IB Physics/Astrophysics SL

F.1 Introduction to the Universe

F.1.1 Structure of the Solar System

Within our own universe, there exist many celestial bodies each with their own unique properties; the planets being no exception.

Planet	Mass/kg	Radius/m	Orbit radius/m	Orbital period (avg.)
Mercury	3.30 x 1023	2.44 x 106	5.79 x 1010	88.0 days
Venus	4.87 x 1024	6.05 x 106	1.08 x 1011	224.7 days
Earth	5.98 x 1024	6.38 x 106	1.50 x 1011	365.3 days
Mars	6.42 x 1023	3.40 x 106	2.28 x 1011	687.0 days
Jupiter	1.90 x 1027	6.91 x 107	7.78 x 1011	11.86 years
Saturn	5.69 x 1026	6.03 x 107	1.43 x 1012	29.42 years
Uranus	8.66 x 1025	2.56 x 107	2.88 x 1012	83.75 years
Neptune	1.03 x 1026	2.48 x 107	4.50 x 1012	163.7 years
Pluto (now a dwarf planet)	1.50 x 1022	1.15 x 106	5.92 x 1012	248.0 years

Note: All planets beyond Mars are gas giants; i.e. Jupiter is a failed sun.

With these given properties and the right equation, you can find the volume and density of any of these planets.

Volume:$V={\frac {4}{3}}\pi r^{3}$
r=radius of the planet or sphere

Density:$D={\frac {M}{V}}$
M=mass
V=volume

F.1.2 Bodies within the Universe

Binary Star: Two stars orbiting a common center.

Black Dwarf: The remnant of a white dwarf after it has cooled down. It has very low luminosity.

Black Hole: A singularity in space-time: the end result in the evolution of a very massive star.

Brown Dwarf: Gas and dust that did not reach high enough temperatures to initiate fusion. These objects continue to compact and cool down.

Cepheid Variable: A star of variable luminosity. The Luminosity increases sharply and falls off gently with a well-defined period. The period is related to the absolute luminosity of the star and so can be used to estimate the distance to the star by using the Cepheid variable as a standard candle.

Clusters of Galaxies: Two or more Galaxies that are close enough to each other that they affect each others through gravitation.

Comet: A small body composed of mainly ice and dust that orbits the sun in an elliptical orbit.

Constellation: A group of stars which are in a particular pattern or design.

Clusters: Gravitationally bound system of galaxies/stars.

Constellations: Group of galaxies/stars given a specific name. The 12 zodiacs are examples - Pisces, Aries, Taurus, Gemini, Cancer, and so on.

Dark Matter: Matter in galaxies that is too cold to radiate. Its existence is inferred from techniques rather then direct visual contact.

Galaxies: Giant assemblies of stars, gas, and dust held together by the gravitational forces they have on each other. Our Galaxy is called the Milky Way.

Interstellar Medium: Gases and dust that are filling the space between stars. Interstellar mass’s density is very low with about one atom of gas for every cubic centimeter of space.

Main Sequence Star: A normal star that is undergoing nuclear fusion of hydrogen into helium.

Neutron stars: A very dense star, consisting only of uncharged neutrons. They are created when very massive stars explode, leaving this neutron 'ball' behind. A neutron star is smaller then a white dwarf and extremely dense. It is microscopic and is a prime example of microscopic quantum physics.

Nebulae: From the Latin word for 'cloud'. Used to label all sorts of stuff in space, that are now known as star cluster or galaxies. It is sometimes still used for a concentration of gas and dust.

Nova: A sudden increase in luminosity of a white dwarf caused by material from a nearby star falling into the white dwarf.

Parallax: The apparent motion of a star against the background of a more distant star, due to the motion of the Earth around the Sun. The angle is measured at different times during the year. The distance of the Sun to the Earth is known. Distances are specified in parallax angles in seconds of arc (parsec). At large distances the uncertainty becomes too large and it can't be applied. Example : angle = (6_10^-5)° = (6_10^-5)° (3600) = 0.22´´ of arc (seconds of arc) in parsecs: 1/0.22´´ = 4.5 pc.

Planetary nebula: The ejected envelope of a red giant star.

Pulsar: Sends out sharp, strong burst of radio waves at regular intervals ranging between milliseconds and 4 seconds. They appear to be rapidly rotating highly magnetic neutron stars. The pulses are very energetically charged particles. The rotation and pulse rates gradually slow down as energy is radiated away.

Quasars: Small, extraordinarily luminous extragalactic objects with high redshifts. They do not seem to conform to Hubble’s Law. They are as bright as nearby stars, but display very large redshifts. According to Hubble’s law the quasars must be either extremely distant and incredibly bright (thousands of times brighter than ordinary galaxies) or that they are closer than the redshift suggests. There is either an unresolved brightness problem or an unresolved redshift problem. One theory is that quasars could be powered by black holes.

Red Dwarf: A very small star with low temperature. It is relatively red in color.

Red Giant: Luminous stars with low surface temperature. These stars are produced when hydrogen in the core of the star has fused into more heavy helium. Gravity forces the star to contract, but at the same time it heats up. The hydrogen around the core now burns more fiercely and causes the outer envelope of the star to expand and thus cool. This low surface temperature produces light at a longer wavelength.

Stellar Cluster: A group of stars that are physically near each other in space, created by a collapse of the same gas cloud.

Supernova: Gigantic stellar explosions. These occur when very massive stars explode. Because of Einstein's law, E=mc², fusion can not take place after that iron is created. The star collapses since there is no force that can outweigh the gravitational force. When it is impossible to compress it further, it explodes violently giving out extremely luminous light.

White Dwarfs: Are developed as all fuel in a star is exhausted. Gravity forces the star (with a mass lower than 1.4 solar masses) to contract and heat up at a high pressure. The atoms lose some of their electrons. It becomes a small hot white dwarf.

F.1.3 Stellar Cluster and Constellations

A stellar cluster is a group of stars that are near each other in space and created from the same gas cloud. On the other hand, a constellation is just a group of stars that seem to be close to each other because they form a recognizable pattern in space. In a stellar cluster the stars are attracted to each other because of high temperatures in the core of the stars, where the protons end up having electrostatic repulsion due to the Hydrogen Nuclei Fusion. A constellation appears to be stars close to each other. The celestial sphere is what makes them appear like the stars are physically near.[LG,JD]

F.1.4 Light Years

A light year is a unit of measurement of Ultra-solar system distances. It's the distance traveled by light in one year. The speed of light is 186,287.5 miles per second. You can find out the number of seconds in a year by multiplying the number of seconds in a minute (60) by the number of minutes in an hour (60). Then multiply that by the number of hours in a day (24), and multiply that by the number of days in a year (approximately 365.25). One light year is equal to 9.46 x 10^15m, which is also equal to 0.3068 Parsecs(Pc).

Example: The distance to the nearest star (Proxima Centauri)from the Earth is 4.31 light years, which is equal to 1.3pc. This means that it would take 4.3 years to send a message to Proxima Centauri.

F.1.5 Relative Distances

The average distance between stars in a galaxy is approximately 1 pc, which is equal to 3.26 light years. The average distance between galaxies within the same cluster ranges from 100 kpc (kilo-parsecs) to several hundred kpc (kilo-parsecs). Galaxies in different clusters can be up to a few Mpc (mega-parsecs) apart. 1 Mpc is equal to 1000 kpc.

F.1.6 Movement of Constellations and Stars over time

During the night, stars and constellations seem to be moving from east to west, but the relative position of the constellations do not change. The celestial sphere is the area around earth where the stars and constellations are located. The North Star, Polaris, is on the north celestial pole and doesn't seem to move at all. The rest of the stars and constellations seem to rotate around the North Star. Stars seem to move less the further north or south they are located. As the Earth rotates, the hemispheres receive different views of the stars and constellations.

For the Earth to make one revolution per year there must be a small change every day. The change per night is 0.986 degrees and is not easily detected.

F.2 Stellar Radiation and Stellar Types

F.2.1 Energy Source

All stars follow a simple proton-proton cycle in order to maintain an equilibrium between gravity and pressure. When the star is expending fuel it rises in temperature and therefore rises in pressure. This is required in order to keep a balance between the great force of gravity that is trying to compress the star. At the beginning of a star's life cycle the star consists mainly of hydrogen; in fact they are made of 98% hydrogen. There are three basic stages of the Proton-Proton cycle:
1.) Two hydrogens fuse to form a deuterium, plus a positron and a neutrino. Each positron is annihilated to create 2γ(gamma) particles [they are in turn absorbed and re-emitted as 200,000 photons of light per γ(gamma) particle.]
2.a.) A deuterium and a hydrogen fuse to create helium and a γ(gamma) particle.
2.b.) Another deuterium and a hydrogen fuse to create helium and a γ(gamma) particle. [Thus far there have been 4γ(gamma) particles created, those are 800,000 photons of light. That's one bright star!]
3.) Two helium atoms fuse to create a heavy helium atom.
Once the hydrogen in the star runs out, it begins to consume the created helium from the hydrogen reactions. Based on the star's color, you can find out what type of fuel it's consuming. This is where the Hertzsprung-Russell diagrams were derived from.

F.2.2 Star stability and Equilibrium

The stability of a star depends on the equilibrium between two opposing forces. The equilibrium depends on the gravitation which can collapse the star and the radiation pressure which can make the star expand. This equilibrium is gained through nuclear fusion which provides the energy the star needs to keep it hot so that the star's radiation pressure is high enough to oppose gravitational contraction.

F.2.3 Luminosity of a star

"Luminosity is the amount of energy radiated by the star per second; that is, it is the power radiated by the star. As shown in the next section, luminosity depends on the surface temperature and surface area of the star." In simple words: Luminosity depends on temperature and the radius (surface area) of the star.

Luminosity is the result of a chemical reaction within a star.

F.2.4 Brightness and Measurement of Luminosity

Brightness is the received energy per second per unit area by apparent brightness.

b=L/4πd^-2 Wm^-2

Apparent brightness is directly proportional to intrinsic luminosity, where apparent brightness is measured using a charge coupled device (CCD) that releases one electron when hit by a photon.

F.2.5 Black Body Radiation

A black body is an object that absorbs all electromagnetic radiation that falls onto it. The Stefan-Boltzmann Law can be used to determine the radiation being emitted in the form of electromagnetic waves by the black body. To understand the Stefan-Boltzmann Law in further depth refer to section F.2.8. The Wien displacement law relates the wavelength to surface temperature. The Wein constant is 2.90 x 10^-3 Km. This means that the higher the temperature, the lower the wavelength at which most of the radiation is emitted as energy.

F.2.6 Spectra of black bodies

• 
  wavelength (nanometers) against Intensity (arbitrary)

As the overall intensity increases, the relative intensity of the peak of the curve (most intense wavelength) decreases in intensity, and the most intense wavelength gets longer and longer (less and less energetic).

F.2.7 Wien's Law

Wein's displacement law states that the higher the temperature the lower the wavelength at which most of the energy is radiated.

Wein's displacement law is stated as: λ_max= b/T Where λ is the maximum wavelength (meters), T is the temperature of the blackbody (Kelvin), and b is a constant for 2.90 x 10^-3Km

F.2.8 Stefan-Boltzmann Law

The Stefan-Boltzmann Law helps to find luminosity of a star's distance from an observer. The luminosity is then used to find the absolute brightness of the star. The equation to find luminosity is L=σAT⁴.

The Stefan-Boltzmann constant: σ = 5.67 x 10^-8Wm^-2K^-4

A= Surface Area

T= Temperature

Once the luminosity is found, place it into the equation b=L/4πd² for L.

d= distance for this equation.

This will allow you to find the absolute brightness of the star.

F.2.9 Stellar Spectra

The surface temperature of a star is determined by measuring the wavelength at which most of the radiation is emitted. Most stars have essentially the same chemical composition., yet show different absorption spectra. The reason is that different stars have different temperatures. Absorption spectra gives information about the temperature of the star and its chemical composition. Stars are divided into seven spectral classes according to their color.

Spectral Class Summary

Spectral Class Effective Temperature (K) Colour H Balmer Features Other Features M/MSun R/RSun L/LSun Main Sequence Lifespan

O 28,000 - 60,000 Blue weak ionised He+ lines, strong UV continuum 20 - 60 9 - 15 90,000 - 800,000 1 - 10 Myr

B 10,000 - 28,000 Blue-white medium neutral He lines 3 - 18 3.0 - 8.4 95 - 52,000 11 - 400 Myr

A 7,500 - 10,000 White strong strong H lines, ionised metal lines 2.0 - 3.0 1.7 - 2.7 8 -55 400 Myr - 3 Gyr

F 6,000 - 7,500 White-yellow medium weak ionised Ca+ 1.1 - 1.6 1.2 - 1.6 2.0 - 6.5 3 - 7 Gyr

G 4,900 - 6,000 Yellow weak ionised Ca+, metal lines 0.85 - 1.1 0.85 - 1.1 0.66 - 1.5 7 - 15 Gyr

K 3,500 - 4,900 Orange very weak Ca+, Fe, strong molecules, CH, CN 0.65 - 0.85 0.65 - 0.85 0.10 - 0.42 17 Gyr

M 2,000 - 3,500 Red very weak molecular lines, eg TiO, neutral metals 0.08 - 0.05 0.17 - 0.63 0.001 - 0.08 56 Gyr

It is known that hydrogen is the predominant element in normal main sequence stars, making up 70% of their mass followed by helium with 20%. The rest is made up of heavier elements.

F.2.10 Stellar Spectra classification system

Class	Temperature	Colour
O	30 000 - 60 000	Blue
B	10 000 - 30 000	Blue-white
A	7 500 - 10 000	White
F	6000 - 7500	Yellow-white
G	5000 - 6000	Yellow
K	3500 - 5000	Orange
M	2000 - 3500	Red

F.2.13 Hertzsprung-Russell diagram

Determines the fate of stars.

F.3 Stellar Distances

F.3.1 Parallax

Parallax is a term used to describe the distance between two objects in space. When an observer on Earth photographs a relatively nearby star against a background of distant stars on two different occasions six months apart, the target star image will appear to have shifted against the more distant stellar background. The baseline shift of the observer on Earth is 2 astronomical units (AU). By convention, calculations are normalized to one AU, the radius of the Earth's orbit, so one half of the measured shift in apparent position is deemed the "parallax" of the target.

F.3.2 The Parsec

A parallax of one arcsecond is called a parsec. Since we know, the radius of the Earth's orbit, simple Euclidean geometry allows us to calculate that a star exhibiting a one arcsecond shift is 3.26 light years or one parsec away from Earth.

F.3.3 Limits of the Stellar parallax method

Hubble's law: v = Hd, where v is the speed, H is the Hubble parameter, and d is the distance. It describes Hubble's observation, that most lines in the spectra of other galaxies were redshifted, and the amount of shift was approximately proportional to the distance of the galaxy from us. So the velocity is proportional to the distance. H is approximately 75km/s/Mpc. Note that this does not work well for nearby galaxies because of two reasons. One the error margin for calculating distances is somewhere around 15% and we have more accurate ways to measure distances for close distances and two earth is still a part of many of the closer systems so the difference in velocities is negligible. It's also important to note that this is a value that changes with new data coming in. Hubble's law really describes the speed at which celestial bodies move away from each other and changes because the further away a galaxy gets the faster it goes.

F.3.4 Determining distance with Stellar parallax method

Determining distances with stellar parallax is done by measuring a star periodically against a fixed background. Then you do some simple algebra with the change in area. Much more accurate at close distances than Hubble's law but Hubble's law is much better for long distances despite its error margin.

F.3.5 Apparent Magnitude Scale

The U.S. physicists A. Penzias and R. Wilson detected (in 1956) microwave radiation coming equally from all directions in the sky, day and night. This radiation is like the one radiated by a black body at a temperature of 3 Kelvin, therefore the name 3K radiation.

This discovery supports the theory of Big Bang, where strong shortwave radiation was supposed to be sent out. The radiation spread filling the expanding universe uniformly. With time it cooled, to the now observed temperature of 3K, and now strikes the Earth as microwaves.

F.3.6 Concept of Absolute Magnitude and Determining Absolute Magnitude

The Absolute Magnitude of a star relies on the apparent magnitude initially. The apparent magnitude of a star is a measure of brightness of a star seen from earth in a relative system of classification. The higher the numerical value of apparent magnitude, the dimmer the star. This relates to absolute magnitude because the absolute magnitude is the apparent magnitude a start would have if observed from a distance of 10pc. To determine absolute magnitude, you use the following equation : M=m+2.5log(100/d²)

F.3.7 Apparent Brightness and Apparent Magnitude Examples

An example for apparent magnitude is as follows: The star Capella has an apparent magnitude of +0.05 and its distance from Earth is 14pc. Estimate its absolute magnitude.

To solve this problem we would use the following equation: M=m+2.5log(100/d²)

Next we would plug in the variables and get the following: M=+0.05+2.50log(100/14²)

M=2.55log(100/196)= -.745252982

So the estimated absolute magnitude is -.745252982

F.3.8 Finding Luminosity from the Spectrum

Luminosity is the amount of energy radiated by the star per second. Basically put, it is the power radiated by the star. Luminosity depends on the surface temperatue and the surface area of the star. To relate luminosity to the spectrum, the higher the luminosity a star has, the higher up on the spectrum it will be.

F.3.12 Cepheid Variables and Cepheid Nature

Cephied Variable: A star thats luminosity increases sharply and falls gently in a period of time, the period of time is related to the luminosity of the star. The Cephied variable is used to estimate the distance of the star. There is a relationship between the period of the light curve and the peak luminosity. On Cephieds have periods of one to fifty days. The variable provides information of the star structure developing theories for stellar structures.

F.3.13 Cepheid Variables and relation to Period and Average Absolute Magnitude

In absolute and apparent magnitudes astronomers would classify stars according to how they appeared to be to the observer on earth based on its brightness. The higher the magnitude then the dimmer the star, brightness of stars were defined in assigned numbers that varied the magnitude of the start from 1-6. Not all stars are the same distance, which is a factor in how the magnitude of a star is classified, while the brightness also varies as opposed to two starts in the same place, direction, and same brightness.

F.3.14 Cepheid Variables as "Standard Candles"

"Standard Candles," are those that are described as Cepheid Variables because the way that the Cepheid stars are described, it occurs the same way as candles. The farther away a candle is from the other the less luminosity and brightness is seen from the observer. Distance is what takes away the luminosity and brightness of a star the farther away it is from the observers stand point of view.

F.3.15 Cepheid Variables on the Luminosity-Period Graph

Cepheid Variables on a Luminosity-Period Graph due to their brightness increase and gradual fade offs, curves on the graph. It gives a sine graph type of picture because of it's periodic behavior in brightness. The outer layers of the star go through contractions and expansions periodically. When it expands outward it's because the star is brighter at a surface at high velocity, and when it's dim at a surface it moves inward.

F.4 Cosmology

F.4.1 Newton's Theory of the Universe

If the universe is infinite, how come then that we observe a black night sky? Shouldn't such a universe provide an infinite number of stars so that wherever we looked it would be bright?

F.4.2 Olber's Paradox

The Olbers Paradox questions why the night sky is dark. If the universe was infinite and thereby contained an infinite amount of stars, then theoretically there would be an infinite amount of energy radiating from the stars making the night sky infinitely bright. The Olbers Paradox applies to all infinite models, but does not apply to finite models. This is due to the fact that:

1. There is a finite number of stars and each star has a finite lifetime.
2. The universe has a finite age and stars that are beyond the event horizon have not yet had time for their light to reach Earth.
3. The radiation received is redshifted and so contains less energy.

F.4.3 Solutions to Olber's Paradox

The matter and radiation of our present time was initially all packed together into an extremely hot and dense fireball, that exploded giving rise to the Big Bang. Within seconds, matter was accelerated through 3 dimensions, expanding and developing very rapidly. Time became a measure of the rate of that expansion, the necessary 4^th dimension.

F.4.4 Doppler Red Shift and Expanding Universe

Curvature of space (also called four-dimensional space-time) : According to Einstein's general theory of relativity light is also affected by gravity. This means that light can be bent 'around' planets, and follow a curved path. Light must travel through the shortest distance available between two points, meaning that the curved path is the shortest distance, hence space itself is curved.

Open universe : The universe will continue to expand forever because the curvature of the universe is negative.

Flat universe : The curvature is zero, the universe is infinite.

Closed universe : The curvature is positive, so the universe is finite.

This can all be showed through non-Euclidean mathematics, where the sum of angles in a triangle either subtends of exceeds 180°.

F.4.5 Big Bang Model

Big Bang model: About ten to twenty billion years ago,all matter and energy in the universe was concentrated in one area from which it expanded quickly. This expansion began when an explosion took place somewhere in space. There is no gravity in space acting on the debris of this explosion so it moves away from the site of the explosion at a velocity that is proportional to its distance from the spot where the explosion took place. Therefore, the farther away the debris is from the explosion, the faster it moves and likewise the closer it is the slower it moves.

The big bang model is related to the four fundamental forces which include: gravity, strong force, weak force, and electromagnetic force. It is assumed that at first these forces started out as one and later separated into the four.

F.4.9 Critical Density

The density that is required to create a flat universe is called critical density. It is round about 5×10^-26kgm^-3.