E7 General Relativity

Gravitational Mass: The gravitational mass of an object is the ratio of the gravitational force applied to an object and the strength of the gravitational field.

Inertial Mass: The force applied to an object divided by the acceleration it produces.

These two are identical for all intents and purposes on earth and would only become significant if we accelerate in deep space where the gravitational forces are negligible.

Einstein's Thought Experiment:

Consider two scenarios: If there is an astronaut doing an experiment of dropping a ball in a stationary elevator on earth or in an accelerating elevator. Einstein states that the observations would be identical and the astronaut would not be able to determine whether he is in an accelerating frame of reference or in a gravitational field. This is called the principle of equivalence.

Bending of Light:
Light bends in the presence of bodies with large masses. Evidence: During a solar eclipse in 1919, Arthur Eddington found that stars behind the sun were shifted outwards as shown:

This proves that light from the stars bends due to the presence of a body of large mass like the sun which provides evidence for general relativity.

Gravitational lensing:
Light from distant galaxies behind other galaxies reach the earth because of bending of light. This allows us to see them and is evidence for general relativity.

The distant galaxy appears as a ring around the galaxy in front which is called the Einstein ring.

Time ticks slower in accelerating frames of reference. Consider an beam of light being shone at a wall inside a room that is moving at constant velocity and in a room that is accelerating:

To an observer in the room accelerating, the light appears to be hitting the same spot as he is stationary with respect to the event. But to an observer outside the room, the light appears to be taking a curved path. As the speed of light is constant and the distance that light travels increases according to the observer (curved distances are longer than linear ones), time must be slowing down.

This is why light ticks slower closer to bodies with large masses, as light bends around them.

E6- Relativistic Kinematics

H5 - Evidence to Support Special Relativity

Muon Experiment:

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Evidence:

Muons are produced 10 km above the surface of the earth when cosmic rays from the sun interact with the atmosphere giving them speeds of 0.99c. They have a half life of 2.2 * 10^-6 seconds which means that they should only be able to travel about 1 km before decaying and we should not be able to detect any muons at the surface of the earth. But we do.

gamma factor = 7.1

Conclusions:

As the muons is travelling close to the speed of light, special theory of relativity applies. This means that time is dilated; to an observer on earth, the muon's half life will appear to be 7.1 * 2.2 * 10^-6  seconds. Time dilates. Also, the distance that they have to travel appears to be 10/7.1 km = 1.4 km. The distance contracts. As the time dilates and the distance contracts, the muons appear to survive long enough to reach the surface of the earth, which supports the special theory of relativity.

IB Question:

Michelson Morley Experiment:

 The aim of this experiment was to test for the presence of ether (which is what scientists thought was the medium through which light travels). 

So for sound, if the wind was moving in the direction of the sound wave, sound would travel faster and if the wind was moving in the opposite direction, then sound should travel slower. They thought the same was true of light and ether. The animation below shows the predicted results of the Michelson Morley experiment:

From:http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/mmexpt6.htm

Method:
In the centre is a half mirror that splits the incident beam of light into two (one reflected and one passed through) that have equal intensities that hit mirrors, converge back at the half mirror and strike the detector.
Depending on  the direction of the ether, one beam was thought to arrive faster than the other.
The experiment was repeated by rotating the apparatus and measuring differences in speed of the return beams.
Interference patterns of the two light beams were used to detect differences in their return time)
But this did not happen, and they both arrived at different times. This proved firstly that ether doesn't exist and that the speed of light is constant.

Pion Decay experiment:
A pion was accelerated close to the speed of light. When it decayed it produced two gamma-ray photons. The speed of these photons was not the sum of the speed of the pion and the photon as Galilean transformations would suggest but was the speed of light. This once again proved the second postulate of the special theory of relativity that the speed of light is constant in all inertial frame's of reference.

H4 - Consequences of Special Relativity

Twin Paradox:
If Nikita was on earth and her twin Atikin was on a very fast journey into space, then from Atikin's perspective Nikita's clock is ticking slower and from Nikita's perspective Atikin's clock is ticking slower. Consequentially they both believe that the other will be younger when they meet on Atikin's return. But they both cannot be younger; this is the twin paradox.

In reality, Atikin will be younger because, for Atikin, to attain that velocity, she must accelerate and before turning to return, she must decelerate. Again, on the return journey she must accelerate to attain that velocity and decelerate on returning to earth. Therefore, she is no longer in an inertial frame of reference. In the accelerating frame of reference, time appears to be ticking slower to an observer in it; therefore, Atikin will be younger on returning to earth.

In order to resolve the twin paradox the general theory of relativity was developed.

Haefele Keating Experiment:

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from: http://www.youtube.com/watch?v=cmPebZA2ZdI

Questions:
from Hienneman:

IB:

H3 Relativistic Kinematics

Light Clock: A beam of light reflected between two parallel mirrors that is used to measure time is called a light clock.


From: http://www.patana.ac.th/secondary/science/anrophysics/relativity_option/commentary.html

As can be seen, to Jill, her light clock appears to have spanned exactly 10 seconds no matter what her speed. This is because the light clock is stationary with respect to her and the beam of light only has vertical motion not horizontal motion. But if Jack observes Jill's clock, the beam of light from the source is not only moving upwards but also to the right. This means that it has a greater distance to travel than the beam of light approaching the mirror from his light source. As the speed of light is constant, this must mean Jill's clock must be moving slower according to Jack to compensate for the increased distance that  the light beam has to travel.

Proper Time is the time as measured in the frame of reference where the event it occurring. This happens to be the shortest possible time that can be recorded for an event to occur.
Using the above example, both Jack and Jill will measure proper time using their clocks as they are in the frame of reference in which the event is occurring.

Time Dilation Derivation: 

Yeah...you can't really see what's written.
This is what is written:

HL Physics Hienemann, Chris Hamper


Gamma (in the video) is called the Lorentz factor.
The speed of light must be less than c, as otherwise the denominator would be 0 and time would be infinitely long. Therefore, we always measure the time taken for an event to occur in different frame of reference that has relative motion to be longer than the time taken for that event as measured by someone in that frame of reference. Time is said to dilate.

The variation of the Lorentz factor as the velocity of the object varies is shown below:

Graph Data

Time Dilation problems:

Proper Length:Proper length of an object is the length  of an object as measured by an observer in the same frame of reference as the object. This is the longest possible distance of the object that can be measured. As on object approaches the speed of light, its length appears to decrease in the direction of its motion.

If an astronaut is moving between stars close to the speed of light, then the distance between the stars appears to contract which means that the astronaut feels that the distance be travels is shorter and he covers it in a shorter time than what it appears to be from earth. 

From: http://www.youtube.com/watch?v=xvZfx7iwq94&feature=relmfu

Questions on time and distance dilation:

IB Questions:

H2 - Concepts of Special Relativity

Inertial frame of reference is one which is moving at constant speed where Newton's Laws of motion apply. Accelerating frames of reference, like circular motion is a non inertial frame of reference ( there is acceleration towards the centre). Acceleration must be 0 for a frame of reference to be inertial.
Special Relativity: The two postulates for special relativity are

1. The speed of light is constant for all observers in an inertial frame of reference
2. The laws of physics are the same for all inertial observers.

Simultaneity: If two events happen together, they are said to be happening simultaneously.
 Consider

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To a person inside the train moving at constant speed, if both the pulses of light are sent out at the same then the time taken for both beams to hit mirrors on either end and reach back to him is the same. The light pulses return simultaneously. 

But consider a stationary observer on the platform observing this event:

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When the train moves forward, the light travelling towards the left has a shorter distance to travel as the back of the train has moved forward towards the pulse of light. Similarly, the light travelling towards the right has a longer distance to travel as the front of the train is moving forward away from the pulse of light. Speed of light is constant in all inertial frames of reference irrespective of the speed of the frame of reference of the observer. Therefore, to an observer on the platform, the light pulse on the right reaches the mirror after the one on the left.

This event is not simultaneous to an observer on the platform. Two events occurring at different points in space cannot be simultaneous for two different frames of references.

From: http://www.youtube.com/watch?v=AZ6N85lNgHY

H1: Introduction to Relativity

H1: Frame of reference: It is a system of coordinates that allows the position of various objects to be defined and measurements to be made.



From: http://www.patana.ac.th/secondary/science/anrophysics/relativity_option/commentary.html#intro_to_SR

If object A is moving due east at 20 ms^-1 and object B is moving towards it due west at 15 ms^-1 then

  V_{a rel B} means the velocity of A relative to B i.e. B is the observer and A is the object. 

V_{a rel B} = V_a - V_b = 20 – (-15) = 35 ms^‑1. 

This is an example of a Galilean transformation where time is assumed to be independent of the observer.

Consider:

As they are moving towards each other their relative velocities with respect to each other are twice their original velocities by Galilean transformation = 0.9800c + 0.9800c = 1.960c. This is faster than the speed of light which is not physically possible; therefore at speeds greater than 1c, Galilean transformations break down and consequentially the Theory of Relativity was 'invented.'

But in our everyday life we do not deal with speeds close to the speed of light; therefore the Galilean transformations hold.

The photon moving in the same direction as the pion (photon R) will have a speed of 1c (speed of light/photon)+ 0.900c which is 1.90c. 

The photon moving in the opposite direction as the pion (photon Q) will have velocity 1c - 0.900c = 0.100c.

E6 Galaxies and Expanding Universe

E6.1 A galaxy is a cluster of stars. A galactic cluster is a cluster of galaxies. Galactic clusters grouped together is called a super galactic cluster.

E6.2 As the universe is expanding there spectra are red shifted. Galaxies that are the furthest away are red shifted the most.

E6.3 

Heinneman HL Physics, Chris Hamper

E6.4 Hubble's law is that the distance of celestial bodies is proportional to the recession velocity.

Heinneman HL Physics, Chris Hamper

The gradient of this line is called Hubble's constant. The value of this constant changes depending on the number of data points we obtain and plot.

Heinneman HL Physics, Chris Hamper

E6.6 The recession velocity is calculated by observing the red shift and using the equation above to obtain the recession velocity. The distance is calculated by using the three methods (Stellar parallax, Spectroscopic parallax and Cepheid variable method). Using this, we can plot the data points on the graph to obtain Hubble's Constant. Units: Recession Velocity in Kms^-1 and Distance in Mpc^-1

E6.8

Hubble's equation...

Hubble's constant is about 72kms^-1Mpc^-1

E6.5 The data points are scattered about the line of best fit which means that we have random errors in our measurement. Also the gravitational attraction slows down the recession speed of the galaxies but we are assuming that the recession speed is constant.

 We don't have data points near the origin for the graph because the red shift is so small that the errors in the apparatus become significant.

E6.7 Hubble's constant's units are kms^-1Mpc^-1 which is ditancetime^-1distance^‑1 which is time^-1 Therefore the inverse of Hubble’s constant is the age of the universe. 

Heinneman HL Physics, Chris Hamper

But we are assuming that the velocity is constant but we know that the universe is slowing down. Therefore the speed we measure today is higher than what it actually is which means that the time measured as the age of the universe is an upper limit.

E5 Stellar Processes and Evolution

E5.1 Stars are formed when huge clouds of dust and gas are compressed. This cannot occur on its own as the gravitational attraction between them is not strong enough. Star formation is initiated by either a supernova explosion or by two dust clouds' collision. As gas clouds are pulled closer together, the gravitational potential energy converts to kinetic energy causing an increase in the temperature. Also as the particles get closer together the force between them increases causing a rise in the pressure of the dust cloud. The cloud therefore collapses into an increasingly hotter body.

The centre of the dense core rapidly contracts leading to high temperatures and pressure. This is called a protostar but cannot be seen as it is surrounded by a cloud of dust.

After 100000 years the radiation from the star blows away the dust and its mass stabilizes. It now forms a pre-main sequence star. The abundance of hydrogen in the universe suggests that the star is made mainly of hydrogen. When fusion reaction starts taking place, the radiation pressure eventually equals the gravitational pressure to form a body of stable mass. This is now a main sequence star like the sun.


E5.2 5.3Eventually when the hydrogen runs out, the gravitational pressure starts increasing causing the star to collapse. As this occurs the temperature and pressure of the star increases and it becomes possible to fuse helium into higher elements. However, these fusion reactions release an even greater amount of energy which causes the start to expand. When the star expands, the outer layers become cooler. This therefore forms a red giant star. The process of fusing higher and higher elements ensues till be reach iron which has the greatest binding energy per nucleon. After this fusing iron to form higher elements would be endothermic and the star cannot continue to shine. At this point, when the star has run out of nuclear fuel, it starts collapsing due to gravitational force. From here it may either turn into a neutron star or, if it is big enough, into a black hole.

E5.4
The initial mass of the dust cloud determines the size of the star that forms from it which helps us determine where different stars join the main sequence.

Hinemann HL Physics, Chris Hamper

E5.5

E4- Cosmology

E4.1Newton's model of the universe: Newton believed that the universe is infinite in space and time, uniform and static (otherwise it would collapse under its own gravity). This would means that the universe has an infinite number of stars spreading out to infinity.

E4.2Olber's Paradox: If Newton's model was true then this would imply that the night sky ins infinitely bright as there are an infinite number of stars in the sky.

Suppose stars are distributed in an infinite number of thin shells. If each has a Luminosity of L, related to the apparent brightness (b) and distance d, this would imply that the apparent brightness is inversely proportional to the square of the distance.

If there was a thin shell of stars with thickness T at distance d then the volume of the shell would be:
d

Heinneman HL Physics, Chris Hamper

E4.3 On observing galaxies through the Hubble telescope we have observed a red shift in a majority of their spectra which suggests that they are moving away from us, supporting the notion that the universe is not static but is expanding.

E4.4: If galaxies are moving away from each other, logically, this must mean that at some point in the past they were closer together. Extending this idea, we know that once, the entire universe must have been concentrated in a single spot called the singularity. We believe that about 13 billion years ago, the universe must have begun with an explosion called the Big Bang. Space and Time began following the Big Bang.

E4.5 4.6 4.7: Cosmic Microwave Background radiation (CMB radiation) was observed by Penzias and Wilson when they received excess noise from their radio. This is an echo of the Big Bang still echoing in the universe. Penzias and Wilson found that the intensity of the radiation that they received corresponded to that of waves in the microwave region. Substituting this in Wien's displacement law equation they found the average temperature of the universe to be 2.7 K. 

The Big Bang model resolves Olber's paradox as if the galaxies are moving away from each other and are red shifted then this means that we won't be able to see them. This is why the night sky appears dark.

E4.8

Heinemann HL Physics - Chris Hamper

Open Universe: This will continue to expand. Gravity will slow it down but not stop the expansion

Closed Universe: This will eventually collapse back on itself; this is called the big crunch and the reverse of the big bang will occer.

Flat universe:This will

E4.9 Critical density  is the mass per unit volume  that would create a flat universe.

E4.10 If the actual density of the universe is lower than critical density then this would create an open universe, but is the actual density of the universe is higher than the critical density then this would create a closed universe

E4.11 

Dark matter is matter that is not visible as it emits little or no radiation.

WIMPs  are Weakly Interacting Massive Particles. These are hypothetical particles serving as one possible solution to the dark matter problem. They are held together by weak forces and gravity. As they do not interact through electromagnetic radiation it is not possible to detect them and as they do not interact with strong nuclear force, they do not create atomic nuclei. (http://en.wikipedia.org/wiki/WIMPs)

MACHOs are massive astrophysical compact halo objects. This is another hypothesis as to what dark matter is which claims that dark matter is made of the same substance as 'ordinary matter' but just happens to emit little or no radiation making them invisible. (http://news.bbc.co.uk/2/hi/8508662.stm)

It is difficult to determine the density of the universe because we cannot detect dark matter as it does not emit electromagnetic radiation; therefore we know neither its exact volume nor its mass. Knowing the density of the universe in important because this will allow us to determine the fate of the universe; whether it is open, closed or flat.

E12. Current scientific evidence suggests that we have an open universe.

E13. International projects include the study of Drake's equation which consists of a series of variables that, when multiplied together, give the probability of us finding extra terrestrial life. 

Other links:

http://insciences.org/article.php?article_id=8795- Finding dark energy

http://insciences.org/article.php?article_id=6839- Black holes

E14. 

Arguments for astrophysics:

• Understanding the nature of the universe and attempting to answer questions like why we are here and whether or not there is intelligent life other than us
• Develops technology which may improve the quality of life in the future
• To colonise new planets if, in the future, the earth becomes inhabitable
• To understand the fate of the universe and what our future might be

Arguments against astrophysics:

• Money could be used for economic development instead
• Money could be used to conduct more pragmatic research  like medical or energy sources
• Is the information really worth the cost?

E3 - Stellar Distances

As we move, objects appear to change their relative distances. Objects closer to us appear to move more than those further away that might appear stationary. This apparent movement is called parallax and this effect can be used to measure stellar distances as the closer the star is to the earth the greater the parallax shift will be.

Angles: 3600 arc seconds (3600") = 1° = 2π/360 radians

Using distant stars as a reference the angle that the moving star makes with the earth is observed over a six month period (why? it gives the maximum angle) as shown:

Here we can see that:

tanq = 1 Au/ d

using small angle approximation we can see that this means:

q = 1 Au/ d

If q was 1 second ( (2π/360 * 60 * 60) radians ) then the distance d would be given by 1 parsec. 

  p = 1/d

where p = angle in arcsecs 

               d = distance in parsecs

This method can be used for stars up to a few 100 parsecs away after which the angle becomes so small that the uncertainties become significant.

Apparent magnitude: This is a scale to measure the apparent brightness of a celestial body where each level is equivalent to a rise of 2.51 times the brightness of the previous one and the brightness increases with the negativity of the numbers. This means that the brightest star ( sun) will have the largest negative value of all the stars.

The ratio of the apparent brightnesses of two stars bA and bB is related to their apparent magnitudes MB adn MA by:

Absolute magnitude: This is the apparent magnitude of the celestial bodies at a distance of 10 parsecs from earth. 

E2- Stellar Radiation

Fusion is the main source of energy for fusion.

When fusion occurs this leads to the production of heat which gives the molecules greater kinetic energy. As a result they exert an outward force that forces that would force the star to expand. As this force acts over the surface area of the star, it is called radiation pressure. However, the gravitational force caused of the star also exerts an opposing force that ensures that the start does not expand. This is referred to as gravitational pressure as it acts over the surface area of the star. In stable starts like the sun, the gravitational pressure is equal to the radiation pressure which ensures that the size of the star remains constant.


Luminosity (L) is defined as the total power emitted radiated by the star (units: W). It is affected by the star and its surface temperature. Stars with higher surface temperatures and radii will have greater luminosities. This is however  not the power received by, us the observer. The per received per unit area by an observer is called the apparent brightness (b) of a star. As power is emitted as a sphere, the apparent brightness is given by:

The units are W m^-2.

If two stars are different distances apart then, for them to have the same apparent brightness, the one further away needs to have a greater luminosity.

The apparent brightness of a star can be measured using a charge coupled device (CCD) which works on the principle of the photoelectric effect. 

Wien's and Stefan-Boltzmann Law

Radiation from a perfect emitter is known as black-body radiation. 

Wien's displacement law states that the peak wavelength of the emission of a black body is inversely proportional to the temperature.

Energy emitted per unit time is known as power but also luminosity and is given by:

E2.7: Absorption spectra for stars are observed and using the characteristic absorption spectra for the different elements, the elements that the star is composed of are discerned.  Observing the absorption spectra of stars can give us the peak wavelength which can be used to calculate the surface temperature using Wien's law. This temperature can be used to calculate the luminosity of stars using the Stefan-Boltzmann law. This can be used to calculate the distance between the star and the earth if the apparent brightness is known.


However depending on whether the star is moving away from us or towards us, the entire spectrum maybe be shifted towards higher or lower frequencies due to the Doppler effect. As stars move away, their wavelengths increase causing the frequency to decrease. This is called a red shift. If stars are coming towards us, their wavelength is compressed causing an increase in frequency causing a blue shift. A red shift in the majority of the spectra observed, is evidence for the expansion of the universe.

E2.8: As different stars give out different spectra of light, this allows us to classify stars on the basis of spectral classes. Stars emitting the same types of spectra are allocated to the same spectral class. The seven main classes are: OBAFGKM

As the temperature of the sun is around 5800 K, this means that  it is a group G star which are yellow.

Mnemonic: Oh be a fine guy, kiss me.

For looking at spectra:

http://astro.unl.edu/classaction/animations/light/spectrum010.html 

Sun's spectra:

Different types of stars:

Red Giant:  Large size, red colour, comparatively cooler as red light has a lower frequency, source of energy is fusion but of elements other than hydrogen.

White Dwarfs: Small in size, white in colour, comparatively hot, fusion is no longer taking place. Eventually it will cool down to make a brown dwarf.
Cepheid variables: They are observed to have regular variation in brightness and luminosity which is due to oscillations in the size of the star. They are rare, but useful in determining large distances in galaxies as there is a link between the period of brightness and average luminosity.


Binary Star systems: consisting of two stars revolving around a common centre of mass
Spectroscopic Binary systems:  As stars move around one star appears to be receding while the other appears to be approaching which means that there is a  red/blue shift in their adsorption spectra

Eclipsing Binary systems: This occurs due to a periodic variation in brightness as one stars obscures of overlaps the other. When the less bright star comes before the brighter star there is a decrease in the apparent brightness:

Position 2 = less apparent brightness


Hertzsprung-Russell diagram:


(memorize...)

Astrophysics: Introduction to the Universe

Planets Mnemonics:
My  : Mercury
Very  : Venus
Educated  : Earth
Mother  : Mars
Just  : Jupiter
Served : Saturn
Us  : Uranus
Nothing :Neptune

Relative Planet Sizes

hyperphysics.phy-astr.gsu.edu

Asteroids: A small rocky body that drifts around the Solar system

Meteorite: An asteroid on a collision course with the planet is known as a meteoroid. Small meteors can be vaporized due to friction with the atmosphere. The bits that arrive are called meteorites.

Comet: are mixtures of rock and ice

Stellar Constellation: Are a group of stars that are physically close to each other, created by the collapse of the same gas cloud

Constellation: These are patterns of stars that have been identified.

Light Star: The distance light travels in one year.

Comparing Distances:
Distance of the visible universe: 10^26m

Distance between local galaxies: 10^22 m

Distance of our galaxy: 10^21m

Distance of our solar system: 10^13 m 

Movement of constellations:

Over one night: They appear to be rotating in the sky around a fixed point (the pole star) due to the rotation of the earth

This shows the movement of Orion over one night.

Over a year: The pivot of rotation (i.e.) the pole star appears to have changed position in the sky which is due to the revolution of earth. This means that the constellations appear at different places in the sky depending on the time of the year.