Introduction

Charge-coupled devices, or CCDs, were invented at Bell Laboratories, New Jersey, in 1969. The advantages of CCDs for optical astronomy over the previous technologies were quickly realized and the use of CCDs revolutionized astronomy in the 1980s due to their sensitivity and linear brightness response. CCD cameras are now the most common detector at optical observatories around the world and are the sensing element in nearly all commercial digital cameras.

It was also recognized early on that CCDs were sensitive to X-ray radiation as well as optical light, although optimizing the technology for X-ray use took longer. The first suborbital rocket flight equipped with an X-ray CCD camera was launched in 1987 to observe SN 1987A. Japan's ASCA (Tanaka et al., 1994), launched in 1993, was the first satellite with an X-ray CCD camera. Since that time, CCDs have become ubiquitous in X-ray astronomy and are part of the focal-plane instrumentation in almost all recent past, current, and planned missions. The CCD detectors on the largest of the currently operating missions are ACIS (Garmire et al., 2003) on Chandra, EPIC (Turner et al., 2001; Strüder et al., 2001) and RGS (den Herder et al., 2001) on XMM–Newton and XIS (Koyama et al., 2007) on Suzaku. Other CCD detectors on currently operating missions are XRT (Burrows et al., 2000) on Swift and SSC on MAXI (Matsuoka et al., 2009).

Understanding the basic principles of CCD operation and some of the resultant features encountered in data analysis is important for accurate interpretation.

The X-ray waveband contains atomic and ionic transitions for nearly all astrophysically abundant elements – with the notable exception of H and He. These arise primarily from transitions involving electrons in the 1s shell but for heavier elements (i.e. Fe, Ni), there are transitions involving higher shells as well. This appendix contains a short discussion of spectroscopic notation combined with information on a selection of particularly strong transitions, including those from hydrogen-like and helium-like ions, Fe XVII–Fe XXIV, as well as fluorescent transitions from neutral atoms and ionization edges for all of the abundant elements. More information about atomic data useful for X-ray astronomy can be found at http://www.atomdb.org.

Spectroscopic notation

A complete discussion of spectroscopic notation is beyond the scope of this handbook; we suggest the short but highly informative text by Herzberg (1945) for a more detailed review; another useful source is the X-ray Data Booklet published by the Center for X-ray Optics and Advanced Light Source at Lawrence Berkeley National Laboratory (http://xdb.lbl.gov). It should be noted that X-ray astronomy is rife with poorly used spectroscopic terminology, so following the form used by an earlier refereed paper does not guarantee proper usage.

Introduction

Why do X-ray astronomers need statistics? Wall and Jenkins (2003) give a good description of scientific analysis and answer this question. Statistics are used to make decisions in science, evaluate observations, models, formulate questions and proceed forward with investigations. Statistics are needed at every step of scientific analysis. A statistic is a quantity that summarizes the data (mean, averages etc.) and astronomers cannot avoid statistics.

Here is a question asked by an X-ray astronomy school student:

What does “a good fit” or “a decent fit” mean, and what constitute “low counts” data? These expressions carry a definite meaning, but taken out of context are not precise enough. Is the question whether the total number of counts in the spectrum is “low” or whether the number of counts per resolution element is “low”? For example, a total number of counts such as 2000 could be viewed as “high,” but it might be considered “low” if the total number of counts is divided by the number of resolution elements (for example there are 1024 independent detector channels in a Chandra ACIS spectrum).

Introduction

It may be obvious why visible astronomy utilizes images, but it is illustrative to consider the value of focusing to X-ray astronomy. A list of advantages offered by the best possible two-dimensional angular resolution would include:

1. (i) Resolving sources with small angular separation and distinguishing different regions of the same source.

2. (ii) Using the image morphology to apply intuition in choosing specific models for quantitative fits to the data.

3. (iii) Using as a “collector” to gather photons. This is necessary because X-ray-source fluxes are so low that individual X-ray photons are detected; the weakest sources give less than one photon per day.

4. (iv) Using as a “concentrator,” so that the photons from individual sources interact in such a small region of the detector that residual non-X-ray background counts are negligible.

5. (v) Measuring sources of interest and simultaneously determining the contaminating background using other regions of the detector.

6. (vi) Using with dispersive spectrometers such as transmission or reflection gratings to provide high spectral resolution.

The Earth's atmosphere completely absorbs cosmic X-rays. Consequently, X-ray observatories must be launched into space; so size, weight, and cost are always important constraints on the design. In practice this leads to a trade-off between the best possible angular resolution and the largest possible collecting area.

Differential equations of second order occur frequently in applied mathematics, particularly in applications coming from physics and engineering. The main reason for this is that most fundamental equations of physics, like Newton's second law of motion (2. 7), are second order differential equations. It is not clear why Nature, at a fundamental level, should obey second order differential equations, but that is what centuries of research have taught us.

Since second order ODEs and even systems of second order ODEs are special cases of systems of first order ODEs, one might think that the study of second order ODEs is a simple application of the theory studied in Chapter 2. This is true as far as general existence and uniqueness questions are concerned, but there are a number of elementary techniques especially designed for solving second order ODEs which are more efficient than the general machinery developed for systems. In this chapter we briefly review these techniques and then look in detail at the application of second order ODEs in the study of oscillations. If there is one topic in physics that every mathematician should know about then this is it. Much understanding of a surprisingly wide range of physical phenomena can be gained from studying the equation governing a mass attached to a spring and acted on by an external force. Conversely, one can develop valuable intuition about second order differential equations from a thorough understanding of the physics of the oscillating spring.

Archives

Unlike other branches of astronomy that have proprietary telescopes, X-ray astronomy is of necessity done using satellites and suborbital rockets. These are funded by national governments, which typically insist that all data be made public within a reasonable time. As a result, practically all X-ray-astronomy observations are available on-line; the challenge is to find the right data. Fortunately, X-ray-astronomy data are concentrated in a small number of archives and it is usually clear which website to try.

The best way to think of an archive is as a collection of tables, some of which have data sets attached. Some of the tables are simple catalogs, e.g. a list of stars with positions, spectral types, and fluxes. Other tables come with considerable data attached, e.g. the observation catalog for some mission that lists pointing position, exposure time and so forth but also provides links to all the publically available data for the observation. These data may include basic event files, cleaned event files, auxiliary information such as housekeeping and orbit files, and product files such as spectra, images, and lightcurves. The archive Internet interface will generally allow the astronomer to choose which categories of data to download for the selected observation.

As an example, consider finding Chandra observations of the Perseus cluster. Its name can be entered in the Chandra data archive search page and the NED or SIMBAD servers used to translate the name into a position.

This section is included as an aid to a beginner in X-ray astronomy who wishes to start the learning process by using a “good” source – one where adequate data can be guaranteed and no unusual circumstances make analysis difficult. For example, although Sco X-1 was the first X-ray source beyond the Solar System ever detected, it is so bright that it can be observed with modern detectors only in extremely unusual modes, making it a poor choice for today's beginner. The sources listed here have been regularly observed by numerous satellites in normal modes of operation and should provide good “test” cases for beginners. That said, there is nothing stopping observers from requesting observations even of common sources in unusual modes, so care should be exercised when selecting an observation.

Point sources

Although they may have some intrinsic extent, the sources in Table A3.1 are all point sources as far as past and current X-ray telescopes are concerned. Observations of these sources may or may not involve gratings; this must be determined on an observation-by-observation basis.

Diffuse sources

All of the sources in Table A3.2 are diffuse sources of varying extent. Some (such as the Cygnus Loop) will fill the FOV of any X-ray detector, while others (e.g. Cas A) will generally fit within the FOV of most instruments. Sources within the Solar System, such as Jupiter, move too rapidly for X-ray satellites to track them.

Introduction

This chapter describes X-ray data reduction for extended sources; the process required to isolate an image of the source from the background or to isolate the source spectrum from all of its backgrounds. We begin by describing the ways in which data are less than ideal and what general methods are employed to work around these problems. We then walk through a typical reduction, posing and discussing the questions that need to be considered with each step.

Study of diffuse sources is usually done with imaging spectrometers such as position-sensitive proportional counters and CCDs. Combining spectral and imaging (photometric) data increases the versatility of the analysis, but that versatility comes at the price of more complex procedures, which require simultaneous analysis of both the spectra and the photometry.

A brief consideration of photometry demonstrates why this is the case for non-ideal detectors. First, a region is defined that includes the source and a second “background” region that is free of sources. For an ideal detector, the surface brightness in the background region (in counts/pixel) can simply be determined and subtracted from the source region. Several modifications are required for non-ideal detectors where the response and the point spread function (PSF) may vary with position.

This chapter contains five projects which develop the ideas and techniques introduced in this book. The first four projects are all based on recent research publications. Their purpose is to illustrate the variety of ways in which ODEs arise in contemporary research - ranging from engineering to differential geometry - and to provide an authentic opportunity for the reader to apply the techniques of the previous chapters. If possible, the projects could be tackled by a small group of students working as a team. The fifth project has a different flavour. Its purpose is to guide the reader through the proof of the Picard-Lindelöf Theorem. At the beginning of each project, I indicate the parts of the book which contain relevant background material.

Ants on polygons

(Background: Chapters 1 and 2, Exercise 2. 6)

Do you remember the problem of four ants chasing each other at constant speed, studied in Exercise 2. 6? We now look at two variations of this problem. In the first, we consider n ants, where n = 2, 3, 4, 5 …, starting off on a regular n-gon. Here, a 2-gon is simply a line, a regular 3-gon an equilateral triangle, a 4-gon a square and so on. In the second, more diffcult, variation we consider 4 ants starting their pursuit on a rectangle with side lengths in the ratio 1:2. This innocent-sounding generalisation turns out to be remarkably subtle and rich, and is the subject of recent research reported in Chapman, Lottes and Trefethen [4].

Introduction

This chapter describes some of the data-analysis methods used by X-ray astrophysicists. Any data analysis must begin with careful consideration of the physics underlying the emission before starting to progress through a series of software tools and scripts. After confirming that existing observations could (at least potentially) answer the question at hand, the first step is to determine what observations of the desired source(s) exist. Recent observations are often the best starting point, but even old data are better than nothing. Once some usable data are available the analysis, either spectral, imaging, timing, or some combination of the three, can proceed.

Low-resolution spectral analysis

General comments

Most recent and current X-ray observations are performed using detectors which provide imaging combined with relatively low spectral resolution. Early missions such as the Einstein Observatory or ROSAT used X-ray mirrors with good imaging capability combined with microchannel plates or position-sensitive proportional counters that had limited spectral sensitivity, typically R ≡ E/ΔE ∼ 1-10. More recent missions, starting with ASCA, and current missions, such as Chandra and XMM–Newton, use X-ray-sensitive CCDs. These tend to have somewhat higher backgrounds, small pixels, and substantially better spectral resolution, R ≡ E/ΔE ∼ 10-50, than proportional counters. For comparison, the standard “UBVRI” optical filter system is equivalent to R ∼ 4.