Critical Flashcards
Lorentz transformations
Lorentz transformations are mathematical equations that relate the space and time coordinates of two inertial reference frames moving at a constant velocity relative to each other, ensuring the constancy of the speed of light in all inertial frames as described by special relativity.
Lorentz Factor
Beta = v/c
Length Contraction
Length L’ in S’ measured
in S with Dt=0:
L = L’/ɣ
(BB looks smaller/more compact than it is bc of length contraction)
S’ is observed frame, S is the source frame
Time Dilation
Time interval DT’ in S’ with
Dx’=0 measured in S:
ΔT = ɣΔT’
(redshift (velocity) leads to time dilation
S’ is observed frame, S is the source frame (red line is continuous, photo got cut off)
Abberation of Light
The aberration of light is the apparent shift in the position of stars or other celestial objects caused by the motion of the observer, such as the Earth’s movement around the Sun. This phenomenon occurs because the velocity of the observer changes the angle at which incoming light is received, similar to how raindrops appear to fall at an angle when you’re moving.
Relativistic Beaming Distortion
Relativistic beaming, also known as Doppler beaming, is a phenomenon where the apparent brightness of an object is increased when it moves towards an observer and decreased when it moves away at relativistic speeds. This effect arises due to the relativistic contraction of the object’s emission region in the direction of motion, causing more photons to be directed towards the observer.
Synchrotron Radiation
Synchrotron radiation is electromagnetic radiation emitted by charged particles, such as electrons, when they are accelerated in a curved path by a magnetic field.
(relativistic cyclotron radiation)
Larmor’s Formula
Larmor’s formula describes the power radiated by an accelerating charged particle.
Energy radiated in e- frame, non-relativistic!
Energy loss rate Lorentz Invariance
Energy radiated in lab frame
Magnetic field energy density
Magnetic field energy density refers to the amount of energy stored per unit volume within a magnetic field. It represents the energy density associated with the magnetic field lines and is proportional to the square of the magnetic field strength.
energy/volume
Electron energy loss rate for
specific pitch angle
Electron average energy loss rate
Averaging over an isotropic distribution of pitch angles:
Cooling time
The length of time that an electron can emit synchrotron radiation
The faster the electron is going, the quicker it decays (High energy electrons lose their energy first)
the stronger the magnetic field the shorter it lives as well
Cyclotron
Non relativistic (i.e. a continuous spectrum)!
The non relativistic cyclotron emission (gyro-radiation) is polarised.
Importance of Supernovae:
-Birth events of neutron stars and stellar black holes
- Powerful source of heating for ambient interstellar gas
- Intensive X-ray sources (Bremsstrahlung of hot gas)
- Connected to 𝛾-ray
- Radio sources (Synchrotron of electrons in SNR magnetic fields)
- Possible sources of high energy particles
- Origin of many heavy elements
- Energy radiated ~10^51 erg (10^44 J)
Types of Supernovae
- thermonuclear
- core collapse
- pair instability
What is the Schwarschild radius definition?
The Schwarzschild radius is the radius of the event horizon of a black hole, defined as the distance from the black hole’s center at which the escape velocity equals the speed of light.
Types of Accretion
There are two types of accretion:
- With LMXBs, there is Roche-Lobe overflow accretion:
the primary star evolves filling the roche lobe, matter flows from the primary to the secondary star through L1giving us an accretion disk (around BH)
- With HMXB, there is Bondi-Hoyle Accretion:
the primary star has strong stellar wind, the compact object is fed by the overflow of the primary giving us spherical accretion (around BH)
Eddington Lumonosity definition
The Eddington luminosity is the maximum luminosity that a celestial object (like a star or an accreting black hole) can achieve where the outward force of radiation pressure balances the inward pull of gravity. This balance prevents the object from blowing apart due to its own radiation. (also max luminosity due to accretion)
the radiation exerts a force mainly on the free electrons through thomson scattering
Accretion assumtions to find Eddington Luminosity
- steady spherical symmetrical accretion
- accreting materail mainly hydrogen and fully ionized
What are the definitions for T_BB, T_rad, and T_th?
T_BB: Temperature the source would have to radiate the power as a Black Body
T_rad: Temperature related to the energy of a typical photon
T_th: Temperature that the accreted material would reach if the gravitational energy is turned entirely into thermal energy
Physical mechanism for accretion collecting on the body
- Particles in almost circular orbits across BH that: Loss of energy and angular momentum due to viscous interaction with particle in adjacent radii. (i.e. There is a slow drift to smaller radii until reaching the star surface. Due to the viscous interaction, frictional heat is radiated away)
- The matter in the accretion disk is prevented from falling into the central object by Centrifugal forces, but it can fall into the central object only if it loses angular momentum (from the viscous forces)
- The viscosity transfers angular momentum outwards, causing matter to spread outwards and allowing other matter to spiral inwards. This acts as a frictional force dissipating heat
- E_rot decreases and finally the matter is accreted onto the central object
Disc emission spectra:
- at low frequencies: Rayleigh-Jeans emission from outer layers ~R_max
- at high frequencies: Wien’s law exponential cut-off at disc’s inner hottest layers ~R*
Collapsar model discription
Collapsar’ model (hypernovae) process
1) A very massive star (M>30SM) has lost its H and He outer layers and is rapidly rotating
2) at the end of thermonuclear reactions, we get a core collapse
Kerr BH with accretion disc formed inside the star
3) Some material is ejected in Jets along the axis of rotation
due to magnetic field lines channeling
4) The jets break through the surface of the star, producing shock waves that destroy the star
the jets last for~20sec
this is the time required for the BH to accrete the inner core of the star
5) The relativistic particles in the jets radiates:
if one of the jets is pointed towards the Earth, we get a GRB
Radius of AGN accretion disc
R_disc ~ 10^2 - 10^3 R_S
Which AGN bodies have BLR and NLR spectra in their emissions?
BLR visible in spectra of Quasars and Seyfert 1
NLR visible in spectra Seyfert 2
BLR electron density, clouds velocity, broad line region, and clouds size
electron density: ~ 10^9 - 10^10 cm^-3
clouds velocity: ~10^4 km/s
broad line region: ~0.1-1pc
clouds size (~R_sun): ~10^11cm
NLR electron density, clouds velocity, narrow line region
electron density: ~ 10^2 - 10^4 cm^-3
clouds velocity: ~10^2 - 10^3 km/s
narrow line region: ~50-200pc
