Exploring Energy Transfer and Evolutionof Supernova Remnants
through Year-Scale X-ray Variability

Masamune MATSUDA
Kyoto University

A thesis submitted for the degree ofDoctor of ScienceFeb. 6th, 2024

Acknowledgements
First and foremost, I extend my heartfelt gratitude to my supervisors, Prof. Takeshi
Go Tsuru and Assoc. Prof. Hiroyuki Uchida, for their invaluable guidance and encouragement during my five years at Kyoto University. I am also grateful to Assoc. Prof.
Takaaki Tanaka for his excellent advice, both during his work at Kyoto University andafter his promotion to Konan University. Working with them over the past years has
been an honor and a source of pride.
I would like to express my appreciation to my collaborators, Yuki Amano, Hiroya
Yamaguchi, Shiu-Hang Lee, and Adrien Picquenot, for their insightful comments andcontributions. Special thanks are due to Brian J. Williams for his kindness and valuable
support during my stay at GSFC/NASA, as well as for providing invaluable advice related
to my research. My time there was made meaningful, thanks to him.
I extend my thanks to all members of the Cosmic-Ray group at Kyoto University forengaging in pleasant communications and fruitful discussions.
最後に、私をこれまで育ててくれた両親及び家族に、この場をお借りして感謝を伝えます。本当にありがとうございました。

2023年12月

松田真宗Abstract
Shock waves in the Universe are generally formed as “collisionless” shocks, whose physical scale is much shorter than the mean free path of Coulomb collisions. The collisionlessshock can be observed in ubiquitous astrophysical environments, involving a number of
unsettled problems, such as cosmic-ray acceleration, electron heating, and the evolutionof magnetic field turbulence. Year-scale X-ray variabilities of supernova remnants(SNRs) can provide information on real-time changes in shock and post-shock physics,
although there are few examples of it. This thesis focuses on time variability in SNRs to
explore the shock acceleration, heating, and expansion of SNRs with the Chandra.
Tycho’s SNR is a nearby and young SNR, which we can observe fine structure fromthe high-resolution imaging with Chandra. Therefore, we can expect the detection oftime variabilities even on a small physical scale. A previous study indeed discovered
X-ray flux changes of synchrotron radiation from part of the western “stripe” structure
in Tycho’s SNR, where protons might be accelerated to PeV energy. Expanding the region
and energy band for searching time variabilities from Chandra images obtained in 2000,
2003, 2007, 2009, and 2015, we discovered two significant flux changes: fluctuation innonthermal radiation along each stripe in the entire structure and gradual brightening
in thermal radiation in the northeast rim. Spectral analysis of the former reveals theyear-scale fluctuations of photon indices and brightness in each region, along with atight anti-correlation between them. The fluctuation timescale suggests the magnetic
field amplification up to ∼ 500 𝜇G. Additionally, the stripes have harder spectra thanremnant rims, indicating a more effective acceleration than the rim. In the lattercase of thermal radiation, we discovered an increase in electron temperature from∼ 0.3 keV to ∼ 0.7 keV over 15 years from spectral analysis. The temperature increase
can be attributed to ambient dense gas heated by shock and subsequent thermal energy
acquisitions from heavier ions via Coulomb collisions. Comparison with the calculationof electron temperature evolution can confine the electron-to-proton temperature
ratio immediately behind the shock (𝛽 0 ≡ 𝑇𝑒 /𝑇𝑝 ), whose value is consistent with thatmeasured with previous H𝛼 observations. We propose a new method to measure 𝛽 0
independently of H𝛼 measurements.
To investigate the ambient dense gas of Tycho’s SNR implied above, we measured
the velocity of remnant expansion with newly observed data from 2022 to 2023 with
Chandra, as well as the data from 2003 to 2015 analyzed by a previous study. Our
result exhibits further deceleration in the western shell from 2015 to 2021 beyond thevelocity from 2003 to 2015 measured by the previous study, supporting the existenceof the wall formed by the wind from the progenitor system. Our spectral analysis ofsynchrotron X-rays from the shock front revealed softening trends in some regions. We
would understand the ambient environment and the effect of shock–wall interaction
on the radiation from the shock, combining the theoretical model of shock dynamics
and particle acceleration in future work.
We propose the application of measuring the thermal X-ray variabilities to otherregions of Tycho’s SNR or other SNRs, aiming to investigate the dependence of 𝛽 0 on
parameters like shock velocity and magnetic field. Kepler’s SNR, which is similar inage, distance, and explosion type to Tycho’s SNR, shows an indication of thermal Xray brightening along the distribution of circumstellar medium. This would enable
us to investigate the dependence on factors such as ambient magnetic field. We also
introduce the possible existence of thermal X-rays from heated ambient gas extending
across a northeastern wide area of Tycho’s SNR. The high-resolution spectroscopy withthe recently launched observatory, XRISM, allows us to resolve it and reveal the time
variabilities of ion temperatures behind the shock.

Contents
List of Figuresix

List of Tables

xiii

1

1
1
2
2
5
5
8
10
10
10
13

Shock Waves in Astrophysical Environment
1.1 Overview of Collisionless Shock . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Shock Heating in Astrophysical Plasma . . . . . . . . . . . . . . . . . . .
1.2.1
Rankine-Hugoniot Jump Condition . . . . . . . . . . . . . . . . .
1.2.2 Ionization Equilibrium . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Thermal Equilibrium in Plasma . . . . . . . . . . . . . . . . . . .
1.2.4 Collisionless Electron Heating . . . . . . . . . . . . . . . . . . . .
1.3 Particle Acceleration in SNR . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1
Cosmic Ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Diffusive Shock Acceleration . . . . . . . . . . . . . . . . . . . . .
1.3.3 Acceleration Timescale and Maximum Energy . . . . . . . . . . .

2 Supernova and Supernova Remnant
15
2.1 Supernova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1
Supernova Classification . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Core-Collapse SN . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Thermonuclear SN . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Supernova Remnant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Evolution of SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Thermal Radiation from Shock-Heated Particles . . . . . . . . . . 22
2.2.3 Non-thermal Radiation from Shock-Accelerated Electrons . . . . 25
3 Time Variabilities of Supernova Remnant
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Changes in X-ray Spectrum and Flux . . . . . . . . . . . . . . . . . . . . .
3.3 Expansion of Blast waves and Ejecta . . . . . . . . . . . . . . . . . . . . .

29
29
29
34

4 Instruments
4.1 Overview of Chandra X-ray Observatory . . . . . . . . . . . . . . . . . . .
4.2 High Resolution Mirror Assembly (HRMA) . . . . . . . . . . . . . . . . .
4.2.1 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Effective Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Point Spread Function . . . . . . . . . . . . . . . . . . . . . . . .

39
39
40
40
41
42

vii

viii

Contents

4.3

Advanced CCD Imaging Spectrometer (ACIS) . . . . . . . . . . . . . . . .
4.3.1 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Quantum Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Angular Resolution and Field of View . . . . . . . . . . . . . . . .
4.3.4 Energy Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44
44
45
46
47
48

5 Particle Heating and Accleration Observed by X-ray Time Variability
5.1 Our Objective and Target Selection . . . . . . . . . . . . . . . . . . . . . .
5.2 Observations and Data Reductions . . . . . . . . . . . . . . . . . . . . . .
5.3 Time Variabilities of Non-Thermal X-ray Radiation . . . . . . . . . . . . .
5.3.1 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Time Variabilities of Thermal X-ray Radiation . . . . . . . . . . . . . . . .
5.4.1 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51
51
53
54
54
59
65
65
71

6 Future Works
79
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Probing the Shock–Cloud Interaction from Variability of Expansion Velosity 80
6.2.1 Our Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2.2 Observations and Data Reductions . . . . . . . . . . . . . . . . . 80
6.2.3 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Time Variabilities in Kepler’s SNR . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 Measuring Ion Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 Conclusions

97

Appendices
A Time Evolution of Electron Temperature with Low Shock Velocities

101

References

103

List of Figures
1.1
1.2
1.3

1.6
1.7
1.8

Schematic of the SNR shock waves and the particle behavior . . . . . . .
Ion fraction of Ne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The relation between the ion/proton temperature ratio and the atomic
number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An example calculation of the temperature evolution at the shock . . . .
Temperature ratio 𝛽 as a function of the shock velocity 𝑣𝑠 for Balmerdominated shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The reported detection of low-ionized Fe K𝛽 line . . . . . . . . . . . . . .
Cosmic ray spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of diffusive shock acceleration . . . . . . . . . . . . . . . . . .

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9

Classifications of SNe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Stellar onion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Schematic illustrating the progenitor of Ia SNe . . . . . . . . . . . . . . . 18
Schematic of the cross-section of an SNR and the position of the shocks 20
Thermal radiation model from a plasma composed only of silicon . . . . 21
Examples of ionizing plasma model . . . . . . . . . . . . . . . . . . . . . 23
Centroid energies of the Fe K𝛼 and Fe K𝛽 lines . . . . . . . . . . . . . . . 24
Intensity ratio of Si He𝛽 /Si He𝛼 lines . . . . . . . . . . . . . . . . . . . . . 25
The shape of the spectrum of synchrotron radiation from a single electron 26

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

Flux change of synchrotron X-ray in RX J1713.7−3946 . . . . . . . . . . . .
The spectra of SN 2023ixf in M101 at 4 (Epoch I; black) and 11 (Epoch II;red) days from its explosion . . . . . . . . . . . . . . . . . . . . . . . . . .
X-ray light curves of SN 1987A and intensity ratio in the 0.5–2.0 keV . . .
Time variabilities in Cassiopeia A . . . . . . . . . . . . . . . . . . . . . . .
The expansion of SNR N103B from 1999 to 2017 . . . . . . . . . . . . . . .
Chandra image of G1.9+0.3 and proper motion vectors . . . . . . . . . .
The shock proper motion in RX J0852.0−4622 (Vela Jr.) . . . . . . . . . . .
The decelerations of the blast waves in Tycho’s SNR . . . . . . . . . . . .

31
32
33
34
35
36
37

4.1
4.2
4.3
4.4
4.5
4.6

The Overview of Chandra X-ray Observatory . . . . . . . . . . . . . . . .
Four HRMA mirror pairs and associated structures . . . . . . . . . . . . .
The effective area of HRMA, HRMA/ACIS, and HRMA/HRC . . . . . . . .
The HRMA effective area as a function of the off-axis angle . . . . . . . .
The encircled energy of HRMA versus angular radius . . . . . . . . . . .
The Layout of ACIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40
40
41
42
43
44

1.4
1.5

ix

2
4
6
7
8
9
11
12

30

x

List of Figures

4.7
4.8
4.9
4.10
4.11
4.12
4.13

Schematic view of the front- and back-illuminated type of the CCD sensors 44
The quantum efficiency of ACIS . . . . . . . . . . . . . . . . . . . . . . . . 45
Effects of the molecular contamination of the ACIS OBF . . . . . . . . . 46
The dither pattern of ObsID:8551 . . . . . . . . . . . . . . . . . . . . . . . 47
The energy resolution of ACIS versus X-ray energy . . . . . . . . . . . . . 48
ACIS FWHM as a function of the row number . . . . . . . . . . . . . . . . 49
Non-X-ray background spectrum of ACIS . . . . . . . . . . . . . . . . . . 50

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8

52
53
55
55
56
56
58

RGB image of Tycho’s SNR . . . . . . . . . . . . . . . . . . . . . . . . . . .
The “stripe” structure in Tycho’s SNR . . . . . . . . . . . . . . . . . . . . .
Difference image of the stripe region in 4.1–6.1 keV band. . . . . . . . . .
Chandra ACIS images of the stripe region in Tycho’s SNR . . . . . . . . .
The profile extracted from the stripe structure . . . . . . . . . . . . . . .
The smoothed profile extracted from the stripe structure . . . . . . . . .
Spectra extracted from the stripe regions . . . . . . . . . . . . . . . . . .
The surface brightness of the stripes in each year as a function of thephoton index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Comparison between the stripe and rim data . . . . . . . . . . . . . . . .
5.10 A difference image of Tycho’s SNR between 2003 and 2015 within theenergy band of 0.7–1.5 keV . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 Exposure-corrected X-ray images of Knot1 in three energy bands . . . .
5.12 Comparison of the spectra between the Knot1 and Ref1 region . . . . . .
5.13 Spectra extracted from the Ref1 regions . . . . . . . . . . . . . . . . . . .
5.14 Spectra extracted from the Ref2 region . . . . . . . . . . . . . . . . . . . .
5.15 Spectra extracted from the Knot1 regions . . . . . . . . . . . . . . . . . .
5.16 The best-fit values of 𝑘𝑇𝑒 , 𝑛𝑒 𝑡 , and EM of the soft NEI component as afunction of year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.17 Spacial correlation of the Knot1 region between X-rays and H𝛼 lines . . .
5.18 H-like ion fraction of Ne and Mg . . . . . . . . . . . . . . . . . . . . . . .
5.19 Schematic of the Knot1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.20 The comparison electron temperature as a function of the time between
the observation and the calculation . . . . . . . . . . . . . . . . . . . . .
5.21 Same as Figure 5.20, but varying the start time of shock heating . . . . .
6.1
6.2
6.3
6.4
6.5

The 4.1–6.1 keV flux image of Tycho’s SNR in 2009 and the regions to
extract profiles and spectra . . . . . . . . . . . . . . . . . . . . . . . . . .
Profiles extracted from each region . . . . . . . . . . . . . . . . . . . . . .
The chi-squared in measuring the rim proper motion . . . . . . . . . . .
The shock velocities and accelerations in each region as a function of
the azimuthal angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fitting result of the spectra extracted from shock regions . . . . . . . . .

59
61
65
66
67
68
69
70
71
72
73
74
75
75

82
83
84
85
86

List of Figures

The best-fit parameters of the power-law component as a function of
the observed year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 The images of flux change band of Kepler’s SNR in the energy band of
0.5–0.7 keV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Example of the spectra of time variable region in Kepler’s SNR . . . . . .
6.9 The overview of XRISM and its pre-launch requirements . . . . . . . . .
6.10 An example of the GMCA result applied to SNR N103B Yamaguchi et al.
(2021) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.11 The ratio image made by separating counts of the H𝛼 counterpart component and the total counts . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12 Simulated XRISM spectrum of Tycho’s SNR . . . . . . . . . . . . . . . . .

xi

6.6

87
91
92
93
94
95
96

Compatrison between electron temperature change in Knot1 and calculations under assumptions of low shock velocities . . . . . . . . . . . . . 101
A.2 Same as Figure A.1, but different assumptions of the start time of Coulomb
heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.1

xii

List of Tables
4.1

The properties of major X-ray observatories . . . . . . . . . . . . . . . . .

5.1
5.2
5.3

Observation Log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Best-fit parameters of the stripe regions in 2009 . . . . . . . . . . . . . . . 64
Best-fit Parameters of Ref1 and Knot1 Regions . . . . . . . . . . . . . . . . 77

6.1
6.2
6.3
6.3

Observation Log for Proper-Motion Analysis . . . . . . . . . . . . . . . .
The observed proper motions of shock waves . . . . . . . . . . . . . . . .
Best-Fit Parameters of Spectral Analysis . . . . . . . . . . . . . . . . . . .
Best-Fit Parameters of Spectral Analysis (Contunued) . . . . . . . . . . .

xiii

41

80
84
88
89

xiv

1
Shock Waves in Astrophysical Environment
Contents
1.1
1.2

1.3

1.1

Overview of Collisionless Shock . . . . . . . . . . . . . . . . . . . . 1
Shock Heating in Astrophysical Plasma . . . . . . . . . . . . . . . . 2
1.2.1
Rankine-Hugoniot Jump Condition . . . . . . . . . . . . . . .
2
1.2.2 Ionization Equilibrium . . . . . . . . . . . . . . . . . . . . . .
5
1.2.3 Thermal Equilibrium in Plasma . . . . . . . . . . . . . . . . .
5
1.2.4 Collisionless Electron Heating . . . . . . . . . . . . . . . . . .
8Particle Acceleration in SNR . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1
Cosmic Ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Diffusive Shock Acceleration . . . . . . . . . . . . . . . . . . . 10
1.3.3 Acceleration Timescale and Maximum Energy . . . . . . . . . 13

Overview of Collisionless Shock

Shock waves in the Universe are observed in ubiquitous astrophysical environments,
including the Earth’s bow shock (e.g., Schwartz et al., 1988), supernova remnants (SNRs)(e.g., Laming et al., 1996; Ghavamian et al., 2001), gamma-ray bursts (e.g., Mészáros& Rees, 1997), and merging galaxy clusters (e.g., Markevitch et al., 2005; Russell et al.,2012). In contrast to terrestrial shock waves, astrophysical shock waves have much
shorter thickness than the collisional mean free path, resulting in infrequent Coulomb
collisions. These types of shock waves are generally referred to as “collisionless shock.”Let us consider, for example, the bow shock front of Earth’s magnetosphere in the solar
wind. Its thickness is ∼ 100 km while the mean free path in this region extends toapproximately the distance between the Sun and Earth (∼ 1.5 × 109 km). The physics ofthe collisionless shock closely involves a number of intriguing topics, such as cosmicray acceleration and electron heating. This chapter presents the basic backgroundof collisionless shocks in SNRs.
1

2

1.2. Shock Heating in Astrophysical Plasma(a) Observer rest frame(b) Shock rest frame

ISM

Downstream

Upstream

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vd⇢u , wu , Pu

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vd )

vsh

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Figure 1.1: Schematic of the SNR shock waves and the particle behavior around the shock inobserver rest frame (a) and shock rest frame (b).

1.2 Shock Heating in Astrophysical Plasma
1.2.1

Rankine-Hugoniot Jump Condition

The shock waves compress the downstream gas. It follows that the shock waves heat
the particles downstream. Let us consider a scenario where a shock wave in SNR
propagates through the interstellar medium (ISM) at a velocity 𝑣sh , while the particles
behind the shock front possess a velocity of 𝑉 (Figure 1.1). In the rest frame of theshock front, the particles are injected with 𝑣𝑢 (= 𝑣sh ) and exit with 𝑣𝑑 (= 𝑣sh − 𝑉 ). The
jump conditions can be expressed as follows, considering the conservation laws ofmass, momentum, and energy:
𝜌 𝑢 𝑣𝑢2

𝑃𝑢 + 𝜌𝑢 𝑣𝑢= 𝜌 𝑑 𝑣𝑑 ,

(1.1)
2

= 𝑃𝑑 + 𝜌 𝑑 𝑣𝑑 ,

(1.2)

1 2
1 2
𝑣𝑢 + 𝑤𝑢 =𝑣 + 𝑤𝑑 ,
2
2 𝑑

(1.3)

where 𝜌 , 𝑃 , and 𝑤 represent mass densities, pressures, and enthalpy densities for the upstream (subscript 𝑢 ) and downstream (subscript 𝑑 ) regions, respectively. These formulas
are called “Rankine-Hugoniot jump conditions” (Rankine, 1870; Hugoniot, 1887, 1889).
When we assume the behavior of an ideal gas, the enthalpy can be written as follows:
𝑤 = 𝐶 P𝑇 =𝛾𝑃.
(𝛾 − 1)𝜌

(1.4)

Here, 𝛾 (≡ 𝐶 P /𝐶 V ) represents the adiabatic index (≡ 𝐶 P /𝐶 V ), where 𝐶 P and 𝐶 V denote theheat capacities at constant pressure and volume, respectively. The ratios of densities,
pressures, and temperatures between upstream and downstream can be derived from

1. Shock Waves in Astrophysical Environment

3

Equations (1.1), (1.2), (1.3), and (1.4):
𝜌 𝑢 𝑣𝑑=𝜌 𝑑 𝑣𝑢𝑃𝑢
𝑃𝑑
𝑇𝑢
𝑇𝑑(𝛾 + 1)𝑃𝑢 + (𝛾 − 1)𝑃𝑑,
(𝛾 − 1)𝑃𝑢 + (𝛾 + 1)𝑃𝑑
(𝛾 + 1)𝜌𝑢 − (𝛾 − 1)𝜌 𝑑,(𝛾 + 1)𝜌 𝑑 − (𝛾 − 1)𝜌𝑢
𝑃𝑑 𝜌𝑢 𝑃𝑑 [(𝛾 + 1)𝑃𝑢 + (𝛾 − 1)𝑃𝑑 ]=.
𝑃𝑢 𝜌 𝑑 𝑃𝑢 [(𝛾 − 1)𝑃𝑢 + (𝛾 + 1)𝑃𝑑 ]

=
=
=

(1.5)
(1.6)
(1.7)

The velocities of the gas can be expressed as follows:[(𝛾 − 1)𝑃𝑢 + (𝛾 + 1)𝑃𝑑 ] 2,2𝜌 𝑑 [(𝛾 + 1)𝑃𝑢 + (𝛾 − 1)𝑃𝑑 ](𝛾 + 1)𝑃𝑢 + (𝛾 − 1)𝑃𝑑
.
2𝜌 𝑑

𝑣𝑢2 =
𝑣𝑑2 =

(1.8)
(1.9)

In the case of a strong shock, in which the upstream pressure is negligibly small compared to its downstream pressure (𝑃𝑑 /𝑃𝑢 ≫ 1), Equations (1.5), (1.7), (1.8), and (1.9)
can be simplified to:
𝛾 −1
,
𝛾 +1
(𝛾 − 1)𝑃𝑑
𝑃𝑑 𝜌𝑢
=
=
,
𝑃𝑢 𝜌 𝑑(𝛾 + 1)𝑃𝑢(𝛾 + 1) 2 2(𝛾 + 1) 2𝑃𝑑=𝑐 ,=2 (𝛾 − 1)𝜌 𝑑 2𝛾 (𝛾 − 1) s
(𝛾 − 1)𝑃𝑑 𝛾 − 1 2
=
=
𝑐 ,
2𝜌 𝑑
2𝛾 s

𝜌 𝑢 𝑣𝑑=𝜌 𝑑 𝑣𝑢
𝑇𝑢
𝑇𝑑

=

𝑣𝑢2
𝑣𝑑2

(1.10)(1.11)(1.12)(1.13)√︁

where 𝑐 s = 𝛾 𝑃𝑑 /𝜌 𝑑 is the downstream sound velocity. Especially for monoatomic gas(𝛾 = 5/3), the ratios of densities and velocities are
𝜌𝑢
𝜌𝑑
𝑣𝑢
𝑣𝑑

=

1
,
4

(1.14)= 4.

Moreover, the downstream average temperature can be expressed from Equations (1.12)
using the shock velocity 𝑣sh (= 𝑣𝑢 ):𝑘 B𝑇𝑑= 𝜇𝑚 H=𝑃𝑑 2 (𝛾 − 1)2=
𝜇𝑚 H𝑣 sh𝜌𝑑(𝛾 + 1) 2

3
2𝜇𝑚 H𝑣 sh.
16

(1.15)(1.16)

where 𝑘 B , 𝜇, and 𝑚 H represent the Boltzmann constant, the averaged molecular mass,and the hydrogen mass, respectively. Using the Equation (1.16), we can estimate downstream temperature from the shock velocity. This can be interpreted as the conservationfrom the kinetic energy of the shock to the internal (thermal) energy of the plasma.

4

1.2. Shock Heating in Astrophysical PlasmaBareHe-likeH-likeBare

He-likeH-likeO-likeBare

N-likeF-likeH-likeHe-likeC-likeFigure 1.2: (Top panel): Ion fraction of Ne in CIE plasma as a function of the electron temperature(𝑘𝑇𝑒 ). (Middle panel): Ion fraction of Ne in ionization plasma of 𝑘𝑇𝑒 = 3 keV as a function of 𝑛𝑒 𝑡 .
(Bottom panel): Ion fraction of Ne in recombining plasma of 𝑘𝑇𝑒 = 0.01 keV as a function of 𝑛𝑒 𝑡when the initial state is the fully ionized state. The calculation of each panel is conducted using
PyAtomDB1

1. Shock Waves in Astrophysical Environment

1.2.2

5

Ionization Equilibrium

After the electron temperature of the plasma changes rapidly for some reason like theshock heating, the ionization state is driven out of equilibrium. The time evolution of
the ion density in the collisional ionization plasma can be written as:→−

1 𝑑𝑁 𝑍
𝑛𝑒 𝑑𝑡→−= 𝔸(𝑍 ,𝑇𝑒 ) 𝑁 𝑍 ,(1.17)

where
→
−
𝑁𝑍

= (𝑛 𝑍 ,0 , 𝑛 𝑍 ,1 , · · · , 𝑛 𝑍 ,𝑍 )𝔸 =(1.18)

𝛼 𝑍 ,10
0···
0©−𝑆 𝑍 ,0
ª
­𝑆
®
−(𝑆
+
𝛼
)
𝛼
0
·
·
·
0
𝑍 ,1
𝑍 ,1
𝑍 ,2
­ 𝑍 ,0
®
­
®
..
..
­ 0
®
.
𝑆
−(𝑆
+
𝛼
)
·
·
·
.
­
®
𝑍 ,1
𝑍 ,2
𝑍 ,2
­ .
® . (1.19)
..
..
..
..
..
­ ..
®
.
.
.
.
.
­
®
®
­
­ 0
®
···
···
𝑆 𝑍 ,𝑍 −2 −(𝑆 𝑍 ,𝑍 −1 + 𝛼𝑍 ,𝑍 −1 )𝛼 𝑍 ,𝑍
­
®
0
·
·
·
·
·
·
0
𝑆
−(𝑆
+
𝛼
)
𝑍
,𝑍
−
1
𝑍
,𝑍
𝑍
,𝑍
«
¬

Here, 𝑛 𝑍 ,𝑧 , 𝑆 𝑍 ,𝑧 and 𝛼𝑍 ,𝑧 are the density, the ionization rate coefficient, and the recombination rate coefficient of the ions with the atomic number 𝑍 and the charge number
𝑧 . These coefficients solely depend on the electron temperature 𝑇𝑒 . Thus, 𝑁®𝑍 depends∫
on 𝑛𝑒 𝑑𝑡 = 𝑛𝑒 𝑡 under the assumption of the constant electron temperature. The
parameter of 𝑛𝑒 𝑡 is called “ionization timescale,” which is used to describe the plasma
equilibrium state in general analysis codes. The ionizing state can be divided into
the following three states:
1. Collisional Ionization Equilibrium (CIE):
The equilibrium state between collisional recombining and ionizing in the plasma.
Typically, the plasma reaches CIE when 𝑛𝑒 𝑡 = 1013 cm−3 s. Figure 1.2 (top) shows
the ion population of Ne under the CIE state as a function of 𝑘 B𝑇𝑒 .
2. Non-Equilibrium Ionization (NEI):
The state in which either collisional recombining or ionizing is dominant. The
plasma state of NEI can be divided into ionizing plasma (IP; Figure 1.2 middle)
and recombining plasma (RP; Figure 1.2 bottom).

1.2.3

Thermal Equilibrium in Plasma

The post-shock plasma reaches thermal equilibrium initially within each ion and subsequently between different ions. As can be seen from Equation (1.16), the temperature(𝑇𝑖 ) of ion with mass 𝑚𝑖 can be written as:
𝑘 B𝑇𝑖 =

3
2
𝑚𝑖 𝑣 sh.
16(1.20)

LETTERS

6

1.2. Shock Heating in Astrophysical Plasma

2007 and 2011 combined011)

o the dataFe60

el (thermal broadening)

Ti = ATp
Ti /Tpel (no thermal broadening)

40

Si
Ne Mg
20

0
0.827

0.828

Ti = Tp

p
0

10

20

30
Atomic number

40

50

60

Figure 1.3: The relation between the ion/proton temperature ratio and the atomic number
4 | Ion to proton temperature ratios measured by combining ther 2011. Fe XVII line(Miceli et Fig.
al., 2019). The red line shows the function predicted from Equation (1.20). The black2007
and 2011 observations of SN 1987A for Ne, Mg, Si and Fe lines.
with (red curve) andline shows the best-fit function, and the green lines show the confidence levels of 90%.
The red line marks the mass-proportional trend predicted by equation (1),
ith the corresponding
while the black line shows the best-fit trend, with the corresponding 90%bars at 1σ) and theEquationconfidence(1.20) showsthatindicatedthe ion temperature
range
by green lines.is proportional to the ion mass. Actually,
D, hydrodynamic.

Miceli et al. (2019), who measured the temperature of several ion species of shockheated plasma in SN 1987A, revealed a linear increase of the ion temperature depending
2011 data), so this process is indeed negligible. For the ion–proton interactions,on the ion
mass (Figure 1.3).
instead, the e-folding time is comparable to the time elapsed after the shockh observations perHowever,for theveryenergy
turbulentmagnetic fieldbetween
that we expect
in the post-ion species
Let usimpact.
considerthe caseof theequipartition
two different
interaction, we also
shock region31, the thermalization timescale increases32 by a factor of about 5, thus
(test
and field particles). We assume that the test and field particles alreadymechanism holds
forparticles
making this process negligible for our case.
reach thermal
statesof 𝑇 andratio
𝑇𝑓 , respectively,
andattheTheequilibrium
electron to proton
temperature
has been measured
the particles
reverse follow the33
shock of SN 1987A
by analysis ofobservations
performed
in 20111962):(that is, at
Maxwell-Boltzmann
distribution
depicted
as follows(Spitzer,t = 24 yr) by the Cosmic Origins
 3/2 of the Hubble Space Telescope Spectrograph2change(HST-COS),
showingthatT/T≈
0.14–0.35
(this
value
can
slightly with
𝑚
𝑣
𝑚
e p
model. The model𝑒𝑒exp− to explain,(1.21)
𝑃but(𝑣 )T=e/Tp > 0.1 is always
different
assumptions,requiredthedata33). We
n code of the supernova
2𝜋𝑘 B𝑇
2𝑘 B𝑇
calculated Te/Tp at the reverse shock from our hydrodynamic model at t = 24 yrtime evolution of the(corresponding
to 2011).
We considered
computational
cellsbethat
g the first 250 dayswhereof𝑘 B is the Boltzmann
constant.
The only
timethoseevolutionof 𝑇 canwritten as:
were more than 99% filled with hot (kT ≥ 0.3 keV) ejecta material. The ejecta
duces the expansion
𝑇𝑓 −tracer
𝑇 included in the calculation24.
𝑑𝑇a passive
material can be identified thanks tonova24. The model=.
(1.22)
This selection allows us to isolate thelity of reproducing
𝑑𝑡 shocked𝑡eq(high-temperature) ejecta in theimmediate post-shock region (in the post-shock flow, the ejecta are rapidly
lsewhere24. We focus
with relaxation
the shocked circumstellar
medium,andby
the percentage of ejecta in
Here, 𝑡eq mixed
is calledtime, which
is givenshocked plasma. The
the computational cells rapidly drops below 99%), thus selecting a narrow sheet
calculated through the density-weighted

3/2 computedof plasma behind the reverse shock.
We
thenated at the shock front
3𝑚𝑚
𝑇𝑓 3/2𝑇
𝑓 𝑘B
average value ofelectron to proton temperature
ratio for+ these cells, finding(1.23)
𝑡 eqthe =, as suggested11 for
1/2 𝑛 𝑍 2 𝑍 2𝑒 4 ln 𝛬 𝑚
𝑚 𝑓 the observations.
8
(
2
𝜋)
𝑓
T
/T
=
0.155,
inremarkableagreementwiththatinferredfrom𝑓e p
r simulations (this is 3/model.
This provides a further indication of the reliabilityof our2
ons does not allow us to
𝐴𝐴 𝑓
𝑇 𝑇𝑓
= 5.87+s,(1.24)
calculate the evolution
2 𝑍 2 ln 𝛬 𝐴
𝐴 𝑓 synthesized the Chandra𝑛 𝑓 𝑍 theSynthesis of X-ray spectra. Fromal cell of the post-shock𝑓model results, we
HETG X-ray spectra from the values of Te, density and ionization timescale
ons in the time Δt = t −1ts
(Fosteral., 2017).
(computed on the basis of the plasma density
andettime
elapsed after the shockma in each cell was https://atomdb.readthedocs.io/en/master/
heating) by adopting the ATOMDB V3.0.8 database. We adopted the VNEI model
ilibrium of ionization
within XSPEC which requires, as input, Te and the plasma ionization timescale toure ratio Te/Tp therefore
compute the continuum and line emission, and followed the same approach as iner the shock impact.
previous studies24,34,35. The X-ray spectrum from each cell was filtered through theission measure of thephotoelectric absorption by the interstellar medium, with the appropriate column
ng the same volume)
density36. All synthetic spectra are folded through the Chandra instrumental
orresponding to year

1. Shock Waves in Astrophysical Environment

7

1 High-Resolution X-Ray Spectroscopy of Supernova Remnantsneonheliumproton1

kT (keV)10oxygen7

0.1electron

108

109

1010
net (cm−3 s)

1011

1012

Fig. 1.2: Temperature evolution of shock-heated plasma as a function of ionisation
Figure 1.4: An example calculation of the temperature evolution−1
at the shock waves, assuming atimescale. The shock speed
is assumed to be 1000 km s . The equilibration pro−1
shock velocity of 𝑣sh = 1000 km s (Katsuda, 2023). The heating of each ion at the shock front
ceeds with Coulomb collisions.
is assumed to follow Equation (1.20), and then the temperature changes in accordance with
Equation (1.22).
at the Earth is the emissivity ratio modified by a factor describing the relative interstellar absorption at the two lines with energies of E1 and E2 , and is given by:where 𝑛 , 𝑍 , and 𝐴 are densities, atomic number, and mass number of ions, respectively.
Ω1 (∼ 25–30) is defined as
F1 ln
The Coulomb logarithm= 𝛬exp[(E(1.2)
2 − E1 )/kTe ]exp[(σE2 − σE1 )NH ],
F2
Ω2


4𝜋
3
ln 𝛬strength≡ ln or𝑛oscillator𝑒 𝜆D ,
where Ω is the effective collision
strength, σE gives the photo-(1.25)
3electric absorption cross section at energy E, and NH is the hydrogen column density
the intervening
Becauseoscillator
strengths,
line energies,
and34–35
the 2 .
where thefor
Debye
length 𝜆Dmaterial.
is available
in theNRL Plasma
Formulary,
pages
cross sections of the intervening material are given in the literature, we can estiFigure
antemperature
example calculationof thethelinetemperature
evolutiontheleftshock.
mate1.4
theshows
electronby measuring
flux ratio and
NH . Fig.at1.3
clearlydemonstrates
that
the
Si
He
β
/
Si
He
α
ratio
strongly
dependsontheelectron
The timescale of the thermal equilibration depends on the density and the square oftemperature.
the particle
charge (Equation 1.23). Additionally, the temperature at the shock depends
Another useful clue to inferring the electron temperature is the forbidden-toon the ion mass. Thus, thermal equilibrium is reached first between ion–ion and thenresonance ( f /r) ratio in He-like ions. The G(≡ ( f + i)/r) ratio has been often used
betweenequilibrationis broadly
into three
for electron–ion.
this temperatureThe
diagnostics,
but it canbe usuallydivided
approximatedto thephases:
f /r ratio,
because the i lines are usually much weaker than f and r lines. The collisional− 3 s):
1. Full
non-equilibration(𝑛𝑒 𝑡 ≲temperature
5 × 1010 cm
excitation
rates have different
dependence
between f and r lines; theintensity
of
the
r
line
increases
more
rapidlywiththetemperaturethan the f line
The temperatures of all species are different.
[120]. Therefore, the f /r ratio decreases with increasing temperature, as can be seen
− 3 beenin Fig. 1.3
right. These line ratios(α10/β≲, β𝑛/𝑒γ𝑡, and
/r)cm
have
2. Partial
non-equilibration(5 × 10≲ 10f12s): measured for many
SNRs,
providing
us
with
their
electrontemperatures.
Equilibration within all ions is reached, but electron–ion equilibration has not yet.

3. Full equilibration (𝑛𝑒 𝑡 ≳ 1012 cm−3 s):
All particles are equilibrated.
2 https://library.psfc.mit.edu/catalog/online_pubs/NRL_FORMULARY_13.pdf

8

1.2. Shock Heating in Astrophysical Plasma

Figure 1.5: Temperature ratio 𝛽 as a function of the shock velocity 𝑣𝑠 for Balmer-dominatedshock in several SNRs (van Adelsberg et al., 2008). Data points are fitted by calculations withthe methodology by van Adelsberg et al. (2008). Each symbol shape of the data point shows thecorresponding SNRs. The solid curve denotes the dependence 𝛽 (𝑣𝑠 ) ∝ 𝑣𝑠−2 .

1.2.4 Collisionless Electron Heating
Under the typical environment around SNRs (𝑘 B𝑇𝑒 = 1 keV, 𝑛𝑒 = 1 cm−3 ), the relaxation time between electron and proton can be calculated to be ∼ 100, 000 years from
Equation (1.23). Since the typical age of an SNR is 1,000 years, the temperature ratio
immediately behind the shock (𝛽 ≡ 𝑇𝑒 /𝑇𝑝 ) is expected to be:
𝛽≡

𝑚𝑒
1
𝑇𝑒
=
≃
.
𝑇𝑝 𝑚𝑝
1836

(1.26)On the other hand, several studies observed 𝑇𝑒 /𝑇𝑝 to be ∼ 1 at the shocks in young SNRs
with age of hundreds of years (e.g., Laming et al., 1996; Ghavamian et al., 2001; Rakowski
et al., 2003). This deviation between the expectation and the observation is generally
attributed to processes other than Coulomb collisions, called “collisionless electron
heating.” The mechanism of the process is often thought to be related to the energy
dissipation at the shock front and precursor due to strong plasma turbulence (e.g.,
Shimada & Hoshino, 2005; Bohdan, 2023). However, the physics of collisionless electron
heating is still unsettled because there are several complex instabilities in forming thecollisionless shock, which should be directly associated with the process (e.g., Cargill &Papadopoulos, 1988; Ghavamian et al., 2007; Rakowski et al., 2008; Laming et al., 2014).
The electron-to-ion temperature ratio immediately after shock (𝛽 ≡ 𝑇𝑒 /𝑇ion ) is commonly used to describe the equilibration between electrons and ions at shock transition,consequently indicating the efficiency of collisionless electron heating. The ratio 𝑇𝑒 /𝑇ion
is determined with various methods, including “in situ” observations of solar wind bow

1. Shock Waves in Astrophysical Environment

9

Figure 1.6: (Left): The detected Fe K𝛽 emission in the reverse shock in Tycho’s SNR (Yamaguchi etal., 2014). (Right): The dependence of the centroid energy of Fe K𝛼 and K𝛽 lines and their flux ratioon 𝑇𝑒 /𝑇ion . The plots in the right panel correspond to the different ambient densities. The redregions represent the observed value with statistical (dark) and systematic (light) uncertainties.

shocks (Schwartz et al., 1988; Masters et al., 2011). Optical observations can measure
𝑇𝑒 /𝑇𝑝 using H𝛼 line diagnostics in Balmer-dominated shocks (e.g., Ghavamian et al.,
2001; Medina et al., 2014). The H𝛼 lines at the shock wave consist of two components: anarrow component, which emits from hydrogen atoms collisionally excited at the shocktransition layer, and a broad component, which emits from hydrogen atoms producedby charge transfer collisions with downstream protons. The ratio of the broad-to-narrowcomponents can estimate 𝑇𝑒 /𝑇𝑝 since it depends on the charge exchange, ionization,and excitation rates, which are sensitive to the electron and proton temperatures, andthe width of the broad component directly reflects the proton temperature (Laming,
2000). H𝛼 lines come from a confined region downstream of the shock within ∼ 1014 cm(van Adelsberg et al., 2008) because hydrogen atoms in the deeper region are ionized
and unable to emit H𝛼 lines. Thus, this method is often used in SNRs because it allows
the selective measurement of 𝑇𝑒 /𝑇𝑝 at a stage where Coulomb collisions have not yet
significantly influenced the plasma. Figure 1.5 shows the 𝑇𝑒 /𝑇𝑝 (≡ 𝛽 ) in several SNRs
estimated from the broad-to-narrow component ratio of H𝛼 lines.
Another method to measure 𝑇𝑒 /𝑇ion is proposed by Yamaguchi et al. (2014). They
estimated 𝑇𝑒 /𝑇Fe from Fe K diagnostics in Tycho’s SNR using the X-ray data obtainedwith the Suzaku satellite. Their spectral analysis detected K𝛼 and K𝛽 lines from highlyionized and low-ionized iron, as seen in Figure 1.6 (left). From the comparison with thesimulation of the Fe ionization state, the observed centroid energy of the K𝛽 line requires

10

1.3. Particle Acceleration in SNR

the 𝑇𝑒 /𝑇Fe ≃ 0.01 at the reverse shock (Figure 1.6 right). They presented a method thatallows us to diagnose the plasma state and measure 𝛽 with low-ionized iron, which ispresent in a limited region of immediate postshock ejecta.

1.3 Particle Acceleration in SNR
1.3.1

Cosmic Ray

ISM particles are accelerated at collisionless shocks and gain relativistic energy after
injections into the shocks. These particles are referred to as cosmic rays (CRs), whichreach Earth from cosmic space. Since the discovery of CR by Hess (1912) using balloons,they have been observed through various experiments, including balloon missions(e.g., Fleisher et al., 1975), satellites (e.g., The Fermi-LAT Collaboration et al., 2017), andground-based detectors (e.g., Telescope Array Collaboration et al., 2023). Figure 1.7
illustrates the CR spectrum observed by different experiments. The spectrum exhibits
distinct breaks mainly at two energy points: ∼ 3 × 1015 eV (referred to as the “knee”)
and ∼ 3 × 1018 eV (referred to as the “ankle”). CRs with energy below 3 × 1015 eV are
believed to be produced in our Galaxy (the Galactic CR), while those with higher energy
are thought to come from extra-galactic sources. Nevertheless, the origin of CR and
its underlying physics remain open questions.
SNRs have been thought to be the main candidate of the Galactic CR accelerator for
decades (Baade & Zwicky, 1934). One reason is that it has a large enough energy budget toaccelerate CRs. The required energy to keep the CRs is estimated to be 𝐿 CR ≃ 1041 erg s−1 .
Because the energy of supernovae (SNe) is 𝐸SN ≃ 1051 erg, the explosion energy provided
to the Galaxy as kinetic energy is 𝐿 SN ≃ 𝑓SN 𝐸SN ≃ 1042 erg s−1 under an assumptionof the SNe rate to be 𝑓SN ≃ 0.03 yr−1 . If 10% of the total kinetic energy is used for theacceleration of CRs, the CR energy density can be explained feasibly. Another reason
is the acceleration mechanism of the first-order Fermi acceleration or Diffusive ShockAcceleration (DSA) (e.g., Axford et al., 1977; Blandford & Eichler, 1987; Drury, 1983;
Bell, 1978a,b). The acceleration mechanism predicts the energy spectrum of CRs to be
distributed in power law, which is consistent with the observations.

1.3.2 Diffusive Shock Acceleration
Let us consider the behavior of individual particles around the shock transition. Figure 1.8 presents the schematic view of the DSA. Here, we make three assumptions: (i)steady state, (ii) plane parallel non-relativistic shock, and (iii) test particle approximation. Under the test particle approximation, we ignore the effect on the macroscopic
plasma by non-thermal particles.
Around the collisionless shock, the plasma has magnetic field turbulence, leading
to charged particle scattering via magnetohydrodynamical (MHD) waves such as Alfén
waves. The scattering alters the direction of the particle motion, resulting in particle

1. Shock Waves in Astrophysical Environment

11

Energy [J]

10−8

10−6

10−4

10−2

1/
m

2

/y

1/
m

e+

e− +e+

10−1

Knee

γ

p̄

Ankle
2

/y
r

10−3

1/
km

E2 Intensity [GeV m−2 s−1 sr−1]

p

101

r

2

AMS-02
AUGER
BESS
CALET
CREAM
DAMPE
FERMI
HAWC
HESS
ICECUBEICETOP+ICECUBEKASCADEKASCADE-Grande
NUCLEONPAMELA
Telescope Array
Tibet-III
TUNKA
VERITASgithub.com/carmeloevoli/The CR Spectrum

103

102

1

/s

10−10

γ IGRB10−5

ν + ν̄
LHC

10−7

103

1

106
Energy [GeV]

109

1012

Figure 1.7: Cosmic ray spectrum (Evoli, 2020).

acceleration as the particle transverses the shock multiple times. When an upstream
particle with the energy 𝐸 injects downstream, the downstream energy in the rest frameof the upstream observer (𝐸 ′ ) can be expressed as:
𝐸 ′ = 𝛾 (𝐸 + 𝑉 𝑝𝑥 ),(1.27)√︁
where 𝑉 = 𝑣𝑑 −𝑣𝑢 , 𝑝𝑥 = (𝐸 /𝑐 ) cos 𝜃 , and 𝛾 ≡ 1/ 1 − (𝑉 /𝑐 ) 2 ≃ 1 are the downstream veloc-

ity in the upstream frame, the momentum perpendicular to the shock, and the Lorentz
factor, respectively. The angle 𝜃 is between the momentum vector and the normal shock
vector. From Equation (1.27), the particle, after crossing the shock, gains energy of:
Δ𝐸 = 𝐸 ′ − 𝐸 =𝑉
𝐸 cos 𝜃 .
𝑐

(1.28)

The averaged gain energy is derived as follows:


Δ𝐸
𝐸



∫ 𝜋/2
=
0

Δ𝐸
2𝑉
2 sin 𝜃 cos 𝜃 𝑑 𝜃 =.
𝐸
3𝑐(1.29)

Also, the particle downstream undergoes multiple scatterings and eventually crosses
the shock. As the upstream gas appears to approach with a velocity 𝑉 in the downstream

121.3. Particle Acceleration in SNR

E = E0<latexit sha1_base64="hD/h+L/rGA5N0Nl7dKXbkArzdH0=">AAAB7HicbVBNSwMxEJ2tX7V+VT16CRbBU9kVvy5CUQoeK7htoV1KNs22odlkSbJCWfobvHhQxKs/yJv/xrTdg7Y+GHi8N8PMvDDhTBvX/XYKK6tr6xvFzdLW9s7uXnn/oKllqgj1ieRStUOsKWeC+oYZTtuJojgOOW2Fo7up33qiSjMpHs04oUGMB4JFjGBjJb9+U++5vXLFrbozoGXi5aQCORq98le3L0kaU2EIx1p3PDcxQYaVYYTTSambappgMsID2rFU4JjqIJsdO0EnVumjSCpbwqCZ+nsiw7HW4zi0nTE2Q73oTcX/vE5qousgYyJJDRVkvihKOTISTT9HfaYoMXxsCSaK2VsRGWKFibH5lGwI3uLLy6R5VvUuqxcP55XabR5HEY7gGE7BgyuowT00wAcCDJ7hFd4c4bw4787HvLXg5DOH8AfO5w/QX44M</latexit>

MHD wave✓◆
4V
E = E0 1 +3c<latexit sha1_base64="84FgJcqc/hwKLs0t5oOM50vXHd4=">AAACCnicbVDLSsNAFJ3UV62vqEs30SJUhJJofWyEohRcVrAPaEKZTCft0MmDmRuhhKzd+CtuXCji1i9w5984bbPQ1gMXDufcy733uBFnEkzzW8stLC4tr+RXC2vrG5tb+vZOU4axILRBQh6Ktosl5SygDWDAaTsSFPsupy13eDP2Ww9USBYG9zCKqOPjfsA8RjAoqavv165qXdPm1IOSdWx7ApOk0kyTU5LagvUHcNTVi2bZnMCYJ1ZGiihDvat/2b2QxD4NgHAsZccyI3ASLIARTtOCHUsaYTLEfdpRNMA+lU4yeSU1DpXSM7xQqArAmKi/JxLsSznyXdXpYxjIWW8s/ud1YvAunYQFUQw0INNFXswNCI1xLkaPCUqAjxTBRDB1q0EGWMUBKr2CCsGafXmeNE/K1nn57K5SrF5nceTRHjpAJWShC1RFt6iOGoigR/SMXtGb9qS9aO/ax7Q1p2Uzu+gPtM8ffqaZiw==</latexit>✓◆24V
E = E0 1 +3c<latexit sha1_base64="IVPH60WDnXMO1yxun42pnqyQBpo=">AAACDHicbVDLSsNAFJ34rPVVdekmWISKUJJaHxuhKAWXFewDmlom00k7dDIJMzdCCfkAN/6KGxeKuPUD3Pk3TtsstPXAwOGcc7lzjxtypsCyvo2FxaXlldXMWnZ9Y3NrO7ez21BBJAmtk4AHsuViRTkTtA4MOG2FkmLf5bTpDq/HfvOBSsUCcQejkHZ83BfMYwSDlrq5fPWy2rUcTj0o2MeOJzGJy40kPiGJI1l/AEf3JZ2yitYE5jyxU5JHKWrd3JfTC0jkUwGEY6XathVCJ8YSGOE0yTqRoiEmQ9ynbU0F9qnqxJNjEvNQKz3TC6R+AsyJ+nsixr5SI9/VSR/DQM16Y/E/rx2Bd9GJmQgjoIJMF3kRNyEwx82YPSYpAT7SBBPJ9F9NMsC6END9ZXUJ9uzJ86RRKtpnxdPbcr5yldaRQfvoABWQjc5RBd2gGqojgh7RM3pFb8aT8WK8Gx/T6IKRzuyhPzA+fwC685ov</latexit>

Shock front

Figure 1.8: Schematic of diffusive shock acceleration.

frame, the particle gains the energy of 2𝑉 /3𝑐 . Consequently, the average energy gain
over a single round-trip through the shock is calculated to be (Δ𝐸 /𝐸 ) = 4𝑉 /3𝑐 ≪ 1.
Therefore, when the particles undergo 𝑛 round-trips, their energy is given by:

4𝑉
𝐸𝑛 = 𝐸 0 1 +
3𝑐

𝑛





4𝑉
≃ 𝐸 0 exp𝑛 ,3𝑐(1.30)

which indicates that the particle energy increases exponentially. Here, 𝐸0 is the initial particle energy.
The particle escapes from the shock with the probability of 4𝑣𝑑 /𝑐 per round-trip. As
a result, the number of accelerated particles decreases as the number of round trips
increases. The probability of the escape at the 𝑛 -th time step (𝑃𝑛 ) is expressed as follows:
𝑃𝑛 = 1 −

4𝑣𝑑 𝑛


𝑐

×

4𝑣𝑑

.

𝑐

(1.31)

From Equation (1.30) and (1.31), the energy spectrum of the accelerated particles follows
a power-law distribution:𝑑𝑁∝ 𝐸 − ( 3𝑣𝑑 /𝑉 ) −1 = 𝐸 − (𝑟 +2 )/(𝑟 −1 ) ≡ 𝐸 −𝑠 ,𝑑𝐸

(1.32)

where 𝑟 ≡ 𝜌 𝑑 /𝜌𝑢 = (𝛾 + 1)/(𝛾 − 1) is the shock compression. Using 𝑟 = 4 in Equation (1.14),
the spectral index is 𝑠 = 2, in agreement with the observed CR spectrum.
In order to inject DSA, the particles have to reach high energy enough to cross theshocks. However, there are large gaps between thermal and injection energy, especiallyfor electrons. This crucial problem is referred to as “injection problem”, which is still
open despite various theories (e.g., Bohdan, 2023).

1. Shock Waves in Astrophysical Environment

1.3.3

13

Acceleration Timescale and Maximum Energy

The acceleration timescale (𝑡acc ) can be defined as the time that it takes for the accelerated particle to double in energy:𝐸𝑡acc = 𝑡cyc
Δ𝐸

,(1.33)

cyc

where 𝑡cyc is the time of a round trip, and ⟨𝐸 /Δ𝐸 ⟩cyc = 3𝑐 /4𝑉 is the reciprocal of theenergy gain of a round trip. Using diffusion coefficient 𝔇, 𝑡cyc can be expressed by:𝑡 cyc =

4𝔇𝑢

+

4𝔇𝑑

.

𝑣𝑑 𝑐

(1.34)


𝔇𝑑
+
,
𝑣𝑢
𝑣𝑑

(1.35)

𝑣𝑢 𝑐

Therefore, 𝑡acc can be written as:𝑡acc =

3 𝔇𝑢
𝑉

which means that it is necessary to get the diffusion coefficient 𝔇 in order to calculate𝑡acc . The coefficient 𝔇 is expressed by:
𝔇=

𝜆 mfp

𝑐=

𝜂𝑟g

3

𝑐,

3

(1.36)

where 𝜆mfp is the mean free path of the scattering, which can be written as 𝜆mfp = 𝜂𝑟g in
the DSA situation using the gyro radius 𝑟g . 𝜂 = (𝐵/𝛿 𝐵) 2 is the parameter called “gyrofactor”, which depicts the magnitude of magnetic field turbulence. Generally, 𝜂 ≥ 1, so
the mean free path is not smaller than the gyro radius. The state of 𝜂 = 1 is the so-called“Bohm limit,” which corresponds to the state with the highest magnetic field turbulence.
Assuming 𝔇𝑢 = 𝔇𝑑 for the sake of simplicity and using 𝑣𝑢 = 4𝑣𝑑 = 𝑣sh , the timescale 𝑡acc
can be rewritten as followings using Equation (1.35) and (1.36):
𝑡acc =

20 𝑐𝑟g𝜂.
23 𝑣sh(1.37)

Under the environment of the magnetic field 𝐵 , the gyro radius of the particle withthe charge of 𝑍 𝑒 and the energy of 𝐸 can be expressed as:
𝑟𝑔 =

𝐸𝑒
.
𝑍 𝑒𝐵

(1.38)

Thus, when the maximum energy of the particles accelerated by DSA is written as
the following using 𝑡acc :2

3 𝑣sh 𝑍 𝑒𝐵𝑡acc
𝐸 max =.
20
𝜂𝑐(1.39)

The acceleration time 𝑡acc is limited mainly by three scenarios: the age of the SNR, synchrotron cooling loss, and escaping (Reynolds, 1998). We provide detailed explanations
for each scenario in the following.

14

1.3. Particle Acceleration in SNR

Age-limited When particles continue to be accelerated with little escape or coolingsince the formation of shock waves, the acceleration time is limited by the remnant
ageage. This scenario is called age-limited case. The maximum energy 𝐸max
can be written
as the following using the remnant age 𝑡age :age𝐸max =
 2 𝑡 age
3𝑍 𝑒 −1 2𝑣 sh𝜂 𝑣 sh𝑡 age 𝐵 ≃ 40𝜂 −120𝑐3000 km s−1
103 yr




𝐵
TeV .
10 𝜇G

(1.40)The maximum energy is generally limited by this scenario because this timescale istypically smaller than the other two timescales. Thus, to produce particles accelerated
to the “knee” energy, it is necessary to increase the magnetic field to the order of mG.
However, the mechanism of the magnetic field amplification is not well established(e.g., Inoue et al., 2012; Caprioli & Spitkovsky, 2014).
Loss-limited The accelerated particles lose their energy by radiating synchrotron
radiation. When the cooling by synchrotron loss is dominant, the maximum energy
loss is limited by the synchrotron cooling time. Using the definition of cooling time𝐸max
𝑡 loss ≃ 9 (𝑚𝑐 2 ) 4 /( 4𝑍 4𝑒 4 𝐵 2 𝐸𝑒 ) proposed by Zirakashvili & Aharonian (2007) (see also
Equation 2.26), the maximum energy can be expressed as:
loss
𝐸max
≃ 70𝜂 −1/2





𝑣 sh

3000 km s−1

𝐵
10 𝜇G

 −1/2TeV .

(1.41)

This case is applicable when the magnetic field is amplified and the time scale is smaller
than the remnant age, as can be seen from 𝑡loss ∝ 𝐵−2.
Escape-limited At the late time of the remnant lifetime, particles escape upstream
from DSA without being scattered by MHD waves. The maximum energy in the escapelimited case is roughly given as:

esc
𝐸max
≃ 100

𝜆max

3 × 10−2 pc




𝐵
TeV ,
10 𝜇G

where 𝜆max is the maximum wavelength of MHD waves to scatter particles.

(1.42)

2
Supernova and Supernova Remnant
Contents
2.1

Supernova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1
Supernova Classification . . . . . . . . . . . . . . . . . . . . .
2.1.2 Core-Collapse SN . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Thermonuclear SN . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Supernova Remnant . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Evolution of SNR . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Thermal Radiation from Shock-Heated Particles . . . . . . . .
2.2.3 Non-thermal Radiation from Shock-Accelerated Electrons . .

2.1

Supernova

2.1.1

Supernova Classification

15
15
16
17
19
19
22
25

Based on our comprehension of the explosion mechanism, there are two major groupsof supernovae (SNe): core-collapse SNe and thermonuclear SNe. The following sections
describe the mechanisms and characteristics of each explosion type. Before describing
the core-collapse and thermonuclear explosions, we will briefly overview anotherclassification scheme based on two observational criteria: optical spectroscopy andlight-curve shape (Figure 2.1). The origin of the observational classification can dateback to Minkowski (1941), who observed some SNe without hydrogen absorption in
their spectra (Type I), while others showed hydrogen absorption. While Type II SN isconsistently a core-collapse event, Type I SN encompasses both thermonuclear andcore-collapse explosions. Further subcategories of Type I SN can be defined basedon the presence of silicon and helium absorption lines. Type Ia SNe, identified by thepresence of silicon absorption lines, are attributed to thermonuclear explosions (Eliaset al., 1985). Within the Type I group, those without silicon absorption lines are thought to15

16

2.1. Supernova

on Astrophys Rev (2012) 20:49

Page 5 of 120

2 The classification ofernovae, based on optical
ctroscopy and light-curvepeFigure 2.1: Classifications of SNe based on optical spectra and light-curve shape (Vink, 2012).

ctroscopy and light-curve shape (Fig. 2). Type IIP are the most common type ofbe associatedwith core-collapse
They are classified
as Typethat
Ib SNe (without
e collapse supernovae,
and opticalstudies of explosions.
potential progenitor
stars confirm
Si butinitialwith He
absorption
andM
Type
Ic SNe and(without
both Siand He absorption
ir progenitors havemassesin thelines)∼8–17
that they
explode" range,lines).
Meanwhile,
II category
can be divided
based
on light-curve
he red supergiant
phase,
while thestillTypehavinga substantial
hydrogenenvelope(see shape into
a
Type
IIP
with
a
plateau
phase
and
a
Type
IIL
with
a
lineardeclinefromdiscussions in Smartt 2009; Chevalier 2005). Type IIL progenitors probably have the peak.
ubstantially less massive envelope, either due to stellar wind mass loss, or due toary interaction.2.1.2
Type IIb
supernovae are
Core-Collapse
SNa class intermediate between Type Ib andpe II, in that their spectra would initially identify them as Type II explosions, but at
Core-collapse
SNe occur
result Alsoof corethis
implosions
at the end ofas
thethelives of massive
times their spectra
evolve into
Type asIb thespectra.
can be understoodstars:but
typically≳ 10𝑀 ⊙ inremoval
the mainof
sequencephase (see a envelope
review by Woosley& Janka,ult of substantial,
not complete,
the hydrogen-richdue2005).
Below,
we provide
a brief explanationof the mechanism
core-collapse SN. As astellar wind massloss,
or binaryinteraction.
The prototypalType IIb of
supernova
massivestar
evolves,thecompositionof
its
elementschangesvia
stellarnucleosynthesis.
SN 1993J (Podsiadlowski et al. 1993; Woosley et al. 1994). Interestingly, recent
first, a helium-rich core is formed through hydrogen burning, followed by a carbonntification andAtsubsequent
spectroscopy of the light echo of the supernova thatrich core through helium burning. Subsequently, as nucleosynthesis progresses, cores
sed the extensively studied SNR Cassiopeia A (Cas A) shows that it is the remnantof heavier elements are formed sequentially. Then, an iron core is formed at the centera Type IIb supernova,
as the spectrum shows both hydrogen and weak helium lineof the star as the depicted configuration of each elemental layer in Figure 2.2. Whenorption (Krauseal. 2008a).
theetcenter
temperature increases to 5 × 109 K by the heating driven by contraction, theNot listed in Fig.
Type
IInstableclass.nucleus,
Type IInsupernovae
are characterized
core2ofis56theFe, themostis formed,
and heavier
elements are no longer
narrow hydrogen
emissionlines,
whicharethought
to
come
from
a
dense
circum- resulting insynthesized. It follows that the core is heated further by the contraction,
lar environment,by substantial
mass
the progenitor. Its
the probably
absorptioncausedof gamma
rays from theverylost
hotby
core:
ce in the diagram is not quite clear, as at least one Type IIn supernova, SN 2001ic,
56
Fe + 𝛾spectrum→ 13 4He subsequently+ 4𝑛 − 124.4 MeV.
(2.1)
s observed to be a Type Ia supernova whose
evolvedinto ape IIn supernova(Hamuyet al.created2003).by this photo-disintegration process decays into neutrons andThen,the helium
protons:

Core collapse supernovae

4

He → 2𝑝 + 2𝑛 − 28.3 MeV .

(2.2)

Subsequently, after protons capture electrons and gradually transform into neutrons,

re collapse supernovae
mark the loses
end oflivesof supported
massive stars;
that is, thosestars
the core eventuallyitstheability
to beby processes
like nucleosynthesis
andh main sequencemassesM!
8
M(see
Woosley
and
Janka
2005,
for
a
review).
" a result, the core collapses toward the center, forming a protoelectron degeneracy. Ast prior to collapse
thestarconsistsof different
with theproducts
theneutron star or a black hole.
Most of thelayers
gravitationalenergy(∼ 1053oferg)
is transformed
erent consecutive burning stages. From the core to the outside one expects: ironup elements in the core (silicon-burning products), then silicon-group elements
ygen-burning products), oxygen (a neon burning product), neon and magnesiumrbon-burning products), carbon (a helium-burning product), helium (a hydrogen-2. Supernova and Supernova Remnant

17

Figure 2.2: The configuration of a massive star nearing SN. The timescales are computed for astar with a mass of 25𝑀 ⊙ (Deaconu, 2008).

into neutrinos, as confirmed by the observations of SN 1987A with the Kamiokande(Hirata et al., 1987), IMB detector (Haines et al., 1988), and the INR Baksan Underground
Scintillation Telescope Alekseev et al. (1987). The rest of the energy (∼ 1051 erg) is used as
the explosion energy of SN. While the detailed conversion mechanism from gravitationalenergy to explosion energy is still under debate, neutrino heating, initially proposedby Wilson (1985), is a widely supported mechanism to drive the shock revival.

2.1.3

Thermonuclear SN

Type Ia SNe are widely believed to originate from the explosive thermonuclear burningin white dwarfs with no hydrogen envelope. It is supported by the fact that the SNe arealso found in all galaxy types, including old elliptical galaxies, which have abundant
low-mass stars (Wang et al., 1997). When the mass of a carbon-oxygen (CO) whitedwarf approaches or exceeds the Chandrasekhar limit (∼ 1.4𝑀 ⊙ ) for some reason, theelectron degeneracy pressure can no longer support its own mass. After the core density
reaches ∼ 2 × 109 g cm−3 , the ignition and detonation of 12C + 12C fusion is triggered,following the ejection of the entire star, including its core (Arnett, 1969). The ejecta
of Type Ia SN has a metal-rich characteristic; the abundances of heavy elements likeintermediate-mass elements (IMEs: silicon, sulfur, argon, and calcium) and iron are
higher compared to those of the solar abundance.
Although many astrophysicists accept that the thermonuclear SNe occur from binary
systems, there are a variety of theories for the progenitor system that enables the whitedwarf to obtain mass, leading to ongoing debates (Liu et al., 2023, for a review). Figure 2.3
shows the various theories of the progenitor system. We note that they are not all of
the theories. The theories can be divided mainly into two groups: single-degenerate

18

2.1. Supernova
Z. Liu et al.: SN Ia explosion in binary systems

5Fig. 2 Schematic illustration of binary evolutionary paths for SNe Ia in the SD and DD scenario (see also Wang 2018).
Figure 2.3: Schematic illustrating the progenitor of Ia SNe (Liu et al., 2023).
Note that evolutionary channels here are not complete and that new channels may still be proposed in the future.
A number of candidate progenitors have been sugrial (Paczynski 1976; Fujimoto 1982a,b; Livio et al. 1989;
gested for the SD scenario, including cataclysmic variable
Nomoto et al. 2007; Shen & Bildsten 2007; Wolf et al.
stars like classic novae, recurrent novae and dwarf novae
2013; Piersanti et al. 2014; Wang 2018), which causes dif(Webbink et al. 1987; Hachisu & Kato 2001; Warner
ficulties for explaining the observed nearby SN Ia rate(SD)and4).double-degenerate(DD) scenarios.
The former
assumesthe binaryofHeuvela whiteMoreover, some recent observations
seem
supersoft
X-ray sources(van denet al.
(see
Section2003)),
to pose a challenge to the SD scenario (see Section 5)1992), symbiotic systems (Webbink et al. 1987;
dwarf and a non-degenerate companion star, such as a main sequence or red giant.
such as the missing of surviving companion stars
Yungelson & Livio 1998) and WD + hot-subdwarf
in Thesupernova
binaries star(see via
Sectionet al.
3.2.3; (Whelan
Iben & Tutukov
1994;
whiteremnants
dwarf(SNRs)
gains (Kerzendorf
mass fromthe2012;companion
accretion& Iben,Schaefer & Pagnotta 2012; Ruiz-Lapuente et al. 2018), theNelemans & Tauris 1998; Geier et al. 2013).
1973).ofIn
the latter
whitedwarfs merge and explode (Webbink, 1984; Iben
absenceswept-up
H/Hescenario,
in their latetwo
spectra(Leonard
In the SD scenario, a WD accretes and retains com2007; Tucker et al. 2020) and low X-ray flux from nearby& Tutukov, 1984). Because each scenario has
its advantages and disadvantages, thepanion matter that carries angular momentum. As aelliptical galaxies (Gilfanov & Bogdán 2010; Woods et al.
the WD
spinstowitha short period the
which
2017;
Kuuttila et system
al. 2019).of
In Ia
addition,
the SDOneconsequence
progenitor
SNe isalthoughstill open.
observational
guide
distinguishingleads to an increase of the critical explosion mass. If
scenario makes the explosion rather homogenious, it turns
is probing
circumstellar
by thethe critical(CSM)
mass iscreatedhigher than
the wind
actual frommass ofouttwoto bescenarios
difficult to cover
the observedthe
rangesin brightness medium
the WD, the SN explosion could only occur after theand decline rates in this scenario. However, to conclude
a progenitor
white
dwarf duringmasstheaccretion.
WD increases the spin period with a specific spinwhether
the SD scenario
is promisingfor producing
down timescale. This scenario is known as the “spinmajority of SNe Ia requires comparing a full range of preup/spin-down
model” their(Di Stefanoal. 2011; Justham
dicted observational
consequences
this scenario
withType Ia SNe
are also fromknown
as standard
candles because
peaket brightnesses
2011; Hachisu et al. 2012b). In this model, if the spinthe observations of SNe Ia (see discussions in Section 5).
have much less variation than the other SNedown
types.
Owingis tothis,thanType
Ia SNeis theoften
timescalelongerabout
106 yrs,
CSM

used as distance indicators in cosmology, leading to the findings of the accelerating
expansion of the Universe (Perlmutter et al., 1998; Garnavich et al., 1998; Riess et al.,1998) and measurement of the Hubble constant 𝐻0 (Riess et al., 2019).

2. Supernova and Supernova Remnant

2.2

Supernova Remnant

2.2.1

Evolution of SNR

19

Free Expansion Phase After SN, most of the explosion energy (𝐸SN ) transfers to thekinetic energy and 2–3% of 𝐸SN to thermal energy. The exploded ejecta expands with thevelocity of ∼ 10, 000 km s−1 , much faster than the sound velocity in ISM (∼ 10 km s−1 ). It
follows that forward shock (blast wave) at the front of the ejecta. The blast wave sweeps
up the surrounding ISM. Since the mass of swept-up ISM (𝑀ism ) is negligibly smaller
than the mass of ejecta (𝑀ej ) in the early phase after the SN, the ejecta expands without
decelerating. Therefore, this phase is called the “free expansion phase.” The expansion
velocity (𝑣sh ) and the radius of the supernova remnant (SNR) can be denoted as follows:
√︄
𝑣s =

2𝐸SN
𝑀 ej


= 1.0 × 104

 1/2 𝐸 SN

1051 erg𝑀 ej −1/2
km s−1 ,𝑀⊙(2.3)

𝑅 sh = 𝑣 sh𝑡 ,(2.4)

where 𝑡 is the time from the SN. 𝑀ism can be calculated as follow:
𝑀 ism =

4 3
𝜋𝑅 𝜇𝑚 H 𝑛 0 ,
3 sh

(2.5)

where 𝑛 0 is the hydrogen density in ISM. The free expansion phase continues until
𝑀 ism is comparable to 𝑀 ej . From Equation (2.3), (2.4), (2.5), the timescale of this phase
is a few hundred years:𝑡 ∼ 1.9 × 102

 −1/2 𝐸SN1051 erg𝑀 ej 5/6  𝜇  −1/3  𝑛 0  −1/3
yr.
𝑀⊙
1.4
1 cm−3(2.6)

Adiabatic Expansion Phase (Sedov-Taylor Phase) When 𝑀ism becomes much larger
than 𝑀ej , the shock velocity decelerates due to swept-up ISM. In this phase, where
the radiative cooling is still negligible, the SNR expands adiabatically. This phaseis called the “adiabatic expansion phase (Sedov-Taylor phase),” where a self-similarsolution (Sedov-von Neumann-Taylor solution) can give the shock radius, velocity, andtemperature under an approximation of point source explosion (Sedov, 1946; Taylor,
1950; von Neumann, 1976):𝑅 sh = 4 × 10
𝑑𝑅 sh
𝑣sh =
𝑑𝑡

 2/5 𝑡

14

104 yr= 5 × 102

𝑇sh = 3 × 10

6

𝑡

𝐸SN

1051 erg

 −3/5 

104 yr𝑡

104 yr 1/5 𝐸 SN

𝑛0

1 cm−3
 1/5 𝐸 SN

1051 erg𝑛0 −1/5

1 cm−31051 erg
 −6/5  −1/5

 2/5 𝑛0

1 cm−3 −2/5km,(2.7)

km s−1 ,

(2.8)

K.

(2.9)

After the adiabatic expansion phase over ∼ 10, 000 years, 70% of the explosion energy
transfers to thermal energy (Chevalier, 1974).

20

2.2. Supernova Remnant(a)Contact discontinuity
Heated ISM

Unshocked ISM

Heated ejecta

Unshocked
ejectaForward shock
300

(b)

Reverse shock
314 & MCKEE
TRUELOVETRUELOVE
Vol. 120& MCKEE(c)speed in the ambient medium was essentially inÐnite given t* + 2.3. In Figure 6, w

that the density there was neglected but the pressure was blast-wave shock nevernot) resulted in a description of the deceleration of the the end of the time ran
ejecta surface under the action of external resistance and below 6% before t ,revinternal collisions. It represents the Ðrst self-consistent solu- t* + 2.2. We note thattion for the deviation from pure free expansion of a Ðxed- small at early times, itenergy, ejecta-dominated SNR. Although the ambient period. In Appendix Dpressure in their model was assumed to be due to an under typical condition
ambient magnetic Ðeld, it was taken to be isotropic. Bern- The preceding simulastein & Kulsrud (1965) and Kulsrud et al. (1965) further remnant to the ST simil
pursued the problem of magnetohydrodynamic evolution, the HS similarity solut
and many workers have followed these early studies with time limit, we carried ou
more sophisticated models (see Jun & Norman 1996b and range of t. We found threferences therein). Rosenberg & Scheuer (1973) turned to self-similar solution by
numerical simulation to study self-consistently the problem ““ cold ÏÏ startup at t haof a piston expanding into a uniform ambient medium. evolution of /(t) in 0the
Their Ðnite-energy piston was decelerated by the pressure of methods to determine
the shocked ambient material. From an astrophysical from the pressures in s
FIG. 1.ÈIllustration of the primary regions within a nonradiative SNR
the key
improvement
over shocks
the workof Kruskal begins as unity because
5.ÈShock positions
for blast-wave
and reverse
in remnants
during the transition from ED- to ST-stage evolution. The unshocked FIG. standpoint,of ann \ 0et(uniform)ejecta.
Solidinclusionline : Numericalsolution.
Dashed:al. (1965)
wastheof a blast
wave
infreetheirline
model.
ejecta freely
expands
met by the reverse
shock.cross-section
At the time shown, theFigure2.4:(a)until
Schematicof the
of
SNR
at
the
transition
state
from
the
to
and it takes time for th
reverse shock
always moves
in Lagrangianshells of shocked ejecta and shocked ambient gas move at approximatelyAnalytic solution.
Major The
advancesin studying
theinward
deviationfrom free contact interface, andcoordinate,but,asshown,
itisinitially
carriedoutwardin
Euleriancoordithesamevelocity.
This
distribution
is
taken
from
a
simulationof
uniform
adiabatic expansion phase. (b) The radius of thenate
forwardand reverseshock
a function
time
expansionin the ED
stage asweremade onceoftheinternal shocked gas. This me
while moving slowly relative to the unshocked ejecta ahead of it.
ejecta expanding into a uniform ambient medium. The slight smearing ofstructure of the ejecta was solved for simultaneously with shock smearing due tothe shock fronts
a normal e†ectof artiÐcial viscosity employed by the(Truelove& isMcKee,1999).
the ambient Ñow. Gull (1973) and Ardavan (1973) both fol- to repeatedly measurenumerical method to mediate the shocks. The scaling of axes is discussed in° 2.
numerical
Figures 5 andof6.theThe
resolutiondoing
lowedsimulation
the internalinhydrodynamics
ejectaÈGull
points behind the shocused insothis
and otherandsimulations
discussed in Boththis paper
is thenumericallyArdavan analytically.
assumed1.10 to calculate / oejectabe initiallyand bothmadediscussed
in toAppendix
B. hot,In theseÐgureswe theplotnovel
shockobserED
for R*) for the shock m
vationthe formationof ½aM0,shock
positionsandofvelocities
over t*3N inwithinordertheto ejecta
focus at a
r
The
shock
positionvari
Taylor(1950a,1950b) into phasethe physics
of terrestrial
atmorelatively
latetime. A result
more relevant
astrophysicaltheperiodexpansionof transition
surrounding\to0.495.
We
At thetransition
fromthe freetouponadiabaticphase,
thet*STswept-up
polation procedure use
spheric explosions led to a self-similar solution for thehave plotted
SNRs wasobtainedwhen(1974)
adoptedinitially
the reverse
shock
upMcKeeto the timeof thebounce.
very smooth plot of inf
spherical
Ñow dueto theof awhichlarge amountofIn Figure
cold,5,uniform
ejecta.errorHethewhichformationofa reversetheseenin theblast-wave
shock
ISM
pushesback
onpoint
thereleaseejecta,is decelerated.
It residual
creates
thepredicted
shock,
moves
direct measurement is lathermal energy. Their work inaugurated the Ðeld of blast-position
shock
ejecta at aas
very
early timeandat t*within\ 3 isthedecreasing
a fractionof R*. produced
It is an anbwave physics.
Shklovskii(1962) recognizedapproximate
analytic
expression
forpowerthemotionof the the direct measuremene†ectof the constantterm
in the2.4
o†set
law thethattowards
theSomewhat
inner later,
direction(reverse
shock)(McKee,1974).
Figure(a)shows
the applicability of adiabatic blast-wave theory to thevanishes
reverseshock.
Using cold ejecta,
Gull (1975)
extended his shock front. From met
as thetime-dependent
term becomesdominant.
and from method (2) w
problemofSNR
evolution.
Extensive
reviewsofblast-wave
previousworkandconÐrmed
prediction
schematic of the SNR at the phase, and Figure
2.4(b)peaks
shows
positionand thevelocityThis
error
nearthet andis never McKeeÏs
greaterthan
5%. Theof the
ST thephysics and astrophysics are given by Sedov (1959, 1992),error in
earlyofshock.
MansÐeld& Salpeter below the value /ED +theformation
solution forthereverse
positionof thereverse shock& Raizerand(1966),
Ostrikershock.
& McKeeandbecomes(1974)
simulated
anexplosion
along
the
lines
Gull (1975) ODEs of Hamilton & S
ofZeldovich
the forward
reverseAs(1988),
shown
in Figure
2.4 (c),theshocks
move
over
time.
signiÐcant
only
once
the bounce
process
isofunderdix C). Since our inten
Bisnovatyi-Kogan & Silich (1995).
andalsoconÐrmedthereverseshock,
but
their
way at t [ t , at which point we no longer expectfocus
our wasanalytic solution, we ad
rev ISM
thecanself-similar
limits of early
andthelate forward-shocked
times is aconstant-acceleration
TheBetween
X-raysbe detectedfrom
not uponthe EDand
stage.reverse-shocked
Instead,they followed
the
model
to
be accurate.
Prior toejecta,
t evolution, therevuntil
period of nonÈself-similar transitional Ñow represented byerror in
past
ST stageto timesenoughthat radiative
cooling In Figure 8 we plot th
R theremains
below
2% ; itlate
does
not exceed
10%
r boundary
whichheatedseen
to in
high
enoughtemperatures.
The
betweenthe heatedISM
the sortareof structure
Figure1. Deviationfrom freebecameimportant, andthey discovereda scaling
relation region containing the s
expansion is the Ðrst step in the nonÈself-similar evolution.
for solutions in that regime. Cui (1980), by making theandejecta is(1960),
calledconsidering“contactexplosion
discontinuity.”
youngapproximation
SNRs, suchthat
as Tycho’s
SNR and
Stanyukovich
physics inMany
untenable
the shocked-ejecta
densitygeneral, described how a Ðnite ambient pressure worked toremains uniform, developed very approximate analyticSNare ofthought
toproducts
be categorizedin contact-discontinuity
this phase.
stall1006,
the motion““ explosion;ÏÏ his analysis as
rep-a remnant
shock andtrajectories to extend
resents an early realization of the importance of the presMcKeeÏs analytic solution to later times.
sure of the ambient medium in the deceleration of the ejecta.
SigniÐcant progress in understanding the early-time evolParker (1963), exploring the problem of solar Ñares driving
ution of SNRs was made with the discovery of self-similar
shocks into the solar wind, extended the work of Taylor
solutions that applied between the free expansion and(1946) by developing a solution for a spherical piston
Sedov-Taylor limiting solutions. Chevalier (1982) andexpanding at a power-law rate into a pressureless, powerNadyozhin (1985) independently found a self-similar solulaw density ambient medium. In both TaylorÏs and ParkerÏs
tion for the Ñow in a remnant with ejecta comprised of a

2. Supernova and Supernova Remnantn Astrophys Rev (2012) 20:49

Page 33 of 120

21

10 The emissivity of asilicon plasma out ofation equilibrium= 1 keV,
5 × 1010 cm−3 s). Showne contributions ofhoton emission (red solid
free–bound continuum
dashed line) andsstrahlung (free–freeion, red dotted line). The
emissivity is also shown,ding Si-L- and Si-K-shell
mission (based onlations made with theral code SPEX, Kaastra et
03)

Figure 2.5: The emissivity of a plasma composed solely of silicon and not in ionization
equilibrium (with parameters 𝑘𝑇𝑒 = 1 keV and 𝑛𝑒 𝑡 = 5 × 1010 cm−3 s). The plot illustrates thecontributions from two-photon emission
by the red solid line), free-bound continuum−3 , the(depicted
e ion in its final (indicatedstate. Since
σ
(hν)
∝
ν
spectralshape for hν $ χ reby the red dashed line), and bremsstrahlung (free-free emission, nrepresented bybles that of thermal
However,
if emissivity,
kT % hν,a situationthatthe bremsstrahlung.
red dotted line). Additionally,
the total
encompassing
Si-Land will
Si-K-shell line
is displayed plasmas,(Vink, 2012). free–bound emission results in narr in photo-ionizedemissions,or overionizedemission peaks near the series limits of lines. These line-resembling features ared radiative-recombination continua, which is usually shortened to RRCs. RRCs
Radiativein
CoolingPhasespectra
When 𝑇of
too low SNRs,
to ignoresuggesting
the radiative cooling,
sh becomesrecently been identified
the X-raya few mature
the expansion of the remnant is no longer adiabatic. This phase is called “radiative
presence of overionizedplasmas (Sect. 10.3).
cooling phase.” In the early phase of it, the outer region with high density is cooled bywo-photon emission results from electrons in metastable states, such as the 2sradiation. On the other hand, the inner region keeps adiabatical expansion because itof a hydrogen-like atom. Since decay to the 1s level is forbidden (because $s =has low density and high temperature. This phase is also known as “pressure-driven
can either be collisionallyde-excited (unlikely in the rarified plasmas of SNRs),
snowplow phase” because the remnant expands by the pressure of the inner ejecta
can de-excite by(McKee
emitting
two photons,
withthe associateddistributed& Ostriker,
1977). For thenon-relativistic
single energy
atomic gas(𝛾 = 5/3), the shock
two photons.
expansion of this phase has the dependency of
he different types of thermal continuum process are illustrated in Fig. 10 for the∝=𝑡 2/71. keV.
(2.10)
of a silicon-rich plasma out of equilibrium with 𝑅kT
sh e
When the ejecta is cooled further, the shock expands only by the momentum conser-

2 Non-equilibrium ionization

vation. This phase is also known as “momentum-conserving snowplow phase” (Cioffi
et al., 1988). The shock expands with the dependency of

discussion of thermal emission so far pertains to all coronal plasmas, whether∝ 𝑡 1/4 .
are found in clusters of galaxies, SNRs, or cool𝑅shstars,
with the additional comation that young SNRs can have very metal-rich plasmas. But there is anotherortant difference between the optical emission from SNRs and other hot astroical plasmas: SNR plasmas are often out of ionization equilibrium. This is usu-

(2.11)

22

2.2.2

2.2. Supernova Remnant

Thermal Radiation from Shock-Heated Particles

2.2.2.1 Contiuum Emission
Thermal X-ray spectra comprise continuum emission originating from bremsstrahlung(free-free emission), recombination continuum (free-bound emission), and two-photon
emission, the latter arising from the radiative electron transition from a metastable
quantum level. Figure 2.5 shows the three components from the plasma composed solelyof silicon. The following gives an explanation of the radiative process of bremsstrahlung,which this thesis mainly relates to, in SNRs.
Bremsstrahlung (Free-free Emission) Bremsstrahlung radiation is produced as theresult of decelerations or deflections of charged particles colliding with other particles (Rybicki & Lightman, 1985a). Since an ion with an atomic number 𝑍 is 1836𝑍
times heavier than electrons, the ions are resistant to deceleration. Thus, most of theobserved bremsstrahlung radiation originates from electrons. The emissivity of thebremsstrahlung from the plasma with the electron density 𝑛𝑒 , ion density 𝑛𝑖 , and fixed
electron velocity 𝑣𝑒 can be written as:∑︁
24𝜋𝑒 6
𝑔 ff (𝑣 , 𝜈)𝑛𝑒𝑛𝑖 𝑍𝑖2 ,3 3𝑐 3 𝑚𝑒2𝑣𝑒𝑖𝜀ff (𝑣𝑒 ) = √(2.12)

where 𝑒 , 𝑐 , and 𝑚𝑒 are the elementary charge, the light speed, and the electron mass,
respectively. 𝑔 ff (𝑣 , 𝜈) is a gaunt factor, which is defined as:√
𝑚𝑒 𝑣 3𝑔 ff (𝑣 , 𝜈) ≡
ln
.
𝜋
𝑒 2𝜈

3

(2.13)

Observing SNR plasma in practice, the velocity 𝑣𝑒 does not adhere to a single value
as previously assumed but follows a Maxwell–Boltzmann distribution as expressed in
Equation (1.21). Therefore, the emissivity from the plasma with the electron temperature
𝑇𝑒 can be obtained by integrating the radiation from electrons with different velocities:∑︁
25𝜋𝑒 62𝜋 1/2 −1/2
ℎ𝜈
𝑇
𝑔
exp
−
𝑛
𝑛𝑖 𝑍𝑖2𝑒ff
𝑒
𝑘 B𝑇𝑒
3𝑚𝑒 𝑐 3 3𝑘 B 𝑚𝑒
𝑖


𝜀ff

=

= 2.0 × 10−41





𝑘 B𝑇𝑒



 −1/2 

𝑛𝑒  ∑︁

cm−3

keV


ℎ𝜈
× 𝑔 ff exp −
𝑘 B𝑇𝑒



𝑖



 𝑛 
𝑖
𝑍𝑖

cm−3

(2.14)(2.15)

erg s−1 Hz−1 cm−3 .

Here, 𝑔 ff is a velocity averaged gaunt factor, whose approximate value can be given froma review by Brussaard & van de Hulst (1962). The typical value in the X-ray emitting
plasma can be expressed as:
𝑔 ff =

3𝑘 B𝑇𝑒 −1/2


𝜋ℎ𝜈

.

(2.16)

2. Supernova and Supernova Remnant

23

Photons cm 2 s 1 keV 1

1012

kTe = 0.5 keV
kTe = 1.0 keV
kTe = 2.0 keV

107
102
10 3
10 8
0.5

1

2

Energy (keV)

5

10

Figure 2.6: Examples of ionizing plasma model with parameters 𝑛𝑒 𝑡 = 1 × 1010 cm−3 s anddifferent 𝑘𝑇𝑒 . The model is made by the vnei model in the XSPEC tool. The abundance of eachelement is assumed to be solar composition. Solid, dashed, and dash-dotted curves represent
the model when 𝑘𝑇𝑒 = 0.5, 1.0, and 2.0 keV multiplied by 1, 5 × 104 , and 1 × 109 , respectively.

∫

Since the luminosity 𝐿 of the plasma in a certain energy given by 𝑑𝑉 𝜀ff , it is pro∫portional to 𝑛𝑒 𝑛𝐻 𝑑𝑉 when elemental abundances are known. This factor is calledemission measures (EMs), often used in spectral fitting codes.
Examples of radiations from X-ray emitting plasma with the solar composition are
given in Figure 2.6. In the case of these abundances, The bremsstrahlung by electronsinteracting with protons and helium ions is dominant. So, we observe the superposition of the radiation originating from these two particles. The emissivity depicted in
Equation (2.15) can be approximated as the following:(𝜀∝(ℎ𝜈) −0.4
exp (−ℎ𝜈/𝑘𝑇𝑒 )(ℎ𝜈 ≤ 𝑘𝑇𝑒 )(ℎ𝜈 > 𝑘𝑇𝑒 ).

(2.17)

Thus, the slope and the cutoff energy depend on the electron temperature of the plasma,
as can also be seen from Figure 2.6. Conversely, using this, it is possible to estimate
the electron temperature by analyzing the shape of the spectrum.
2.2.2.2

Line Emission

Emission lines are radiated from atoms whose electron bounded at a certain energy
level transits to a lower energy level prompted by, e.g., collisional excitation, inner-shellionization, and cascade decay following radiative recombination. The photon energy
corresponds to the difference between the before and after states. Bohr’s model can

24

2.2. Supernova Remnant
The Astrophysical Journal, 780:136 (6pp), 2014 January 10
9

10

11

6700
6600

(a) Kα centroid

6500

Energy [eV]

6400
7800

Flux Ratio

Energy [eV]log(ne t [cm-3 s]) 8

0.15(b) Kβ centroid

7400
7000

(c) Kβ/Kα flux ratio

0.1

Theoretical Values of
Rat
z

EKα
(eV)

EK
(eV

0
1
2
3
4
5
6
7
8
9
10
11

6402
6402
6402
6401
6400
6399
6399
6399
6398
6401
6405
6410

705
706
706
705
706
707
707
708
709
710
711
712

0.05
0

the theoretical v
During the calcul
Charge Number
5 keV. The temp
3. Expected
centroidof (top),the (a)andFe Fe
Kα𝛽 and
Fe Kβ andemission,
FigureFigure
2.7: Centroid
energiesof the energies
Fe K𝛼 lines
lines(b)(middle)the flux ratio only for z ! 2and(c) the Kβ/Kα
flux ratio
as a functionof thetheFechargenumber z of
Fe ions,of K𝛽 /K𝛼 (bottom)
as a functionof charge
number ofion (Yamaguchi
et al.,
2014).
becomes domina
with the corresponding ionization ages (ne t) indicated at the top. The best-fit
population of sucvalues for Tycho’s SNR are shown with the solid green lines. The dashed green
lines in panel(c) indicate
thefrom
1σ lower
upper
limits of (H-likethe observed
approximate
emissionenergies
ionsandwithan electronions).value.
When an SNR, as discusse
Thetransits
black squares
red circles
are the𝑛 values
electronfrom andquantum
numberto 𝑛 ′ , calculated
the energyusing
is: the atomic data
essentially indepof Palmeri et al. (2003) and Mendoza et al.
(2004),
respectively.
We also use the1
1
FAC to calculate the rate coefficientsof collisional
ionization and excitation. (2.18) ejecta.
𝐸 ≃ ℎ𝑐𝑅 ∞ 𝑍 2 2 − ′ 2 ,As found in Fi
The blue triangles are obtained by full calculations𝑛𝑛 using the FAC (K. A. Eriksen
et al., in preparation).
to the charge sta
where 𝑅 ∞ is Rydberg constant (= 1.00 × 107 m−1 ). Complex calculations are needed of ∼1 × 1010 cm(A color version of this figure is available in the online journal.)
to obtain the line energies from the ions in other ionization states. We can access theand the time elaline energies from the atomic database such as AtomDB (Foster et al., 2017) and SPEX
Table 1
result is consiste(Kaastra et al., 1996). Figure
shows Parameters
the centroid
Fe K𝛼 and Fe K𝛽 lines inBest-fit2.7
Spectralforenergies
the NW of
Rim
Hwang et al. 19
different ionization states and the flux ratio between the two lines. As can be seen fromfrequently been athe figure,the centroidCentroid
energies generally
increase as the ionization
Emission
FWHM
Fluxstate increases.
for the Fe K-sh
−6 photons cm−2 s−1 )(eV) of plasma can(eV)(10using
The electron temperature
be measured
the line intensity ratio
find in Figure 3+35 ratio of two lines with
+0.72
emittedion.+10The
observed
flux
energies𝐸andCr from
Kα the same5482
141
5.05
1
−12
−0.69
−45
corresponds to s
+0.47
𝐸2 canbeKαdepicted as:6012+25
Mn
141
1.73
−26
−0.46
with an ionizatio 107 ± 1
Fe Kα
𝐹6435
Ω±1 1
𝐸2 − 138
𝐸1 ± 2 
1=exp
exp (𝜎𝐸2 − 𝜎𝐸1 )𝑁H , +0.61(2.19) lower than that in
𝐹7104
Ω
𝑘𝑇160
2
𝑒
Fe Kβ
±210
± 42
5.62−0.56
flux ratio (R) is a
+0.42
Ni
Kα
7478
±
32
138
1.82
where Ω and 𝜎E are the effective oscillator strength and photo-electronic−0.43 absorption in the range z = 8
cross section at energy 𝐸 , respectively. This equation represents the modified emissivity experiences a dra
Notes. The uncertainties are the statistical component in the 1σ confidence
shell electrons (wrange. The Gaussian widths (FWHM) of the Mn Kα and Ni Kα lines are linked
as z increases. T
to those of Cr Kα and Fe Kα, respectively.
ratio for z = 13,
and Kβ centroids
0

5

10

15

20

[120]. Therefore, the f /r ratio decreases with increasing temp
in Fig. 1.3 right. These line ratios (α /β , β /γ , and f /r) have bee
SNRs, providing us with their electron temperatures.
25

0.5
0.4
0.3

Si Heα−f / Si Heα−r

0.01
5×10−30

Si Heβ / Si Heα

0.015

0.6

2. Supernova and Supernova Remnant

0.5

1

1.5

2

0.5

kTe (keV)
Figure 2.8: Intensity ratio of Si He𝛽 /Si He𝛼 lines as a function of the electron temperature 𝑘𝑇𝑒
when the ionization timescale of 𝑛𝑒 𝑡 = 1011 cm−3 s (Katsuda, 2023).

Fig. 1.3: Line intensity ratios of Si ions in an ionising plasma.
as a function of electron temperature at a fixed ionisation tim
ratio by aRight:
factor of therelative
theαtwoinstance,Same
asinterstellar
left butabsorption
for Si atHe
- flines.
/ SiForHe
α -r. theline intensity ratio of Si He𝛽 /Si He𝛼 varies strongly depending on electron temperature
𝑘𝑇𝑒 as shown in Figure 2.8.

In reality, it is not rare that the temperatures from f /r ratio
those from β /α ratios or global fittings [133, 150]. This could
In the X-ray band, the synchrotron radiation is mainly observed from electrons thattaminations of additional emission processes that will be desare accelerated to the relativistic velocities by DSA, indeed observed in many SNRs
Therefore,
weetshould
keep etinal.,mind
electrontemperature(e.g. Koyama
et al., 1995; Theal., 1996; Koyama
1997). Itthat
is emitted
due to the
influenceratiosof the Lorentz(see also& Syrovatskii
and Rybicki &(or Gforce
ratios)areGinzburg
subject
to some(1965)
systematic
uncertaintie
Lightman (1985b) for reviews). The power per frequency 𝜔 from an electron with a
can be estimated from line broadening
given velocityIon
𝛽 = temperatures
𝑣 /𝑐 ≃ 1 is:√ 3 be of 1 eV and thus canaccessible
only with high-resolutio
3𝑒 𝐵 𝛽 2 sin 𝛼𝜔𝑃 (𝜔) =
𝐹,
(2.20)
2
𝜔 cdistribution of thermal particl
The2𝜋𝑚velocity(E/∆ E ! 100).
𝑒𝑐
where 𝛼 Maxwellian
and 𝐵 are the pitchdistribution:
angle (the angle between the velocity and field) and themagnetic field, respectively. The critical frequency 𝜔 c is defined as:
!
2
m/2
π
kT
exp(−mv/2kT ).
3𝑒𝛾 2f𝐵(v)sin 𝛼=
𝜔 ≡.
(2.21)
2.2.3

Non-thermal Radiation from Shock-Accelerated Electrons

c

2𝑚𝑒 𝑐

The formuladefined as thefollowing
using
Bessel functionof orderfollowing
5/3c ) isemitted
The𝐹 (𝜔/𝜔
lines
from
ions
aretheDoppler
shiftedthis(𝐾 5/3 (𝜉 ) ):∫∞ be represented by a Gaussian distri
so that the line profile𝜔 can𝐹 (𝜔/𝜔 c ) ≡𝐾 5/3 (𝜉 )𝑑𝜉 .
(2.22)
𝜔 c 𝜔/𝜔
!
σ = E0 /c kT /m,cwhere E0 is the line energy and c is the speed of light. Combin

26

2.2. Supernova Remnant

1.0

F( / c)

0.8
0.6
0.4
0.2
0.0 0

/ c = 0.29
1

2

3

/ c

4

5

Figure 2.9: The spectral shape of synchrotron radiation from a single electron. The dotted line
shows the position of the peak energy.

The synchrotron spectrum from an electron has a shape as depicted in Figure 2.9. The
spectrum has a peak at the frequency of 𝜔 ≃ 0.29𝜔 c . The averaged photon energy fromthe electron with energy 𝐸𝑒 can be approximated as:


𝐵
𝜀≃2
10 𝜇G



2

𝐸𝑒

100 TeVkeV .

(2.23)The total emitting power can be given from the Lamor formula:𝑃synch =4
𝜎T𝑐 𝛽 2𝛾 2𝑈𝐵 ,3(2.24)

where 𝜎T = 8𝜋𝑟02 /3 is the cross-section of Thomson scattering with the classical electron
radius 𝑟0 ≡ 𝑒 2 /(𝑚𝑒2𝑐 ) , 𝑈𝐵 = 𝐵 2 /( 8𝜋) is the energy density of the magnetic field 𝐵 .
Equation (2.24) also can be given by the integral of Equation (2.20) over 𝜔 . When bothelectron and proton have the same energy 𝐸 , the power ratio is
4

𝑃𝑝
𝑚𝑒
=
≃ 9 × 10−14 ,𝑃𝑒
𝑚𝑝(2.25)

using 𝜎T ∝ 𝑚 −2 , 𝛾 = 𝐸 /(𝑚𝑐 2 ) . Therefore, the synchrotron emissivity of protons ismuch less than that of electrons. Otherwise, the electrons have high emissivity, making
them more susceptible to energy loss through radiation. This energy loss is called“synchrotron cooling,” whose timescale is given by 𝑡loss = 𝐸 /|𝑑𝐸 /𝑑𝑡 | . Because |𝑑𝐸 /𝑑𝑡 |is equivalent to Equation 2.24, the timescale 𝑡loss can be expressed as:𝑡synch =3𝑚𝑒 𝑐 2
4𝜎T𝑐𝑈𝐵 𝛾≃ 12.5𝐸𝑒

100 TeV −1 

𝐵
100 𝜇G −2yr.

(2.26)

2. Supernova and Supernova Remnant

27

When the energy distribution of particles is expressed power law as depicted in
Equtation 1.32, the photon energy spectrum can derive from the integral of the radiation from electrons:
∫

𝑑𝑛
𝜀
𝑑𝜀

𝑃 (𝜔)𝑁 (𝐸 )𝑑𝐸(2.27)= 𝜀 − (𝑠 −1 )/2 ≡ 𝜀 −𝛼 .

(2.28)=

Thus, photons also have a power-law spectrum. Its photon index Γ, which is definedas 𝑑𝑛/𝑑 𝜀 ∝ 𝜀 −Γ has a relation of:Γ =𝛼+1=

𝑠 +1

.

2

(2.29)

We can estimate the cutoff energy by combining Equation (2.23) and Equation (1.40),(1.41), (1.42):
𝜀 cutoff

age

4 3 2
∝ 𝜂 −2𝑣 sh
𝐵 𝑡 age ,(2.30)

loss
𝜀 cutoff2∝ 𝜂 −1𝑣 sh,(2.31)

esc
𝜀 cutoff∝

𝜆2max 𝐵 3 , ,(2.32)

28

3Time Variabilities of Supernova Remnant
Contents
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Changes in X-ray Spectrum and Flux . . . . . . . . . . . . . . . . . . 29
3.3 Expansion of Blast waves and Ejecta . . . . . . . . . . . . . . . . . . 34

3.1

Overview

Observations of time variabilities are typically used in studies of transient sources,
such as novae (e.g., König et al., 2022), SNe (e.g., Foley et al., 2014; Jacobson-Galánet al., 2023), gamma-ray bursts (e.g., Abbott et al., 2017), and fast radio bursts (e.g.,
Enoto et al., 2021). These studies can offer insights into the real-time changes in theirradiation processes or other states. On the other hand, it is difficult to observe rapid time
variabilities in radiation from SNRs, which have passed several hundred years since their
explosions. Considering their shocks with a high Mach number, it seems natural for theplasma to undergo rapid time variabilities. One possible reason why the variabilities
cannot be observed is that we might have missed them even though they actuallyoccur. Observatories with high-resolution imaging, such as Chandra and Hubble Space
Telescope (HST), can help us detect time variabilities in SNRs on small scales.

3.2

Changes in X-ray Spectrum and Flux

The first detection of X-ray time variability in SNR is nonthermal flux changes in theshell of RX J1713.7−3946 discovered by Uchiyama et al. (2007). Figure 3.1 shows the nonthermal X-ray brightening and decay that they discovered from high-resolution images
taken by Chandra. They found the time variabilities on a one-year timescale with the29

30

3.2. Changes in X-ray Spectrum and Flux

Figure 3.1: Flux change of synchrotron X-ray in RX J1713.7−3946 reported by (Uchiyama etal., 2007). The contours in panel (a) show the TeV gamma-ray emission obtained from HESS
measurement. Panel (b) and (c) is the zoom-in view of the yellow regions in panel (a).

sub-pc scale. The flux decay was interpreted as a result of a decrease of the electron
cutoff energy due to the rapid synchrotron cooling, while the flux increase can be causedby the boosted DSA efficiency and/or amplified magnetic field. Thus, the decay andbrightening timescale can be equivalent to the synchrotron loss and DSA acceleration
time. From the Equations (1.39) and (2.23), the acceleration time can be described as:𝑡acc ≃ 1𝜂 𝜀  0.5  𝐵  −1.5 

keV

 −2

𝑣 sh

3000 km s−1

mG

yr.

(3.1)

Whereas the cooling loss time can be written as follows using Equation (2.23) and (2.26):
𝑡synch ≃ 1.5

𝐵mG −1.5 𝜀  −0.5

keV

yr.

(3.2)

Therefore, they suggested that the magnetic field in the region where the variability
was observed is required to be amplified to 𝐵 > 1 mG, considering the measured shock
velocity 𝑣sh < 4500 km s−1 . Conversely, the time variabilities of nonthermal radiation
can be a probe for the magnetic field amplification that induces fast acceleration andsynchrotron cooling. Such rapid time variabilities in synchrotron X-rays are subsequently discovered in several other young SNRs, such as Cassiopeia A (Uchiyama &3. Time Variabilities of Supernova Remnant

31

The Astrophysical Journal Letters, 952:L3 (6pp), 2023 July 20

Grefenstette et al.

Figure 2. (Top) The background-subtracted spectrum for Epoch I (δt ≈ 4 days, black) and Epoch II (δt ≈ 11 days, red) showing the best-ﬁt model (solid lines) and
the Gaussiancomponent(dottedspectralines) demonstrating
spectral evolutionof theResiduals
to themodel. (GrefenstetteFigure3.2: Theof SN the2023ixf
in M101
at source.
4 and(Bottom)
11 daysfromitsbest-ﬁt
explosion

et al., 2023). The solid and dashed curves show
Tablethe2 best-fit model and a Gaussian component
Spectral Fits for tbabs(nlapec+Gauss)for the Fe K𝛼 line, respectively.
Epoch
Epoch I
Epoch II

NHintakT (keV)

NormbLine (keV)

Width (keV)

Normc

Wstat/dof+5267
5.6 ± 2.7

>25
+22
3412

1.06 ± 0.13
+0.2
1.30.16.45 ± 0.08
6.57 ± 0.17 +<0.2
0.45 ± 0.2

6.6 ± 214 ± 5888/843892/842

Aharonian, 2008; Patnaude & Fesen, 2009), G330.2 1.0 (Borkowski et al., 2018), and(Okuno,Matsuda,2020).
Note. Tycho’s
Uncertainties SNR
indicate the
90% conﬁdence
intervals basedeton al.,
the MCMCrun.
a1022 atoms cm−2.
nlapec normalization [10−3].
c10−6 ph cm−2 s−1; frozen since the line is narrower than the energy resolution of the NuSTAR detectors.
b

In the case of thermal X-rays, time variabilities in SNRs are limited to a few examples,
whereas they are often discovered in earlier phases after SNe. Grefenstette et al. (2023)+2.4-4 counts s−1.
10(EpochIn detected
XSPEC, our model is tbabs(nlapec+Gauss). Here
2–10
keV bandwith ata count
rate of 7.7-2.1 ´11
hard X-rays from SN 2023ixf in a nearby
galaxyM101
4 (EpochI) andthe tbabs component accounts for both Galactic absorption
We ﬁt the stacked spectrum with our ﬁducial model ﬁxing
days
from
As seenFigure 3.2,
II get Epochbrighterthan using
in aalongII)
thisline ofsight its
andexplosion.
intrinsic absorption
in theinsource.
all counts
parameters intoEpoch
the NuSTAR
I values,
The Galactic component is low enough5 (8 × 1020 cm−2) thatmultiplicative constant to allow for ﬂux variability. We ﬁnd an
Epoch
I,
which
they
interpretedasaresultoftherapidly
decreasing
absorption
based
+10
-14
we neglect it in the discussion below. Below we adopt NHint toobserved 0.3–10 keV ﬂux of 6.6erg cm−2 s−1,
6.6 ´ 10refer onto theintrinsic
absorbinganalysis.
material in the
ThelatternlapectheirspectralInSN.
thephase,≈9theearliestof the
SNR evolution,
× lowerthan stage
measuredby NuSTARduring Epoch I,
model is a “no-line,” or continuum-only, emission from a
implying some X-ray ﬂux evolution. If we limit our analysis to
SN 1987A
has
observedwith X-ray
and detected in X-rays fromcollisionallyionizedgas.been
We usethis to approximate
the observatories
the six observations that occurred during the NuSTAR Epoch I+2.9-13
bremsstrahlung
continuum
emission.
observation,
theobserved
0.3–10 et
keVal.,
ﬂux2004;
is 3.4Miceli
shock-heatedISM(e.g., Burrows et al., 2000; Michaeletal.,2002; Park
-2.5 ´ 10−2 −1
To estimate the uncertainties on the ﬁt parameters we use theerg cm s , which is consistent with the NuSTAR Epoch I
et al.,2019).
al. (2005)
reporteda rapid
theUnfortunately
0.5–2 keVthere
band
Markov
Chain
MontePark
Carloet(MCMC)
method(emcee;ﬂuxincrease
extrapolatedin
intoflux
this inband.
were noForeman-Mackey et al. 2013) as implemented in XSPEC to
Swift/X-ray
Telescope(XRT)ROSAT
observations
thatChandra.
took place during
approximately
6000
days
after
the
explosion,
based
on
data
from
and
estimate 90% conﬁdence intervals for all parameters. Table 2
NuSTAR Epoch II so we cannot repeat our analysis for thatprovidesthe also
best-ﬁtobserved
parameters andtheir conﬁdence
Theya transition
inintervals.
the soft X-ray
distribution from being localized toobservation.
No source
is listed atas
theevidence
position of thesupernova
becoming
morewidespread.
Thesefindings
wereinterpretedof theblastin the
3.2. Swift/XRT Observations
Chandra Source Catalog (CSC2; Evans et al. 2010), andwave
reaching
main bodyof thein dense
Some recent
studiesalso reported
theof thethe sensitivityof the Chandra
observations
at the positionFor
the Swift
data, whilethe
the supernova
is not detected
the full CSM.
−16
−2 −1
supernova
islisted
by
CSC2
as
8
×
10
erg
cm
s
in the0.3–10
keV
band
in
a
stack
of
the
ﬁrst
25
observations
that
span
ongoing spectral changes (e.g., Sun et al., 2021; Ravi et al., 2021). −15−2 −1
6 days with a total exposure time of 36.6 ks, it is detected in the5

0.5–8 keV band and 1.2 × 10

erg cmsin the 2–8 keV

band,
well belowinthe
NuSTAR
andin
Swift/XRT
ﬂuxes. We
Year-scale X-ray time variabilities are rarely
detected
SNRs,
even
young ones.
https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl
examined the deepest archival Chandra image of the region,
As the only example known to us, we can mention Cassiopeia A, a young Galactic SNR(Rutherford et al., 2013; Patnaude & Fesen, 32007, 2014). Brightening features identified
by Patnaude & Fesen (2014), with a size of 1015–16 cm, show correlations with optical Si
XIII images (panel a and b in Figure 3.4). Through a temporal and spatial comparison
between the X-ray and optical emissions, they discovered (1) time delays of appearance

32
No. 1, 2005

the model prediction that had successfully described the data
remarkable brightening
obtained over the previous ∼13 yr. It is remarkable that this soft
The optical/IR observat
X-ray upturn is accompanied by several other notable obserpansion measurements,
vational events: (1) The blast wave shock kinematics obtained
curves are all consistent
with our recent Chandra gratings observations indicated signifinteracting with dense g
icantly slower velocities (v ∼ 300–1700 km s"1) for the X-ray–
On the other hand, th
3.2. Changes in X-ray Spectrum and Flux
less rapidly
than the sof
emitting hot gas than shockSNR
velocities
previously estimated by
1987A
L75
than the extrapolated RO
X-ray and radio images (v ∼ 3000–4000 km s"1) (Zhekov et al.
2005). (2) The fractional contribution to the total observed Xcurve). We note that thray flux from the decelerated shock (based on the two-shock
the radio light curves (Fspectral modeling; S. Park et al. 2005, in preparation) has steadily
inates from synchrotron
hind the reverse shock (increased over the past 5 yr and became dominant after day∼6200 (Fig. 3). (3) Until day ∼5800, the radial expansion rate
between the light curveof the X-ray remnant was v ∼ 4000 km s"1 (P04). The latest
sion suggests that the
Chandra data show that the X-ray radial expansion rate decreases
behind the reverse shoc
substantially to v ∼ 1600 km s"1 after day ∼6200 (J. Racusin etshock front. However, t
al. 2005, in preparation). (4) The entire inner ring became dombetter correlation with t
inated by the optical hot spots by day ∼6000 (e.g., plate 1 in
X-ray image (Fig. 4). A
McCray 2005). (5) Recent mid-IR observations of SNR 1987A,
in the hard X-ray light cu
most likely indicating dust emission in the inner ring, show asoftening of the X-ray

Fig. 1.—Composite light curves of SNR 1987A. The ROSAT data are taken from Hasinger et al. (1996). The radio fluxes were obtained with the Australian
Telescope Compact Array (ATCA; provided by L. Staveley-Smith) and are arbitrarily scaled for the purpose of display. The solid curve is the best-fit model for
the 0.5–2 keV band light curve (ROSAT!Chandra) as determined in this work (§ 3). The short-dashed curve is the best-fit model from P04, which is extrapolated
after day 6000. The long-dashed curve is the linear model fitted to the ROSAT data only and then extrapolated until day 7000. The two dotted vertical lines markday 3700 and 6000, respectively. The inset is a log scale presentation showing the details of the ROSAT, radio, and hard X-ray light curves.

the model prediction that had successfully described the data
remarkable brightening since day ∼6000 (Bouchet et al. 2005).
obtained over the previous ∼13 yr. It is remarkable that this soft
The optical/IR observations, X-ray spectral results, X-ray exX-ray upturn is accompanied by several other notable obserpansion measurements, soft X-ray images, and soft X-ray lightvational events: (1) The blast wave shock kinematics obtained
curves are all consistent with our picture that the shock is now
with our recent Chandra gratings observations indicated signifinteracting with dense gas all around the inner ring.
icantly slower velocities (v ∼ 300–1700 km s"1) for the X-ray–
On the other hand, the hard X-ray flux is increasing muchless rapidly than the soft X-ray flux, although it is still steeper
emitting hot gas than shock velocities previously estimated bythan the extrapolated ROSAT light curve (Fig. 1, long-dashed
X-ray and radio images (v ∼ 3000–4000 km s"1) (Zhekov et al.
2005). (2) The fractional contribution to the total observed Xcurve). We note that the hard X-ray light curve is similar to
ray flux from the decelerated shock (based on the two-shock
the radio light curves (Fig. 1). The radio emission likely origspectral modeling; S. Park et al. 2005, in preparation) has steadily
inates from synchrotron emission from the shocked ejecta behind the reverse shock (Manchester et al. 2005). The similarity
increased over the past 5 yr and became dominant after day∼6200 (Fig. 3). (3) Until day ∼5800, the radial expansion rate
between the light curves in the radio and the hard X-ray emisof the X-ray remnant was v ∼ 4000 km s"1 (P04). The latest
sion suggests that the hard X-rays might also be produced
Chandra data show thatFig.
the X-rayradial0.5–2
expansion
ratebanddecreases
behindtheSNR
reverseshock (a)ratherthanDethe decelerated forward
2.—ThekeVintensity
ratio of
1987A.
2002
substantiallyto v3.3:∼ 1600kmlights"1 after
day ∼6200
Racusin
et intensity
front.
However,thewith
radio
map
does
a clearly
cember(day
5791)
toof2000
December(dayshock5038)
ratio,
overlaid
contours
FigureX-ray
curvesSN(J.1987A
and
ratio
in
the
0.5–2.0
keV
(Park
etnot
al.,show2004).
al. 2005, in preparation). (4) The entire inner ring became dombetter correlation with the hard X-ray image than with the softof the 0.5–2 keV image taken in 2002 December.
(b) 2005 July (day 6716)
Fig. 3.—Fractional contribu
inated by the optical hot spots by day ∼6000 (e.g., plate 1 in
X-ray image (Fig. 4). Alternatively, the low rate of brightening
to 2002
December(dayof5791)
ratio, overlaid
the may
0.5–2
keVresult fromthe
0.5–2.0 keV band fl
McCray 2005). (5) Recent
mid-IRobservations
SNR 1987A,
in thewith
hard contours
X-ray lightof
curve
simplythetotal
overall
image
July.ring, and(2005,
betweenopticaland on
X-ray
on thearcsecbetweenmost likely
indicating
dust taken
emissionin2005the features
inner
show (2)a displacements
softening of the X-ray
spectrumas scalean increasing
fractioninofpreparation).

the optical and X-ray features. They explained these behaviors as originating fromejecta with highly inhomogeneous density, as depicted in Figure 3.4 (c). From theirresults, we can expect to detect time variabilities on a small scale in SNRs with alarge apparent diameter.

Fig. 2.—The 0.5–2 keV band intensity ratio of SNR 1987A. (a) 2002 December (day 5791) to 2000 December (day 5038) ratio, overlaid with contoursof the 0.5–2 keV image taken in 2002 December. (b) 2005 July (day 6716)
to 2002 December (day 5791) ratio, overlaid with contours of the 0.5–2 keVimage taken on 2005 July.

Fig. 3.—Fractional contributions of the slow and fast shock components tothe total 0.5–2.0 keV band flux, based on the spectral fits of S. Park et al.
(2005, in preparation).

The Astrophysical Journal, 789:138 (21pp), 2014 July 10

Patnaude & Fesen

3. Time Variabilities of Supernova Remnant

33

The Astrophysical Journal, 789:138 (21pp), 2014 July 10

Patnaude & Fesen

Figure 5. Comparative emission changes of Cas A’s X-ray and red optical emission over 31 and 60 yr, respectively. Upper panels: Einstein, ROSAT, and Chandra
ACIS-S images of Cas A showing an apparent increase of clumpy emission features from 1979 to 2011. The Chandra image has been smoothed by a 9 pixel Gaussian to
Figure7. Comparison
X-ray
and
emission for Feature “A.” Upper panels: Einstein, ROSAT, and Chandra X-ray images with a circle 10!! in diameter centeredapproximate the resolutions of the Einstein and ROSAT images. Lower four rows: Cas A in optical broadband
red emission
images fromof
1951
to 2011.
Theoptical
1951–1989
images are Palomar Hale 5 m plates PH563B, PH3033S, PH5254vB, PH6249vB, PH7252vB, PH7766vB,
PH8192vB,
PH8206vB,
the 1992location.
and 1998 images
on Feature
A’s
approximate
Lowerarepanels: images of coincident optical emission for this feature. Note: the circles are centered on the emission feature and
MDM 1.3 m and 2.4 m images, while the 2004 and 2011 are HST ACS F625W+F775W and WFC3 F098M images (see Table 1 for details). Note the considerable
followthefeature’s
propermotion.
In
the1976
and 1992 images, the optical knot is seen to cross in front of, or behind, a brighter stationary QSF emission knot.
brightening of the remnant’s optical emission along northern and southern limbs, most dramatically seen between the early 1950s and the 1970s, but continuing up to
the present.
(A color version of this figure is available in the online journal.)

(a)

(b)

The Astrophysical Journal, 789:138 (21pp), 2014 July 10

Patnaude & Fesen

Figure 6. 2004 Chandra ACIS-S image of Cas A with five regions marked that showed significant brightening.
Figure 8. Comparison of X-ray and optical emission for Feature “B.” Upper panels: 2000–2013 Chandra X-ray images of Feature B. The red circles (dia. = 10!! )(A color version of this figure is available in the online journal.)

9(c)

are centered on Feature “B.” Lower panels: optical images of the area around Feature “B” at the same scale as the X-ray images. The ejecta knot responsible for the
X-ray emitting Feature “B” brightened considerably between 1999 and 2002 in the optical and between 2000 and 2004 in X-rays. The optical image sections shown
are approximately centered on the emission feature but do not follow the feature’s proper motion.
(A color version of this figure is available in the online journal.)

Figure 9. Comparison of X-ray and optical emission for Feature “C.” Upper panels: X-ray images of emission Feature C. The red circles have a diameter of 10!! on
the X-ray images. Lower panels: optical images. The red circle shown here has a diameter of 10!! . The optical images are: Palomar 5 m images PH3033S, PH5107,
PH6249vB, PH7252vB, PH8192vB, PH8206vB, an MDM 1.3 m image taken using a broad [S ii] interference filter, and an MDM 2.4 m image taken with an R filter.
The optical image sections shown are approximately centered on the emission feature and follow the feature’s proper motion northward. These images show a dramaticincrease in X-ray brightness between 2000 and 2011 and a sharp increase in optical brightness between 1989 and 1992.
color
versionof thisis available
in theonlinejournal.)downstream from X-ray emission, which appears closer to the
Figure 21. Schematic representation of(A
dense
knot
ablationthatfigure
leads todense opticalknots
positioned
reverse shock. In panel a, the unshocked knots are overtaken by the reverse shock. In panel b, a radiative shock is driven into the knots, and material is ablated off thesides of the knots, due to Kelvin–Helmholtz instabilities. In panel c, the ablated material appears as tails in optical emission
10and as diffuse trailing emission in X-rays.
Due to the lower velocity of the ablated material, it remains closer to the reverse shock position.

Figure 3.4: (a): The Chandra images of Cassiopeia A. (b): Zoom-in view of the time variablefeature marked as “B” in panel (a). (c): Schematic of the density profile of the time variable
regions do not seem to have obvious positional or temporalmaterial. In both sets of cases, a positional offset occurs due
offsets which might be attributed to ablation of dense ejecta
mass ablation of&lessdense material
from the ejecta knots
features to(Patnaude
Fesen,
2014).
(bullets) which is then shock heated to higher X-ray emitting
temperatures.
4.5. Regions with Coincident X-Ray and Optical Emission
As described in Section 3.2, certain regions are bright inboth optical and X-ray emission. These cases may be instanceswhere optical clumps are embedded within or associated with anextended, lower density medium like that seen in Figure 12. Boththe dense and less dense regions are shocked at the same time.
The optical emission becomes bright and as it advects away fromthe reverse shock, the lower density region becomes bright as
well, in X-rays. This is shown schematically in Figure 20. Whatdifferentiates this scenario from that presented in the previous
section may be that the maximum density of the opticallyemitting material is possibly lower than in the very dense knots,
and the contrast between the dense and diffuse components isless (or the transition between dense and diffuse components
more gradual). It could also occur if we are viewing shocked
ejecta with significant mass ablation along our line of sight. We
note that Figure 20 can qualitatively describe both regions thatshow coincident X-ray and optical emission, as well as regionsthat show a time delay between X-ray and optical emission, if thedensity contrasts and knot envelope structure are appropriately
chosen.
4.6. Regions with Neither Temporal or Spatial Correlations
Just as some regions have coincident optical and X-ray
emission, there are other regions in the Cas A remnant thatexhibit emission only in either optical or X-ray bands. Theseknots.
One such region lies in the remnant’s northwest limb
(Figure 17) where the advance of Cas A’s reverse shock front
can be readily identified in optical emission (Morse et al. 2004).
Between 1976 and 1992, the optical emission from this region
has brightened substantially. This is in contrast to the X-ray
emission (upper panels of Figure 17), which show no clearly
discernible change between 1979 and 2011. Morse et al. (2004)
estimated preshock densities in this region of 25–2500 cm−3 .
Ram pressure equilibrium thus requires that the velocity of thisshocked high-density component to be ≈20–200 km s−1 . With
these densities, it is not surprising that there is no correlated
X-ray/optical emission in this region especially since it has
evolved into a nearly continuous filament of optical emission.
In contrast to the optically bright region in the northwestwith no corresponding X-ray emission, a completely opposite
situation exists in the northeast near the base of the remnant’s
NE jet. Shown in Figure 18 is a region of brightening X-ray
emission with no corresponding optical emission. Between 2000
and 2010, the filament has brightened substantially, with little
or no corresponding optical emission seen. Using Chandra
observations between 2000 and 2012, we modeled both the
Si–K 1.75–2.0 keV line flux and line centroid and found thatthe flux has increased by nearly a factor of three in 12 yr, andthe line centroid has increased by 10 eV over the same period,
indicating an ionizing plasma (see Table 3).
While these two cases are quite different, the underlying causeof the differences may again be attributed to ejecta density. In
the case of the X-ray emission in the northeast, lack of opticalemission here could be due to the ejecta density being too low

The Astrophysical Journal Letters, 865:L13 (5pp), 2018 October 1

Williams et al.

unquestionable, and it appears in nearly the same location inboth epochs. But Figure 2 shows why any localization
algorithms would not be able to report a location to theaccuracy that we require. This source contains only about
15–25 photons
34 in total, depending on the epoch, which is
3.3. Expansion of Blast waves and Ejecta
nowhere near enough counts to get a two-dimensional centroid
accurate to the sub-arcsecond level. For example, using
Journal Letters, 865:L13 (5pp), 2018 October 1
Williams
Equation (13) from Kim et al. (2007), The
we Astrophysicalﬁnd positional
uncertainties in the source shown in Figure 2 of 0 44 and 0 35
in the ﬁrst and second epoch, respectively. The accuracy of thelocation is of utmost importance here, as the signal we aresearching for is so small. For reference, even with Chandra anda time baseline of 17.4 years, a 5000 km s−1 blast wave at a
distance of 50 kpc would move only ∼0 37 during that time,or about 3/4 of a Chandra pixel.
We explored other options as well. There are a few knots ofemission near the center of the remnant that could, in principle,
serve as markers for alignment.11 However, not only are thesesources relatively diffuse (several arcseconds in extent), there isalso no way of knowing that these knots do not have their own
motion or slightly varying surface brightness proﬁles over the17.4 years of evolution of the remnant. We also opted not to
Figure 2. One of the sources detected in the ﬁeld of view, loc
use CIAO’s sub-pixelization algorithms. There is too muchapproximately 1 5 from the remnant. The counts image from theobservationis shown on the left, while the 2017 image is shown on the rpotential for the introduction of a signiﬁcant systematic
Eachimage is shown at the native Chandra pixel scale of 0 492. The souncertainty by reducing the pixel size on the same length scale
contains 16 counts in the 1999 data and 25 counts in the 2017 data.
as the signal that we are trying to measure.
Ultimately, the uncertainties involved in obtaining image
in identical fashion with the chandra_repro script in CI
registration down to a fraction of a pixel led us to focus on5pp), 2018 October 1
Williams et al.
version 4.9 (using version 4.7.3 of the CalDB).
measuring one thing that does not require knowledge of the
Figure1.
Chandra
X-ray
image
of
N103B,
with
0.5–1.2
keV
in
red,
WCS
registration:theremnant’s
diameteralong
variousaxes.
the same location in
1.2–2.0
in green, ofand 2.0–7.0 keV in blue, overlaid with our four proﬁle2.1. Image Registration and Alignment
As we show below, we chose to measure
the keVdiameterhy any localization
extraction regions, as described in the text. Each region is 10 pixels wide,
formingt a locationN103B
to thein four directions (see Figure 1), effectively
where a pixel is the native Chandra pixel scale of 0 492. The scale bar at the
The standard methods for aligning images in the w
twoabout
orthogonal cardinal coordinate systems.
Beforebottomis 30″ inmaking
length.
contains only
coordinate system (WCS) for a proper motion measurem
we made one ﬁnal ﬁltering of the data.
he epoch, thesewhichmeasurements,isinvolve either registering point sources detected in the im
ACIS array has suffered signiﬁcant degradation
low in image units to the more useful physicalmeasurementsatmade
dimensionalThecentroidwith known sources from external catalogs or aligning−1
energies
launch.
At slow, taking advantage of the known distance toquantityof km
For example,
using due to contaminant buildup sincecommon point sources within the images from each epo
energies,thedifference
betweentheeffective
area
in
1999
and
the LMC of 50 kpc (Pietrzynski et al. 2013).
we ﬁnd positional
allowing for at least a relative alignment. While the for20170 is35quite signiﬁcant. Thus, we only considered counts at
method is obviously preferred, X-ray analysis often relies
e 2 of 0 44 andenergies
the latter, due to the relative paucity of point sources in
The accuracyof theabove 1.7 keV (up to 7 keV), high enough to ensure a 2. Observations
similar effective area to the 1999 observations while still
X-ray band with known optical counterparts. N103B iss the signalnearly
we arecapturing
LMC remnant; however, its location (R.A.=05h08m
en with Chandra
and the strong ∼1.8 keV Si Kα line. We conducted a new epoch of Chandra imaging observadecl.=−68°43′34″, J2000.) is well outside the main bations of N103B in the spring of 2017, with a total of 400 ks
s−1 blast wave at a
the LMC and unfortunately, the number of point sources in
spread over 12 separate observations between March 20 and
37 during that time,
3. Measurements and Discussion
ﬁeld of view is quite low. We restricted ourselves to p
June 1. We used the ACIS-S array for these observations,sources on the S3 chip (within 4′ of the remnant), beca
placing the remnant (only ∼30″ in diameter) close to the centerre are a few knotsofWe measured the diameter of the remnant using the radial
Chandra’s point-spread function (PSF) degrades quicklyof the optical axis of the telescope on the S3 chip, whereat could, in principle,
proﬁles (obtained by using projection regions in ds9) alongfunction of off-axis angle.
Chandra’s spatial resolution is best. The 1.7–7 keV image fromer, not only four
are these“diameter” regions, shown in Figurethese
1. These
regions
To search for point sources in the events ﬁles from the2017 observations is shown in Figure 1. To measure theds in extent),
there
issample the entire brightness proﬁle of theremnant
alongepochs, we ﬁrst used the CIAO task wavdetect, as rec
expansionof theremnant, we compared our 2017 observationso not have theirown covering position angles 0–180, 45–225, 90–270,
diameters
mended by the Chandra X-ray Center. This task “found” awith the earliest epoch: a 1999 December 4 observation foress proﬁlesandover135–315.
theWe made no attempt to scientiﬁcallyadozen point sources in the image, but most were false positi
40 ks (PI: deﬁne
G. Garmire).
Our method for ﬁtting the proper
We also opted
nottocenter of the remnant, because the site of
the explosionisA relatively simple search by eye conﬁrmed that only ﬁv
motioninvolves shifting
one epoch with respect to another,
There is toomuch We simply drew the diameter regions
unknown.
to run forthroughthese sources were real and detected in both epochs. Us
accounting
the uncertainties in each epoch. This technique
gniﬁcant systematicthe geometric center of the circular structureof N103B.
these ﬁve sources as input, we created a transformation ma
is described
moreTofully in Williams et al. (2017) and Katsuda
he same lengthscale
obtainenough signal for a robust proﬁle, eteachregionbut
is brieﬂy,
10
ﬁle using the wcs_match task, then used wcs_update to aal. (2008),
we extract brightness proﬁles from thee.
pixels wide, or ∼5″. The normalized image
brightness
proﬁlesthe 1999 epoch 1 image to the 2017 epoch 2 image.
in unitsof counts, with the square root of the number ofin obtaining
image from each diameter in the two epochs
extracted
shownin
Unfortunately, the results of this alignment were not accu
countsareas theuncertainty
on each pixel. One epoch, generally
l led us to Figure
focus on3. We are not concerned with thethesmallchangesin as the “reference” epoch, with the otherenough for a robust measurement. When we attempted
second,is usedre knowledgeof thestructure, only the change in theinternal
diameterthethe total χ2 value is minimized.
measure the proper motion of the leading edge of the emiss
epoch
shiftedofuntilalong various
axes.as marked by the sharp rise of the shock
remnant
do we are
The front.
effect Tothatmeasuringis proﬁlesquite ofsmall(presumably
shock
varied substantia
Figure3. Brightness
epoch(sub1 (1999) in blueand epoch 2the(2017)
in front), our results
ure the diameter ofarcsecond, see Section 3),redso
tookregions
care tominimizeor for display
with results
15,000 km s−1 in some places
for wediameter
1 and
2, normalizedpurposes.approaching
The gray−1
shadedmark (Williams
the regions ininet
which
ﬁts were
performed.
effectively11forming
negative
5000 kmsThe in others! We re-did the alignment useliminate
anymatter
potential
sourcesofregionssystematic
uncertainty
oural.,the2018).
From the Figure
point of viewof image
it isofa SNR
simpleto from
3.5:
Thetransformation,
expansion
N103B
1999
to 2017(Left):
Chandranormalization
adjusted thefor each
The proﬁles
run north-toarbitrarily deﬁne any point one wishes as a “ﬁxedsource” for alignment
ems. Beforesimply
makingCIAO
tool, srcextent, but the results were simil
measurements.
First, we opted
not towas
combinedatashaded
from region.anotherregionﬁle,
1 keV
and
(left1.2–2.0
to right) forregions
2–4.
the twoepochs.
X-ray
image of N103B withtheenergy bandof for
0.5–1.2(red),
keV(green),and 2.0–on location within the remnant.
wildlyvarying depending
our 12
observations
into a south
singleevent
aseast–west
this
could
createﬁltering of between
the data.
Upon further inspection, we concluded that most of the pbiases in the resulting FITS ﬁles at the sub-pixel level which
degradation at low 7.0 keV (blue). Green boxes
are the regions for extracting profiles. (Right): The
extracted profiles
sources (all but one) used for alignment in both the wavde
would be difﬁcult, if3 not impossible, to quantify. For our
nce launch. At low
andsrcextent methods do not have a strong enough detec“reference”
frame,
weusedthedeepestsingleobservation
fromive area in 1999 and in 1999 (blue) and 2017 (red).
for these algorithms to ﬁt a PSF and determine an accu
our 2017 observations: a 60 ks observation (ObsID 19923)
considered counts at
location. As an example of what we mean by this, we show
begun on April 26. These data, along with the 1999 data (which
h enough to ensure aof our sources in Figure 2. That this source is rea
combine for a time baseline of 17.4 years), were both processed
servations while still.
2

ussion3.3

Expansion of Blast waves and Ejecta

Since a typical young SNR expands with a velocity of about 103–4 km s−1 , we can observe

nant using the radial
the expansion of a nearby (∼ 1–10 kpc) SNR over ten years in arcsec scale. Recent
gions in ds9) along
re 1. These regions observatories with excellent angular resolution (e.g., Hubble Space Telescope, Very
the remnant along
0, 45–225, 90–270, Largy Array, and Chandra X-ray Observatory) have allowed us to measure the proper
cientiﬁcally deﬁne a motions in parts of SNRs. The proper motions of metal-rich ejecta over many yearsof the explosion isgions to run throughcture of N103B. Toe, each region is 10
brightness proﬁles
epochs are shown inhe small changes in
he diameter of the3. Time Variabilities of Supernova Remnant

35

The Astrophysical Journal Letters, 837:L7 (6pp), 2017 March 1

Borkowski e

Figure 3.6: Chandra image of G1.9+0.3 in the 1.2–8.0 keV band and proper motion vectors of eachfeature (Borkowski et al., 2017). The color bar represents the deviations from radial in degrees.
The length of the white allow corresponds to 0′′. 25 yr−1 .

enable us to estimate the center of the explosions and explosion age. These values
play an important role in studying the compact objects (Katsuda et al., 2018; Tsuchioka
et al., 2021) or companion stars (Kerzendorf et al., 2019; Li et al., 2021). For example,
using optical observations, Banovetz et al. (2021) estimated a center of explosion of
1E 0102.2−7219 from a measurement of 120 O-rich ejecta knots.

Although tracing the proper motions of fine structures is challenging in the extragalactic SNRs, Williams et al. (2018) estimated expansion velocity through changes inthe diameters of SNR 0509−68.7 (N103B) in the Large Magellanic Cloud (Figure 3.5). The−1measured expansion velocity is 4170+−1280
1310 km s , suggesting an undecelerated age of 850
years and a somewhat younger real age. Expansion measurements are also reported in
Figure 3. Proper motion vectors overlaid on the 2011 (top), 2015 (bottom right), and the transformed 2015 (bottom left) images. They are color-coded according to
other
SNRs,
suchcenteras 1Eof0102.2−7219marked(Xi etbyal.,
0509
67.5 (Hovey
deviations in direction fromradialextra-galactic(with respect to thegeometrical
the remnant,the2019),
red cross),in −degreesaccordinget
to al.,
the vertical scale. The w.
arrows indicate 0. 25 yr−1
2015; Helder et al., 2010; Roper et al., 2018; Guest et al., 2022), 0519−69.0 (Guest et al., 2023).
(The data used to create this ﬁgure are available)

We can obtain further insights from the expansion of Galactic SNRs because we canaccurately measure shock velocities and discuss their spatial variation. The study ofthan the overall median of 0. 28 yr−1 (11,000 km s−1). Among(0. 29 yr−1). The latter corresponds to a very bright X-ray k−1
all regions with m < 0. 17 yr , most (12 out of 14) are withinwith motion strongly deviating from radial (by 42°), while
the northern rim, predominantly toward the north and northeast
former is the northernmost of ﬁve fast-moving knots/ﬁlame
where radio emission is particularly bright. This includes
that are mostly located near where G1.9+0.3 is the brightest
regions f (0. 16 yr−1), g (0. 13 yr−1), h (0. 14 yr−1),
radio.
i (0. 095 yr−1), k (0. 11 yr−1), l (0. 089 yr−1), and n
The motions along the northwest outer radio contour (regio

e absolute astrometric accuracy is achieved to2
angulardistancev at
epochsand 2005,
2, andandj1 (v)andj2 (v) repspectively.
Using
the criteriaof xmin" 2.7
shiftedprofilewith theprofilesof 12003,
2007.
nInour
X-rayanalysis,
dataassets
(MOS1
and
MOS2
such
bad
columns
or of
gaps
ofthe
CCD
chips.
the fourfollowing
we takeaccount22
resentl theuncertainties
at and
each
bin.
The
minimumforx than(l), 2!, at 90
We
examine
p
0!,
2!,
4!,
6!,
8!,
10!.
The
x
valuesasthevaluesoftheshiftslessrewell withinaroundtheirmean posiposition
accuracy2!ascircles
theconservative
systematic
Next,
we quantitatively
measurethe2shift of the X-ray fil, isaroundsince we40thebinsroughlyxmin
a functionofl areshown39,
in Figure
3. Insumfigure,threekindsintheallcases.
Witharoundtakingthe
into accountore, without extraamentcorrections
ofthecoordinates
based on one-dimensional
profilesacrossthefilament.
of
data
pointswithrectangular,
circular,ortriangular
marks
shock
front.
Weshift
theradialprofileof
2001
andcomparecertainty, we derive the shifts of the X-racethe 2001astrometric
andWe
2007
imagesof 1.5–8
keV
theofabsoluteaccuracy
is achieved
selecta northern
portion toofare,thenarrow
NW
filament
respectively,
responsible
forasthreecases in ofwhich
we focus
the shifted
profilewiththe profiles
2003,
2005, and 2007.
inFigure1(right).
We
canclearlyseeablack
2!. In the following
analysis,
we takeaccount
ofon few
shown
in Figure
1, since
we findbadexamine
columns
there.
Wea different
two epochs,i.e.,2001-2003,
2001-2005,
andx 2 values as
We
l
p
0!,
2!,
4!,
6!,
8!,
and
10!.
The
ow
line runningfrom asthethenortheast(NE) tosystematicthe2
e position
accuracy
conservative
2001-2007.
We findoccursatlp
0!,4!,andandEjecta
6! for three kinds
a function
ofthatl arexmin
shown
Figure3. In
thefigure,3.3.
Expansion
ofinBlastwaves) as a sign of the 36
expansionof the shock front
3
each
two
epochs2001-2005,
and
2001-2007,
reVizier Online Data Catalog, I/297 (N.
Zacharias
et al., 2001-2003,
2005).
of data points with rectangular,
circular, or triangular
marks
black or white lines are due to artificial effects2
enceof the 2001 and 2007 images of 1.5–8 keVspectively.
Using
the
criteriaofxmin " 2.7 , we can determine
lumns or gaps of CCD chips.
are,respectively,
responsible
for
three
cases
in
which
we
focus
n in Figuremeasure
1 (right).
seefila blackthe values of the shifts less than 2!, at 90% confidence level
antitatively
theWeshiftcanof clearly
the X-rayonadifferent
twoepochs,i.e.,
2001-2003,
2001-2005,
andthe systematic unline runningprofiles
from thenortheast(NE) to thein all the cases. With taking into account2
nrrow
one-dimensional
acrossthe filament.
2001-2007.
We
find
that
x
occurs
at
l
p
0!,
4!,
and
6!
for
we derive the shifts of theminX-ray filament in 2003,
W)
as aportion
sign ofthe shockrthernof thethe expansion
narrow NWoffilament
as frontcertainty,
each two epochs 2001-2003, 2001-2005, and 2001-2007, reerlinesbad
arecolumns
due to artificial
e 1,blacksinceorwewhitefind few
there. Weeffects
2
spectively. Using the criteria of xmin
" 2.7, we can determine
columns or gaps of CCD chips.
thevaluesoftheshifts
less
than
2!,
at 90% confidence level
Data Catalog, I/297measure(N. Zachariasal., 2005).
quantitatively
theet shiftof the X-ray filin all the cases. With taking into account the systematic unon one-dimensional profiles across the filament.
certainty, we derive the shifts of the X-ray filament in 2003,
northern portion of the narrow NW filament as

ure 1, since we find few bad columns there. We

e Data Catalog, I/297 (N. Zacharias et al., 2005).

Fig. 1.—Left: XMM-Newton 1.5–8 keV band image obtained in 2001. The
image is binned by 5! and has been smoothed by Gaussian kernel of j p
10!!. The intensity scale is square root. Point-source positions which we use
to examine the astrometric accuracy are indicated as P1, P2, and P3. We
investigate the radial profile of the X-ray filament in the rectangular area.
Right: Same image but subtracted one obtained in 2007. The intensity is linearly
scaled from !1.5 # 10!4 to "1.5 # 10!4 counts s!1 pixel!1.

MM-Newton 1.5–8 keV band image obtained in 2001. The
y 5! and has been smoothed by Gaussian kernel of j p
scale is square root. Point-source positions which we use
trometric accuracy are indicated as P1, P2, and P3. We
ial profile of the X-ray filament in the rectangular area.
but subtracted one obtained in 2007. The intensity is linearly# 10!4 to "1.5 # 10!4 counts s!1 pixel!1.

Fig. 2.—Radial profiles at each epoch in the 1.5–
a 2! scale. Four profiles from top to bottom are re2001, 2003, 2005, and 2007. [See the electronic edi
color version of this figure.]

Fig. 2.—Radial profiles at each epoch in the 1.5–8 keV band, binned witha 2! scale. Four profiles from top to bottom are responsible for the one in2001, 2003, 2005, and 2007. [See the electronic edition of the Journal for acolor version of this figure.]

XMM-Newton 1.5–8 keV band image obtained in 2001. The
by 5! and has been smoothed by Gaussian kernel of j p
Figure3.7: Thepositions
shock propermotionin RX J0852.0−4622 (Vela Jr.) (Katsuda et al., 2008a). Lefty scale is square root.
Point-source
which weuse
top:in thekeV bandobtainedprofiles
in 2001.atThe
represents
thekeVregionastrometric accuracy
areXMM-Newton
indicated as P1,image
P2, andP3. 1.5–8.0
We
Fig. 2.—Radial
eachboxepochin the 1.5–8
band, binned withadial profile of the for
X-rayfilament thein thearea.
a An
2! scale.
profilesfrom top fromto bottom
are2007.
responsible
extracting
fluxrectangularprofile. Top-bottom:
imageFourin 2001
subtracted
one inRight:for the one inge but subtracted one obtained in 2007. The intensity is linearly
2001, 2003, 2005, and 2007. [See the electronic edition of the Journal for a
!4
!1
The
each
year
boxversion
in theofleft-top
panel.
5 # 10!4 to "1.5 #10profilecountsins!1
pixel. extracted from thecolor
this figure.]

G1.9+0.3 by Borkowski et al. (2017) can be cited as an example that revealed large spatial
variation in the remnant expansion (Figure 3.6). They suggested that the slow shocks
are caused by the collision of SN ejecta with a dense CSM, indicating the asymmetric
distribution of CSM. Such indications of ambient material are also discovered from the
expansion measurements of other SNRs, including Kepler’s SNR (Katsuda et al., 2008b),
Cassiopeia A (e.g., Sato et al., 2018), SN 1006 (Katsuda et al., 2013), and RX J1713.7−3946(e.g., Acero et al., 2017; Tanaka et al., 2020).
Accurate measurement of the shock proper motion also enables us to determine theremnant age and distance. Katsuda et al. (2008a) reported an expansion measurement in
RX J0852.0−4622 (Vela Jr.) with XMM-Newton image in the 1.5–8.0 keV band (Figure 3.7).
Their result indicated its expansion rate of 0.023% ± 0.006% per year, which is five times
lower than those of young SNRs less than 1000 years. Such a slow velocity suggested that
Vele Jr. was younger (1700–4800 years) than had been commonly expected. Additionally,
they estimated the distance to the remnant to be ∼ 750 pc under an assumption ofhigh shock velocity ∼ 3000 km s−1 .
Tanaka et al. (2021) reported the proper motion of the synchrotron X-rays at theforward shock of Tycho’s SNR using Chandra data over 12 years. Figure 3.8 shows the3. Time Variabilities of Supernova Remnant

37

The Astrophysical Journal Letters, 906:L3 (7pp), 2021 January 1

Tanaka et al.

The Astrophysical Journal Letters, 906:L3 (7pp), 2021 January 1

Tanaka et al.

Figure 2. Velocity and acceleration of expansion of Tycho’s SNR. The horizontal axis shows the azimuthal angle in degrees from the north in counterclockwise

direction.
numbers
top correspondto observations
region identiﬁers
deﬁnedinin 2009.
FigureNorth
1(a). The
topandpanel
Figure 1. (a) Chandra ACIS image of Tycho’s SNR in the energy
band ofThe4.1–6.1
keVonobtained
from theperformed
is up,
eastshowsis to velocities measured for the intervals of 2003–2007 (black(red ﬁlled
circles),(blueTheﬁlled
For aredirectwith the result by Williams et al. (2016), velocities from thephoton2003–2009
cm−2 s−1 witha pixelsize ofand
0 2009–2015
492 × 0 492.
redcircles).
rectangles
thecomparison
regions used
the left. The color scale indicates ﬂux from each pixel in units ofﬁlled
10−9circles),are also
plottedare
(cyan
small
open circles).
Plottedthe bottom
panel areaccelerations
estimated under an assumption that they are constantfor the expansion measurements. (b) Radial proﬁles observed ininterval
Regionof1.2003–2015
The black andred points
fromobservationsin 2003andin2009,
respectively.
The
origin
over
the
time
periodbetween
2003
and
2015.
The
distanceto
Tycho’sSNR
isassumed
to
be
2.5
kpc
in
converting the angles to physical lengths.
of the horizontal axis corresponds to the location of the shock front in 2003. (c) The same as (b) but for Region 8.

Figure 3.8: The decelerations of the blast waves in Tycho’s SNR reported by (Tanaka et al., 2021).
The red regions in the left panel showTable
the2 regions for the shock velocity analysis. The right-top
found thatmeasurementsof the radiating electrons. Such an effect may be able to
Statistics
from theExpansion
Measurements agree well with each other.
panel shows the shock velocity
in eachwetwo
interval
ofrapid
thedeceleration
years. The
right-bottom
showstheradio and X-ray results.
Furthermore,
conﬁrmed theeven when
account for panel
the differences
betweenχ(DegreesofFreedom)
weappliedtheanalysis
methodbyWilliamsetal.
(2016).
It Signiﬁcanceis notable that, after the deceleration, the expansion
Region ID
acceleration 2003–2007
rate in each region.
−1
Expansion of Tycho’s
SNR has also been measured
velocities(σ)
reached ∼0 3 yr regardless of direction (Figure 2).
2003–20092009–2015
2003–2015in the
2
min

a

radio band by several authors such as Strom et al. (1982),

This can 0.1
be interpreted as a consequence of the blast wave
hitting a 0.1
dense gas wall surrounding the SNR with approximately uniform
density, i.e., a cavity wall structure. We
0.6
performed4.6one-dimensional spherically symmetric hydrodynamical simulations
to test this assumption using the Virginia
3.7
Hydrodynamics
1 (VH1) code (Blondin & Ellison 2001).
3.4
Following5.4a previous successful multi-wavelength model forthis SNR6.7
by Slane et al. (2014), we initiated the simulations
using an SN
6.4 ejecta with a mass of 1.4 Me, an explosion energy
51
of 10 erg,
4.2 and an exponential density proﬁle. The ejecta ﬁrst
expands 2.9
into a cavity with a uniform number density−3
1.2 . The deceleration of the blast wave begins as it
nb = 0.3 cm
2.6 gas cloud, which is modeled as a density jump at ahits a dense
radius Rc from the explosion center with a spatial density
the blast waves.
gradient dnc/dr. We assume a uniform density nc = 100 cm−3
The systematics between the radio and X-ray results could be
Note.
a
Signiﬁcance with which a hypothesis of constant velocity between
2003
and
2015
is
rejected.
inside the wall, but our results are not sensitive to this because
arising from observational or analysis methods, but could also
the blast waves at all regions experiencing a deceleration are
be ascribed to physics related to the radio and X-ray emissions.
found to be still climbing up the density gradient currently.
Electrons emitting radio synchrotron photons
have a much
4.
Discussion
4–10 as presented in Table 2, indicating signiﬁcant decelerationin
We present the simulation result compared with thelonger loss timescale than those emitting synchrotron X-rays.
these regions. To our knowledge, this is the ﬁrst Thus,
detectionofWe
discovered
rapid
deceleration
of
the
blast
waves ofobservational
data in Figures 3(a)–(d). Both Rc and dnc/dr atradio-emitting electrons would accumulate in the downongoing deceleration of blast waves of SNRs. We notethatthe
Tycho’sSNR
byanalyzing
X-ray
data
obtainedwith
Chandrathe interface
between the cavity and the wall are treated asstream region, which creates a plateau-like proﬁle there (Slane
2003–2007 and 2003–2009 velocities are not etindependentin
2003–2015.
Similarexpansion
measurements
werepermodel parameters to ﬁt the observational data. We tried the ﬁve
al. 2014). As the shock expands, the number of electrons= 0, 30, 100, 300, and 1000 cm−3 pc−1, for eachcases, dnc/dr
downstream
wouldetbeal.
increased,because they were both derived using the same accumulated
2003 data. in the formedby region
Williams(2016),andwho obtained
shock
region,inandfound that the data from Regions 1–4, 5–9, andthe location
radio emission
peakcomparisons
would graduallybe
Likewise, the 2003–2009 and 2009–2015 measurements
are not of thevelocities
based onof Chandra
images2000/
11–13
are best
shiftedtowardthe Asother
hand, check,
X-rays we
would
independent for the same reason. However, we emphasizethatthe downstream.
2003 and On
2015.
a sanityplotted thevelocities
for ﬁtted with the dnc/dr = 1000, 300, and−3−1
be emitted
right at thethe shockthe muchshorter lives
2003–2007 and 2009–2015 measurements do not use
the sametime becauseintervalof of
2003–2015
in Figure100
2 cm
for pcdirect, respectively. In each region, we interpret that

1
2
3
4
5
6
7
8
9
10
11
12
13

18.0 (22)11.3 (20)33.7 (35)68.3 (47)38.3 (40)48.3 (40)38.2 (40)42.1 (40)45.4 (25)
13.5 (15)20.1 (15)29.6 (40)25.5 (27)20.4 (22)18.9 (22)17.0 (22)Reynoso et al. (1997), 18.5
and (20)
Williams et al. (2016).
The radio
14.0 (20)
13.5 (20)results
lower velocities
than our
76.5(35) tend to point to systematically51.7 (35)47.9 (35)
X-ray
show
high as
68.1(47) results, and do not
73.2(47) expansion velocities
71.6as(47)
what
we
observed
before2007.
Accordingtotheresultsby45.7 (40)87.6 (40)52.2 (40)
Reynosoetal.
(1997),
theshock
velocities
measured
between58.3 (40)62.5 (40)47.1 (40)−1−1
1983/1984
and 1994/1995
41.1
(40)
46.5 are(40) 0 38 yr and 0 35
45.9yr(40) in thelocationscorresponding42.8
to Regions
5 and 8, respectively.
36.0
(40)
(40)
32.0 (40)These
values
2003–2009
results than the
2003–2007
45.6(25) are closer to our 31.9
(25)
44.7
(25)
values.
the(15)possible
14.6(15) However, even without
13.5 (15)taking into account
15.5
systematiceffects between
23.2
(15)
12.6 the(15)radio and X-ray measurements,
19.7 (15)34.7
65.3 (40)61.9 the(40) radioour(40)2009–2015 velocities
are still lower than
29.7
(27)
(27)
34.5 (27)velocities
and thus are 40.6
consistent
with recent decelerationoftime variabilities of the blast waves. As can be seen in the figure, a rapid decelerationis revealed in the southwest blast waves (especially Regions 6–11). They interpreted
it as the result of a recent hit on a dense wall around the remnant. The SD scenario
can explain the dense wall, considering that the wind from a progenitor white dwarf
during mass accretion can create the cavity wall, while there is no plausible interpretation in the DD scenario. Their results provided constraints on the progenitor
activity before the expansion.

comparison with the result by Williams et al. (2016). We

data set in common, and still indicate rapid deceleration.

4

3

38

4
Instruments
Contents
4.1 Overview of Chandra X-ray Observatory . . . . . . . . . . . . . . .
4.2 High Resolution Mirror Assembly (HRMA) . . . . . . . . . . . . . .
4.2.1 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Effective Area . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Point Spread Function . . . . . . . . . . . . . . . . . . . . . .
4.3 Advanced CCD Imaging Spectrometer (ACIS) . . . . . . . . . . . . .
4.3.1 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Quantum Efficiency . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Angular Resolution and Field of View . . . . . . . . . . . . . .
4.3.4 Energy Resolution . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

39
40
40
41
42
44
44
45
46
47
48

Overview of Chandra X-ray Observatory

The Chandra X-ray Observatory is a telescope, which is still active, launched aboard
the Space Shuttle Columbia by NASA on July 23rd, 1999 (Weisskopf et al., 2000). Its orbit
is an elliptical shape with a perigee altitude of 16,000 km and an apogee altitude of
134,000 km. This orbit makes the period 64.3 hours, enabling the telescope to observe
55 hours continuously without interruption from the Earth’s radiation belts (Chandra
X-ray Center, 2023b).
Figure 4.1 shows the overview of the Chandra X-ray Observatory. Its dimensions,excluding the solar arrays, are 13.8 m × 19.5 m, and it weighs 4800 kg. The High Resolution
Mirror Assembly (HRMA) is responsible for generating images on the focal plane withthe half-power diameter (HPD) of the point spread function (PSF) of < 0′′. 5. The
imaging capabilities are supported by two types of cameras: the Advanced CCD Imaging
Spectrometer (ACIS) and the High Resolution Camera (HRC). Between the HRMA and39

40

4.2. High Resolution Mirror Assembly (HRMA)

design of the spacecraft and after completion of the HRMA, careful analysis revealed that the reaction
wheel disturbances would have excited resonant modes in the HRMA and ruined imaging performance.
Damped isolators, which were not originally planned in the program were implemented and solved theissue.
Risk Retirement

Figure 4.1: The Overview of Chandra X-ray Observatory (Credit: NGST & NASA/CXC).

In any development effort, especially when new technologies, materials or applications are involvoed,
Development Units, Engineering Units, and/or Qualification Units are produced to ensure the as built
hardware behaves the way the design engineer envisioned it. This is a very valuable process; however,whenfocal-planebuilding a new,cameras,
complex system
likeareChandra,modelsare necessary,
but not alwaystransmission
the
there
two these
gratingsystems:the low-energy
sufficient. It is very important to demonstrate integration, assembly, and test processes, procedures, andgrating
and thehigh-energy
transmission
grating(HETG).
In this thesis, we
equipment (LETG)
before attempting
themon the flight hardware.
The Verification
Engineering
Test Articles(VETA),
producedfor
Chandra,
were
key
to
the
successful
developmentoftheChandra
Telescope.
explain the HRMA (Section 4.2) and the ACIS (Section 4.3), as these instruments areusedin early
ourstages
research.
During theof the Chandra design development, it was realized that Kodak’s traditional mirror
mount design would not work for the configuration and design parameters of this x‐ray telescope. Early
funding (prior to official program start) was provided and a series of analyses and tests were conducted to
identify a high quality bonding material and mount configuration that would satisfactorily support the optics
while meeting their performance requirements. The program’s decision to provide the funding based on
the proposed study was key in jump starting the design efforts for the HRMA. Additionally because thisdesign parameter was evaluated early on, it provided further opportunities to refine and qualify the concept
with the test articles.

4.2

High Resolution Mirror Assembly (HRMA)

4.2.1 Layout

Kodak established an early contract with Hercules to develop and design the graphite epoxy sleeve conceptthat along with a new flexure and mount pad design would support the HRMA’s mirrors at their neutral axis.
The
X-ray
telescope,
Chandra
X-ray
Observatory
istheir
highlyrenowned for its
This design
providedvery littleHRMA,
distortionof
to thethe optics
during boththe assembly
phase andtransition
to an on‐orbit environment. Many interactions, interfaces, and designs
were bantered about between the′′
exceptional
resolution, measuring less than 0. 5. The HRMA comprises a nested set ofcompanies that tied in each company’s expertise to make the concept successful. Establishing the contract
early andworkingpairs
the issues
in a team oriented
approach
was criticalto its optics
success. (Figure 4.2). Each mirror pairfour
mirrorconfigured
to form
Wolter-I
type

consistsof a andparaboloid
as thefrontasmirror
hyperboloid
as the backDue to the naturesize of the Chandra
telescopewell as allandof theacoordination
and “handshakes”
that mirror. The
were required amongst the involved companies supporting this program, a well‐defined and agreed oninterface set was required and needed to be developed. TRW had the lead on the interface development
and control.
Their trust in theirsubcontractors and the reciprocal trust of
TRW helped develop a well thought out and awell flushed out interface and requirement
control document. The document was very
definitive about the telescope and the facility
requirements and interfaces. Because of thejoint efforts, it was clear as to the roles eachmember needed to support and which roleeach member needed to take control.

Figure 4:(HRMA)High

Resolution

MirrorAssembly

Figure 4.2: Left: Four HRMA mirror pairs and associated structures (Arenberg et al., 2014). Right:
Schematic of grazing incidence of Wolter-I type optics (Credit: NASA/CXC/S. Lee).

4. Instruments

41

Chapter 4. High Resolution Mirror Assembly (HRMA)

40

Figure 4.4: The HRMA/ACIS and HRMA/HRC e↵ective areas versus X-ray energy

Figure 4.3: The effective arealinear-linearof HRMA, HRMA/ACIS,
and HRMA/HRC against X-ray energy in(top) and log-log (bottom) scales. The structure near 2 keV is due tolinear-linear (left) and log-logiridium(right) scale.
Theeffectiveof HRMA/ACISand HRMA/HRC
M-edge.
The
HRMAarease↵ectivearea is calculatedby the raytrace simulation ba
are determined by multiplying
area of
HRMA
theXRCFquantum
efficiency
ofThe
eachHRMA/ACIS e↵ec
onthe
theeffective
HRMA model
andscaledwith
by thecalibration
data.
area
is
the
productoftheHRMAe↵ective
area
and
the
Quantum
Efficiency (QE) of AC
detector at its aimpoint (Chandra X-ray Center, 2022).

I3 (front illuminated) or ACIS-S3 (back illuminated). The HRMA/HRC e↵ective are
the product of HRMA e↵ective area and the QE of HRC-I or HRC-S at their aimpoi
UV/IonShields (UVIS).
length of each mirror is 84 including
cm, and thethee↵ect
totaloflength
betweenpre- and post-collimatoris 276 cm. The focal length is 10.070 ± 0.003 m.

4.2.2

Effective Area

The HRMA unobstructed geometric aperture measures 1145 cm2 . Supporting struts
partially block the HRMA aperture, accounting for less than 10% obstruction. Consequently, the HRMA throughput varies with X-ray energy due to the dependence of
Figure 4.4: The HRMA/ACISand HRMA/HRC
versusenergyin Figure 4.3 (solid lines)
optical reflectivity
on bothe↵ective
photonareas
energyandX-raygrazing
angle.
linear-linear (top) and
log-log(bottom)scales.
The
structure
near
2
keV
is
due
to
the
shows the HRMA effective area released in CALDB 4.1.1. The values change aroundiridium M-edge. The HRMA e↵ective area is calculated by the raytrace simulation
based
the iridium M-edge near 2 keV, with ∼ 800 cm2 on the low-energy side (0.1–2 keV)on the HRMA model and scaled by the XRCF calibration data. The HRMA/ACIS e↵ective
area is the product of the HRMA e↵ective area and the Quantum Efficiency (QE) of ACISI3 (front illuminated) or ACIS-S3 (back Table
illuminated).
HRMA/HRCe↵ectiveis4.1: TheThepropertiesof major
X-ray area
observatories
the product of HRMA e↵ective area and the QE of HRC-I or HRC-S at their aimpoints,
Observatories
Mirrors Detectors Energy Range Effective Area
HPD
including the e↵ect of UV/Ion
Shields (UVIS).
(keV)(cm2 )(arcsec)
ChandraHRMA
ACIS
0.1–10400 @ 5 keV
0.5
XMM-Newton—
EPIC
0.1–101500 @ 2 keV∼ 15
Suzaku
XRT
XIS
0.2–12
440 @ 1.5 keV
120
HitomiSXT
SXI
0.4–12590 @ 1 keV
∼ 70
HXT
HXI
5–80
174 @ 30 keV∼ 100
NuSTAR—FPM
3–78.4100 @ 30 keV
58
XRISM
XMA
Xtend
0.3–12∼ 435 @ 6 keV∼ 1.′ 3

References
[1]
[2,3]
[4]
[5]
[6, 7][8][1] Chandra X-ray Center (2022), [2] Jansen et al. (2001), [3] Aschenbach (2002), [4] Mitsuda et al. (2007)[5] Takahashi et al. (2018), [6] Harrison et al. (2013), [7] Brejnholt et al. (2012), [8] XRISM Science Team (2020)

4.2.

Calibration and Performance42

41

4.2. High Resolution Mirror Assembly (HRMA)

FigureThe
HRMA
e↵ectivearea
areaasversuso↵-axisofangle,
averagedangle
over offour
azimuthal
Figure4.4:4.5:
The
HRMA
effective
a function
the off-axis
theposition of thedirections, for selected energies, normalized to the on-axis area for that energy.
source (Chandra X-ray Center, 2022).

and ∼ 300 cm2 on the high-energy side (2–6 keV). As the photon energy increases, theeffective area decreases, and it becomes almost insensitive around 10 keV.
The effective area also depends on the off-axis angle, as can be seen in Figure 4.4(also known as vignetting). This is attributed to the reduction in the effective apertureof the telescope due to off-axis displacement, as well as the increased area where X-rays
cannot be reflected due to the larger incidence angle. The effective area decreases more
at off-axis angles as the photon energy increases. We analyze the X-ray data considering
this effect and apply corrections accordingly.

4.2.3

Point Spread Function

As mentioned above, the angular resolution of the HRMA is superior to other X-ray observatories, constituting one of the notable attributes of the Chandra X-ray Observatory.
Figure 4.5 shows the fractional encircled energy function (EEF) of the HRMA calculated
for an on-axis point source. EEF is calculated by dividing the integrated energy of thepoint spread function (PSF) within a given radius by the total energy of the PSF. The

4. Instruments
4.2.

Calibration and Performance

43
43

FigureFractional
encircled
energyas a function
angularradius, calculated
for X-ray
Figure4.5:4.6:
TheTheencircled
energyof HRMA
versusangularof
radiusgeneratedfor selectedanon-axispoint
source,atselected
X-ray
energies.
The
curvesarethecombined
response
energies, assuming an on-axis point source (Chandra X-ray Center, 2022).
and centered at the common focus of the full HRMA, i.e. four nested mirror pairs. For
higher energies (8.638 keV and 9.700 keV), the curves are broadened at small radii. Thisis because
the focus
of higher
energies
not coincide
with
the HRMA
commonfocus,
diameter
where
the EEFreaches
50%does
is called
the half
power
diameter(HPD),
which
but
is
o↵set
by
about
0.2
arcsec,due
toaslight
tilt
of
the
HRMA
mirrorpair
6.
is often used to represent the angular resolution. Table 4.1 provides a reference for theproperties
majortilted
X-raywithobservatory
can beConsequently,
seen that HRMA
has anangularpair 6 isofslightly
respect tomirrors.
the otherIt three.
the imagefrom′′mirror pair
not isasatsymmetricalas theproducedby thetootherThe
resolutionof 06. 5isthat
least an orderof images
magnitude
superior
othershells.
observatories.
e↵ect of this asymmetry on images depends on energy because of the di↵erent relative
contribution of mirror pair 6.
Figure 4.8 shows simulated HRMA/HRC-I images at several energies. The e↵ect ofthe mirror pair 6 alignment errors can be seen in the higher energy images as then mirror

444.3. Advanced CCD Imaging Spectrometer (ACIS)

4.3

Advanced CCD Imaging Spectrometer (ACIS)

4.3.1 Layout

Chapter 6. ACIS: Advanced CCD Imaging Spectrometer

100

ACIS FLIGHT FOCAL PLANE
not constant with Z~22 pixels ~11"
I0

I1

w203c4r
0

22 pixels= 11"

w193c2
1x

I2
w158c4r}

I3

ACIS−I(aimpoint on I3 = (953.74, 958.74)

w215c2r

2

3
330 pixels = 163"

S0

S1

S2

S3

S4

S5

w168c4r
4

w140c4r
5

w182c4r
6+
w134c4r
7

w457c48

w201c3r9}

ACIS−S(aimpoint on S3 = (193.74, 520)+∆Y
18 pixels = 8".9.

columnthree

Bottomtwo

Image Regionone

BI chip indicator

node zero

+Z
Top

.

row

TargetOffsetCoordinates+Z
Pointing+∆Z

Coordinates−Z

Pixel (0,0)+Y

Sim Motion

Frame StoreCCD Key

NodeDefinitions

Row/Column
Definition

Coordinate
Orientations

Figure 6.1: A schematic drawing of the ACIS focal plane; insight to the terminology is
given in the lower left. Note the aimpoints: on S3 (the ‘+’) and on I3 (the ‘x’). Note
Figure 4.6: The Layout of ACIS
(Chandra X-ray Center, 2022).
the di↵erences in the orientation of the I and S chips, important when using subarrays(a) Front-Illuminated CCD
Low Energy
X-rayHigh Energy
X-ray(Section 6.13.1). Note also the (Y, Z) coordinate system and the target o↵set convention(see Chapter 3) as well as the SIM motion (+/ Z). This view is along the optical axis,
from the sky toward the detectors, (-X). The numerous ways to refer to a particular CCD
are indicated: chip letter+number, chip serial number, and ACIS chip number (CCD ID);see also Table 6.1. As indicated, S3 and S1 are back-illuminated (BI) CCDs, and the(b) Back-IlluminatedCCD
rest are front-illuminated(FI) CCDs. The node
numbering scheme and the row/column
directions are illustrated lower center. The row and column pixel indices run from 0 to1023. The row direction corresponds to CHIPX (1 to 1024 pixels). The column direction
corresponds to CHIPY (1 to 1024 pixels); the row at CHIPY = 1 is closest to the frame store.

Low Energy
X-rayHigh Energy
X-ray

Gate
Insulator

Depletion
Layer

Field-free region

Figure 4.7: Schematic view of the front- (a) and back-illuminated (b) type of the CCD sensors.

The Advanced CCD Imaging Spectrometer (ACIS) can capture high-resolution images and acquire spectra of moderate resolution simultaneously. Figure 4.6 shows thelayout of ACIS. ACIS comprises a total of 10 CCD chips: four ACIS-I chips arranged ina 2 × 2 configuration and six ACIS-S chips set linearly in a 1 × 6 arrangement. There
are two types of chips on ACIS: front-illuminated (FI) and back-illuminated (BI) chips.
Figure 4.7 shows a cross-section view of a CCD pixel, which consists of a depletion
layer made primarily of silicon, an insulating layer including SiO2 , and electrodes foreach pixel (Townsley et al., 2002). We note that the front side is the side on which theelectrodes are arranged. While two chips (ACIS-S1 and S3) are back-illuminated (BI)106

Chapter 6. ACIS: Advanced CCD Imaging Spectrometer

observation of G21.5-0.9, a bright supernova remnant/pulsar wind nebula. The detector
was appropriately o↵set for each o↵-axis angle so that the data were obtained at the same
4.focalInstruments
position, minimizing the e↵ects of any spatially-dependent variations in the CCD45
response.

Figure4.8:
6.3:
The
quantum
efficiency(convolved
withtransmissionof contamination
the appropriateon
FigureThe
quantum
efficiency
of ACIS,
includingthetheeffectof molecular
opticalblocking
filter)of theFI CCDs (from a row nearest the read-out) and the two
OBF(Chandra
X-ray
Center,2022).
BI CCDs as a function of energy. S3 is somewhat thicker, hence more efficient, than S1.
These curves include the e↵ects of molecular contamination, as discussed in the text.

type, the other eight chips are front-illuminated (FI) chips. BI chips of ACIS have higher
effective area in the soft band and better energy resolution than FI chips. The low-energy
X-rays reaching the FI CCD sensors experience attenuation due to the electrodes at thegate, leading to reduced quantum efficiency, as depicted in Figure 4.3.

4.3.2

Quantum Efficiency

Quantum efficiency (QE) is the proportion of the incident photons that are converted
to electrons, generally used to measure a CCD sensitivity. Figure 4.8 shows the QE as
a function of the photon energy, considering the effect of molecular contaminations
on optical blocking filters (OBFs). Given that the ACIS chips are sensitive not only to
X-rays but also to optical/UV light (Lumb et al., 1991), OBFs are positioned between thechips and HRMA to prevent optical light from affecting the CCDs. The filters consistof polyimide placed between two thin aluminum layers. Based on astrophysical dataand observations of external calibration sources, it is apparent that the ACIS effective
area has degraded since launch due to the molecular contamination from out-gassed
material on the cold ACIS OBFs. For calibration of the contamination, Chandra has

46

4.3. Advanced CCD Imaging Spectrometer (ACIS)
Chapter 6. ACIS: Advanced CCD Imaging Spectrometer110

5τ (E = 0.66 keV)

4

6.5.

3
Quantum Efficiency and E↵ective Area

111

2

1

0
2000

2005

2010
Year

2015

2020

Figure 6.7: Optical depth of the contaminant measured at 0.66 keV from observations of
the galaxy cluster Abell 1795. Red squares denote observations with the target on ACIS-S
and blue disks denote observations with the target on ACIS-I. The solid curve shows the
optical depth of the contaminant as modeled in the most recent contamination model.

6.5.1

Molecular Contamination of the OBFs

Astronomical observations and data acquired from the on-board ACIS external calibration
source (ECS) show that the ACIS e↵ective area below 2 keV has continuously declined
since launch due to the build-up of out-gassed material on the cold ACIS optical blocking
filters. The HRC operates at a warmer temperature and shows no sign of contaminationbuild-up. The build-up of contaminant on the ACIS filters has been monitored with ECS
observations, LETG/ACIS-S observations of the blazars Mkn421 and PKS2155-304 andthe isolated neutron star RXJ1856-3754, and ACIS imaging observations of the rich clusterof galaxies Abell 1795 and the oxygen-rich supernova remnant E0102-72.3.
Typically the calibration team releases updates to the contamination model near
FigureACIS-S
spectrumthegalaxyAbell 1795atany
six di↵erent
epochsrangFigure
4.9:(Top):
The
opticaldepth
0.66
keVcluster
as accounts
a function
thetime (year).
The
blue circles
theendof 6.8:each
calendaryear.ofatThe
updateforofchangesin thecharacing from2000
to 2021. The decreasing
countbyrates
in the spectrumof aoverstablecalibration
teristics
ofthecontaminationasmeasuredobservations
taken
the
same
calenand the redsource
squaresshow thethe ACIS-I
respectively.
The data on
is measured
demonstrate
thedataloss ofof e↵ective
areaanddue ACUS-S,
to the build-upof the contaminant
dar year. Most updates do not a↵ect the analysis of data taken prior to that calendar
from observationsof Abell 1795. (Bottom): The effects of the molecular contamination of thethe ACIS OBF.
year. The contamination model is distributed as part of the CALDB, and is described
ACIS OBF
onthe
spectra.
The spectra are extracted from the ACIS-S
dataareof Abelltaken inat: https://cxc.cfa.harvard.edu/ciao/why/acisqecontam.html.
Therethree 1795
compo2000 (black),(red),contamination2010 (green), model:
2015 (blue),(cyan),
and(magenta). on thenents to 2005
the ACIS
1) the2018
build-up
rate
of 2021the contaminant
ACIS filters, 2) the spatial distribution of the contaminant on the ACIS filters, and 3)

been observing Mkn 421 and PKS 2155−304 (blazars), and RX J1856−3754 (a neutron star)with LETG/ACIS-S and 1E 0102.2−7219 (SNR) with ACIS. Figure 4.9 shows the effect of
the ACIS OBF contamination on the effective area. We note that the accumulation of
OBF contamination has increased faster since 2014.

4.3.3

Angular Resolution and Field of View

The angular resolution of Chandra is determined by its pixel size of CCDs since HRMA
has the high angular resolution as mentioned in Section 4.2. The ACIS chip has an

4. Instruments

47

Aspect Solution
29401264.13929401164.138Time from Observation Start (ksec)

294010Dec

64.13729400929400864.13629400764.13529400664.13429400564.133294004
6.346

6.344

6.342

6.340
RA

6.338

6.336

6.334

Figure 4.10: The dither pattern of ObsID: 8551. The color corresponds to the time from the startof the observation.

imaging area of 25 mm (∼ 8.′ 3) square consisting 1024×1026 pixels. Thus, the pixel sizeis 23.985 𝜇m square, which corresponds to 0.4920 ± 0.0001 arcsec squares (Chandra
X-ray Center, 2023a). The EEF reaches 90% at 4 pixel and 5 pixel when the X-ray energyis 1.49 keV and 6.4 keV, respectively.
Unless there is a specific request, the spacecraft dithering is performed in a Lissajous
pattern (Figure 4.10) during all observations. For ACIS observations, the default dither
pattern currently encompasses 32 arcsec peak-to-peak range in both Y and Z directions.
There are two major purposes of dithering: (i) to provide some exposure in the area corresponding to the CCD gaps and (ii) to smooth variations in the response in each pixel.

4.3.4

Energy Resolution

The pre-launch energy resolution approached the theoretical limit across most energy
levels for the ACIS FI chips. In contrast, the BI chips showed lower performance, as
shown in Figure 4.11. However, the resolution has degraded since the launch and theorbital activation due to higher charge transfer inefficiency (CTI), which is the ratioof transferred charges that are trapped and lost. The increased CTI was caused by
low-energy protons (also known as soft protons) that reached the focal plane throughHRMA by Rutherford scatterings during the radiation belt crossing. Thus, the resolution
degrades as the row is far from the readout. A correlation algorithm for the lost energy
resolution has been developed. The ACIS energy resolution on board can be monitored

118

48

Chapter 6. ACIS: Advanced CCD Imaging Spectrometer
4.3. Advanced CCD Imaging Spectrometer (ACIS)

6.14:
The
ACIS pre-launchenergy resolution
as aenergyfunctionof energy.
FigureFigure
4.11: The
ACIS
pre-launchenergy resolution
versus X-ray(Chandra
X-ray Center,2022).

6.7.1

Correcting the Energy Resolution of the CCDs

The
ACIS
instrument
team hasdeveloped
correction
algorithm
muchAsofcanby theexternal
calibration
sources,
Al K𝛼a (1.49
keV) andMn K𝛼that(5.9recovers
keV) lines.
thelost energy
resolution
theresolution
FI CCDs. function
The correction
recovers
a significant
fractionofbe seenin Figure
4.12, theof CHIPY
improves
after thecorrection.
the CTI-induced loss of spectral resolution of the FI CCDs at all energies. The algorithm
has been incorporated in the CIAO tool acis process events since CIAO 2.3. Figure 6.15
illustrates
the improvement that the tool provides. As of 2006-Dec, data for the two BI
4.3.5 Backgroundchips can also be corrected in the same way, including a correction for serial CTI for theBIchips, thoughe↵ects areon
more
The resulting
responsevery nearly
uniform
ChandraX-ray theobservation
thesubtle.
orbit path
is affectedby the isnon-X-ray
backgroundacross the BI chips once this correction is made. As of December 15 2010 (CIAO 4.3),(NXB), which originates from external particles (e.g. CRs). In addition to this, ACIS
temperature dependence was added to the CTI correction in acis process events for bothalso
instrumental fluorescence lines generated from collisional excitations ofFI
anddetects
BI CCDs.
e.g.,
Al,
Au,
and
Si. resolution
These backgrounds
the theanalysisin theathighband
The ACIS energy(here takenaffectto meanfull widthhalf energy
maximum(≳ 5 keV),thephotoncounting
rate from
thesquarestellarroot
objectsis energy,
relatively[FWHM]
of awhere
narrow
spectralline)
varies roughly
as theof theandlow.
increaseswithdistancefrom spectra
the read-out.
OnACIS.
FI chips
(the
array
and
all atheS array
Figure 4.13
shows
the NXB
taken by
While
theI BI
chips
have
peak
aroundchips
except
S1
and
S3),
the
increase
with
CHIPY
(row
number)isdramatic,
ascan∼ 10 keV originating from the minimum ionizing particles, the FI chips do not havebe
such
seen
in FigureThis
6.15.is The
spatialon partially
BI chips (S1and S3)much grade
weaker,filtering.
but
a structure.
becausethedependence
FI chips canretrieve
NXBis using
depends on both coordinates. This can be illustrated by measuring the resolution at the
Because NXB events enter ACIS without passing through HRMA, their incident angle isaimpoints and default o↵sets in e↵ect 2009 (see Table 6.4), using the Al-K↵ (1.49 keV)
generallywider
NXB distribution
on the focal
Grade filteringand
Mn-K↵large,(5.9 resultingkeV) linesinina theexternal
calibration source.
Data plane.
were summed
over
is
an
imaginganalysistechniquethatclassifies
eventsfrom
their
distributions.
three months in early 2012, and over an area of 128 ⇥ 128 pixels. Note that these valuesThewidths of the depletion layer of the FI and BI chips are 50–75 𝜇m and 45 𝜇m, respectively(Garmire et al., 2003). Thus, the NXB distribution of the BI chips is smaller than thatof FI chips, leading to a challenging reduction of NXB by the grade filtering method.
On the other hand, the NXB of FI can be reduced well by this method because of thehigh depth of the depletion layer.

4. Instruments

49

6.7.
6.7.

119
119

Energy Resolution
Energy Resolution

Figure 6.15: Energy resolution (FWHM in eV) versus row number (CHIPY) for several
cases
plotted.
The
top
plot
shows
Aluminum(1.49 row
keV),
while
the
bottomfor
plot
isThe
for
FigureACIS
FWHM
asa functionof thenumber(Chandra
X-ray
Center,2022).
Figureis4.12:
6.15:
Energyresolution(FWHM
in roweV)K↵
versusnumber(CHIPY)severalManganese
K↵
(5.9
keV).
The
resolution
of
the
I3
chip,
with
and
without
CTI
correction)
upper
and
lower
panelsarethedatameasured
using
the
Al
K
𝛼
and
Mn
K
𝛼
lines,
respectively.
cases is plotted. The top plot shows Aluminum K↵ (1.49 keV), while the bottom plot is for
is
shown
by K↵
the
black
curves,andthe S3resolution
is shown
bycorrespondthe
red curves.
Manganese
(5.9
keV).
The
resolution
ofthe I3
chip,
with
and
without
CTI
correction)Blackand redcorrespond
to I3
and
S3 data,
and
solid
and
dottedlines
to thecorrected
is
shown
by
the
black
curves,andtheS3
resolution
is
shown
by
the
red
curves.
and non-corrected data, respectively.

6.17.

On-Orbit Background50

143
4.3. Advanced CCD Imaging Spectrometer (ACIS)

Figure 4.13: Non-X-ray background spectrum of ACIS (Chandra X-ray Center, 2022).

Figure 6.27: Energy spectra of the charged particle ACIS background with ACIS in thestowed position (a 50 ks exposure taken in 2002-Sep; standard grade filtering, no VF filtering). Line features are due to particle-induced fluorescence of material in and surrounding
the focal plane. The “I023” points are from combined I0, I2, and I3 data.

5
Particle Heating and Accleration
Observed by X-ray Time Variability
Contents
5.1 Our Objective and Target Selection . . . . . . . . . . . . . . . . . . .
5.2 Observations and Data Reductions . . . . . . . . . . . . . . . . . . .
5.3 Time Variabilities of Non-Thermal X-ray Radiation . . . . . . . . .
5.3.1 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Time Variabilities of Thermal X-ray Radiation . . . . . . . . . . . .
5.4.1 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1

51
53
54
54
59
65
65
71

Our Objective and Target Selection

Year-scale time variabilities in SNRs can provide information on real-time changes
at the shock and post-shock, which is related to a number of unsettled problems,such as cosmic-ray acceleration and electron heating. Aiming to understand particle
acceleration and electron-heating mechanisms of shock, we search for time variabilities
in nonthermal and thermal X-rays in SNRs. Tycho’s SNR (also known as G120.1+1.4)
is a nearby and young SNR whose origin is a Type Ia SN in 1572 (e.g., Baade, 1945;
Ruiz-Lapuente et al., 2004). Since the apparent diameter of the remnant is ∼ 8′ , we
can observe fine structures with the high-resolution imaging of Chandra (0′′. 5 HPD).
Therefore, we can expect the detection of time variabilities even on a small physicalscale. Throughout this thesis, we assume the distance to be 2.5 kpc based on the workby Zhou et al. (2016) when calculating the actual scales and velocities.
Tycho’s SNR is bright in nonthermal and thermal X-ray radiation; the former andlatter dominate in the energy band of ≳ 4 keV and ≲ 4 keV, respectively (e.g., Sato51

52

5.1. Our Objective and Target Selection

2003

2007

2009

2015

Figure 5.1: RGB image of Tycho’s SNR taken by Chandra in 2003, 2007, 2009, and 2015. Red, green,
and blue correspond energy bands of 0.7–1.5, 1.7–2.6, and 4.1–6.1 keV.

& Hughes, 2017). Figure 5.1 shows a three-color image. As can be seen in the figure,nonthermal radiation is bright in the outward shell, and thermal radiation is in theinside clump. Each radiation is widely thought to originate from synchrotron radiationand thermalized ejecta (e.g., Hwang et al., 2002).
The nonthermal radiation in the southwestern region has a peculiar structure called“stripe” (Figure 5.2). Eriksen et al. (2011), who discovered this structure, estimatedthe energies of the accelerated protons in this region to reach PeV, assuming that the
gap between the stripes (8′′ ) equals twice its gyroradius. Although some theoreticalmodels were proposed to explain the structure, the origin is still open. Okuno, Matsuda,
et al. (2020) reported the year-scale variable features of synchrotron X-rays in thesouthwestern regions of the remnant, which correspond to the part of stripes. Section 5.3
reports spectral and imaging studies of the whole of the stripe structure. We aim to5. Particle Heating and Accleration Observed by X-ray Time Variability
The Astrophysical Journal Letters, 728:L28 (5pp), 2011 February 20

(d)

53

Eriksen et al

(a)

(c)

(b)

Figure 1. Chandra X-ray 4.0–6.0 keV image of the Tycho supernova remnant, smoothed with a ∼0."" 75 Gaussian and displayed with an arcsinh scaling, showin
FigureChandraX-ray
image fromin 4.0–6.0
keV
band(Eriksen
et al.,
2011).
Zoom-inviews
various regions of striping
in the5.2:
nonthermal
emission.
Clockwise
the upper
right:(a) themain western
stripes
discussedin thisLetter;on(b) a fainter ensemble o
stripes; (c) a previouslyboth
knownbrightarcofnonthermal
emission,
with
our
newly
discovered
streamers;
and(d)filamentsof“rippled
sheet”
sides show characteristic structures, including “stripe” features in the panels (a) and (b).morphology common i
optical observations of middle-aged SNRs.

study
the temporaland spatial
variabilities
X-raysspectralandof plasma and field,
enclosedby “wandering
filaments”
of in synchrotron
the stripes, rulingoutfromanythedramatic
flux
variability like thahigh density and frozen-in
field.
One
goal
ofobservationsisobserved
inthenon-thermal
X-ray
filaments
imaging analysis, whereas Okuno, Matsuda, et al. (2020) focused on two specific features. of the SNR RX
to discover evidence for structure in SNR shocks on this spatial
J1713.7−3946 (Uchiyama et al. 2007). Our preliminary prope
The thermal radiation from the SNR is dominatedby shock-heatedejecta are(e.g.,
scale.
motion measurements
for the stripes
consistent with th
overall
expansion
of
the
blast
wave
and,
in
particular, show no
Hwang et al., 2002), where Yamaguchi et al. (2014) represented the evidence of collievidence
for
non-radial
flow.
There
are
no
obviouscounterpart
2. OBSERVATIONS
sionless
electron heating as mentioned in Section 1.2.4. On the other hand, previous
to these features in the radio (Reynoso et al. 1997) nor in thstudiesreportedthetheinteractionof forwardshock
We observed Tycho
in 2009
April with
Chandra X-ray
Obmid-IR.with ISM with other wavelengths,
servatory Advanced
CCD
Imaging
Spectrometer
imagingarray,
including H𝛼 lines (Ghavamian et al., 2000; J.-J. Lee et al., 2010), infrared light (Ishihara
as part of a Cycle 10
Program(LP), using
the fouret Largeal., 2010).
Moreover,Tanakaet al.ACIS-I(2021) shows the recent shock-cloud
collisions from
3. ANALYSIS
front-side illuminated CCDs, operated in faint mode. The total
thepropermotionsofshockwaves.
These
results
indicate
that
the
shock
wave heatedprogram was split into nine individual ObsIDs, which we repro3.1. Location of the Stripes
cloudrecently.
5.4charge
searches for the time variabilities in shock-heatedcessed with CIAO theversion
4.1,veryusing
the gainSection
tables andtransfer inefficiency
correctionin theCALDB
4.1.3. Examination
ISM
to reveal
evolutionof heated plasma. Before investigating the nature of the stripes, we first musof the light curves revealed no significant background flares.
locate them within the three-dimensional volume of the remnant
The registration of the individual pointings was improved using
The canonical picture of a young SNR consists of three distincthe measured relative positions of background point sources,
fluid discontinuities: the blast wave, which marks the shock
5.2shiftsObservations
Data Reductions
though the necessary
were typically #1""and. For imaging
propagating into the ambient medium, a Rayleigh–Taylor (R–T
analyses, the registered event lists were merged using the stanunstable contact discontinuity (CD) at the ejecta-interstella
dard CIAO tools and have a total average livetime of ∼734.1 ks.
material boundary, and a reverse shock that propagates into
The Chandra telescope observed Tycho’s SNRthein stellar
2000, 2003,
2007, 2009, and 2015. The
For spectroscopy, counts were extracted and RMFs and ARFs
remains. Warren et al. (2005) set an upper limit fo
logObsID,can bewhich
seen wein Table
5.1. The thedata
in 2000 andin the other
years
were
were generated forobservationeach individual
fit jointly
azimuthally
dependent
projected
radiusof the reverse shockin XSPEC versionobtained
12.6.0. with ACIS-S and ACIS-I, respectively.
in Tycho
the locationof theandFe Kαemission. Adopting
The usingobservationsin 20072009
In Figure 1 we wereshow conducted
the image fromtheand4 toeight
6 keV
band,
their
centerof expansion, withthe western
stripes peak at a radius o
twice
times
All of thedata
is reprocessed
the Chandra
which is dominated by the synchrotron component of the X-ray
220"" , well outside the 190"" position of the reverse shock at tha
Calibration Database (CALDB) version 4.10.7. We note that the observation in 2000 was
spectrum. Apart from the well-known limb-brightened shell,azimuth. While the position of the stripes does coincide withobserved
the back-illuminatedchip (ACIS-S3).
a number of brightregionswith
are seen
toward the projected
the Warren et al. estimate of the CD, the regularly spaced, linea
interior of the remnant.
While
the
strongestof
thesefeaturesmorphologyof thestripes posidoes not correspond to
We corrected the coordinates of the observed
data based
onnon-thermal
the point source
are visible in earlier Chandra data (Warren et al. 2005) our
any features in the R–T plumes of thermal emission tracing th
to aimprove
the accuracy
ofregularly
imaging analysis.
the coordinatessignificant
deeper observationtions
reveals
striking patternof nearly
ejecta First,
boundary,
nor is the CD ofa prominent
feature elsewhere in""spaced stripes. The brightest group, centered ∼30 interior to
the 4–6 keV band. Conversely, the blast wave is a bright sourcof 4–6 keV emission, and we identify the stripes as projected
the western limb (see Figure 2), has a peak surface brightness
features of this forward shock.
twice that of the brightest sections of the rims and is the
Tycho’s blast wave is traced by a very thin shell of X-ray
primary subject of this communication. A second, fainter pattern
emission, with a typical thickness only 1%–2% of its radiu
extends east–west, 55"" –75"" inside the southern rim, and there is54

5.3. Time Variabilities of Non-Thermal X-ray Radiation
Table 5.1: Observation Log

ObsID
115
3837
7639
8551
10093
10094
10095
10096
10097
10902
10903
10904
10906
15998

Start Date
2000 Oct 01
2003 Apr 29
2007 Apr 23
2007 Apr 26
2009 Apr 13
2009 Apr 18
2009 Apr 23
2009 Apr 27
2009 Apr 11
2009 Apr 152009 Apr 172009 Apr 13
2009 May 03
2015 Apr 22

Effective Exposure (ks)
49
146
109
33
118
90
173
106
107
40
24
35
41
147

Chip
ACIS-S
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I

Astrometry Correction
Corrected
Corrected
Corrected
Not corrected
Corrected
Corrected—
Corrected
Corrected
Corrected
Not corrected
Not corrected
Not corrected
Corrected

point sources in the field are determined using the CIAO task wavdetect and are aligned
based on the source coordinates detected in the dataset with ObsID 10095, whose
effective exposure time is the longest, using the task wcs_match. All of the event files
are reprocessed by the tasks wcs_update. Due to the dependence of frame alignment
accuracy on photon statistics, short-time observations (ObsID 8551, 10903, 10904, and
10906) are excluded from the group of corrected observations. These observations
are used only for spectral analysis.

5.3

Time Variabilities of Non-Thermal X-ray Radiation

5.3.1 Analysis and Results
5.3.1.1

Imaging Analysis

First, we search for notable time variabilities of the stripe structure from a difference map
created by subtracting an exposure-corrected image taken in 2003 from one taken in 2015as shown in Figure 5.3 (a). We select an energy band of 4.1–6.1 keV where the non-thermal
emission is thought to be dominant (e.g., Eriksen et al., 2011). Figure 5.3 (b) shows a zoomin view of the western region. Most stripe features seem to move outward, corresponding
to the remnant expansion shown from the proper motion of the outer rim (Warren et al.,2005). However, focusing on the stripes, some flux changes cannot simply be explainedby the expansion along the red curves in Figure 5.3 (c), which generally brighten from2003 to 2015. Note that there are too faint features within the brightened stripes to appear
in the exposure-corrected images shown in Figure 5.4. We also detect indications of theproper motion of the faint stripes oriented perpendicular to the shock normal.

5. Particle Heating and Accleration Observed by X-ray Time Variability

55

Figure 5.3: (a) Same as Figure 5.10, but in the energy band of 4.1–6.1 keV. The yellow box
corresponds to the region delineated in panels (b), (c), and Figure 5.4. (b) Zoom-in view of theyellow box in the panel (a). (c) Similar to panel (b), with superimposed guides aiding in theidentification of noteworthy features. The color scale is presented in units of photons s−1 cm−2 .

Ref1
S1
S2
S4 S3
S5
S6
S7
S8
S9

2003
0.0

1.6

R2
R3
R4
R5

2007
3.2

R1

2009

4.8

Flux (photon cm 2 s 1)

2015
6.4

8.0
1e 8

Figure 5.4: Chandra ACIS images of Tycho’s SNR taken in 2003, 2007, 2009, and 2015, exposurecorrected and confined to the 4.1–6.1 keV band. The regions denoted as S1–9 (green ellipses) andR1–5 (cyan boxes) are earmarked for spectral analysis. The Ref region (white circle) serves the
purpose of the parameter estimation for the thermal component of the stripe emissions.

We extract the 1D profile projected along the azimuthal direction from the rectangular region shown in Figure 5.5, where the brightest stripe (S7 in Figure 5.4) exists.
We can detect the sharp and broad peaks corresponding to the rim and stripe. It isclear that both peaks move outward with time, as already indicated by the difference
image in Figure 5.10. We measured the proper motion in the same way as (Tanaka et al.,2021) (see Section 6.2 for detailed explanations). A velocity of the rim is obtained as0′′. 29 ± 0′′. 01 yr−1 , which can be translated into 3400 ± 100 km s−1 . This result is consistent
with one reported by Williams et al. (2016); the region we analyzed roughly coincides

縞状構造の固有運動についてより定量的に評価を行うため，縞状構造の中で最もフラッPublications of the Astronomical Society of Japan (2020), Vol. 72, No. 5

クスの大きい縞 (図 7.2d の S7 領域) について 2003 年と 2015 年の射影プロファイルを作成した．図 7.3(a) が射影プロファイルを作成した領域，(b) がそのプロファイルである．左3.2 Spectral analysis
We performedof each側の広がったピークが縞状構造，右側の鋭いピークがのリムにあたる．これら
2つ
56
5.3. Time Variabilities ofSNR
Non-Thermal
X-ray
Radiation spectroscopy
extracting spectra from the nine regionsの年のプロファイルの比較から，縞状構造及び順行衝撃波のピーク位置が移動しているこ

in figure 1d. Backgrounds are extracted fr
the SNR within the ACIS-I array. The spe
so that each bin has at least 10 counts and(a)(b)
the 0.5–10 keV energy band with XSPEC$"8"1996). Following the works by Sato and!""#Yamaguchi et al. (2017), and Okuno et al.
!"$%
the spectra with a model consisting of;8"thermal components. We modeled the nonnent with a power law. To the thermal em
be attributed to the supernova ejecta, w:8"component non-equilibrium ionization (Nthe vnei model in XSPEC. We employed
Boulder absorption model (TBabs; Wilm
98"
interstellar absorption.
One of the two NEI components repres
from Fe whereas the other is for intermedia!8"(IMEs: Mg, Si, S, Ar, and Ca). We treated t
IMEs and Fe as free parameters, and linke"8"of Ni to$""Fe. Since Tycho’s SNR is of type Ia
"
!"
9"
:"
;"
dances of H, He, and N were fixed to zero
/14<2<13*+=>54?57
of O and Ne with respect to C were fixed abecause C has the lowest atomic number inFigure5.5: (a): The region used for theof the radial profile of the brightest
stripe
図 7.3:
縞状構造の射影プロファイル．
(a)extraction
プロファイル作成に用いた領域．
(b) be(a)present
で示した領
would
in the ejecta. Emission m(S7) region. We extracted the profile from the!! green box region. (b): The profile projected in theand IME components [≡ (1/[C/H]# )(1
域から抽出したプロファイル．1 ビンは長さ 0. 5 にあたる．左側が SNR の中心方向となっている．
green box along the azimuthal direction. The zero point is the inner edge of thewere
box.linked to each other, where d is the diFig. 2. (a) Regions used for the projection in panel (b). (b) Profiles
SNR, ne and nC are the number densitie
縞状構造と衝撃波速度を比較するにために，まず，衝撃波の固有運動を測定した．測定
extracted from the regions shown in panel (a). The widths of each bin
carbon, and V is the volume of the emittin
are 0.!! 5. The geometrical center of the SNR is to the left. (Color online)
方法は Tanaka et al. (2020) で用いられた方法と同様の方法を用いた．つまり，
the spectra, we found residuals at ∼1.2 keseen in spectra of Tycho’s SNR (e.g., Sato
! (fi,15 − fi,03 )2
2
Okuno et al. 2020)(7.1)
as well as other SNRsχ ≡2 + (df
2
(df
)
)
i,15
i,03
2019).
Althoughthecause of the residualia discussion by Okon et al. 2019), we aと定義した χ2 値が最小になる移動量を調べた．ここで，fi,15 と fi,03 はそれぞれ，番目 the fits. The ceto the model to iimprove
the Gaussian was allowed to vary for spe
regions except for S9. We fixed it to 1.23
it cannot be well constrained.
Strong non-thermal emission of the
difficult to determine the parameters fo
ponents. To constrain the parameters, thlyzed a spectrum extracted from thecontribution from the non-thermal com
labeled as “Ref” in figure 1d. Since theat a similar radius of the SNR to stripthat ionization ages (ne t) of the NEI
common between the “Ref” region and s
mined them by fitting the “Ref” spec
Fig. 3. Comparison
betweenthe profile
profile in with2015 andmoved
profileshifted by 3′′
Figure 5.6: Comparisonof !!the 2015
(blue)
thethe2003
profile. 0 (red).
tral fitting
yielded ne t = 4.52 × 1010 s
in 2003 by 3. 0. Oblique lines correspond to the range of 1σ . Solid lines
Oblique lines representthe1
𝜎range.
Solid
lines
depictsmoothed
profiles.
0.74 × 1010 s cm−3 for the IME andshow smoothed profiles with a bandwidth = 3. (Color online)

2003

2015

&'()*+,*$"-.*/01213*4-$*56-!7とが明らかになった．

5. Particle Heating and Accleration Observed by X-ray Time Variability

57

with their Reg 13. On the other hand, it is difficult to measure the proper motion of the
stripe due to its time-variable shape. Examination of the profiles indicates that the peakcorresponding to the stripe exhibits a narrower width in 2015 compared to 2003. Thisphenomenon is more evident in the smoothed profiles presented in Figure 5.6. In thisfigure, the profile in 2003 is moved by 3′′. 0, which corresponds to the travel distance ofthe shock in 12 years. The peak location exhibits rough alignment, indicating that theproper motion of the stripe is comparable to that of the rim.
5.3.1.2

Spectral Analysis

We analyze spectra of each bright stripe extracted from the nine regions labeled as S1–S9
depicted in Figure 5.4. We select a blank region outside of the remnant as a backgroundregion. We do not use ObsID 115 because its chip is back-illuminated, which makes
the high background in the high-energy band, as mentioned in Chapter 4. We conduct
spectral analysis using an X-ray spectral fitting package XSPEC version 12.13.1 (Arnaud,1996). The spectra are binned so that each bin count is at least ten. The model forthis analysis consists of a nonthermal component from synchrotron radiation and athermal component from the ejecta. A Tüebingen-Boulder model, power law, and two
non-equilibrium ionization (NEI) models are employed as the model of the interstellar
absorption, synchrotron radiation, and thermal ejecta radiation, respectively.
Within the two components of the ejecta model, one characterizes the emission
originating from iron and nickel (Fe component), while the other is dedicated to IMEs,
namely magnesium, silicon, sulfur, argon, and calcium (IME component). The abundances of hydrogen, helium, and nitrogen are presumed to be absent since Tycho’s SNR
is a Type Ia remnant. We fix the abundance of oxygen and neon at the solar composition
with respect to carbon, which has the lowest atomic number in the ejecta element. The
abundances of other elements
are free parameters. We also set the emission measure∫(EM), which is defined as 𝑛𝑒 𝑛 C𝑑𝑉 /( 4𝜋𝑑 2 [C/H] ⊙ ) , as a free parameter. Here, 𝑛𝑒 and 𝑛 C
are the number densities of electron and carbon, respectively, and 𝑉 is the volume of theemitting plasma. The abundances of all elements are allowed to vary freely except for
nickel, which is linked to one of iron. We link the emission measure of the Fe component
to that of the IME component. The spectral fitting reveals an appearance of residuals
at ∼ 1.2 keV, which are also seen in spectra of Tycho’s SNR (e.g., Sato & Hughes, 2017;
Okuno et al., 2020) as well as other SNRs (e.g., Okon et al., 2020). This residual is thought
to derive from the uncertainties of the atomic data of Fe XXII relating to the emissivities
of Fe L lines (e.g., Gu et al., 2022), so we add a Gaussian to the model for improvementof the fit. We set the centroid energies of all regions as free parameters except for thatof the S9 region, which is fixed to 1.23 keV due to the impossibility of the constraint.
Eventually, the analysis model is described as:
Interstellar Abs. × [NEI(IME) + NEI(Fe) + Gaussian + PowerLaw].

(5.1)

Since the stripe regions have strong nonthermal emission, it is difficult to determine
the parameters of the NEI component. Therefore, for a constraint of the parameters,

58

5.3. Time Variabilities of Non-Thermal X-ray Radiation

85-5

Publications of the Astronomical Society of Japan (2020), Vol. 72, No. 5

respectively. We fixed ne t to these values when fitting theparameters as well as time variability of each stripe. Anotherspectra of the stripe regions.
finding to note here is the strong anti-correlation between
weWe
investigate
the NEI component’s parameterby analyzing spectra extracted from thefirst fitted spectra from observations in 2009, whichsurface brightness and photon indices of the stripe emission.
regioncontribution
nonthermal component, which is labeled as “Ref”
have the withhighestlessstatistics
thanks to thefrom
longestthe
exposuretime.
Thespectraareplottedwith
thebest-fitting
modelsin Figure 5.4. Since the distance from the remnant
center to this region is comparable to
4 Discussionin figure 4, and the best-fitting parameters are summarized
that
of
the
stripe,we
presumecommonionization
ages (𝑛𝑒 𝑡 ) between
the “Ref” region
in table 2. We then fitted spectra from 2003, 2007, andOur imaging and spectral
analyses have revealed the time2015 theobservations
to Thesesee the time
variability ofthe stripe
variable nature
the stripes
Tycho’s SNR.
Althoughandstripes.
ionizationages
are determined
byoffitting
theinspectrum
obtainedemissions. The parameters for the thermal components were
Okuno et al. (2020) reported time variability
in −
two
10
3 and
from the “Ref” region. We obtain the best-fit value of 𝑛𝑒 𝑡 as 4.52 × 10 s cmonly
fixed to those obtained for the spectra in 2009 except for the
structures, including S1 in our definition, similar variability
10 s cm − 3 for the IME and Fe components, respectively. When fitting the spectra
0emission.74 × 10measures.
In figure 5, we plot surface brightness of
seems to be rather universal in this part of the SNR. Figure 5
thenon-thermal
componentasafunctionofphotonindex,
indeed indicates significant flux variability of S2, S3, S7,of the stripe regions, 𝑛 e𝑡 is fixed to these values.
which reveals a significant stripe-to-stripe variation of theand S9, in addition to S1. The fluxes of the stripes S2 andWe first fit spectra obtained from the observations in 2009, whose statistic is thehighest as a result of the longest exposure time. Figure 5.7 shows the spectra andtheir best-fit models, and Table 5.2 lists the best-fit parameters. We then fit the spectra
extracted from the observations in 20003, 2007, 2009, and 2015 to see the time variabilitiesof the stripe emissions. Figure 5.8 depicts the surface brightness of the nonthermal
component as a function of the photon index (Γ), revealing a significant stripe-to-stripe

Downloaded from https://academic.oup.com/pasj/article/72/5/85/5892602 by 08749000 user on 02 July 2021

Fig. 4. Spectra and the best-fitting models for regions S1–S9 obtained from the data taken in 2009. The black curves are the sums of all the

Figure5.7:
Spectraextracted
from theregions
in 2009.
The redandmagentacomponents
whereasthe other colors
show contributions
fromstripe
each component.
The (S1–9)
red and orange
curves represent
the NEI
modelsfor the Fe
and IME components, respectively. The magenta-dashed curves indicate the Gaussian added to the model. The model curves for the power-law
curvesshow the NEI component of IME and Fe, respectively. The yellow curve is the Gaussiancomponent are drawn in blue. (Color online)
component. The blue curve is the power-law component.

5. Particle85-7
Heating and Accleration Observed by X-ray Time Variability

Publications
59 of the Astrono

equations assume
monochromatic
reality and hence
The 3D locati
in the discussion
netic field streng
the stripes are oIn this case, pro
to be slower tha
transverse veloci
velocity is calcul
shell expansion
3.1). We found th
to be measured wthat the morphol(figure 3). On thethe shell, or far dverse velocity of
Figure 5.8:Fig.
The5.surface
brightness
thesurface
stripes brightness
in each yearandas athefunctionof indicesthe photon index.
Relation
betweenofthephotoncase. The proper
The color of
each
label
corresponds
toindividual
regions.
The
brightness
of
the
plots
correspondsin the stripes. The data points from each stripe are plotted in different
to the shock nor
to the respectiveof thedata.
colors.epochs
The colortint
indicates the epochs of the data points. (Color
scenario. Models
online)different location
variation of the parameters as well as time variability of each stripe. An additional
2012; Caprioli &noteworthy
is theinrobust
relationshipbetweenthe surface
S3 finding
are lower
2003inverse
than inthe otherobserved
years. The
stripebrightness
the photonof the stripefrom
emission.
Chandra observa
S7andunderwenta indices
flux brightening
2003 to 2007 andmeasure the prop
then decayed. In the case of S9, a continuous flux increase
5.3.2 Discussions
to give meaningfu
was observed from 2003 to 2015. The images in figure 1
which is a key tofurthermore indicate that not only the bright stripes but
5.3.2.1 Accelerating and Cooling Timescaleof the stripes.
also much fainter structures are also variable.
etrevealedal. (2007),
Uchiyama
Our imagingFollowing
and spectralUchiyama
analyses havetime-variablefeaturesandin all of theOur spectros
stripes weanalyzed. The
analyzedsimilarvariabilities
those which
revealed anti-cor
Aharonian(2008),and stripes
Okunohave
et al.
(2020),
we cantoestiOkuno, Matsuda et al. (2020) reported (corresponding to the S1 region), suggestingphoton
theindices (mate the magnetic field strength of the emitting region if we
universality of the variabilities in this part of the SNR. Figure 5.8 indeed shows especiallythe synchrotronthe brighteningof as
relativistic
significantattributeflux variabilitiesof the S2, S3,toS7,productionand S9 regions,well as theelecS1 region. The
correlation as wetrons
through
acceleration
andthecomparedflux decay
decrease years,of while
fluxes of theS2 andS3 regionsin 2003 are
lower
toto
subsequent
the S7 region
experienced
a fluxincrease fromX-rays.
2003 to 2007,
followedby a subsequent
R5 defined in fig
electrons
emitting
synchrotron
Assuming
diffusive
decline. In
the
case
of
S9,
a
consistent
flux
augmentationisobserved
from
2003
to
2015.
gible in these regi
shock acceleration, we can write the acceleration timescale
Additionally, the images in Figure 5.3 not only highlight the prominent bright stripes
power law and p
asbut also reveal that much fainter features exhibit variabilities.
those from the s$−1.5 !
! ε "0.5 #
"
−2
vsh
B
the stripes, the dyr, (1)
tacc = 4η
keV
400 µG
3400 km s−1
significant anti-cothan the stripes w

60

5.3. Time Variabilities of Non-Thermal X-ray Radiation

The magnetic field strength of the emitting region can be estimated from the timescaleof variabilities as explained in Chapter 3. We can attribute the flux increase to the generation of relativistic electrons through acceleration and the flux decay to the reductionof the electrons through synchrotron cooling. Assuming diffusive shock acceleration,
the acceleration timescale can be written as𝑡acc = 4𝜂 𝜀  0.5 keV𝐵
400 𝜇G −1.5 

 −2

𝑣 sh

3400 km s−1

yr,

(5.2)

from Equation (1.37). Here, we assumed the shock velocity as 𝑣sh = 3400 km s−2
according to our measurement in Section 5.3.1.1. Following Equation (2.26), the synchrotron cooling timescale is𝑡syn = 4 𝜀  −0.5 keV𝐵
500 𝜇G −1.5yr.

(5.3)

We here note that the above equations assume the monochromatic emission of synchrotron photons from electrons with a certain energy for simplicity. Given that theobserved flux changes are several years, the above equations suggest that the magnetic
field in the stripe region is ∼ 500 𝜇G.

5.3.2.2 3D location of the stripe structure
Revealing the 3D location of the stripes would provide clues to their physical origin.
In the magnetic field strength estimation in Section 5.3.2.1, the stripes are implicitly
assumed to be located on the projected blast waves of the SNR. In this scenario, thestripes are expected to apparently move with slower velocities than the expansion, giventhat only a transverse velocity component is observable. The calculated transverse
velocity is 0′′. 25 yr−1 , assuming a spherical shell expansion with a velocity of 0′′. 29 yr−1
(see Section 5.3.1.1). The difference between these two values is too small to be measured
using the accessible data since the morphology of the stripe has also been changing
with time (Figure 5.6). On the other hand, when we assume that the stripes are locatedinside the shell or far downstream from the blast waves, the transverse velocity of the
stripes should differ from the above case. This scenario should be in favor of the propermotion of some structures perpendicular to the shock normal. Models proposed inprevious studies (e.g., Bykov et al., 2011; Malkov et al., 2012; Caprioli & Spitkovsky, 2013;
Laming, 2015) assume different locations. A future observation at another epoch would
significantly enhance our ability to measure the proper motion of the stripes accurately.
This precision is crucial for meaningful comparisons with the proper motion of theblast waves, ultimately aiding in determining the line-of-sight positions of the stripes.

5. Particle Publications
Heating and Accleration
Observed bySociety
X-ray Time
Variabilityof the Astronomicalof Japan(2020), Vol. 72, No.615

in Tycho’s SNR
Chandra. One osynchrotron spec
chrotron photon
synchrotron photron energy (Ee )make the synchr
chrotron spectration, if compres
stripes, stochast
spectra would b(2015) studied su
In either casebe very challeng
between surface
The result would
ters are responsib
the stripe emissiowould not appea
Figure 5.9: Fig.
Same
Figure 5.8, buttherelation
stripe data
only from2009 (red)areandshown,and the data
6.asComparisonof thebetweenthe stripes(red)
the rimin figure 5 is thatextracted fromthe Therim regions(blue)plotted
are added
thefromcomparison
between the
stripe and rim.
(blue).
data points
hereforarethe observationsin 2009.
stripes. This sugg
(Color online)
are similar to eac
5.3.2.3 The origin of the stripe structure
stripes have simil
by Lopez et al. (2015), the highest photon energy of
If the synchrotron
radiationfrom thestripe structure
has a universal
mechanism,
theelectrondensities
synchrotron
emissionin Tycho’s
SNR seems
to be limitedbyemission fromthesynchrotron-dominant
rim
is
also
expectedtoexhibittheantisider that the str
its age (= acceleration time). Given that, the cutoff energy
correlation between surface brightness and photon indices, similar to what was seen inthin sheets.
depends both on the shock velocity and on the magnetic
Figure 5.8. To see this, we perform theanalysis of the spectra extracted from the R1–5
4 3
et al. 2015). Thus, thefield strength as ε 0 ∝v shweBcan(Lopez
regions defined in Figure 5.4. Since
neglect thermal emission in these regions,hard
spectraof thewith
stripescan bepower
ascribed
to fastshock
the spectral
fitting
is conducted
an absorbed
law. The
resultsare plotted instrongfield inof2009.
the Inregion.
ConFigure 5.9, velocity
alongside and/or
the data fromthemagnetic
stripes observed
contrast
to the stripes,
5 Conclusiona significant
anti-correlation
does notappear withinpointsitfrom
the rim. Our
sidering
the peculiar
morphologyof thethedata
stripes,would
spectroscopy also revealed that the photon indices for the rim emission (Γ = 2.7–2.9)Usingare
Chandra d
be rather unlikely that only a fast shock velocity accounts
comparatively softer than those of the stripes (Γ = 2.1–2.6). This aligns with the findings
2015, we searche
for the hardness. Instead, it would be more probable thatby Lopez et al. (2015), who indicated the emission with the highest roll-off energy using
the themagnetic
fieldremnant
is amplified
in thethrough,
NuSTAR fromwest of thecoinciding
withstripes
the locationof theforstripes.chrotron radiatio
of Tycho’s SNR
example,the resonant(Skilling hardness1975) orΓnon-resonant(Bell
Then, what
contributes
to the increasedof the spectra in
the stripes?
by Okuno et al. (2004) cosmic-ray
streaming
instability.
Thechange
problemabout
One interpretationis that the hardness
changereflects theof thecutoff energy,
considering
thescenario
analyzed energy
corresponds
to thecutoffregiona synchrotron
variability of thethisis thatband
we cannot
explaintheshorttimeofvarispectrum (Lopez
al.,the2015).
In thisatcase,hardertheThesynchrotron
radiationof nine bright stabilityetof
stripes
the thesame
time.
timescaleof thebecomes,
ties not only of thvariability expected in the age-limited case would be of the
et al. (2020) but
order of the age of the SNR, ∼100 yr, which is much longer
If we attribute t
than observed. One of the possible solutions that can recon-

62

5.3. Time Variabilities of Non-Thermal X-ray Radiation

the higher it means the cutoff energy is. We begin by considering the scenario in which
the stripes share the same origin as the blast waves, although the discrepancy in spectra
between the stripes and the blast waves (Figure 5.9) implies a lower likelihood of thishypothesis. Given the NuSTAR result by Lopez et al. (2015), the energy of synchrotronemitting electrons is limited by the accelerated time corresponding to the age of theremnant (the age-limited case in the Section 2.2.3). The cutoff energy 𝜀max thus depends
both on the magnetic field strength and on the shock velocity (𝜀max ∝ 𝑣sh 4𝐵 3 ) as
described in Equation (2.30). Therefore, the hard spectra observed in the stripes can be
explained by the high shock velocity and/or the intense magnetic field within the region.
Considering the unique morphology of the stripes, it is improbable that the increased
hardness can be solely attributed to the high shock velocity. A more likely scenario isthe enhancement of the magnetic field in the stripes, possibly through mechanisms
such as the resonant (Skilling, 1975) or non-resonant (Bell, 2004) cosmic-ray streaming
instability. In this scenario, explaining the observed variability on a yearly timescale
is challenging, as the acceleration time scale is comparable to the age (∼ 450 years).
One possible explanation compatible with both the result by Lopez et al. (2015) and thefluctuating emissions in the stripes is the local amplification of the magnetic field at the
stripe region: the acceleration time in most regions is limited by the remnant age, whilethat at the stripe region is limited by synchrotron cooling loss as the result of enhanced
magnetic field. Since the analyzed energy range in this study is very limited, we can allow
for various possibilities to be considered. We would constrain the origin of the hardening
by future missions with both the wide energy range and high angular resolution, such
as HEX-P (Madsen et al., 2019) and Lynx (Gaskin et al., 2018). Observation with Chandra
indeed discovered such softening in Tycho’s SNR by Cassam-Chenaï et al. (2007).
When we assume that the stripe location is far downstream of the shock, it issomewhat puzzling to understand the hardness of the spectra. In order to deliver thesynchrotron-emitting electrons to the stripe region, the electrons need to be transported
downstream through diffusion or advection after acceleration at the blast waves. The
transported ultra-relativistic electrons undergo substantial synchrotron cooling losses,
leading to a softening of the electron spectrum and, consequently, the synchrotron
X-ray spectrum. A possible mechanism causing the hardening involves an enhanced
magnetic field. As Equation (2.23) describes, the synchrotron photon energy 𝜀 depends
on the parent electron energy 𝐸𝑒 . It follows that the stronger magnetic field leads to thehigher synchrotron cutoff energy, resulting in the observation of the harder synchrotron
spectra with Chandra. Moreover, the presence of compressible waves/turbulence
within the stripes might cause stochastic acceleration, resulting in a harder electron
spectrum, as theoretically studied by Zhang (2015) and Wilhelm et al. (2020). Threedimensional magnetohydrodynamic simulations by Inoue et al. (2012) suggested thepossibility of magnetic field amplification downstream shock due to interactions between shock and clumps in ISM.
In either scenario discussed above, explanations of the anti-correlation shown in
Figure 5.8 would pose a challenge. The findings suggest that only a few parameters are
related to the observed temporal and spatial variation; otherwise, such a pronounced
anti-correlation would not be observed. It is noteworthy that the surface brightness

5. Particle Heating and Accleration Observed by X-ray Time Variability

63

is equivalent in each region. This equivalence implies that the line-of-sight depth
is comparable between each stripe, assuming similar magnetic field strength andrelativistic electron densities across the stripes. If this holds true, it is more plausible
to consider the shape of the stripes as spheroids rather than thin sheets.

2.37+−00..04
03
0.77 ± 0.01
1.48+−00..4143
1.65+−00..0305
7.2 ± 1.4
161+−6835
130+−6728
105+−4012202+−7226
1.48
4.30+−10..30
98
5.4+−11..92
0.33 ± 0.05
1.28+−00..05
02375 ( 331)

3.31+−00..4238
1.49+−00..0403

4.8+−00..64
84+−10977+−108
85+−1411
156+−28
25

3.31
5.54+−00..9183

3.6+−00..54

0.53 ± 0.05
1.25+−00..02
01
405 ( 291)

† The parameter of [ Fe/C]/[ Fe/C]

0.60 ± 0.05
1.238+−00..002
007
593 ( 302)

1.40+−00..08
07
1.253+−00..006
001
1153 ( 433)

9.2+−01..80

2.11
2.36+−00..35
39
0.74 ( fixed)
7.6+−11..41

2.11+−00..70
40
1.58+−00..03
02
4.53 ( fixed)
5.2+−01..33
183+−4137
134+−3028
99+−249159+−24
23

2.44 ± 0.03
0.50 ± 0.01

5.92+−00..24
20

S5
161.7

4.97
4.63+−00..4026

8.1+−00..47
238+−2322
185+−1120
175+−811
163+−2029

4.97+−00..86
26
1.55 ± 0.01

2.15 ± 0.01
2.35 ± 0.02

5.83+−00..05
06

S4
505.3

S6
141.1

0.47 ± 0.05
1.234 ± 0.003
451 ( 299)

5.5 ± 1.2

1.51
3.00+−10..8483

6.4+−11..76
189+−3846
139+−3936
106+−3122
258+−8677

1.51+−00..54
38
1.57+−00..04
03

2.40 ± 0.04
0.46 ± 0.01

6.32+−00..30
19𝑛 e 𝑛 C 𝑑𝑉 /4𝜋𝑑 2 · [ C/H] ⊙ and linked to each other.

⊙ is linked to one of [ Ni/C]/[ Ni/C] ⊙ .

∫0.36 ± 0.04
1.24 ± 0.01
463 ( 338)

5.4+−10..45

1.77
3.85+−10..2080

9.2+−10..98
190+−4027
143+−3119
110+−1310
158+−55
25

1.71+−00..24
36
1.68+−00..01
04

2.06 ± 0.03
0.76 ± 0.01

5.23+−00..2112

6.37+−00..26
106.80 ± 0.17

2.39 ± 0.04
0.43 ± 0.01

S3
172.7

S2
210.2

S1
143.8

∗ Emmission measures of Fe and IME component are defined by

Flux (10−13 erg s−1 cm−2 )IME componentEM∗ ( 109 cm−5 )𝑘𝑇𝑒 ( keV )𝑛𝑒 𝑡 ( 1010 s cm−3 )[Mg/C]/[Mg/C] ⊙[Si/C]/[Si/C] ⊙[S/C]/[S/C] ⊙[Ar/C]/[Ar/C] ⊙[Ca/C]/[Ca/C] ⊙Fe componentEM∗ ( 109 cm−5 )𝑘𝑇𝑒 ( keV )𝑛𝑒 𝑡 ( 1010 s cm−3 )[Fe/C]/[Fe/C] ⊙Gaussian
Norm. (10−5 s−1 cm−2 )Centroid (keV)
𝜒 2 (d.o.f.)ΓParameters (unit)
Solid angle ( arcsec2 )Absorption
𝑁H ( 1021 cm−2 )Power law

Table 5.2: Best-fit parameters of the stripe regions in 2009

0.44+−00..06
07
1.232+−00..022002
598 ( 418)

8.0+−31..25

1.50
2.04+−00..6226

10.6+−62..91
279+−19469
194+−14448
202+−14065
298+−9377

1.50+−00..49
61
1.71+−00..03
04

2.12+−00..02
03
2.08+−00..02
01

6.03+−00..17
12

S7
338.8

0.35+−00..04
05
1.254+−00..007005
392 ( 321)

3.8+−00..85

2.22
2.41+−00..2655

5.5+−00..89
112+−33
20
91+−251677 ± 9142 ± 24

2.22+−00..5042
1.45+−00..01
04

2.25+−00..02
03
0.64 ± 0.01

6.56+−00..20
28

S8
183.4

0.61+−00..08
07
1.23 (fixed)
606 ( 373)

4.8+−00..65

0.37
1.53+−00..1223

4.8 ± 0.6
73+−10981 ± 1186+−1413
168+−3330

0.35+−00..06
04
1.41 ± 0.03

2.55 ± 0.02
1.05+−00..01
02

5.26+−00..14
12

S9
311.6

64
5.3. Time Variabilities of Non-Thermal X-ray Radiation

5. Particle Heating and Accleration Observed by X-ray Time Variability

5.4Time Variabilities of Thermal X-ray Radiation

5.4.1

Analysis and Results

5.4.1.1

Imaging Analysis

Declination (J2000)

64°12'

3
2

10'

1
0

08'

1

06'

2
3

04' 0.7 1.5 keV
0h26m00s 25m40s 20s
00s
Right Ascension (J2000)(b)

1e 7 4Flux change (2015 CE 2003 CE)

(a)

65

24m40s4

Figure 5.10: (a): A difference image of Tycho’s SNR between 2003 and 2015 within the energy
band of 0.7–1.5 keV. The color bar means the flux changes over the years in units of photons
s−1 cm−2 . Positive values represent brightening, and negative values represent darkening. The
box corresponds to the region delineated in panels (b) and Figure 5.11. (b): Zoom-in view ofthe box in the panel (a). The noteworthy feature is superimposed by a circle for aiding in theidentification.

Figure 5.10 (a) shows a difference image made in the same way as Figure 5.3 but in theenergy band of 0.7–1.5 keV, where the thermal radiation is dominant (e.g., Warren et al.,2005; Sato & Hughes, 2017). The thermal emission can be seen in the interior of the shell,
while the nonthermal emission is dominant in the shell. In the difference maps, we canobserve adjacent increases and decreases in most features. This phenomenon arises
from the movement of bright structures between 2003 and 2015 due to the expansionof ejecta and the radial proper motion of the blast waves. In the northwest, however,there is a bright spot (hereafter, Knot1) where we discover a monotonical increaseof photon counts over time without signs of proper motion as can be seen in theenlarged image of Figure 5.10 (b).
We present visual comparisons of Knot1 flux images in the energy bands of 0.7–1.5
keV (soft band), 1.6–2.5 keV (middle band), and 4.1–6.1 keV (hard band) in Figure 5.11.
We note that thermal radiation predominantly contributes to the soft and middle bands,
while nonthermal radiation dominates in the hard band (e.g., Warren et al., 2005). The
gradual brightening of Knot1 is evident in the soft band images from 2000 through2015 (Figure 5.11 a), consistent with the suggestion in Figure 5.10. In contrast, the66

5.4. Time Variabilities of Thermal X-ray Radiation(a) Soft band (0.7 1.5 keV)
Knot1

Ref1

Ref2
2000

2003

0.0

0.3

2007
0.6

2009
0.9

2015
1.2

Flux (photon cm 2 s 1)

1.5
1e 6

(b) Middle band (1.6 2.5 keV)
Knot1

Ref1

Ref2
2000

2003

0.0

0.3

2007
0.6

2009
0.9

2015
1.2

Flux (photon cm 2 s 1)

1.5
1e 6

(c) Hard band (4.1 6.1 keV)
Knot1
Ref2
2000
0.0

Ref1

2003
0.2

2007
0.4

2009
0.6

Flux (photon cm 2 s 1)

2015
0.8

1.0
1e 7

Figure 5.11: Exposure-corrected X-ray images of Knot1 in the soft (a), middle (b), and hard (c)bands taken in 2000, 2003, 2007, 2009, and 2015. The flux of the soft-band X-rays is shown withwhite contours. The Knot1, Ref1, and Ref2 regions used for spectral extraction are shown by
green ellipses.

middle-band images (Figure 5.11 b) show no significant flux fluctuation, except for theejecta expansion. Moreover, there is no correlation with the Knot1 structure in the softband images. These findings suggest that Knot1 originates differently from the middleband X-rays. The hard-band images in Figure 5.11 (c) reveal relatively faint and stablesynchrotron emission in Knot1, in contrast to the pronounced flux changes observed in
the stripe regions in the southwest (Section 5.3; Okuno et al., 2020; Matsuda et al., 2020).
5.4.1.2 Spectral Analysis
For investigation of the nature of Knot1 and quantitative measurement of its time
variabilities, we conducted the spectral analysis of the radiation from the region shownin Figure 5.11. We merged the datasets obtained in each year; thus, five spectra are
obtained from five different epochs (2000, 2003, 2007, 2009, and 2015). The background5. Particle Heating and Accleration Observed by X-ray Time Variability

67

−2

-1

counts s keV

-1

10

−3

10

−4

10

−5

10

ratio

−6

10
10
1
0.1

2000

2003

2007

2009

−2

-1

counts s keV

-1

10

−3

10

−4

10

−5

10

ratio

−6

10
10
1
0.1

0.5

−2

-1

counts s keV

-1

10

2

5

10

Energy (keV)

−3

10

−4

10

−5

10

−6

ratio

1

10
10
1
0.1
0.5

2015
1

2

5

10

Energy (keV)

Figure 5.12: Comparison between the spectra of the Knot1 region (black) and the best-fit modelof the Ref1 region (red). We apply the response of Knot1 to the Ref1 model. The bottom panelin the box of each year represents the ratio calculated by dividing the Knot1 counts by the Ref1
counts.

region is the same as that used in Section 5.3. The Knot1 region apparently has thecontribution of X-rays unrelated to the brightening structure in the middle band, as
can be seen in Figure 5.11 (b). In order to estimate that contribution, reference spectra
are extracted from a nearby region (noted as “Ref1” in Figure 5.11), whose middle-band
flux is almost the same as Knot1.
Figure 5.12 shows comparisons between the Knot1 spectra and the best-fit model ofRef1 in each year. It becomes evident that the Knot1 emission in the ≲ 1.5 keV band isnotably more intense than that of Ref1, whereas this distinction is not observed in thehigher energy band. A plausible interpretation is that the thermal radiation in Ref1 has
the same origin as the southeastward diffusing ejecta since the thermal radiation in mostregions inside is generally dominated by the ejecta emission (e.g., Cassam-Chenaï et al.,2007; Miceli et al., 2015). It can also be confirmed by its spectrum, which can be replicated68

5.4. Time Variabilities of Thermal X-ray Radiation−2-1

counts s keV

-1

10

−3

10

−4

10

−5

10

−6

chi

10

Ref1

3
0
−3
0.5

1

2

5

10

Energy (keV)
Figure 5.13: Spectra extracted from the Ref1 regions taken in 2000 (black), 2003 (red), 2007
(magenta), 2009 (green), and 2015 (blue). The dashed and dotted curves represent componentsof the NEI model originating from ejecta and power law, respectively.

using an NEI model with pure-metal composition. Meanwhile, the excess emission
can be interpreted as the radiation emitted from the brightening structure. Figure 5.12
also suggests the energy range characterized by a high Knot1/Ref1 ratio is broadened
toward higher energies year-by-year. It provides further support for the flux increase of
the soft-band radiation from Knot1. For a more quantitative assessment of the temporalvariability of the excess in the soft band, we conducted spectral fitting for Knot1 using
the model of a soft NEI component added to the model used for the Ref1 region.
We perform simultaneous fitting for the Knot1 and Ref1 spectra taken in 2000, 2003,
2007, 2009, and 2015. This analysis uses XSPEC version 12.13.1 (Arnaud, 1996) with
AtomDB version 3.0.9 (Foster et al., 2017). We bin the spectra so that each bin has
at least ten counts and fit it over the 0.5–10 keV energy. The model of the Ref1 spectrum iscomposed of an absorbed metal-rich NEI and a power law, in accordance with previous
studies (e.g., Yamaguchi et al., 2017) and expressed as:
Interstellar Abs. × (NEI + PowerLaw).

(5.4)

The metal-rich NEI component represents the radiation from the ejecta. We assume
electron temperature (𝑘𝑇𝑒 ), ionization timescale (𝑛𝑒 𝑡 ), and abundances of the component to be shared for each year. The abundances of magnesium, silicon, sulfur, argon,
and iron are set as free parameters, while those of calcium and nickel are linked to argon
and iron, respectively. The settings for the abundance of other light elements are assame as those of the NEI model in Section 5.3. We also allow to vary EM freely. We fix thephoton index (Γ) of the power-law component to 2.79, which is obtained by the spectral
analysis of a nearby nonthermal-dominated region. The power-law flux is linked eachyear. These components are multiplied by the Tübingen-Boulder model (Wilms et al.,

5. Particle Heating and Accleration Observed by X-ray Time Variability

69

−1

10

2000
2003
2007
2009
2015

−2

-1

counts s keV

-1

10

−3

10

−4

10

−5

10

chi

5
0
−5
0.5

1

2

5

10

Energy (keV)
Figure 5.14: Spectra extracted from the Ref2 region. Each component represents the same
one of Figure 5.13. The parameters, except for the normalizations, are the same as the best-fit
parameters of the Ref1 region. The normalizations of ejecta and power law components are freeparameters.

2000), which represents interstellar absorption. Figure 5.13 and Table 5.3 show fitting
results and best-fit parameters of the Ref1 region, respectively.
The Knot1 spectra are fitted with a model consisting of the ejecta component, whichis also employed for the Ref1 region, and an additional soft component. So the modelfor the Knot1 region can be expressed as:
Interstellar Abs. × (NEI(Soft) + NEI(ejecta) + PowerLaw).

(5.5)

We fixed each elemental abundance to the solar value. We allow 𝑘𝑇𝑒 , 𝑛𝑒 𝑡 , and EMs tovary freely over years except for 𝑛𝑒 𝑡 in 2000. Only 𝑛𝑒 𝑡 in 2000 is fixed to that in 2003
minus 4 × 109 cm−3 s (= 42 cm−3 × 3 yr) since it cannot be determined due to a lackof statistics. Note that the other parameters did not vary beyond the 1𝜎 confidence
level as the result of fixing 𝑛𝑒 𝑡 . We linked the parameters of the ejecta component to
the Ref1 spectra, but the EMs are free parameters.
The uncertainty in 𝑛𝑒 𝑡 for the year 2000 could potentially arise from X-ray contamination originating from the southwestern ejecta. Consequently, we explore the prospectof southwestern emission extension by examining the spectrum of an inner region of
Knot1 directed towards the southwest (designated as the Ref2 region in Figure 5.11). In
Figure 5.14, we present the Ref2 spectra along with models whose parameters, excluding
EM, are fixed to those of Ref1. As depicted in the figure, the Ref2 spectra lack the softband excess observed in Knot1. This outcome indicates that the southwest extension is
negligible and that the soft thermal emission is exclusively attributed to Knot1.

70

5.4. Time Variabilities of Thermal X-ray Radiation−2-1

counts s keV

-1

10

−3

10

−4

10

−5

10

−6

chi

10

2000

2003

2007

2009

3
0
−3
−2

-1

counts s keV

-1

10

−3

10

−4

10

−5

10

−6

chi

10

3
0
−3
0.5

−2

-1

counts s keV

-1

10

2

5

10

Energy (keV)

−3

10

−4

10

−5

10

−6

10

chi

1

2015

3
0
−3
0.5

1

2

5

10

Energy (keV)

Figure 5.15: Spectra extracted from the Knot1 regions. The solid curve is the NEI component
originating from ISM. The other curves represent the same components as Figure 5.13.

Figure 5.15 presents the spectra of Knot1 and the results of the spectral fit. Table 5.3
lists the best-fit parameters. It is confirmed that we can ascribe the time variability solely
to the additional soft component. Given that the NEI model with the solar abundance
effectively reproduces the Knot1 spectra in the soft band, the soft component is likely
associated with ISM heated up by the blast wave. For a more detailed examination of thevariability, Figure 5.16 depicts the value of 𝑘𝑇𝑒 , 𝑛𝑒 𝑡 , and EM in each year. The results with𝑛𝑒 𝑡 in 2000 as a free parameter are also shown in the same figure. The figure indicates+0.16
that 𝑘𝑇𝑒 increases significantly from 0.30+−00..05
07 to 0.69 − 0.12 , which is almost equal in theother case. We further validate this by considering the 𝑘𝑇𝑒 change when 𝑛𝑒 𝑡 is fixed to
8 × 109 cm−3 s, positioned between the best-fit value (9 × 1010 cm−3 s) and the fixedvalue (4 × 109 cm−3 s). In this scenario, the best-fit value of 𝑘𝑇𝑒 in 2000, 2003, 2007, 2009,
+0.10
+0.13
+0.03
+0.14
and 2015 are 0.47+−00..20
28 , 0.42 − 0.04 , 0.59 − 0.10 , 0.53 − 0.06 , and 0.70 − 0.13 keV, respectively.
Considering the observed 𝑘𝑇𝑒 increase, we also employ the gnei model as the softcomponent. The gnei model allows the ionization timescale averaged temperature to5. Particle Heating and Accleration Observed by X-ray Time Variability
1.0

71

4

net (cm s)

0.5

0.3

10

-3

kTe (keV)-5

EM (× 10 cm )

5

10

10

'09 '15

year

'15

2
1

0.2

2000 '03

3

2000 '03'09 '15

year

'15

0

2000 '03'09 '15Year'15

Figure 5.16: The best-fit values of 𝑘𝑇𝑒 , 𝑛𝑒 𝑡 , and EM of the soft NEI component as a function ofyear. The black and gray plots correspond to the case in which 𝑛𝑒 𝑡 is fixed to 4 × 109 cm−3 s, andis free, respectively.

deviate from the current temperature. The resulting value of 𝑘𝑇𝑒 in gnei are 0.26+−00..07
05 ,
+
0
.
09
+
0
.
13
+
0
.
05
+
0
.
17
0.37−0.07 , 0.57−0.10 , 0.52−0.07 , and 0.70−0.13 keV in 2000, 2003, 2007, 2009, and 2015, respectively, virtually identical to those obtained with the NEI model. Notably, there are no
significant changes in 𝑛𝑒 𝑡 and EM over time. It can be interpreted that the observed fluxchange is a result of an increase in electron energy induced by shock heating.

5.4.2

Discussions

5.4.2.1

Origin of Knot1

As outlined in Section 5.4.1, Knot1 in Tycho exhibits a significant increase in soft-band
X-ray flux. The result, coupled with the year-scale increase of the electron temperature,suggests that the blast wave recently heated a compact and dense clump. The modelwith the solar abundance successfully explains the spectra (see Table 5.3), indicating thatthe shock-heated gas originates from the ISM. However, we do not rule out the possibilityof CSM origin, considering the known interaction with a cavity wall in southwesternshell (Tanaka et al., 2021). Note that Knot1 represents the first example of ISM/CSM
X-ray emission in Tycho’s SNR, which is known as an ejecta-dominated SNR. Future
observations with improved statistics or spectroscopy could facilitate the measurementof elemental abundances, shedding light on the true origin and potentially providing
clues to the progenitor system of Type Ia SN of Tycho’s SNR.
Previous H𝛼 observations have unveiled the presence of Balmer-dominated filaments in the northeast region of the remnant, interpreted as forward-shocked neutral
gas and shock precursors (e.g., Ghavamian et al., 2000; J.-J. Lee et al., 2007). Figure 5.17
(the left and middle panels) compares the soft-band X-ray image from 2015 with theH𝛼 image from 2012 (Knežević et al., 2017). This figure demonstrates a strong spatial
correlation between the soft X-rays and a bright H𝛼 structure in the Knot1 region,
supporting the inference that the origin of Knot1 is associated with the ISM or CSM. It is

72

5.4. Time Variabilities of Thermal X-ray Radiation

Knot1

Soft X-ray (2015)

α contours (2012)

H

Hα (WHT, 2012)

Hα (HST, 2008)

Figure 5.17: Spacial correlation of the Knot1 region between X-rays and H𝛼 lines. In the left
panel, the Chandra X-ray image captured in 2015 (same as the rightmost panel of Figure 5.11(a)) is superimposed with contours from an H𝛼 image obtained in 2012 (refer to the middle
panel). The middle panel shows the H𝛼 image, acquired in 2012 using GH𝛼 FaS on the William
Herschel Telescope (WHT; Knežević et al., 2017). The right panel is the H𝛼 image taken in 2008
with WFPC2 on HST (J.-J. Lee et al., 2010).

worth noting that we can observe the bright and complicated shell structure only around
Knot1 in the entire H𝛼 image on the northeast of the SNR taken with the HST (the rightpanel of Figure 5.17 and cf. J.-J. Lee et al., 2010). Similar localized filaments are foundin other SNRs. For instance, Patnaude & Fesen (2007, 2014) found the region in which
X-ray and H𝑎𝑙𝑝ℎ𝑎 are spatially coincident (“XA” region) in SNR Cygnus Loop. Moreover,
Patnaude & Fesen (2007, 2014) reported the time-variable X-ray knots originated fromejecta in Cassiopeia A, which spatially coincide with optical light, as mentioned in
Section 3. Its physical scale is 0.02–0.03 pc, roughly consistent with the estimated size of
Knot1: ≃ 0.04 pc. While these structures in Cassiopeia A are interpreted as dense ejecta
clumps engulfed by the reverse shock, Knot1 is likely to originate from a small-scale
clumpy ISM/CSM heated by the forward shock.
The density of Knot1 can be estimated using the best-fit parameters as follows.
Assuming that the emitting region of Knot1 takes the form of an oblate spheroid withlong and short radii of 0.05 pc and 0.02 pc, respectively, we calculate its volume tobe 𝑉 ≃ 6 × 1051 cm3 . We derive a proton density of 𝑛 H = 35+−64 cm−3 utilizing the EM
best-fit parameter of the soft component in 2015, 𝑛𝑒 𝑛 H𝑉 /4𝜋𝑑 2 = ( 1.2+−00..43 ) × 1010 cm−5 .
Contrastingly, the post-shock density of Tycho’s SNR is estimated to be 𝑛 H = 0.1–2 cm−3
based on the flux ratio of the 70 𝜇m to 24 𝜇m infrared emission (Williams et al., 2013),
leading to the Knot1 density to be roughly 10–100 times higher than its surroundings.
5.4.2.2 Time Variability of Knot1
Time variability of Ionization Sate Given that the parameter 𝑛𝑒 𝑡 in XSPEC represents
the ionization timescale in case of constant 𝑘𝑇𝑒 , the ionization timescale 𝑛𝑒 𝑡 cannot
simply be considered as a product of density and actual time passed from the time in5. Particle Heating and Accleration Observed by X-ray Time Variability

5.0
net (×1010 cm 3 s)

5.0
net (×1010 cm 3 s)

73

2009 2007

2.0
1.0

2003

0.5

2015

2000
0.2

0.3
0.5
kTe (keV)

0.72009 2007

2.0
1.0

2003

0.5

2000
0.2

1.0

2015

0.3
0.5
kTe (keV)

0.7

1.0

10 5
10 1
10 3
11
+
Ion fraction of Mg

10 4 10 3 10 2 10 1 100
Ion fraction of Ne9 +Figure 5.18: H-like ion fraction of neon (Ne9+ ; left) and magnesium (Mg11+ ; right) as a functionof electron temperature (𝑘𝑇𝑒 ) and ionization timescale (𝑛𝑒 𝑡 ). The contours represent theconfidence level of Δ𝜒 = 1.0 (dotted) and 2.3 (solid) in 2003 (blue), 2007 (red), 2009 (green),and 2015 (white). The points show the best-fit values in each year. The data in 2000 show only
the uncertainty of 𝑘𝑇𝑒 because we fix 𝑛𝑒 𝑡 in this analysis. The ion fraction is the result of thecalculation with PyAtomDB (Foster et al., 2017).

which shock passed. We thus discuss the change in the ionization state of Knot1 using
ion fractions of the soft component each year. As depicted in Figure 5.18, H-like neonand magnesium are both increasing, suggesting the progression of ionization from 2003
to 2015. Therefore, we take into account that an SNR shock recently propagating intoa small cloud has heated and ionized Knot1 from year to year.
Cloud Crushing Time The time scale of the property change for clump in ISM is often
characterized by the cloud-crushing time (Klein et al., 1994). Let us consider the clump
with the density 𝜌 c and the pressure 𝑃c in ISM with the density 𝜌 i and the pressure 𝑃i
as depicted in Figure 5.19. Under an assumption of ram pressure equilibrium (𝑃c = 𝑃i ),the relation 𝜌 i𝑢 i2 = 𝜌 c𝑢 c2 can be derived from the Equation 1.2, where 𝑢 is the shock
velocities in the ISM (subscript i) and the clump (subscript c). Thus, we can describe
the velocity of the shock decelerated inside the clump as follows:
𝑢c =

𝑢i
,
𝜒 1/2

(5.6)

where 𝜒 (≡ 𝜌 c /𝜌 i ) is the density contrast between the clump and ISM. The shock velocity
inside the clump 𝑢 c can be determined to be 1500–2500 km s−1 , given that 𝜒 is estimatedto be 𝜒 ∼ 10 from Section 5.4.2.1 and the typical forward-shock velocity 𝑢 i of Tycho’s SNR
is 4000–8000 km s−1 (Tanaka et al., 2021). A cloud-crushing time can be defined as follows:𝑡cc ≡𝜒 1/2 𝑎 0 𝑎 0
= .
𝑢i
𝑢c

(5.7)

74

5.4. Time Variabilities of Thermal X-ray Radiation

ISM

Clump

Blast wave

Shocked
Clump

Shocked
ISM

Figure 5.19: Schematic of the Knot1 region and the shock waves.

Here, 𝑎 0 is the radius of the clumps. The emitting region of Knot1 has a radius of 𝑎 0 ≃0.02 pc, the cloud-crushing time can be calculated as 𝑡cc = 18 × (𝑢 i /2000 km s−1 ) −1 yr.
Note that the result is roughly consistent with the year-scale change of the X-ray flux inKnot1. However, the X-ray-emitting volume should increase over time in this scenario,
leading to the increasing EM. This contradicts the observed result. Considering themultiple filaments discovered with the HST image (Figure 5.17 right), Knot1 is thought toconstrain multiple clumps with radii finer than 𝑎 0 = 0.02 pc, leading to a shorter cloudcrushing time. Thus, the more likely scenario is that the brightening is attributed to the𝑘𝑇𝑒 increase after the shock wave has passed on a short timescale of about one year.
Heating Timescale For an explanation of the observed increase in 𝑘𝑇𝑒 , we first consider the case of thermal equilibration via electron–ion Coulomb collisions without
collisionless heating at the shock transition region. The temperature immediately
downstream for shock velocity 𝑢 c derive from Equation (1.20):
𝑘𝑇𝑖 =3
𝑚𝑖 𝑢 c2 .
16

(5.8)

Given that the electron temperature 𝑇𝑒 is initially lower than the ion temperature 𝑇𝑖 in
downstream plasma, the electron temperature is expected to increase monotonically.
The time evolution is described by Equation (1.22) and (1.23). A convenient expressionof the Coulomb logarithm can be given for X-ray emitting plasma (Masai, 1984):ln Λ = 24.8 + ln
𝑘𝑇𝑒  𝑛𝑒  −1/2

eV

cm−3

.

(5.9)

5. Particle Heating and Accleration Observed by X-ray Time VariabilityYear2000 '03

75

Year'07 '09'15

2000 '03-1Year'07 '09'15

2000 '03-1

uc=1500 km s

'07 '09'15-1

uc=2000 km s

uc=2500 km s

kTe (keV)

1.0

0.5
β0=me/mp
β0=0.05
β0=0.08
β0=0.1

0.2
0

1

2

3

8

4

5

time (×10 s)

6

0

1

2

3

8

4

5

time (× 10 s)

6

0

1

2

3

8

4

time (×10 s)

5

6

Figure 5.20: The comparison electron temperature as a function of the time between theobservation and the calculation assuming the shock velocity in the clump to be 𝑢 c = 1500 (left),2000 (middle), and 2500 km s−1 (right). The plots represent the observed results, and the curves
are the calculation results when 𝛽 0 = 𝑚𝑒 /𝑚𝑝 (red), 0.05 (green), 0.08 (blue), and 0.1 (yellow).
Year
2000 '03

kTe (keV)

1.0

Year

'07 '09'15

2000 '03

1998 as t=0'07 '09'15

1999 as t=00.5

β0=me/mp
β0=0.05
β0=0.08
β0=0.1

0.2
0

1

2

3

8

4

time (×10 s)

5

6

0

1

2

3

8

4

time (× 10 s)

5

6

Figure 5.21: Same as Figure 5.20, but the time in which the shock heating was started is assumed
to be 1998 (left) and 1999 (right)

We adopt an electron number density 𝑛𝑒 = 42 cm−3 based on 𝑛 H = 35 cm−3 (Section 5.4.2.1) and 𝑛𝑒 = 1.2𝑛 H . Neglecting contributions from ions heavier than hydrogenfor simplicity, the time evolution of 𝑘𝑇𝑒 can be calculated as shown in Figure 5.20. The
electron-to-proton temperature ratio (𝛽 0 ≡ 𝑇𝑒 /𝑇𝑝 ) at 𝑡 = 0 without collisionless electron
heating is equivalent to the particle mass ratio, i.e., 𝛽 0 = 𝑚𝑒 /𝑚𝑝 ≃ 5 × 10−4 . Based on
the above estimation, we give the case in which the shock velocity inside the clump𝑢 c is 1500, 2000, 2500 km s−1 . The timescale in which 𝑘𝑇𝑒 reaches around 0.7 keV (thevalue derived from data in 2015) becomes short as the shock velocity increases. This isbecause the greater the shock velocity, the larger the temperature difference between
electrons and protons. Figure 5.20 indicates that the model for 𝑢 c = 1500 km s−1 can
explain the observed data. While the plausible range of the shock velocity 𝑢 c is between
1500 km s−1 to 2000 km s−1 as estimated above, the lower velocity case is discussed inAppendix A. The timescale reaching 𝑘𝑇𝑒 = 0.7 is longer than observed in this scenario.

76

5.4. Time Variabilities of Thermal X-ray Radiation

For astrophysical shocks, collisionless electron heating is often effective at the shocktransition, as explained in Section 1.2.4. Secondly, let us consider this case. In this case,the ratio 𝛽 0 becomes larger than 𝑚𝑒 /𝑚𝑝 (≃ 5 × 10−4 ). We calculate the temperature
evolution when 𝛽 0 = 0.05, 0.08, and 0.1 as plotted in Figure 5.20. When 𝑚𝑒 /𝑚𝑝 ≤𝛽 0 ≤ 0.05, the model is in good agreement with the data. Considering the uncertainty
regarding when the forward shock indeed hit Knot1, we also conduct comparisons using
calculations with different assumptions about 𝑡 = 0, specifically for 𝑢 0 = 1500 km s−1 ,as illustrated Figure 5.21. Even when the year 1998 corresponds to 𝑡 = 0, observed data
can still be explained by the model with 𝑚𝑒 /𝑚𝑝 ≤ 𝛽 0 ≤ 0.05. However, it seems moreplausible when the year 1999 corresponds to 𝑡 = 0.
Optical observations also estimate 𝛽 0 of some SNRs with different shock velocities
using flux ratios of the broad-to-narrow components of the H𝛼 line (e.g., van Adelsberg
et al., 2008, also see Section 1.2.4). Especially with regard to Tycho’s SNR, Ghavamian et al.
(2001) and van Adelsberg et al. (2008) obtained 𝛽 0 < 0.1 and 𝛽 0 = 0.046+−00..007
006 , respectively,′
in “knot g” region, located ∼ 2 southeast of Knot1. In other SNRs with strong shocksof 𝑣sh > 1000 km s−1 including SN 1006 and Kepler’s SNR, the temperature rato 𝛽 0 is
given as 𝛽 0 ∼ 0.05 (Fesen et al., 1989; Ghavamian et al., 2002). From our comparisonof the observations with the calculations, Knot1 has 𝑚𝑒 /𝑚𝑝 ≤ 𝛽 0 ≤ 0.05 with a shock
velocity of ∼ 1500 km s−1 . It suggests that collisionless electron heating in Knot1 has
a comparable efficiency with the result of the previous H𝛼 observations.
The measurement of 𝛽 0 in X-rays has not been definitive, except for the diagnostics
of low-ionized Fe fluorescence emission proposed by Yamaguchi et al. (2014). This
uncertainty stems from the challenges of selective analysis of X-ray-emitting plasmathat has not yet been influenced by Coulomb heating. Therefore, H𝛼 line diagnostics
are often employed for the measurement, as they can measure 𝛽 0 in just the region
where hydrogen atoms have not reached a fully ionized state, coinciding with the area
where collisionless electron heating is effective. However, estimations using the H𝛼line come with systematic uncertainties in shock models (e.g., van Adelsberg et al.,2008). The analysis presented in this thesis introduces a new method for estimating
𝛽 0 independently of the H𝛼 measurements.

∫𝑛𝑒 𝑛 H 𝑑𝑉 /4𝜋𝑑 2 .

𝑛𝑒 𝑛 C 𝑑𝑉 /( 4𝜋𝑑 2 [ C/H] ⊙ ) .

2.4+−10..09

2.3+−11..32
0.70 ± 0.03
5.0+−11..32
1.4+−00..84
11+−5311+−639+−64
1.1+−00..84
2.79 (fixed)
1.7 ± 0.1

2007
1.01+−00..0403
3.0+−11..422009

1.6+−10..16
1.3+−00..64
1.8+−00..43
0.43 ± 0.10 0.57+−00..14
10 0.51 ± 0.05
+
12
.
0
+
1
.
9
8.8−1.9
7.5−1.4
7.9+−11..30
fixed to the solar value
1.4+−00..73
2.3+−10..09
2.9+−11..10
2.79 (linked to Ref1)
1.7 (linked to Ref1)
731 (629)

1.6+−11..42

2003¶ The parameters of the reference components other than the EMs are linked to the NEI component for the Ref1 region.

‡ EMs for the soft components are defined as† The energy flux in the energy band of 4–6 keV.

∫

Flux† (10−15 erg cm−2 s−1 )ΓEM‡ (1010 cm−5 )𝑘𝑇𝑒 (keV)𝑛𝑒 𝑡 (109 cm−3 s)abundanceEM∗ (109 cm−5 )Flux† (10−15 erg cm−2 s−1 )2.5+−21..28
0.30+−00..0507
4.8

2.4+−11..31

EM∗ (109 cm−5 )𝑘𝑇𝑒 (keV)𝑛𝑒 𝑡 (1011 cm−3 s)[Mg/C]/[Mg/C]⊙[Si/C]/[Si/C]⊙[S/C]/[S/C]⊙[Ar/C]/[Ar/C]⊙ (=[Ca/C]/[Ca/C]⊙ )[Fe/C]/[Fe/C]⊙ (=[Ni/C]/[Ni/C]⊙ )

Γ

2000

Parameters (Units)
𝑁H (1022 cm−2 )∗ EMs for the NEI components are defined as𝜒 2 (d.o.f.)

Reference comp. ¶Power law

Knot1 region
Soft comp.

Power law

Components
Absorption
Ref1 region
NEI comp.

Table 5.3: Best-fit Parameters of Ref1 and Knot1 Regions

5.2+−22..41

1.2+−00..43
0.69+−00..1612
7.6+−21..75

5.2+−22..20

2015

5. Particle Heating and Accleration Observed by X-ray Time Variability
77

78

6
Future Works
Contents
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Probing the Shock–Cloud Interaction from Variability of Expansion Velosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Our Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Observations and Data Reductions . . . . . . . . . . . . . . .
6.2.3 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . .
6.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Time Variabilities in Kepler’s SNR . . . . . . . . . . . . . . . . . . .
6.4 Measuring Ion Temperature . . . . . . . . . . . . . . . . . . . . . . .

6.1

79
80
80
80
81
89
90
92

Overview

In Chapter 5, we presented the energy-transfer investigations of the shock-heated orshock-accelerated particles through X-ray time variabilities. These phenomena arefrequently observed in regions where SNR shocks interact with dense clouds (e.g.,
Ghavamian et al., 2000; Ackermann et al., 2013). We can also raise the possibility of
an interaction between shock and dense cloud in Tycho’s SNR from both the magnetic
field amplification suggested in Section 5.3 and the discovery of shock-heated clumps
in Section 5.4. Tanaka et al. (2021) reported indications of recent collisions of shock withthe wall using the remnant expansion (see Chapter 3). In Section 6.2, we measure theshock velocity with newly observed Chandra data to investigate further the surrounding
environments of Tycho’s SNR.
The analysis in Section 5.4 confined 𝛽 0 in Knot1 of Tycho’s SNR, introducing a new
method to measure 𝛽 0 independently of a previous measurement with H𝛼 observation.
To enhance the usefulness and reliability of the method, we propose two applicationsin Section 6.3 and 6.4. In Section 6.3, we search for the time variabilities in Kepler’s SNR,
79

80

6.2. Probing the Shock–Cloud Interaction from Variability of Expansion Velosity
Table 6.1: Observation Log for Proper-Motion Analysis

ObsID
24420
25046
23538
24419
24421
24418
26426

Start Date
2021 May 20
2021 May 21
2021 Sep 27
2021 Oct 01
2021 Oct 172022 May 25
2022 May 26

Effective Exposure (ks)
14.88
14.71
28.70
28.70
29.69
14.88
14.88

Chip
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I
ACIS-I

considering possibilities to understand the spatial dependences of 𝛽 0 . In Section 6.4,
we discuss the application to time variabilities of ion temperatures with high-resolution
spectroscopy to compare the time evolution of electron temperature.

6.2

Probing the Shock–Cloud Interaction from Variability of
Expansion Velosity

6.2.1 Our Objectives
The shock waves of Tycho’s SNR are likely to have various properties, e.g., velocities
and pre-shock densities. Katsuda et al. (2010) estimated the pre-shock ambient density
surrounding Tycho’s SNR to have azimuthal non-uniformity from the proper motion
measurement of the forward shock. Williams et al. (2013) also reported the azimuthal
density distribution from the infrared flux ratio with Spitzer. As explained in Chapter 3,
Tanaka et al. (2021) indicated a recent shock–wall interaction from the time variabilities
of shock velocities. From these studies, we can expect the various shock properties
in the azimuthal angle and the time. It is crucial to comprehend the impact of theshock–wall interaction on the shock properties in terms of not only the shock physics
but also its potential to constrain the characteristics of the progenitor system. Thissection presents the analysis results of new data observed in 2022–2023 with Chandra
added to data used by Tanaka et al. (2021) and future prospects, aiming to study theproper motion after 2015 and the spectral variability of synchrotron radiation as a resultof the changes of shock properties.

6.2.2

Observations and Data Reductions

We use the Chandra data taken in 2021–2022 as a guest observer program proposed byWilliams (2020), in addition to the data used in Chapter 5. Table 6.1 shows the additional
data log. Similar to the analysis in Chapter 5, we conduct astrometry correction forthe data of ObsID 24420, 25046, 24418, and 26426, which are thought to have almost
no annual parallax for point sources. Since the statistic of each observation is too low,

6. Future Works81

we cannot detect enough point sources to correct the astrometry. Thus, after merging
the observations, we detect the sources and correct the astrometry in the same way as
Section 5.2. We used the data after the correction for imaging analysis, and the total
effective exposure time is 59.35 ks. For spectral analysis, we add the data of ObsID
23538, 24419, and 24421, whose astrometries are not corrected, to improve the statistic,
resulting in a total exposure time of 146.44 ks.

6.2.3

Analysis and Results

6.2.3.1

Imaging Analysis

We used the Chandra flux image in the energy band of 4.1–6.1 keV, where the synchrotron
emission is dominant, as mentioned in Section 5.3.1 for the profile extraction. Radial
profiles are extracted from the regions positioned at the rim of the remnant (labeled
as “1–13” in Figure 6.1 (top)). These regions are the same as those used by Tanaka et al.
(2021). The box rotation angles are determined by hand in order to extract the profile
perpendicular to the rim. The bin width of the profile is 0′′. 5. Examples of the profiles
are shown in Figure 6.2. The shock position corresponds to the peak of the profile. The
profiles in the regions generally move over the years, as discovered by Katsuda et al.
(2010), Williams et al. (2016), and Tanaka et al. (2021).
We investigated rim expansion using the same method as Tanaka et al. (2021). We
explored the displacement distance to achieve the best alignment between the rim peaksin the profiles from two epochs through an artificial shifting of one profile. This distancecan correspond to the actual distance traveled by the rim. When the shift distance is
non-integer multiples of the bin, the shifted profile is reconstructed by distributing
the counts between two bins using a weighted binomial distribution. Comparing two
profiles with flux 𝑓𝑖 and 𝑔𝑖 of bin index 𝑖 , we calculate chi-squared defined as:
𝜒2 =

∑︁
𝑖

( 𝑓𝑖 − 𝑔 𝑖 ) 2,(𝑑 𝑓𝑖 ) 2 + (𝑑 𝑔𝑖 ) 2(6.1)

where 𝑑 𝑓 and 𝑑 𝑔 are the standard deviations of each profile. We used the chi-squared
as the quantified degree of matching between the profiles. The obtained chi-squaredis shown in Figure 6.3. The velocity that makes chi-squared the minimum value 𝜒minis obtained by fitting with a quadratic function, as can be seen from the figure. The 1𝜎
confidence range is defined as the range to be 𝜒 ≤ 𝜒min + 1.
The velocity of the shock proper motion is measured in three intervals: 2003–2009(interval A), 2009–2015 (interval B), and 2015–2021 (interval C). The results are represented
in Figure 6.4 and Table 6.2. The decelerations in the interval A and B in Region #6–11 areconsistent with the previous measurement by Tanaka et al. (2021). Focusing on Region#8 and #9, we reveal that the shock waves decelerated further until 2021. Additionally,
the deceleration is shown in Regions #12 and 13, where the analysis by Tanaka et al.
(2021) reported constant velocities. On the other hand, Regions #5–7, where Tanaka
et al. (2021) discovered significant deceleration from 2003 to 2015, show no significant
velocity change between the intervals B and C.

82

6.2. Probing the Shock–Cloud Interaction from Variability of Expansion Velosity

1e 8 1.6

1
13
12

4

09'

11
10

06'

5

9
6

0h26m00s

2003
Declination (J2000)

09'

2009
13

3
4

12
11

0.4

24m30s0.0

1e 8 1.6

2 1

13

3
4

12
11

10
06'

5
6

03'

2015
64°12'

Declination (J2000)

25m30s00sRight Ascension (J2000)

2 1

0.8

8

7

03'

64°12'

1.2

09'

7

5

9

8

10
6

2021

2 1

13

3
4

12
11

7

9

8

0.8

2 1

13

3
4

12
11

10
06'

5
6

03'
0h26m00s

725m30s

800s

Right Ascension (J2000)

9

10
5
6

24m30s 0h26m00s

1.2Flux (photon cm 2 s 1)

Declination (J2000)

3

Flux (photon cm 2 s 1)2

64°12'

725m30s

800s

Right Ascension (J2000)

0.4

924m30s0.0

Figure 6.1: (Top): The flux image of Tycho’s SNR in the energy band 4.1–6.1 keV taken with
Chandra in 2009. The red boxes are the regions to extract profiles. (Bottom): The flux images
obtained in 2003, 2007, 2009, and 2015. The orange boxes represent regions for spectral
extractions.

6. Future Works

1.5

83

Region 1

2003 2.0
20092015 1.5
2021
1.0

1.0
0.5

4

Region 3

0.5

0.5

0.0
5

1.0

Region 2

15

10

5

0

5

0.0

Region 4

15

10

5

0

0.0

5

Region 5

1.0

15

10

5

0

5

Region 6

1.0

3
2

0.5

0.5

Flux (×10 6 cm 2 s 1)

1
0

20

15

10

5

0.0
1.5

0

Region 7
1.0
0.5
0.0
2.5
2.0

15

10

5

0

5

Region 10

20

15

10

5

0

0.0
1.5

5

Region 8

1.0

1.0

0.5

0.5

0.0
2.5
2.0

15

10

5

0

5

0.0
2.5

Region 11

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

20

15

10

5

0

0.0

5

2.5 Region 13

10

5

0

5

15

10

5

0

5

Region 12

2.0

1.5

0.0

15

Region 9

20

15

10

5

0

Position (arcsec)

5

0.0

20

15

10

5

0

Position (arcsec)

5

2.0
1.5
1.0
0.5
0.0

20

15

10

5

0

Position (arcsec)

5

Figure 6.2: Radial profiles of the shocks of Tycho’s SNR extracted from each region. The blue,
red, green, and yellow lines represent the 2003, 2009, 2015, and 2021 data, respectively. The x-axes
show the distance from the rim position in 2021 in the outward direction.

84

6.2. Probing the Shock–Cloud Interaction from Variability of Expansion Velosity

Best-Fit Result of Chi-squared in Region 880

Chi-squared

60

40

20

0

2003-20092009-2015
2015-2021
0.2

0.3
0.4
Velocity (arcsec yr 1)

0.5

0.6

Figure 6.3: The chi-squared in measuring the rim proper motion and the best-fit quadratic
function in Region #8. The chi-squared is calculated from Equation (6.1) by moving a profile
artificially under assumptions of given shock velocities in intervals of 2003–2009 (blue), 2009–2015 (red), and 2015–2021 (green).

Table 6.2: The observed proper motions of shock waves

Region No.
1
2
3
4
5
6
7
8
9
10
11
12
13

2003–2009(arcsec yr−1 )0.25 ± 0.02
0.30 ± 0.02
0.14 ± 0.03
0.22 ± 0.01
0.28 ± 0.03
0.43 ± 0.04
0.35 ± 0.02
0.40 ± 0.03
0.39 ± 0.02
0.39 ± 0.01
0.39 ± 0.01
0.31 ± 0.03
0.30 ± 0.01

2009–2015(arcsec yr−1 )0.26 ± 0.02
0.29 ± 0.02
0.20 ± 0.05
0.23 ± 0.01
0.25 ± 0.03
0.30 ± 0.06
0.27 ± 0.03
0.34 ± 0.02
0.32 ± 0.03
0.37 ± 0.01
0.38 ± 0.01
0.33 ± 0.02
0.31 ± 0.02

2015–2021(arcsec yr−1 )0.31 ± 0.04
0.27 ± 0.04
0.20 ± 0.04
0.25 ± 0.03
0.24 ± 0.05
0.27 ± 0.04
0.25 ± 0.07
0.26 ± 0.03
0.21 ± 0.05
0.31 ± 0.01
0.30 ± 0.03
0.22 ± 0.04
0.22 ± 0.03

6. Future Works

85

1 2

5

Region Number
6 7
8 9 10 1112 13

2003-2009 (Interval A)
2009-2015 (Interval B)
2015-2021 (Interval C)

5

Velocity (×103 km s 1)

0.4

6

4

0.3

3
0.2

2

Acceleration (×10 2 arcsec yr 2)

0.1
2

1

Angle from North (deg)

5

1
0

0

-1

5

-2

10

Interval A-B
Interval B-C

-3
0

60

Acceleration (×10 6 km s 2)

Velocity (arcsec yr 1)

0.5

3 4

120
180
240
Angle from North (deg)

300

360

Figure 6.4: The shock velocities (top) and accelerations (bottom) in each region as a functionof the azimuthal angle. In the top panel, the circle, squared, and triangle plots represent thevelocities in 2003–2009 (Interval A), 2009–2015 (Interval B), and 2015–2021 (Interval C), respectively.
The plots in the bottom panel show the acceleration rate in Interval A–B (square) and B–C(triangle) under an assumption of the constant rate over the interval. The angle is measured
from the north direction based on the geometric center (𝛼 = 00h 25m 19s. 9, 𝛿 = 64◦ 08′ 18′′. 2 (J2000))
measured by Ruiz-Lapuente et al. (2004).

Counts s 1 keV 1

86

6.2. Probing the Shock–Cloud Interaction from Variability of Expansion Velosity

10 4Counts s 1 keV 1
Counts s 1 keV 1

reg01

reg02

reg03

reg04

reg05

reg06

reg07

reg08

reg09

reg10

reg11

reg12

10 2
10 3
10 4
3
0
-3
10 2
10 3
10 4
3
0
-3

Counts s 1 keV 1

2015
2021

10 3

3
0
-3

10 2
10 3
10 4
3
0
-3

Counts s 1 keV 1

2003
2009

10 2

4.0

10 2

5.0
6.0
Energy (keV)

7.0

8.0

4.0

5.0
6.0
Energy (keV)

7.0

8.0

10 3
10 4
3
0
-3

reg13
4.0

5.0
6.0
Energy (keV)

7.0

8.0

Figure 6.5: Fitting result of the spectra extracted from Regions #1–13. The blue, red, green, andyellow plots correspond to the 2003, 2009, 2015, and 2021 data. The dashed and dash-dottedcurves represent the power-law and Gaussian components. The solid steps are the total modelof all components.

6.2.3.2 Spectral Analysis
The shock-velocity changes can change synchrotron emission properties from the rim.
To investigate the changes in the synchrotron radiation, we extract the spectra fromregions represented in Figure 6.1 (bottom). Each region has a width of 0.′ 15, located at thebright rim structure. The extracted spectra are binned so that each bin has at least ten

6. Future Works

87

Time variabilities of Photon Index
reg02

4

Time variabilities of Flux

reg03

3

2

reg01

4

reg04

reg05

reg06

reg07

reg08

reg09

reg10

reg11

reg12

reg13

2003 2009 2015 2021 2003 2009 2015 2021
Year
Year

2

reg07

reg08

reg09

3
2

reg10

reg11

reg12

3

Flux (× 10 3 cm 2 s 1 keV 1)

Photon Index

reg05

0
3

reg06

2

1
0
3

reg04

2
1
0
3
2
1

2
4

reg03

1

3

4

reg02

2

3

4

reg01

reg13

2003 2009 2015 2021 2003 2009 2015 2021
Year
Year

0
3
2

3

1

2
2003 2009 2015 2021
Year02003 2009 2015 2021
Year

Figure 6.6: The best-fit parameters of the power-law component, i.e., photon index (left) andflux (right), as a function of the observed year.

counts. The analyzing range is 3.5–8.0 keV, where the synchrotron emission dominates.
Although radiation from these regions consists mainly of synchrotron emission, some
spectra demonstrate peaks around 6.4 keV, corresponding to the energy of a Fe K𝛼
fluorescence line. Thus, we used a power-law function for synchrotron radiation and a
Gaussian for a Fe K𝛼 line to reproduce the spectra. We allow flux and photon index (Γ)
to vary in different years while parameters of the Gaussian are linked each year. The
centroid energy of the Gaussian component is fixed at 6.4 keV. We multiplied these twocomponents by the absorption model (the Tüebingen-Boulder model; Wilms et al.,2000). The column density for the absorption model is fixed at the best-fit value inSection 5.4, 𝑛 H = 1.0 × 1022 cm−2 , because it can not be determined in the analyzed
energy band. The extracted spectra and the best-fit models are shown in Figure 6.5,
and the best-fit parameters are listed in Table 6.3. We can reproduce the extracted
spectra with the power law model with a Gaussian.
We plot the best-fit parameters as a function of years in Figure 6.6. As seen in the
figure, the spectral analysis reveals no significant time-variabilities that exceed the
90% (= 1.64𝜎 ) confidence interval. However, focusing on Region #10–12, there are
increasing trends in photon indices.

88

6.2. Probing the Shock–Cloud Interaction from Variability of Expansion Velosity
Table 6.3: Best-Fit Parameters of Spectral Analysis

Region No.

1

2

3

4

5

6

7

8

9

Year
2003
2009
2015
2021
2003
2009
2015
2021
2003
2009
2015
2021
2003
2009
2015
2021
2003
2009
2015
2021
2003
2009
2015
2021
2003
2009
2015
2021
2003
2009
2015
2021
2003
2009
2015

Power law
Gaussian [ 1 ]ΓFlux [ 2 ]Norm. [ 3 ]3.4 ± 0.3
1.4+−00..95
3.0+−00..21
0.7+−00..21≤ 1.1
2.9+−00..43
0.5+−00..32
3.5 ± 0.4
1.0+−00..94
3.2 ± 0.4
1.1+−00..64
3.0 ± 0.1 0.8 ± 0.2≤ 1.5
3.1 ± 0.3
0.9+−00..53
3.4 ± 0.4
1.1+−00..95
3.0+−00..76
0.3+−00..52
3.0 ± 0.2 0.4 ± 0.1
1.5+−11..21
2.6 ± 0.4
0.2+−00..21
3.7 ± 0.5
0.9+−10..05
3.0 ± 0.2
2.6+−00..76
2.7 ± 0.1 1.8 ± 0.2≤ 2.6
2.8 ± 0.2
1.5+−00..64
3.0 ± 0.2
2.3+−00..76
2.9 ± 0.3
0.7+−00..43
3.0 ± 0.1
0.8+−00..21
5.1 ± 1.8
2.8 ± 0.3
0.5+−00..32
3.4 ± 0.4
1.2+−00..95
2.9+−00..54
0.3+−00..31
3.0 ± 0.2 0.3 ± 0.1
1.5 ± 1.3
2.8 ± 0.5
0.2+−00..212.5+−00..65
0.14+−00..19
08
+
0
.
4
2.9 ± 0.4
0.5−0.2
2.9 ± 0.2 0.5 ± 0.1≤ 1.4
2.7 ± 0.4
0.3+−00..32
2.6 ± 0.4
0.3+−00..21
3.1 ± 0.3
0.9+−00..63
2.9 ± 0.1 0.6 ± 0.1
1.5 ± 1.5
2.8 ± 0.3
0.6+−00..42
2.9+−00..43
0.6+−00..43
3.2 ± 0.4
0.8+−00..53
3.2 ± 0.2 0.7 ± 0.2≤ 1.1
2.5 ± 0.4 0.2 ± 0.1
Continued on the next page

6. Future Works89

Table 6.3: Best-Fit Parameters of Spectral Analysis (Contunued)

Year PhoIndex Flux [ 2 ]Norm. [ 3 ]2021 3.1 ± 0.3
1.4+−00..75≤ 1.1
2003 2.9 ± 0.3
0.80−.04.3
2009 2.9 ± 0.1
0.8+−00..21
10≤ 1.5
2015 3.3 ± 0.3
1.3+−00..75
2021 3.2 ± 0.3
1.4+−00..75
2003 2.6 ± 0.3
0.6+−00..32
2009 2.9 ± 0.1
0.8+−00..2111
2.9 ± 1.8
2015 2.9 ± 0.3
0.8+−00..43
2021 3.0 ± 0.3
0.8+−00..53
2003 2.6 ± 0.2
0.7+−00..32
2009 2.8 ± 0.1
0.9+−00..21
3.4 ± 2.1
122015 2.8 ± 0.3
0.9+−00..43
2021 2.9 ± 0.3
0.9+−00..53
2003 3.0 ± 0.3
1.0+−00..63
2009 3.0 ± 0.1 1.1 ± 0.2
13≤ 1.72015 2.9 ± 0.3
0.9+−00..53
2021 3.1 ± 0.3
1.0+−00..63[ 1 ] The centroid energy is fixed at 6.4 keV.
[ 2 ] The flux at 1 keV. The unit is 10 − 3 ph keV − 1 cm − 2 s − 1[ 3 ] The total photons in a line. The unit is 10 − 7 ph cm − 2 s − 1
Region9

6.2.4

Discussion

Using Chandra data observed in 2021, our analysis reveals the proper motion of Tycho’s
SNR shock after 2015, where Tanaka et al. (2021) reported a deceleration in Regions#6–11 from 2003 to 2015. Tanaka et al. (2021) and Kobashi et al. (2023) proposed thatshock deceleration is a consequence of interacting with a wall characterized by a spatial
density gradient in the cavity-to-wall boundary. Assuming this hypothesis, it is likelythat the shocks in Regions #6 and 7, which exhibit constant velocities from 2009 to 2021,
have already traversed the gradient boundary. It follows that the shock had alreadyreached the wall with a spatially constant density before 2009. On the other hand, shocks
in Regions #8 and 9 show even lower velocities in interval C than those in interval B.
Notably, the acceleration rate is almost the same between intervals A–B and B–C. This
result indicates that shocks in these regions have been traversing the gradient boundaryfrom 2003 to 2021. The shocks in Regions #10–13, which show larger deceleration frominterval B to C than before, indicate more recent (around 2015) interaction with thewall than in the other regions.
Kobashi et al. (2023) studied the spatial density distribution surrounding Tycho’s SNR
using an updated hydrodynamic shock model considering an effect of a wind-like cavity

90

6.3. Time Variabilities in Kepler’s SNR

with 𝜌 (𝑟 ) ∝ 𝑟 −2 . Their model concluded that the shock in Region 11–13 indicated theinteraction with the boundary before 2003, contradicting our results. This contradiction
likely arises from an implicit assumption in their analysis: the boundary is inside theshock position in 2015. It is anticipated that the incorporation of our results, whichindicate the start and end positions of the cavity-to-wall boundary layer, into their model
will enable a more accurate determination of the cavity and wall locations, leading to amore precise estimation of the mass-loss rate from the progenitor system of the remnant.
The interaction of the shock with the wall might lead to changes in the properties
of synchrotron radiation from the shock. Although the spectra exhibit no significant
changes, we observe a softening from 2003 to 2015 in regions #10, 11, and 12, as canbe seen from the increasing photon indices in Figure 6.5 (left). When attributing thechange in photon index to the variation of the maximum electron energy, the decrease
2 of Equation (1.40),
in shock velocity can cause softening, as suggested by 𝐸max ∝ 𝑣shgiven that the maximum energy of accelerated electrons is generally limited by theremnant age (Lopez et al., 2015). As the shock velocities in interval A become ≃ 79%,
≃ 77%, and ≃ 70% in regions #10, 11, 12, respectively, the rolloff energy of synchrotron
4 of Equation (2.30),
radiation should decrease to approximately 20–40% from 𝜀 ∝ 𝑣shlikely affecting the photon index in the analyzed energy band.
However, the above scenario is rather simple and cannot explain the changes inother regions, especially regions #6, 7, 8, and 9, with the significant velocity decreases.
One possibility is that the amplified magnetic field raises the cutoff energy. Inoue et al.
(2012) indeed suggested that the interaction between shock waves and the clumpy
interstellar medium can amplify the magnetic field after the shock through turbulent
dynamo action by three-dimensional magnetohydrodynamic simulations. If the wallcontains clumpy interstellar clouds, a more complex scenario is required to explain
the photon index change.
In any case, discussing the spectral time variabilities based only on our results withinsufficient statistics is challenging. Future long-duration observations with Chandra
will enable us to study time variabilities in more detail. Additionally, a comparison
with a hydrodynamical simulation of accelerated particle production in SNRs proposed
by S.-H. Lee et al. (2012) will provide clues about the effect of the shock decelerationon the photon index.

6.3

Time Variabilities in Kepler’s SNR

In the previous chapters, we present the discoveries of time variability in thermal X-rays,
nonthermal X-rays, and the proper motions of shock waves in Tycho’s SNR. This analysis
can be applied to other regions in Tycho or other young SNRs. Exploring other thermal
time variabilities can provide some clues to collisionless electron heating because it
might allow us to investigate the dependency of electron–ion temperature ratio 𝑇𝑒 /𝑇ion
on other parameters: shock the velocity, the magnetic field, and the obliquity angle
between shock normal and upstream magnetic field. Although the magnetic field6. Future Works

91

1.0

Flux (photon cm 2 s 1})
0.0

0.5

0.5

1.0

1e 6

Declination (J2000)-21°28'
29'
30'
31'

2004 CE 2000 CE

17h30m50s

45s

40s

Right Ascension (J2000)

2006 CE 2000 CE
35s

17h30m50s

45s

40s

Right Ascension (J2000)

2014 CE 2000 CE
35s

17h30m50s

45s
40s
35s
Right Ascension (J2000)

Figure 6.7: The images of flux change of Kepler’s SNR in the energy band of 0.5–0.7 keV. The
images are made by subtracting Chandra flux images taken in 2000 from 2004 (left), 2006 (middle),
and 2014 (right). The positive values represent the brightening.

dependency is remarkably thought to affect the collisionless electron heating (Bohdan,2023), there is poor observational evidence in SNRs.
We apply the methodology to Kepler’s SNR, a remnant of SN 1604 located at 3–
7 kpc (Kerzendorf et al., 2014). The remnant is a rare Type Ia event (e.g., Kinugasa &Tsunemi, 1999) whose thermal X-rays from CSM are detected (Reynolds et al., 2007;
Katsuda et al., 2015). The CSM densities are estimated to be 10–100 cm−3 , comparable
to that of Knot1 in Tycho’s SNR in Chapter 5 (Matsuda et al., 2022). Since the thermal
equilibrium timescale is proportional to a reciprocal of the plasma density, we cananticipate the detection of the time variability in a similar time scale as Knot1. Moreover,Reynolds et al. (2021) reported the spatial variations of magnetic field amplificationin the shocks of Kepler’s SNR. Thus, we can also expect the potential of comparisons
between the magnetic field and 𝑇𝑒 /𝑇ion .
We conducted a search for significant flux changes in Kepler’s SNR by making flux
difference images using Chandra data from 2000, 2004, 2006, and 2014 (Figure 6.7). We
identify a gradual brightening in 0.5–0.7 keV over a broad region, coinciding with brightthermal X-rays from CSM (Katsuda et al., 2015; Kasuga et al., 2021). It is noteworthy
that the brightening region spatially aligns with an H𝛼+[N II] image (Sankrit et al.,2008). Figure 6.8 shows the spectra extracted from a region exhibiting significant
brightening. The residuals between data and the model reveal an excess in the ≲ 1 keVband only in 2014, with no corresponding excess in 2000, 2004, and 2006. By applying theanalysis outlined in Section 5.3, we would unveil the dependence of the electron-to-ion
temperature ratio 𝑇𝑒 /𝑇ion on the magnetic fields.
The time variabilities in the broad region suggest that we can also observe time
variability in other SNRs with bright thermal radiation from shocked ISM/CSM, such as
N103B (Yamaguchi et al., 2021) and SNR 0519−69.0 (Guest et al., 2023). The application
to several SNRs enables us to investigate 𝑇𝑒 /𝑇ion in different shock velocities.

6.4. Measuring Ion Temperature−1
normalized counts-1s−1 keV-1

Counts (s keV )

92

0.01
0.01
-3
10−3
10

-4
10
10−4

10−5-5
10

2000
2004
2006
2014

(data-model)/error(data−model)/error

10−6
5
5

0
0

0.5
0.5

11

2
2
Energy (keV)

Energy (keV)

5

5

10
10

Figure 6.8: Example of the spectra which are extracted from the part of time variable region in
Kepler’s SNR. Black, red, green, and blue plots correspond to the data taken in 2000, 2004, 2006,
and 2014. The models in each year are the same.

6.4

Measuring Ion Temperature

In Section 5.4, we developed a method to investigate the shock heating process andmeasure 𝛽 0 ≡ 𝑇𝑒 /𝑇𝑝 at the shock transition. To enhance the reliability of this approach, it
is beneficial to compare the temperature variabilities of electrons with those of protons
and other ions, which should have a relation depicted in Figure 1.4. Since H𝛼 emission
occurs immediately behind the shock, we can measure the proton temperature before
being influenced by Coulomb collisions using the H𝛼 line width broadened by thermal
fluctuations. Proton temperature has been successfully measured in the east rim of
Tycho’s SNR through H𝛼 observations (Ghavamian et al., 2000). By conducting a similar
observation in Knot1, we can compare the proton temperature in Knot1 with the electron
temperature change measured in this thesis.
Measuring the temperatures of other ions poses a challenge with currently active Xray telescopes due to their limited energy resolution. The high-resolution spectroscopy
provided by the recently launched XRISM telescope can overcome this difficulty. The
Resolve instrument onboard XRISM has superior energy resolution of ∼ 7 eV FWHM @6 keV (XRISM Science Team, 2020), enabling the measurement of ion temperatures fromemission lines. Meanwhile, analyzing small spatial features like Knot1 is impractical,
given the Resolve angular resolution of ∼ 1.′ 7, which is not as high as that of Chandra.
The ion temperature of this feature would be revealed with future missions of X-ray
observatories with high resolution of angle and energy like HUBS (Bregman et al.,2023) and Lynx (Gaskin et al., 2018).

Science with the X-ray Imaging and6. Future Works
Spectroscopy Mission

93

Science with XRISM

1Introduction

The X-ray Imaging and Spectroscopy Mission (XRISM), formerly known as the X-ray Astronomy Recovery Mission (XARM), is a JAXA-NASA collaborative mission with ESA participation.
XRISM will o↵er non-dispersive, high-resolution X-ray spectroscopy in the soft X-ray bandpass(⇠ 0.3 12 keV), while o↵ering complementary CCD imaging resolution over a wide field of view.
The primary purpose of the mission is to recover the science that was lost after the Astro-H/Hitomi
mission failed in 2016, approximately one month after launch. Four science categories have been
defined for XRISM: (1) Structure formation of the Universe and evolution of clusters of galaxies;(2) Circulation history of baryonic matter in the Universe; (3) Transport and circulation of energy
in the Universe; (4) New science with unprecedented high-resolution X-ray spectroscopy. A fulldescription of the mission is provided in an SPIE proceeding 1 ; the reader is referred there for moredetail.

This white paper was prepared by members of the XRISM Science
Team for the benefit of the general astronomical community. This
paper describes the capabilities of XRISM, oﬀers a sampling of themany science topics that the mission will address, and discusses
the synergies of XRISM with the plethora of planned and existing
facilities in the 2020s and beyond.

Figure 6.9: The overview ofhttp://xrism.isas.jaxa.jp
XRISM and its pre-launch requirements (XRISM Science Team,
Figure 1: The mission requirements for both the XRISM Resolve instrument and the Hitomi Soft X-ray Spectrometer2020).
(SXS), along with the on-orbit performance of the Hitomi mission as measured in 2016. Hitomi met or exceeded
requirements in all areas.

XRISM represents a revolutionary leap forward in X-ray spectroscopy. With a spectral resoluIf the plasma is distributed over a larger spatial scale than the Resolve angulartion 20-40 times better than the CCD instruments that are used on Chandra, XMM-Newton, andresolution,
the asResolvecan extract
X-raysfrom shocked
plasma.
the
Suzaku, as wella substantially
increasedcollecting
area and ISM
bandpassover To
theinvestigate
grating instruments on thosemissions,
XRISM in
represents
new toolfor studying
universe, technique,and will enable
thermalized
ISM
distribution
Tycho’saSNR,
we focuson antheanalyticalthesciencenever
beforepossible.
XRISM
will
do
this
using
two
instruments:
general morphological component analysis (GMCA), introduced by Picquenot et al.

(2019).
GMCAa categorizes
the observed
into <distinctgroupswithout
losingoverthe• Resolve,
soft X-ray spectrometerwith adata
constant
7 eV FWHM
spectralresolution00
the entire bandpass.
Resolveandhas energy)
a field of viewof 2.9 ⇥ 2.9 Consequently,
over an array consisting
a 6⇥6
multidimensional(position
information.
we canof obtainpixel X-ray microcalorimeter (pixels are 30” in size). The array has an operating temperaturesome groups with different morphologies and spectral signatures, such as thermalizedof 50 mK, and must be cooled by a multi-stage adiabatic demagnetization refrigerator. Thisejectasystem
and nonthermal
Yamaguchi
et al. (2021)
GMCA
analysiscan operate inradiation.
both cryogen
and cryogen-freemodes,usedmeaningthat inthethemissioncanof N103B,
retrieving
threecomponents
havingoriginsof theCSM,
thecontinue
to operate
after
the exhaustioninterpretedof the liquidas
heliumcryogensupply.
The
missionrequirementsforResolve
are
given
in
Figure
1,
along
with
the
measuredvaluesfromthe
Fe-rich ejecta, and IME ejecta origins (Figure 6.10).
in-flight performance of Hitomi (all of which exceeded requirements).

GMCA enables us to understand distributions of X-ray components even when their• Xtend, a soft X-ray imager providing simultaneous coverage of the Resolve field and the surcontribution
is relatively low, potentially serving as an indicator to search for thermal Xroundings over a 380 ⇥ 380 field of view. Xtend will provide CCD-quality imaging spectroscopy
rays from ISM in Tycho’s SNR. Applying GMCA to Tycho’s SNR, we successfully identify an
X-ray component that spatially correlates with
1 the distribution of the H𝛼 line, suggesting
X-rays originating from thermalized ISM/CSM. As can be seen in Figure 6.11, the H𝛼counterpart component the H𝛼 counterpart component appears relatively bright not
only in Knot1 but also in the east and northwest regions, indicating the potential to94

6.4. Measuring Ion Temperature

The Astrophysical Journal Letters, 910:L24 (11pp), 2021 April 1

Yamaguchi et al.

Figure 1. Results of the GMCA applied to the SNR N103B. Top: retrieved spectra of the three components. Middle: spatial distribution of the retrieved components.
The pixel values correspond to the photon counts normalized by the count in the single brightest pixel. Panel (b) is given in the linear scale, whereas panels (c) and (d)6.10:
examplethe
GMCA
appliedto SNR
N103B
et al.shown(2021).
The4 andare inFigure
the square root
scale.An
Bottom:fraction ofof
each
component(seeresult
the text) givenin the linear
scale. The
ellipses Yamaguchi
indicate where the spectrain Figures
7 are extracted.

top and middle panels show spectra and images of the separated components, respectively. The
bottompanel shows
the fraction
each component.
Thedetailed
white analysis
ellipsesofshow
the regions
progenitor’s
explosion.
The retrieved
spectrum ofofComponent
3
later with
the X-rayspectra asthat
well as(blue)
dominates
the
line
ﬂux
of
the
intermediate-masscomparison
withopticalobservations.
they used for spectral extractions.
elements (IME: Si, S, Ar, and Ca) and reproduces also a partof the Fe emission detected in the spectrum of the entire SNR.
This characteristic is consistent with the emission fromincomplete Si burning products typically observed in other
SNRs Ia (e.g., Tycho and Kepler). Given the results, wehereafter call Components 1, 2, and 3 “CSM,” “Fe ejecta,” and“IME ejecta,” respectively. This identiﬁcation will be veriﬁed

Figures 1(b), (c), and (d) present the spatial distribution of
the CSM, Fe ejecta, and IME ejecta, respectively. The pixel
value indicates the photon counts of each component. The
CSM image exhibits several clumpy features in the west, which
spatially coincide with the optical nebula knots (Ghavamian
et al. 2017; Li et al. 2017, see also Section 3). The emission
from the Fe ejecta is strongest in the southwest, consistent with3

6. Future Works

95
0.15

Knot1
0.1
0.10

0.05
0.05

Ratio (Halpha counterpart/total)

0.15

0.00

0

Figure 6.11: The ratio image made by separating counts of the H𝛼 counterpart component andthe total counts. The yellow ellipse represents the region of Knot1.

analyze X-rays from shocked ISM or CSM in these areas, even though their counts
constitute only about 15% of the total counts.
To confirm that Resolve can detect the ISM component, we simulate the spectra of
Tycho’s SNR, comprising thermalized ejecta emission and thermalized ISM emission,
with the latter having 10% of the flux of the ejecta component, as illustrated in Figure 6.12.
The simulation suggests that we can retrieve the ISM component from the spectra. The
prominent emission lines from O VII (≃ 0.654 keV) and Ne IX (≃ 0.922 keV) indicate thepotential to measure the ion temperature of oxygen and neon, respectively, with theline width of thermal broadening. The simulation suggests that we can retrieve the ISM
component from the spectra. The prominent emission lines from O VII (≃ 0.654 keV)
and Ne IX (≃ 0.922 keV) suggest the potential to measure the ion temperature ofoxygen and neon with the line width of thermal broadening. As shown in Figure1.4,
the ion temperatures of oxygen and neon decrease from ≃ 30 keV to ≃ 10 keV overa timescale of 𝑛𝑒 𝑡 ≃ 1010 cm−3 s. This timescale corresponds to approximately ten
years when the electron density 𝑛𝑒 is 50 cm−3 . Consequently, if XRISM observes Tycho’s
SNR several times over a few decades, we could measure the time variability of ion
temperature, revealing the temperature changes of electrons and ions behind the shock
through energy transfer. Furthermore, high-resolution spectroscopy will unveil thecomposition of the ISM component, providing insights into the origin of the wallinvestigated in Chapter 6.2.
The measurement of ion temperature can also be applied to other SNRs with bright Xrays from shocked ISM or CSM. Miceli et al. (2015) indeed measured the ion temperature

96

6.4. Measuring Ion Temperature

102

Simulated Resolve/XRISM Spectrum
Ne IX HeCounts s 1 keV 1

O VIII Ly
101

100

10 1
0.6

ISM

0.7

0.8
0.9
Energy (keV)

1.0

1.2

Figure 6.12: Simulated XRISM spectrum of Tycho’s SNR. The dashed red curve is a component
from thermalized ISM whose flux in 0.5–1.0 keV is assumed to be 10% of ejecta. The dash-dottedand dotted curves in grey are the ejecta and power-law components, respectively.

of certain ion species in post-shock plasma. By measuring the ion temperature of SNRs
with different ages, we can track temperature changes in a still longer timescale until
the post-shock plasma reaches thermal equilibrium between electrons and ions.

7
Conclusions
We discovered the year-scale variabilities of X-ray fluxes and the expansions of Tycho’s
SNR with Chandra, aiming to understand the energy transfers between shock waves
and particles and the shock–cloud interaction. We outline the important results below:
1. We investigated time variabilities in X-rays within the western “stripe” structureof Tycho’s SNR using Chandra images from 2003, 2007, 2009, and 2015. The flux
difference image unveiled time variabilities in synchrotron emission along each
stripe, including faint ones. Our spatially resolved spectroscopy also detectedsignificant time variabilities in photon indices and brightness of synchrotron
radiation from most stripes. Assuming that the flux increased and decreased
due to the increase in averaged electron energy through DSA and their decrease
through synchrotron cooling, respectively, we concluded that the magnetic fieldis required to be amplified to ∼ 500 𝜇G to account for the observed variability
timescale. A comparison of synchrotron emission between the stripe and rim
revealed harder spectra for the stripe (Γ = 2.1–2.6) than the rim (Γ = 2.7–2.9),
indicating amplified magnetic fields or stochastic acceleration far downstreamof the shock. The discovery of a tight anti-correlation between brightness andphoton index was also noteworthy, suggesting that a small number of parameters
are related to the spatial and temporal variability of the stripe. Our results indicate
the possibility of an effective acceleration far downstream of the shock.
2. We applied a time variability analysis to the thermal emission of Tycho’s SNR, aiming to understand the processes involved in electron heating within SNR plasma.
By examining thermal X-ray time variabilities using Chandra data from 2000,
2003, 2007, 2009, and 2015, we identified a significant brightening in a knot-likefeature (Knot1) in the northeast region of the remnant. Our X-ray spectral analysis,
combined with the distribution of the H𝛼 line, revealed that Knot1 originated fromthermalized ISM/CSM by the forward shock, with a density of 𝑛 H ∼ 30 cm−3 . Our
97

98

7. Conclusions

spectral analysis firstly detected a significant increase in electron temperature+0.16
from 0.30+−00..05
07 keV to 0.69 − 0.12 keV over the 15-year period. These findings suggestthat the brightening is attributed to electron heating through Coulomb collisions
following an encounter of the forward shock with dense clumps around 2000. The
calculated evolution of electron temperatures required the electron-to-proton
temperature ratio immediately behind the shock (𝛽 0 ≡ 𝑇𝑒 /𝑇𝑝 ) to be 𝑚𝑒 /𝑚𝑝 ≤ 𝛽 0 ≤0.05, assuming a shock velocity of 1500 km s−1 . We introduced a new method todetect the thermal emission from shock-heated gas and measure 𝛽 0 independentlyof the previous estimation using H𝛼 lines.
3. Our analysis of the flux change in Tycho’s SNR suggests the interaction between
shock and clouds surrounding the remnant. To investigate the surrounding
environment in another way, we measured the proper motion of the shock waves
using newly observed data from 2022–2023 with Chandra added to data analyzed
by Tanaka et al. (2021). Our result reveals further deceleration in the western shellfrom 2015 to 2021 beyond the velocity in 2003–2015 measured by Tanaka et al. (2021).
It supports the existence of the wall formed by the wind from a progenitor whitedwarf during mass accretion. We conducted a spectral analysis of synchrotron
radiation from the shock front to study the effect of the shock–wall interaction
on it. Although we could not observe significant changes in the X-ray spectra,
we discovered softening trends in some regions. The result implies a decrease in
cutoff energy due to the decrease in shock velocity.
4. We proposed extending the observation of X-ray time variabilities to other SNRs.
Initially, we verified the application to Kepler’s SNR, which is similar in age, distance, and explosion type to Tycho’s SNR. Flux difference images of Kepler’s SNRshow a flux brightening of thermal X-ray over large areas that spatially coincidewith an H𝛼 image. Extracted X-ray spectra also suggest an increase in flux in lowenergy band ≤ 1 keV from 2006 to 2014. Examining such time variability in thermal
X-rays would enable us to investigate the dependence of 𝛽 0 on the environment,
including factors like the ambient magnetic field.
5. We found an indication of thermalized ISM plasma extending across a wide areain the northeast of Tycho’s SNR using the General Morphological Component
Analysis (GMCA) method. GMCA is an analytical technique that categorizes theobserved data into some groups based on spatial and spectral information jointly(Picquenot et al., 2019). While the X-ray radiation from the plasma is likely to
account for at most 15% of the total flux, the high-resolution spectroscopy with the
XRISM telescope would allow us to resolve it. It could reveal the time variabilities
of ion temperatures, offering insights into the energy transfer between electrons
and ions behind the shock.

Appendices

99

A
Time Evolution of Electron Temperature
with Low Shock VelocitiesYear
2000 '03Year'07 '09'15

2000 '03'15

2000 '03-1

uc=800 km s

0.5

0.5

0.2
0

1

2

3

8

4

time (×10 s)

5

6

'15

uc=1000 km s

0.5

0.2

'07 '09-1

uc=900 km s

-1

kTe (keV)Year'07 '09

β0=me/mp
β0=0.1β0=0.15β0=0.2β0=0.3

0.2
0

1

2

3

8

4

time (× 10 s)

5

6

0

1

2

3

8

4

time (×10 s)

5

6

Figure A.1: Compatrison between electron temperature change in Knot1 and calculations
under assumptions of 𝑢 c = 800 (left), 900 (middle), and 1000 (right). The solid, dashed-dot,dashed, dotted, and dash-dot-dotted curves show results when 𝛽 = 𝑚𝑒 /𝑚𝑝 , 0.1, 0.15, 0.2, and 0.3,
respectively.

We calculate the time evolution of electron temperature under the assumption oflower shock velocities than those discussed in Section 5.4.2.2 since the trend of the𝑘𝑇𝑒 observed in Section 5.4 may indicate it. The time variations of 𝑘𝑇𝑒 in the case of𝑢 c ≤ 1000 km s−1 is shown in Figure A.1. The slower the shock velocity is, the longer
time it takes for the electron temperature to increase from ∼ 0.5 keV to ∼ 0.7 keV. The
observed 𝑘𝑇𝑒 is roughly explained in the case of 𝛽 0 ≃ 0.15 when 𝑢 c is 1000 km s−1 while
it cannot be explained in any case of 𝛽 0 when 𝑢 c is lower.
Figure A.2 shows the temperature evolution under the different assumptions of
𝑡 = 0 for 𝑢 c = 1000 km s−1 . In these case, the case of 𝑚 e /𝑚 p ≲ 𝛽 0 ≲ 0.15 is reasonable to explain the observed data. In summary, the observed 𝑘𝑇𝑒 change can be also
101

102

A. Time Evolution of Electron Temperature with Low Shock VelocitiesYear
2000 '03Year'07 '09'15

2000 '03'07 '09'15

1998 as t=00.5

kTe (keV)

kTe (keV)

1996 as t=0

0.2

0.5

β0=me/mp
β0=0.1β0=0.15β0=0.2β0=0.3

0.2
0

1

2

3

4
8

time (×10 s)

5

6

0

1

2

3

4

5

6

8

time (× 10 s)

Figure A.2: Same as Figure A.1, but the time in which the shock heating was started is assumed
to be 1996 (left) and 1999 (right).

roughly explain 𝑚 e /𝑚 p ≲ 𝛽 0 ≲ 0.15 for 𝑢 c = 1000 km s−1 in addition to the case of𝑢 c ≥ 1500 km s−1 discussed in Section 5.4. If 𝑢 c is indeed slower, it implies a larger
density contrast 𝜒 or a slower forward-shock velocity 𝑢 i . Future observations with better
statistics or high-resolution imaging should constrain these parameters.
Estimation of 𝛽 0 by H𝛼 lines got greater value than 0.1 in SNRs with slow shocks(𝑣sh ≤ 1000 km s−1 ) such as Cygnus Loop, RCW 86 (Ghavamian et al., 2001), and
SNR 0548−70.4 (Smith et al., 1991). Thus, we note that it can be consistent with H𝛼
observations even if the shock velocity is lower.

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