Frontiers in Physics Flashcards
How do you calculate the moments of inertia of the solar system
MOI of sun –> L = Iw
MOI of planets –> L=mvr
Define flux
Flux is the light received per unit area per second.
Flux = Luminosity / (4 pi r^2)
Define the arcminute
1’ = 1/ 60 degrees
Define the arcsecond
1’’ = 1/3600 degrees
Define the parsec
The parsec is the distance at which 1Astronomical unit subtends 1’’
Define the astronomical unit 1AU
The mean distance between the sun and the earth.
1.5x10^11m
What is one solar mass
1 Solar mass = 2x10^30kg
What order of magnitude is the milkyways diameter
10KPc
What order of magnitude is the universe diameter
GPc
What is Keplers first law
Planets travel in elliptical orbits with the sun at one focus
What is Kepler’s second law
a line connecting a planet to the sun will sweep out equal areas in equal time intervals. Resultant from conservation of angular momentum.
L = mvr = constant
as r increases, v decreases.
What is Kepler’s third law
(orbital Period)^2 ∝ (average distance from the sun)^3
Give the generalised equation of kepler third law
P^2 = 4pi^2 a^3 / G(m+M)
This applies to anything in orbit about a point mass. a is the semi major axis of the elliptical orbit
Derive Keplers laws to derive the universal law of gravitation
Consider a planet in a circular orbit
K3: P^2 = kr^3
k2: v = constant for circular orbit
P = 2pi r / v
sub into K3
rearrange for … = mv^2/r by multiplying both sides by m
by N2 Fplanet = Fsun
solve for newtons law of grav
Describe the virial theorem
GMm/r^2 = mv^2/r (circular orbit)
Therefore, GMm/r = mv^2 = 2KE
GMm/r = PE therefore:
2KE + PE = 0
The virial theorem is true for systems in equilibrium. (Bound orbits)
Give the type of orbits when E<0 , E=0, E>0
E<0: Elliptical orbit, bound orbit
E=0: Parabolic trajectory, unbound orbit
E>0: Hyperbolic trajectory, unbound orbit
Describe how to measure the distances to nearby stars
We can measure the distances to nearby stars using parallax: The apparent positions of nearby stars “wobble” by over ±θ a year relative to distant objects.
Small angle approximation: θ = 1AU/D where D is the distance to star. Hence D (Pc) = 1/θ’’
Define apparent magnitude
apparent magnitude – the brightness of a star as seen from Earth
Define absolute magnitude
a measure of how bright a star would be if it were seen from a standard distance. the absolute mag is the same as the apparent mag if the
source is at 10+ parsec.
Give wein’s law
λ_max = 2.9x10^-3 / T(K)
Hence λ_max emitted by a black body in inversely proportional to its temperature.
Give the conditions for atomic fusion
To overcome strong electromagnetic repulsion between charged ions: HIGH ENERGY, HIGH TEMPERATURE
To collide: HIGH DENSITY
Give the proton-proton reaction chain for fusion of H into He
notes
Describe the formation of a star from a collapsing gas cloud in terms of gravitational and thermal energy
virial theorem where ke is internal thermal energy
Intially, as a cloud of gas collapses, the potential energy (PE)
becomes more negative, so U = -Pe/2 becomes more positive, temp increases, conditions for fusion created
star is born
Why do stars not collapse
The star does not collapse because gravity is balanced by
pressure forces.
Stars have high temperature and density at the centre:
P = ρkT/m
The virial theorem can be applied replacing KE with internal thermal energy, U.
2U + PE = 0
E_tot = U-2U = -U <0
–> star is gravitationally bound
Explain how energy travels outwards from the star’s core
1) Radiation - Photons carry energy via collisions with matter
2) Convection due to bulk motion in the gas. Convective cells rise and fall, transferring energy.
3) Conduction due to temperature gradient
What are the 3 characteristics of main sequence stellar evolution
1) Main sequence stars fuse H–>He
2) Main sequence is the longest phase of stars life
3) L ∝ M^3.5 where L is luminosity and M is mass
Explain stellar collapse and the two outcomes
Eventually, all sources of fusion energy are used up.
- the core collapses
- the star explodes
- the outer layers thrown off
If M < 8M_0, explosion is a mild planetary nebula type leaving a small how remnant (white dwarf)
If M > 8M_0 , the explosion is a violent supernova, leaving
behind a neutron star or blackhole.
Describe gravitational waves
- An accelerating mass produces quadrupole perturbations in the gravitational field
- leads to deformation of spacetime, propagating at the speed
of light.
What can produce gravitational waves
1) Mass accelerating in gravitational field
2) Asymmetry in the mass distribution
What are the 5 sources of gravitational waves
1) Quantum fluctuations in early universe due to inflation
2) Binary supermassive black holes in galactic nuclei
3) Compact binaries
4) compact objects captured by supermassive black holes
5) Rotating neutron stars and supernovae (asymmetry)
How are gravitational waves measured at LIGO
- 4 km arms
- largest sustained ultra-high vacuum
in the world (8x the vacuum of space) - laser measures relative length of arms
- sensitive to deformation due to GW
Define irradience
Energy/(unit area)(unit time)
Define a blackbody
a surface that absorbs all radiant energy falling on it. The term arises because incident energy will be absorbed rather than reflected
why are stars blackbodys
Because it absorbs all radiation that falls on its surface and emits radiation according to its temperature. This means that it has a perfect absorption and emission of all wavelengths, making it an ideal black body.
Define the stefan-boltzmann law
P = K T^4
Define the albedo of an object
The albedo of an object is the fraction of incoming radiation that is reflected.
Define the solar constant
The solar constant, s, is the flux as a result of solar radiation measured at a distance of the planets orbit, by geometric argument, the energy received by the planet is S/4.
Explain elliptical galaxy nomenclature
E#
where # = 10(1-b/a) where b/a is the ratio between the semi minor and semi major axis.
Explain spiral galaxy nomenclature
Either SX (spiral galaxy, exhibits circular centre) or SBX (spiral bar galaxy, exhibits a bar at centre)
X is a letter representative of the angle between spiral arm and centre
Late type spirals (Sc, Sd) tend to have smaller central bulges
What is the luminosity function
Φ(L) = Luminosity function –> Used to quantify the number of galaxies of different luminosities
(or magnitudes) in a volume of space.
Give the expression representing the inverse square law
f(X) ∝ A/x^2
The Schechter function is a combination of which types of functions?
power-law (faint end) and exponential (bright end)
Define the Schechter Function
Empirical function describes relationship between luminosity (top axis), absolute magnitude (bottom axis) and number of galaxies (dependent variable).
Φ(L) = K . L^-1 . e^(-L/L)
where K and L are constants
L^-1 describes the first half of the graph by a power law , e^(-L/L) describes the second half of the graph (exponential decrease) , L describes the ‘knee’ of the function where L^-1 and e^(-L/L*) meet.
What equation gives the number of galaxies per unit volume
N = int[ Φ(L) dL ]
limits zero to infinity
What equation gives the total luminosity of galaxies per unit volume
L_tot = int[ L Φ(L) dL] = KL*
limits zero to infinity
Implies finite total luminosity
Define the inclination angle of a galaxy
The angle, i, at which a galaxy is tilted relative to the observer.
An inclination angle of zero represents a galaxy that is viewed face on such that the whole spiral is visible. Higher inclination angles mean that the edge of the galaxy is seen.
V_los = V_true sin(i)
los = line of sight
Explain how the velocity of stars in a galaxy can be measured using the doppler shift
(λ_obs - λ_lab)/λ_lab = V_los / C
Describe the Ruben and Ford graph, its purpose and the result
Spectrogram of rotational velocity against radius for a large sample of galaxies. Showed that stars at the very edge of the galaxy moved just as fast as those in the center, contrary to the Keplerian model which predicts v ∝ r^0.5.
Hence for M(r) = rv^2/G, M(r) still increases beyond the edge of the galaxy and thus there is non-observable mass unaccounted for. 90% of a spiral galaxy is dark matter.
What are the key characteristics of active galactic nuclei
-Central region outshines entire galaxy
-Central nucleus light output varies on timescales
-Emitting region must be «_space;1 light year across
-Most (but not all) show broad line emission from Hydrogen
-Typically, find masses within the central of the galaxy 10^8 M_0 / ly^3 –> very dense compared to solar systems. Implies supermassive black hole at the nucleus.
Why are active galactic nuclei so bright
An object dropped from rest will get close to c before disappearing into a black hole. Therefore before falling in, it will have energy approx = 1/2 ΔMc^2.
If lots of material falls in, it will collide and heat up, making the region around the black hole hot, emitting light.
If all the kinetic energy is converted into light, the luminosity will be E/Δt = (1/2 ΔMc^2)/Δt.
Give the components of the unified model of Active galactic nuclei AGN
Black hole - at the center of AGN.
Accretion disk - a disklike flow of gas, plasma, dust, or particles orbiting the blackhole.
Relativistic Jet - highly collimated, and fast outflows that emerge in opposite directions from close to the disc.
Obscuring Torus - dust and gas distributed in a torus-like structure surrounding the supermassive black hole
What evidence supports the big bang theory
1) Redshift of galaxies - light we observe from galaxies has been redshifted. This redshift is the result of galaxies moving away from us. The redshift of distant galaxies tells us the Universe is expanding.
2) Microwave Background - The short wavelengths of the gamma radiation emitted in the initial explosion are believed to have become stretched due to the expansion of space into longer wavelength microwaves.
Compare type I and type II AGN
Type I refers to AGN whose nucleus is visible (the spectra has both narrow and broad emission lines), while in type II AGN, the broad line region (BLR) is obscured and the lines are very narrow. This may be due either to the viewing angle or some intrinsic difference in structure
What is Hubbles equation
v = H_0 D
v = recession velocity (redshift)
H_0 = hubble constant
D = distance
1/H_0 = t_0 = age of universe
What are galactical peculiar velocities
Galaxies exhibit peculiar velocities due to their motions w/in galaxy clusters
v_obs = v_hubble + v_peculiar
What are the four fundamental forces
Gravitational interactions;
The strong force (responsible for intranuclear interactions);
The weak force (changes one flavour of quark into another);
Electromagnetic interactions.
What is the relationship between force and potential difference
F(x) = - dU/dx
force is the negative gradient of potential energy. The negative sign arises because it dictates the direction of the force.
What function approximates the potential energy for two neutral atoms
The morse curve
What are the two sections of the lennard-jones graph and the physical processes responsible for them where r_0 is the potential minimum
r < r0 In this regime the atom-atom interaction is increasingly dominated by repulsive forces. The fundamental origin of this repulsion is the Pauli exclusion principle. This principle - which lies at the heart of the behaviour of all matter - states that no two electrons can exist in precisely the same quantum state. r^6 term
r > r0 In this regime the attractive force dominates. This attractive force can arise from bond formation but even in the absence of chemical bond formation there is a physical
attraction between atoms (or molecules or nanoparticles): the van der Waals force. r^12 term
What are van der waal forces
The van der Waals force arises from interactions between atoms or molecules which do not involve the sharing or transfer of electrons and is due to the interaction of dipoles. A dipole is a separation of positive and negative charge and can be of three types: permanent, instantaneous, and induced.
Explain the term ‘sticky’ at the nanoscale
objects at the nanoscale are very “sticky” - atoms and molecules will stick to surfaces and to each other due to the ubiquitous van der Waals force
Explain piezoelectricity
Piezoelectricity arises from the displacement of ionic charges within a crystal. If, when a crystal is strained, a net dipole moment, and therefore electric field, is produced the crystal is piezoelectric. Very small stresses applied to piezoelectrics can generate very large electric fields. Conversely, application of an electric field to a piezoelectric will lead to a deformation of the material. This effect is used in STMs to control the tip position with sub-nanometre accuracy.
Describe STM
- Scanning Tunneling Microscope
Basic Principles of STM:
STM tip and sample surface are both conductive.
Bias voltage applied between tip and sample.
Tip is attached to piezoelectric crystal to control height above sample. Crystal responds to changes in current/ deformation such that either tunnelling current or height is constant.
1) constant current, deformation/height measured, creates a trace. (better as avoids damage to tip/sample)
2) constant height, current measured, creates a trace.
Image produced depends on the bias of the potential between tip and surface
Describe AFM
Atomic Force Microscopy (AFM)
AFM Operation:
o AFM uses a cantilever with a sharp tip to scan the surface of a sample.
o Forces between the tip and the sample cause the cantilever to deflect, which is measured using a laser that measures the angle of deflection to generate a topographical map.
* Contact and Non-Contact Modes:
o Contact mode: tip remains in constant contact with the sample surface.
o Non-contact mode: tip oscillates at nat frequency near the surface without touching it, reducing sample damage. Changes in the surface induce frequency changes, can be measured.
What is a soundwave
Sound waves are longitudinal waves which cause the local pressure and velocity (and displacement) of the medium to fluctuate. A high pressure region is known as a compression, while a low pressure region is a rarefaction.
How can sound waves be described by pressure change from equilibrium
p(x,t) = p_0 sin(2πf(x/v - t)) = p_0 sin(2π(x/λ - ft))
where p_0 is the pressure amplitude, while λ and f represent the wavelength and frequency of the sound wave, and v is the wave velocity.
The plane wave equation can be simplified by using the angular frequency and wave number ω = 2πf , k = 2π/λ
p(x,t) = p_0 sin(kx - ωt)
What equation represents the medium velocity of sound waves
u(x,t) = u_0 sin(kx - ωt)
The pressure and medium velocity are related by p/u = Z
What is the acoustic impedance
Z is the acoustic impedance which is given by the product of density, p , and wave speed, v.
Z = pv
The intensity of a sound wave, I, is the energy which passes through unit area in unit time and can be written as:
I = (p_0)^2 / 2Z
What does the speed of sound in a material depend on
The speed of sound depends on the ratio of the “stiffness” and density of the material. This means that the
speed of sound is highest in stiff materials of low density.
How does the ear work
The ear converts weak vibrations of the air into electrical nerve impulses which are interpreted in the brain.
It is sensitive to sound waves varying by a factor of 1000 in frequency (20-20,000 Hz) and 1012 in intensity (0 – 120 dB).
The sensitivity of the ear varies with frequency.
It is most sensitive in the 3-4 kHz frequency range. The sensitivity shows less variation with frequency as the loudness increases.
Describe the decibel scale
The sound intensities that can be perceived by the human ear span a wide range of intensities. A logarithmic scale is therefore generally used to characterise sound intensity.
Intensity level in dB = 10 log_10 ( I / I_0)
or = 20 log_10 ( p / p_0 )
This required the definition of a reference intensity or pressure (I0 or p0). If a value of p0 = 20 x 10-6 Nm-2 is used, the resulting dB value is known as the Sound Pressure Level (SPL).
Define ultrasound
The term, ultrasound, is used to describe sound waves with frequencies above 20 kHz. Medical ultrasound imaging is based on using sound waves with frequencies in the range of 2 – 10 MHz
what are the 3 types of ultrasound scans
A-mode (amplitude mode). This essentially corresponds to an oscilloscope trace.
B-mode (brightness mode), in which the echo intensity at each position is represented using a grey-scale. The beam can be scanned in one or two dimensions to produce a two or three-dimensional image.
M-mode (motion mode), the beam is fixed in position and changes in the time of echo arrival are used to monitor movement of interfaces (e.g. monitoring cardiac movement in echocardiography).
What limits pulse repetition frequency for ultrasound.
Time must be long enough to ensure that the echo from the deepest structure has been detected before the next pulse is applied.
PRF < v/2L
Explain how medical ultrasound transducers work
A piezoelectric material deforms when a voltage is applied to it, and also produces a voltage when it is deformed. Ultrasound pulses are produced by applying pulses of high voltage (~150 V) to a slab of piezoelectric material. Echoes are detected as weak voltages produced at the surfaces of the slab.
A synthetic ceramic, lead-zirconate-titanate (PZT) is most commonly used in medical ultrasound. PZT has a high density and acoustic impedance.
A slab of thickness, a, produces an ultrasound pulse of a few cycles of frequency, f, when a voltage pulse is applied. The boundary conditions ensure that f = v/2a
A matching layer, with acoustic impedance that is intermediate between tissue and PZT, and coupling acoustic gel are used to maximise transmission of the ultrasound into the body. It is important to ensure that there are no air gaps, between the transducer and the body.
What is attenuation
Attenuation is the decrease in amplitude as ultrasound waves travel through soft tissue.
This is a result of absorption and scattering.
What is the relationship between measured intensity and penetration distance as ultrasound passes through tissue
I = I_0 e^(-2ax)
Where a is the attenuation coefficient
How is energy lost by ultrasound waves as they pass through tissue
Some of the energy is lost from the beam due to frictional heating. The remainder is lost due to scattering.
This involves the wave being broken up and partially reflected from structures that are about the size of λ or smaller.
Interference of the scattered waves gives rise to the “speckled” appearance of tissue in ultrasound images. The form of the speckling is different for different tissues and thus gives useful diagnostic information.
What is the relative frequency change for ultrasound as a result of the doppler effect
f_d / f_i = 2(reflector speed) / v + reflector speed
The factor of two arises due to the frequency shifting of the incoming and outgoing waves.
What is the relative frequency change for ultrasound as a result of the doppler effect where the transducer is at an angle relative to the blood vessel.
f_d / f_i = 2(v_B cosθ)/ v
The variation of the number of radioactive
atoms, N, with time is then described by
dN/dt = -rN
where r is the decay rate
hence N = N_0 e^(-rt)
What is the half life of a radioactive sample
The half-life, t1/2 is the time in which half of the sample of radioactive atoms
decay.
t_1/2 = tln2
where t = 1/r , the time for N to decay to N_0/e
Define the activity of a radioactive sample
The activity A is defined as the number of decays per second:
A = |dN/dt|=rN
What is alpha decay
The alpha particle is a 4He nucleus (2 protons, 2 neutrons). The decay of X –> Y by alpha decay can therefore be
written as:
A A-4
X —–> Y + 4He + energy
Z Z-2
The energy can be calculated using E=mc^2
What is Beta - decay
In beta minus decay Z increases by 1 and A doesn’t change.
Electron and antineutrino produced
A A
X —–> Y + e- + V_hat + energy
Z Z+1
What is beta+ decay
In beta minus decay Z increases by 1 and A doesn’t change. Positron and neutrino produced
A A
X —–> Y + e+ + V + energy
Z Z-1
What is gamma decay
A gamma ray is a high energy photon which is emitted when the nucleus decays from an excited state. The high energy results from the fact that the nuclear energy levels are spaced by ~0.1 -10MeV.
Z and A are generally unchanged in gamma decay. alpha or beta decay often leave the nucleus in an excited state. gamma decay often therefore accompanies alpha and beta decay.
How does radiation exposure cause cell damage
Our bodies are mainly made up of water. Consequently the ionisation of H2O to form free radicals is usually a precursor of radiation damage.
When these free radicals damage DNA it can cause cell death or mutation, sometimes leading to uncontrolled cell
replication = cancer.
Single DNA strand lesions can be repaired – double strand lesions are irreparable. A high ionisation density leads to a greater likelihood of cell damage.
What two factors do the biological effects of radiation depend on
How much energy is absorbed by the body
Ionisation density produced by the radiation
Define the absorbed dose of radiation
Absorbed dose is the energy of ionising radiation absorbed per kilogram.
The SI unit is the gray, Gy = Joules per kg.
Define the relative biological effectiveness RBE
The relative biological effectiveness (RBE) is defined as the biological effect compared to an
equal dose of X-rays.
Define the dose equivalent (sv)
Dose equivalent in sieverts (Sv) = absorbed dose in Gy × RBE
What is Brachytherapy
Sealed radionuclide source ingested ( iridium 192 ), placed in/at site of tumour. Produces high localised dose.
What is external beam radiotherapy
High energy localised radiation beam directed at multiple angles towards tumour sight.
How do Gamma cameras work
Tracer sample ingested
gamma photons are detected, collimated and go through scintillation (conversion to light), the light is then converted to electrical signal.
2D image map of source strength created
How does positron emission tomography work
radionuclide source ingested
source undergoes beta+ decay, positron created.
Positron annihilates with local electron creating two gamma rays that are equal and opposite in momentum.
gamma cameras detect gamma ray pairs and uses time of arrival to determine the annihilation site.
Creates an image.
Why do atomic nuclei exhibit nuclear magnetism
Nuclear Magnetism Many atomic nuclei have intrinsic angular momentum, known as “spin”.
The “spinning” nucleus produces a magnetic field, and the nucleus is said to have a magnetic moment, µ.
Only nuclei with an odd number of neutrons or an odd number of protons have non-zero spin
What is the gyromagnetic ratio
The gyromagnetic ratio, often denoted by the symbol γ (gamma) is the ratio of the magnetic momentum in a particle to its angular momentum J.
μ = γ J
What is the magnetic dipole moment
The magnetic dipole moment of an object determines the magnitude of torque the object experiences in a given magnetic field.
When the same magnetic field is applied, objects with larger magnetic moments experience larger torques.
What equation relates the energy in an applied magnetic field to the magnetic dipole moment
Energy = -μ.B
Explain the quantisation of angular momentum in the 1H nucleus
Angular momentum, J, is quantised and in the case of the 1H nucleus, the component along any particular direction can only take values of ±ℏ/2
Therefore, If we consider the situation where a magnetic field, B_0, is applied along the z-direction, then:
Energy = -μ.B = ∓ γ B_0 ℏ/2
Explain equilibrium magnetisation of the nucleus
It is energetically favourable for the nuclei to come to equilibrium pointing parallel to the field, but because the difference in energy of the two states is so small, random thermal fluctuations mean that at normal temperatures there are almost as many nuclei anti-parallel as there are parallel to the field.
There is however a small excess of nuclei in the parallel state (about one in a 100,000 at 1.5 T and 300 K). This gives rise to a small net magnetic moment, M, which is aligned with the field in samples.
What is the Larmor frequency
When the magnetisation is perturbed from alignment with the field, it experiences a torque which acts to realign M with the field. However, since the nuclei have angular momentum which must be conserved, the magnetisation can’t just realign instead it precesses about the field (like a gyroscope).
The frequency of precession, is w = γ B_0, which is known as the Larmor frequency.
Precessing magnetisation produces a magnetic field which fluctuates at the Larmor frequency. This fluctuating field will induce a voltage in a surrounding coil of wire by electromagnetic induction (like a dynamo). The resulting voltage is the NMR signal that is used in MRI.
What is a magnetic field gradient
A magnetic field gradient is a magnetic field which varies linearly with position.
How are magnetic field gradients used in NMR spatial resolution
If we apply an x-gradient then B = Gx,
where G is the gradient strength measured
in teslas per metre, and since w = γ B, the
frequency of precession also varies linearly
with x -position.
This means that the signals from locations in the object at different x-coordinates have different frequencies and so can be distinguished.
What functions do electricity play in living systems
- underlies the operation of sensory
organs (e.g. ears, eyes) - is key to nerve and muscle function –
including cardiac (heart) and brain
function - used as a weapon by some
animals.
Describe the structure of a nerve cell
Inside of nerve cell is separated from extracellular medium by electrically insulating, semi permeable membrane.
Describe k+ ion movement in cell
High intracellular conc means ions tend to diffuse out of cell described by J_diff.
k+ is positive hence diffusion out of cell creates -ve charge inside cell –> p.d created
p.d means E field created which attracts k+ back into cell
What equation describes the flux across the membrane
J_diff = - D dC/dx
where C is the number of ions per unit volume, J_diff is the number of ions crossing unit area in unit time and D is the diffusion coefficient.
The flow of electric charge across the membrane
J = oE = - o dV/dx
where o is the electrical conductivity
Explain depolarisation in nerve cells
Changes in membrane permeability can make the voltage more negative or less negative.
Special chemicals, known as neurotransmitters are released in synapses and cause membrane permeability changes in the post-synaptic neuron.
Increases in Permeability Na+ raise V and can trigger a depolarisation followed by repolarisation which occurs on a time-scale of a few milliseconds.
Give the 3 ECG peaks and their biological presentation
Pwave - Atrial depolarisation
Rwave - venticular depolarisation
Twave - venticular repolarisation
Give the three voltages measured by Einthoven’s triangle
RA–>LA -ve to +ve
RA–>LL -ve to +ve
LL–>LA +ve to -ve
