problems. On one hand, at least in the United States, astronomers receive little or no formal training in statistics. The traditional method of education has been informal exposure to a few familiar methods during early research experiences. On the other hand, astronomers correctly perceive that a vast world of applied mathematical and statistical methodologies has emerged in recent decades. But systematic, broad training in modern statistical methods has not been available to most astronomers.

This volume seeks to address this problem at three levels. First, we present fundamental principles and results of broad fields of statistics applicable to astronomical research. The material is roughly at a level of advanced undergraduate courses in statistics. We also outline some recent advanced techniques that may be useful for astronomical research to give a flavor of the breadth of modern methodology. It is important to recognize that we give only incomplete introductions to the fields, and we guide the astronomer towards more complete and authoritative treatments.

Second, we present tutorials on the application of both simple and more advanced methods applied to contemporary astronomical research datasets using the R statistical software package. R has emerged in recent years as the most versatile public-domain statistical software environment for researchers in many fields.

Relativistic compact systems are potential sources of UHECRs and VHE neutrinos. This, of course, requires that a considerable fraction of the dissipation energy will be tapped for acceleration of baryons to ultra-high energies, which poses a great challenge to any model. Nonetheless, this possibility is strongly motivated by the detection of cosmic rays at energies up to ~1020 eV. While cosmic rays below the knee, at energies of <1015 eV or so, are commonly believed to be accelerated by supernovae blast waves, the origin of the cosmic-ray population above the knee and, in particular, the UHECRs, is yet a mystery. Scenarios in which UHECRs and VHE neutrinos are produced in relativistic outflows, particularly GRB or blazar jets, have important implications for the jet content, the dissipation mechanism and, possibly, the physics of inner engines. In contrast to electromagnetic emission that can have either leptonic or hadronic origin, VHE neutrino emission is a unique diagnostic of hadronic content. Hence, their detection will be an important step in our understanding of compact astrophysical systems. Furthermore, it may lead to a firm identification of the mysterious astronomical origin of UHECRs.

Ultra-high energy cosmic rays

Cosmic rays were first discovered by Austrian Victor Hess in his pioneering balloon experiments in 1912. For this discovery, he was awarded the Nobel Prize in Physics 1936.

In this chapter we shall develop fundamental concepts for describing relativistic astrophysical outflows. In order for a flow to be accelerated to high Lorentz factors the internal energy per baryon at the flow injection point must largely exceed unity. This internal energy may have a thermal origin as, e.g., in hydrodynamical fireball models, or a magnetic origin, as in pulsar winds and outflows from rotating black holes. A basic question in the former case is how to avoid excessive mass loading. A key issue in the latter case is the conversion of magnetic energy to kinetic energy. Observations seem to indicate that collimation is a generic feature of astrophysical outflows, suggesting that confinement by the ambient medium may play an important role in the dynamics of the system. In the following only steady flows will be considered, for which simple analytic solutions can be obtained.

Hydrodynamic fireballs

Let us consider first an unmagnetized spherical wind. In general, the wind may consist of a mixture of baryons, radiation and electron–positron plasma, which in sufficiently compact regions can be taken to be in local equilibrium. The flow is then characterized by the proper baryon density nb, pressure p, temperature T and velocity uμ = (γ, γv). We further assume that the flow is adiabatic and not subject to any external forces, including gravity (the effect of gravity will be considered in the following sections).

Astrophysical flows as discussed in the previous chapter are subject to steepening, especially when coming off a time-dependent or intermittent source. Steepening results in shocks, where energy in bulk motion is partially dissipated into heat. Strong shocks thereby produce radiation, as alluded to in the general scheme pointed out in Section 1.2, further accompanied by entropy creation. In this chapter, we elucidate the physical processes governing various types of shocks.

Nonlinear steepening of relativistic disturbances

The analysis of small-amplitude MHD waves outlined in Section 5.3 indicates that the speed at which a linear disturbance propagates, Eq. (5.43), depends on local conditions. It is naively expected that over sufficiently long times the wave will be distorted, as the phase speed itself changes over the course of the wave trajectory. Waves generated at some location will eventually steepen into shocks, at which point the fluid picture breaks down. Inside the shock transition the wave dissipates, converting bulk energy into heat. The transition occurs over kinetic scales, roughly the collision length in collisional shocks, the skin depth in collisionless shocks, and the Thomson mean free path in radiation mediated shocks.

Riemann invariants and characteristics

A convenient way to analyze wave steepening is to write the MHD equations in terms of the so-called Riemann invariants, developed in compressible fluid dynamics to understand the process of steepening and shock formation.

The ejection of supersonic outflows drives the formation of strong shocks that propagate into the surrounding medium. Examples are blast waves that form in stellar and galactic explosions. The supernovae accompanying the death of a star, the afterglow emission that follows a GRB explosion, and the radio lobes observed in radio galaxies and blazars are clear signatures of those blast waves. When the energy released by the source is large such that E > (Mj + ρiV)c2, where Mj is the mass of the ejecta, ρi the ambient density and V the volume swept by the shock, the blast wave motion is relativistic. The most notable example is the afterglowshell in GRBs.

The structure formed in spherical explosions at early times is shown schematically in Fig. 8.1. It consists of a forward shock that propagates in the ambient medium, a reverse shock crossing the ejecta, and a contact discontinuity separating the shocked ejecta and the shocked ambient medium. This structure is clearly seen in the X-ray image of the SNR DEM L71 in Fig. 8.2. The early phases in the evolution of a blast wave are considered in detail in Section 8.6.

At sufficiently late times a major fraction of the explosion energy is contained in the shell of shocked ambient medium and the effect of the ejecta on the evolution of the forward shock can be ignored. This stage is well described by the impulsive blast wave model discussed in Section 8.2.

General relativity gives a complete description of gravitation as it follows from conservation of energy–momentum, general gauge covariance and causality. While it has unprecedented predictive power towards cosmology, gravitational collapse and gravitational waves, only recently has gravitation become an experimental science beyond Newton's law of attraction [22] and beyond gravitational redshift [489], with the advance of LAGEOS, LAGEOS II [155] and Gravity Probe B [199]. These controlled experiments provide the first direct measurements of geodetic and frame-dragging precessions combined, from which we can gain confidence in astrophysical models involving strong gravitational fields and their radiation processes.

In this chapter, we summarize the most immediate aspects in a geometrical way to facilitate applications to astrophysics. We follow the general idea that the essential properties of general relativity derive from the Riemann tensor [485, 141]. We use the tensor notation of [631] with Latin indices and use geometrical units in which Newton's constant and the velocity of light are set equal to 1 unless otherwise specified.

Curved spacetime

General relativity describes the motion of particles in terms of world-lines xb(τ) in a curved spacetime with coordinates xb, with the eigentime τ commonly used as the parameter of the world-line family.

The prospect of an inner accretion disk or torus around a black hole producing gravitational waves may be anticipated from non-axisymmetries associated with QPOs, as currently observed in the electromagnetic spectrum in some of the X-ray binaries and, at low frequencies, in SgrA* (e.g., [581]). Non-axisymmetries are a natural outcome of various processes including instabilities, as discussed in Chapter 9. Their output in gravitational radiation may well be energetic on the basis of the energy losses inferred from the observed black hole spin down shown in Fig. 11.5. Long GRBs and perhaps some of the CC-SNe, therefore, offer a unique possibility for identifying Kerr black holes as objects in Nature by calorimetry on all their emission channels [604].

Introduction

Taking advantage of the nearly all-sky monitoring capability of gravitational-wave detectors, blind searches for long gravitational-wave bursts (GWBs) might be optimal in view of the beaming factor of long GRBs from fb < 10 (θ > 25 deg) up to a few hundred (θ ~ 4 deg, [214, 608, 258]). Blind searches are also expected to be competitive with current X-ray/optical surveys for detecting the shock break-out associated with an emerging CC-SNe, as they last only tens of minutes to at most a few hours. Blind searches naturally include long GRBs with no detectable supernovae, e.g., the long event GRB 060614 of duration 102 s discovered by Swift and the halo event GRB 070123 discovered by IPN (see Table 1.1 in Chapter 1).

Gravitation according to general relativity remains singularly challenging to understand from first principles in relation to the other forces in Nature. On the other hand, it describes a broad range of phenomena in cosmology and astrophysics, some of which are well constrained by observations of the galaxy redshifts, the cosmic microwave background [338], supermassive black holes in galactic nuclei, radio-pulsars in neutron star–neutron star binaries [297], and, possibly in the near future, binary mergers, that should ultimately lead to new insights on its origin.

Geometrically, general relativity is built on the Riemann tensor as discussed in Chapter 3, which describes spacetime in terms of two-surfaces for which the Planck scale – the smallest length scale in Nature – introduces a unit of area. The event horizons of black holes are null-surfaces that carry entropy [77] and temperature by virtue of their radiation properties [276]. For macroscopic black holes, the entropy represents a large, hidden phase space of an underlying structure of spacetime and matter. It is tempting to think of low energy manifestations in the real world also without black holes, such as Newton's law between two ordinary point particles [625].

In this decade, we anticipate a complete window for observing the Universe with advanced multimessenger survey instruments for electromagnetic radiation, cosmic rays, neutrinos and gravitational waves.

The evolution of the Universe is largely shaped by gravity, giving rise to large scale structure in filaments and voids down to galaxies and their constituents. The associated radiative phenomena indicate an “arrow of entropy” that points to scales generally less than 1 Mpc, where we find interactive and transformative processes such as galaxy mergers, active galaxies, supernovae and gamma-ray bursts (GRBs). On these scales, the Transient Universe serves as a cosmic beacon in the era of reionization to the present. Thus, entropy appears to be increasing, from an initially low value at the birth of the Universe as conjectured by Penrose, with conceivably jumps in some of the brightest and most extreme transient events.

Multimessenger astronomy aims at the measurement of physical and astronomical parameters across various observational windows, in and beyond the electromagnetic spectrum. It promises a probe of gravity with the potential to discover the relationship between large structure formation by dark matter, galaxy formation, star formation and their end products, to unravel the astronomical origin and physical mechanism giving rise to active galactic nuclei, core-collapse supernovae (CC-SNe) and GRBs.

Ultra-high energy cosmic rays (UHECRs) and cosmological GRBs stand out as the most relativistic transient events that may be telling us about gravitation in the strongly nonlinear regime in the spacetime around black holes.