Last Modified: 07/21/2021 11:28:23

Students without prior knowledge of astronomy will learn various high-energy phenomena occurring in the universe,

such as X-ray radiation from black holes and neutron stars.

Students will also learn principles of the X-ray observation from space using artificial satellites and how to analyze X-ray data to derive physical parameters of the X-ray sources.

Intuitive understanding and order estimates are emphasized than rigorous derivation. Practical examples are given as much as possible.

I intend to use **on-line white-board ** (OneNote) more than slides, since I believe writing by hand will help understanding.

I expect students to **take notes by hand **through the lecture. After the lecture, presented materials will be made available on-line from the current page.

Students are suggested to **bring a calculator to carry out simple calculation ** for practical examples during the lecture.

I am going present several questions and/or problems for each topic of the lecture and answer to these questions by myself. Students are expected to be able to answer to these questions by themselves.

Several contents of the X-ray astronomy exerecises are taken from my previous lectures at University of Tokyo. The lecture notes are available as follows; 2016, 2011, 2006 .

Based on my lecture, produce a handbook of "High energy astronomy" aiming for the first year graduate students (like you!). I expect this "handbook" would be useful for yourself in your career of studying (high energy) astronomy. You may use published material, but in that case, always show the references. Submit a PDF file on-line. You may write either in English or Japanese, and use whatever methods of producing documents (hand-writing and scan, Latex, Word etc.).

The deadline of the report is August 17, 2021. Please submit on the ITC-LMS.

The layout may be awkward when displayed with a web-browser.

PDF of each OneNote "page" is linked from each section below, which may be easier to read (the layout is OK), but the resolution is lower.

**Very fundamental physics:**

- What are the three most basic physical parameters of the universe?

- Indicate that the dimension of "length", "mass", "time" can be
reproduced from these three parameters. What are the meaning of these
values?

- In which physical theories, which of the three parameters appear?
Are there physical theories in which all the three parameters appear?
What is the implication of this fact?

- Obtain the relationship between the Planck length and the Planck
mass. Compare the Planck length with the Schwarzschild radius of a particle having
the Planck mass.

**Electromagnetism:**

- Express Coulomb's law in the MKSA unit and Gauss unit.

- What is the dimension of the electric/magnetic charge in the Gauss unit?

- Express the light-velocity
*c*using the vacuum permittivity ε_{0}and the vacuum permeability μ_{0}in MKSA.

- Express Maxwell equations in the MKSA unit and the Gauss unit.

- Indicate the relation between the magnetic energy density ε
and the magnetic flux
density
*B*. Use both the Gauss unit, where ε is in [erg/cm^{3}] and B is in [gauss], and the MKSA unit, where ε is in [J/m^{3}] and B is in [T]. Examine that both agree.

- Light velocity
- Consider X-rays with the energy E [keV] and the wavelength λ
[A]. Obtain the relationship between λ and E.

- Boltzmann constant
*k*. - 1 eV corresponds to approximately ** [K] or ** [erg].
- Electron rest mass (in keV).

Electron-positoron annihillation line from the Galactic Center. - Approximate nucleon (=proton or neutron) mass (in MeV or GeV). Which is heavier, proton or neutron?
- Stephan-Boltzmann constant σ in [erg/s/cm
^{2}/keV^{4}]- Using above, what is the luminosity of the blackbody emitting neutron star at 2 keV with the radius 10 km?

- Fine structure constant (both formula and value).
- ℏ c
- Classical electron radius
*r*(formula and value)_{0} - Express the Thomson scattering cross-section with
*r*, and remember the value._{0} - Bohr radius (both formula and number)
- Lyman edge energy of hydrogen (both formula and number in eV)
- Lyman edge energy (in keV) of heavy atoms with the atomic number Z.
- Using the value of Bohr magneton ℏ
e/(2m
_{e}c) = 9.3e-21 [erg/gauss], derive the relation between the energy of cyclotron absorption line [keV] and B [gauss].

Cyclotron absorption line from X0331+53

**
Remember or derive the following numbers/formulae which often appear in
astrophysics **

- Value of 1 A.U. (astronomical unit), together with its definition
- 1 A.U. in light-seconds
- 1 year in seconds
- Age of the Universe in year
- Age of the Universe in second
*Is there a clock which is precise with the accuracy within a second in the age of the universe?*

Link to Prof. Katori's lab page

- What is the scale of the universe (ratio between the minimum/maximum length or the longest/shortest time)?
- 1 pc (parsec; together with its definition)

- Schwarzschild radius of a star with mass
*M*. - Schwarzschild radius of Sun and Earth.
- Maximum mass of a white-dwarf (Chandrasekhar limit).

- Typical mass and radius of a white-dwarf?

- Theoretical maximum mass of a neutron star (≈ minumum mass of a black hole)?

- Typical mass and radius of a neutron star?

- Compare typical radius of a neutron star and white dwarf with their Schwarzschild radius
- Are heavier white-dwarfs larger or smaller in size? What about neutron stars?

- Typical entral density of a neuron star?

- Typical magnetic field of an X-ray pulsar (in gauss)? What about extremely strongly magnetized pulsars (magnetors)?

- Thomson scattering opacity (assuming only hydrogen)
κ
_{T}in [cm^{2}/g] - Derive the formula of the Eddington luminosity of a star with mass
*M*, and the Eddington luminosity of a solar-mass star. - What is the blackbody temperature of a white-dwarf shining at the Eddington limit with the radius 5000 km? What about a neutron star with a radius of 10 km?
- What are those sources, neutron stars or white dwarfs shining at the Eddington luminosity?
- Are there "super-Eddington sources" whose luminosities exceed the Eddington luminosity?

- In Newtonian mechanis, obtain the radius of a star with mass M, where the escape-velocity is equal to the light velocity. Compare this with the Schwarzschild radius.
- Maximum known mass of "stellar" black holes? (See these article
and paper
by LIGO collaboration)

- Mass of the black hole in the center of our Galaxy (Sgr A*)? (A latest result, Stellar Orbits around the black hole, Nobel Prize 2020! )

- Maximum known mass of super-massive black holes (AGN, quasars)?

- Are there "intermediate mass black holes" with a mass of 100-1000 Msolar? Which sources are the candidates of the intermediate mass black holes?

- Estimate apparent size of a black hole, dividing the Schwarzschild radius
by the distance to the source. Which is easier to "resolve",
stellar black hole in our Galaxy, or super-massive black holes in
other galaxies?

- Estimate spatial resolution of an radio interferometer at 1mm,
where the base-line is 10,000 km (=maximum on Earth). Also, estimate
spatial resolution of an X-ray interferometer at 1 A, with the
base-line is 20 m. Which has better spatial resolution?

- According to General Relativity, "Photon-capture radius" of a black hole is √27 r
_{g}, where r_{g}is the gravitational radius (=GM/c^{2}). In 2020, the Event Horizon Telescope observed a bright photon "ring" around central blackhole M87 (distance = 16.8 Mpc), where the diameter was 42 µas. By identifyng the ring radius as the photn-capture radius, estimate the black hole mass. (see 2019ApJ...875L...1E) - Assume that black hole is a sphere having the Schwarzschild
radius (ignore General relativistic effects). Estimate "density" of the
black hole, simply dividing the mass by the volume.
Is the density smaller or larger for more massive black holes?
Can the black hole density smaller than that of water? If yes, when?

- Estimate light-crossing time of the Schwarzschild radius of a black hole with mass M.
- Which is easier to detect X-ray variability taking place near the Schwarzschild radius, stellar mass black holes or super-massive black hole?

- Innermost Stable Circulr Orbits (ISCO) of non-rotating blackhole
(Schwarzschild black hole)? What about a spinning black hole at the maximum rate (extreme Kerr black hole)?

- Ignoring general relativity, estimate total energy E (potential
energy + kinetic energy) of a mass m
rotating at the ISCO around a black hole with mass M. Consider the Schwartzshild
case and the extreme-Kerr case. Compare with the precise values using
general relativity.

- When the mass m reaches the ISCO from infinisty (where the initial
velocity is assumed to be zero), the energy -E is
released. What will be the energy conversion efficiency, η, where
-E = ηmc
^{2}? - Compare with the efficiency of hydrogen
nuclear burning.

- How can we estimate black hole spin from X-ray observation?

- What happens if two black holes merge in the
*Binary Black Hole Coalescence*? - What about
*neutron star mergers*? - Assume that we detected a gravitational-event due to a blackhole
merger, where amplitude of the gravitational wave is 10
^{-21}. How much the distance between Sun and Earth (1 astronomical unit) varies due to this gravitational event? Answer with the unit of Bohr radius.

- What do you expect in the case of the binary
*super-massive black hole*merger, where both of the super-massive black holes are X-ray active AGN (i.e., they have accretion disks)?

- According to Hawking (1974), esimate the temprature in [K] for stellar mass black holes. Would it be possible to detect the "Hawking ratiation" from these black holes?
- Estimate the black hole mass which would evoporate in the age of the Universe.
- Estimate the temperature (in eV) of those "primordial black holes" which may have been created in the early universe and would evaporate in the age of the universe.
- How can we search for such primordial black holes?

Several satellite orbits (different seminajor axis, eccentricity, argument of perigee) on the two orbital planes with different inlination angles (i=0 and i=60); Python program

Several satellite orbits (different seminajor axis, eccentricity, argument of perigee) on the two orbital planes with the same inclination angle (i=60) and different longitudes of the ascending node (90 deg difference); Python program

** Constellations **

- How many constellations are there?
- Are the constellation names and their boundaries unambiguously defined?
- If "yes", who defined them? How can we know that?
- Is the "Crab nebula" (05 38 56.6, -64 05 03.3, ICRS) located in the constellation Crab?

- Explain the two ways to express values of the coordinates.
- What are the conversion formulae between them?

- The first celestial X-ray source was discovered in 1962. Since then, how are the X-ray sources named?

** Precession of the equinoxes and epoch **

- Explain precession of the equinoxes and epoch.
- Explain the International Celestial Reference Frame (ICRF) or System (ICRS).
- How can we convert B1950 coordinates to J2000 (ICRF) or vice versa?

** Coordinate conversion **

- Explain the three astronomical coordinates, equatorial coordinates, ecliptic coordinates, and Galactic coordinates.
- Find (web) tools to carry our these coordinate conversions?
- Write a simple program by yourself to carry our these coordinate conversions.

** Season and observability **

- We would like to observe the famous black hole binaries Cyg X-1 (19 58 21.7, +35 12 05.8, ICRS) and LMC X-3 (05 38 56.69, -64 05 03.3, ICRS) with ground "optical telescopes". Where and when can we observe them? How many times do we have the "best" observation periods (=the target is closest to the zenith in midnight) per year?
- What if using ground "radio" telescopes?
- What if using ordinary astronomical satellites, where the telescope is pointing perpendicular to the fixed solar panel? How many times do we have the "best" observation periods (Sun is perpendicular to the solar-panel) per year?
- For most astronomical satellites, there are two locations on the sky which are always observable. Where are they?

** Satellite Attitudes and observing targets **

We consider astronomical satellites where the spin axis is Z-axis and the solar-panel is toward Y-axis, and the telescope is along the Z-axis, which is toward the observing target.

The "Euler angles" to describe the satellite attitude are defined as sequential rotation around Z,Y,Z axis, (φ, θ, ψ).

- Express the 3D rotation matrix using the Euler angles (φ, θ, ψ).
- Explain the relationship between the Euler angles and the satellite attitude
- Explain the relationship between the third Euler angle and the "roll-angle" of the observation.
- North Ecliptic Pole(NEP) and South Ecliptic Pole (SEP) are observable all around the year. What are the Euler angles to observe NEP and SEP in Sprint equinox, Summer solstice, Autumn equinox and Winter solstice?
- The Suzaku satellites observed the NEP region several times at different seasons. Using JUDO2 , see how the roll-angle changes with seasons.

**Satellite attitudes and quaternion (q-prameters)**

- Look at an example of the ASCA satellite attitude file, where four numbers are given every 0.5 sec. Calculate the "norm" of these four numbers. What are these numbers?
- Explain that the unit-quaternion describe rotation in the three dimensional space.
- Derive the relationship between the 3D rotation matrix and quaternion
- What is the merit of using quaternion to describe satellite attitudes (or any 3D rotations) instead of Euler angles?
- Carry out a single rotation around a rotation axis from equatorial coordinate to Galactic coordinate using quaternion?
- How can we perform satellite maneuver efficiently from the attitude described by the quaternion "p" to the one described by "q"?
- How can we "average" two satellite attitudes?

**Satellite orbits, orbital six-parameters, Two Line Elements (TLE)**

- Derive Kepler's three laws by solving the two body problem.
- What are the "orbital six-elements"? Explain the meaning of each parameter.
- What are the Two Line Elements (TLE)? Where can we find various satellites?

**Orbits and astronomical objects **

- Where can we find orbital elements of solar planets and asteroids?
- Find orbital elements of your familiar/favorite asteroids.
- Are there asteroids which have your own name?
- Give some examples of binary stars where elliptical orbits affect their emission?

**The circular restricted three-body problem and Lagrange points**

- Explain the circular restricted three-body problem and the Lagrange points
- What are the "Trojan asteroids"?
- Do we have a plan to reach and study Trojan asteroids?
- Consider the Sun-Earth L2 point. Let's put the Solar mass M
_{1}, Earth mass M_{2}, their distance a, and the distance between the earth and the L2 is r. What will be the equation of the motion at L2? - Solve this equation of motion, and estimate the distance between the earth and L2.

**Space-craft orbits around the L2 point**

- Give several examples of the science space-crafts which are launched to the Sun-Earth L2 point?
- What will be the orbit like around L2? How does it called?

**X-rays and the atmosphere **

- What is the common definition of "space" (or "outer space")?
- Why do we have to launch artificial satellites to study X-ray astronomy?
- Estimate the "mean free path" of air for ~1 keV X-rays, assuming the air density 1.2 x 10
^{-3}g cm^{-3}(at 15 ℃, sea-level pressure) and the Nitrogen/Oxygen photoelectric cross section ~10^{-19}cm^{2}? - We saw tha the air in the ordinary condition is "opaque" to ~1 keV X-rays. How can the human body "transparent" in the medical Röntgen imaging?
- Do we really needs astronoical satellites to study universe in X-rays?
- What are the merits of using balloons or sounding rockets for astronomical satellites instead of satellites?

**Principles of the X-ray and gamma-ray instruments **

Explain pcinciples of the X-ray or gamma-ray detection for the following representative energy ranges. Where do we have the "sensitivity gap"?

- 2- 10 keV
- Tens of keV
- ~Hundreds keV
- ~MeV
- ~ 20 MeV to ~300 GeV
- ~TeV to ~PeV

- What are the three most essential peformances for X-ray astronomy instruments?
- How can we design X-ray instruments to maximize the sensitivity (=signal to noise ratio)?
- X-ray astronomy started in 1962, but the first X-ray imaging satellite using X-ray mirrors (Einstein) was not launched until 1978. Why it was so difficult to make X-ray mirros? Give two reasons.
- Einsten and ROSAT can only image up to < 2 keV. X-ray imaging above > 2 keV was made possible with ASCA for the first time. ASCA was much smaller, lighter and cheaper than Einstein or ROSAT. Why this was made possible? What was the trade-off?
- NuSTAR and Hitomi/HXI are the first satellites which can focus hard X-rays up to ~70 keV. How this was made possible?
- In 2023, what would be the best satellites/instruments for the following X-ray observaions?
- Best X-ray imaging and sensitivity with the highest angular resolution in 0.5 - 10 keV.
- High resolution spectroscopy in the iron K-band (6-7 keV)
- High resolution spectroscopy at ~1keV for
*diffuse*sources (e.g., supernova remnants, clusters of galaxies)? - High resolution spectroscopy at ~1keV for
*point*sources (e.g., stars, X-ray binaries)? - Very fast timing study shorter than ~1 msec for bright X-ray sources?
- Hard X-ray imaging above 10 keV
- Most senstive all-sky imaging in 0.5 - 10 keV, but missing the capability to monitor transient sources
- Monitor all-sky, discover bright X-ray transients and immediate detailed X-ray observation.
- X-ray polarimetry

- In early 2030's, which X-ray astronomy satellite will be operating? What will be the main instrument? How would be the performance?
- In early 2040's, which X-ray astronomy satellite is expected to be launched? What would be the performance?
- In early 2050's, plan your original high energy astronomy mission!

- Explain concept of the optical depth
- Express the optical depth with several different combinations of physical parameters
- When photons interact with matter, what will happen? State the two fundamental processes.

- Formulate the simplest radiative transfer equation in the uniform medium, where the specific intensity is I
_{ν}[erg/s/cm^{2}/Hz/str] and the source function is S_{ν}[erg/s/cm^{2}/Hz/str], and solve it. - Explain the physical meaning of the radiative transfer equation intuitively.
- What will be the solutions of the radiative transfer equation for the optically thick limit and the thin limit?
- Explain the difference between the thermal emission and the blackbody emission
- In the optically thin limit, consider the two cases when there is a strong input emission into the medium, and there is no input emission. What do you observe in these two cases?
- Indicate that intensity of any thermal emission does NOT exceed that of the blackbody emission.

- Explain the process of producing absorption lines and emission lines
- What are the energies of the Fe-K emission lines for neutral iron atom (fluorescent line), He-like iron ion, and the H-like iron ion?
- Show examples of the observations where these three Fe emission lines are detected simultaneously.

- Explain different ways of expressing the energy spectra?
- Often " ν f
_{ν}plot" (= E F_{E}plot) is adopted to express energy spectra. What is this? What is the merit of using the ν f_{ν}plot?

**Blackbody radiation**

- In what kind of sources, will we observe the black body emission in X-ray?
- Derive the formula of the Planck function for the low-energy limit (Rayleigh-Jeans law) and high-energy limit (Wien function)
- Obtain the energy at which the black body energy spectrum (Planck function) peaks.
- Calculate the energy density of the blackbody emission.
- Estimate the energy density of the 2.7 K Cosmic microwave background radiation. Compare this with a typical interstellar magnetic field energy density.
- Explain the differenct of the effective temperature, color comperature, and brightness temperature.
- Draw the black body spectrum with the temperature T in a wide energy range in the
νf
_{ν}plot (E F_{E}plot). Observe the energy spectral distribution, if it is widely distributed or narrowly distributed around the energy E=kT. Are there any continumm emission mechanisms whose spectra are more concentrated (narrowly distributed) around E=kT?

** Thin thermal plasma emission **

- From what kinds of sources in the universe, will we observe thin-thermal X-ray emission?
- Explain the mechanism of the continuum emission from thin-thermal plasma? How is it called?
- If the hot plasma is fully ionized (composed by nuclei and electrons), what kind of spectra are expected from thin-thermal spectra?
- Indicate a simple approximate analytical formula for the thermal bremsstrahlung spectra.
- Indicate examples of X-ray energy spectra from thin-tehrma plasma with different temparetures from 1 keV to 20 keV. Observe the energy spectral evolution with temperature, paying attention to the following points: a) Emission lines, b) Slope of the continuum in 1-10 keV, c) Peak energy of the continuum.

**Thermal Comptonization**

- Observed how the "cut-off power-law" spectra are formed by thermal (inverse) Comptonization process
- Observe how the spectral shape changes with the scattering optical depth and the plasma temperature
- In which celestial objects the thermal Comptonization spectra are expected?

**Non-thermal emission**

- When relativistic electrons have power-law energy destribution (∝ γ
^{-p}, where γ = (1-(v/c)^{2})^{-0.5}), the emerging synchrotron emission and the relativistic Compton scattering also show power-law spectra, ∝ ν^{-s}. Derive the relationship between the exponent of the electron energy distribution (p) and the synchrotoron emission/inverse Compoton energy spectra (s). - What kinds of sources (objects) are expected to emit synchrotron emission and/or non-thermal Comptonization?
- When the relativistic electrons, soft-photons (energy density is U
_{ph}) and magetic fileds (energy density is U_{B}) coexist, both the synchrotron emission (total power is P_{sync}) and inverse-Comptonization (total poewr is P_{compt}) take place simultaneously. Indicate that the relation P_{sync}/P_{compt}= U_{B}/U_{ph}holds. For the supernova remnant SNR RXJ 1713.7-3946 (Aharonian et al. 2006; Yang and Liu 2013), confirm this relation. - In Fig. 19 of Aharonian et al. 2006, the relativistic electron energy
distribution has been assumed to be ∝ γ
^{-p}exp (-mc^{2}/E_{c}), where p is 2 and E_{c}=100 TeV. Explain the following values semi-quantitatively: (a) Inverse Compton spectral peak energy, (b) Cut-off energy of the synchrotron spectrum and the inverse Compton spectrum, (c ) Slope of the synchrotron spectrum and the inverse Compton spectrum below the cut-off energies.

**The standartd accretion disk model (Shakura and Sunyaev disk model; α disk model)**

- What is the most important assumption Shakura and Sunyaev made to solve accretion disk equations?

- In the gas-pressue dominated disk, derive the relationship among the disk thickness, temperature, and gravitational potential.
- In the standard disk model, derive the relationships among the viscous parameter (α), disk thickness(h) , radial velocity(v
_{r}), rotational velocity(v_{φ}), and sound velocity(v_{s}).

- More than 10 years after the standard accretion disk model was proposed, Katz wrote in his text book as follows in 1986: "The theory of disks is in a much more primitive state than that of stars.... It may be appropriate to compare our present understanding of disks to Galileo's understanding of sunspots and solar activity." Why people understoold so little on accretion disks in 1986?
- Ginga satellite, launched in 1987, made a critical observation to confirm presence of the standard accretion disk around black holes. What is that observation?
- Explain solutions of the accretion disk on the surface-density vs. mass accretion-rate plot. Which branches are stable? How these branches are named? Indicate correspondence between these branches and observations.
- What is the physical origin of the visicoity in the accretion disk?
- What is it like the local spectrum from the inner part of the optically thick accretion disk? Is the color temperature to the effective temperature radio (hardening factor) dependent on the disk radius and the luminosity?

- Derive the radial dependence of the optically thick accretion disk temprarure (you may ignore the inner-boundary condition).
- Using the result above, calculate the observed energy spectrum from the optically thick accretion disk (local emission is assumed to be black body)
with the inner disk radius
*r*and temprature_{in}*T*at the distance_{in}*d*and the inclination angle*i*. - Calculate the lumionosity of the optically accretion disk (local emission is blackbody) with the inner disk radius (r
_{in}) and temprature (T_{in}). -
Let's assume that the optically thick disk emits at the Eddington luminosity, and the inner disk radius (r
_{in}) is determined by ISCO of the non-rotating black hole (3R_{s}). Derive the mass dependence of the inner-disk temperature (T_{in}). Is the temperature higher or lower when the mass gets higher? - Compare the disk temperatures for AGN (M=10
^{8}M_{☉}), neutron star (M=1M_{☉}) and black hole (M=10M_{☉}), when they are emitting at their Eddington limits? - What if the local emission is "diluted blackbody", instead of blackbody? How the estimated black hole mass is dependent on the spectral hardening factor?

- What is the remarkable characteristic of LMC X-3 which may be a basis for measuring black hole dpin?
- How the disk luminosity is related to the innermost disk temperature (T
_{in})? - How the relativistic effects affect the optically thick accretion disk spectra?
- How the black hole spin is estimated from X-ray observation of the optically thick accretion disk?

- What kind of the "disk oscillation" did Honma, Matsumoto and Kato proposed in 1991?
- Has it been actually observed?
- Indicate that the slim disk luminosity may exceed the Eddington luminosity.
- How the slim disk spectral shape is characterized compared to that of the standard disk?
- Show examples of the X-ray sources where the slim disk spectra are observed.

- "Radiative Processes in Astrophysics", G. B. Rybicki, A. P. Lightman (Wiley)
- "High Energy Astrophysics", Katz, out of print ,but pdf is freely available
- "Handbook of X-ray astronomy", (Cambridge University Press)
- "High Energy Astrophysics", Longair (Cambridge University Press)
- "Black‐Hole Accretion Disks:Towards a New Paradigm", Kato, Fukue and Mineshge (Kyoto Univ. Press)
- シリーズ現代の天文学8、ブラックホールと高エネルギー現象（日本評論社）
- シリーズ現代の天文学17、宇宙の観測(3)、高エネルギー天文学（日本評論社）
- 「人工衛星の力学と制御ハンドブック―基礎理論から応用技術まで」 姿勢制御研究委員会