Option D: Astrophysics

D.1 – Stellar quantities

• Objects in the universe

The solar system is comprised of the Sun, eight planets, several dwarf planets, numerous moons, and hundreds of thousands of other material left over from the construction of the solar system such as asteroids and comets.

However, the solar system in which we live in is merely a speck of dust in the vast universe.

Single star: A luminous sphere of plasma held together by its own gravity.

Binary star: Two stars orbiting a common center.

Black hole: A singularity in space-time.

Cepheid variable: A star with a period of varying luminosity. The luminosity can be determined from the period and along with the apparent brightness can be used to determine the distance of the star from Earth.

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

Constellation: A pattern of stars visible from Earth that are not gravitational bounded.

Dark matter: Matter in galaxies that are too cold to radiate. Its existence is inferred from theoretical physics rather than direct visual contact.

Galaxies: stars, gas, and dust held together by gravitational forces.

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. Nebula: A cloud of dust, hydrogen, helium and other ionized gases.

Planet: A celestial body that orbits a star.

Planetary system: Gravitationally bounded non-stellar objects in orbit around a star or star system.

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

Stellar clusters: A group of stars gravitationally bounded together.

• The nature of stars

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.

Apart from single stars like the Sun, there are many types of stars in our universe (see previous section).

• Astronomical distances

Our universe is composed of mostly empty space with occasional encounters of matter apart large distances.

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 3*10^8m/s. 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 multiplying that by the number of hours in a day (24) and multiplying that by the number of days in a year (approximately 365.25). One light year is thus approximately equivalent to 9.46 x 10^15m, which is also approximately equivalent to 0.3068 parsecs (pc).

Example: The distance to the nearest star other than the Sun (Proxima Centauri) from the Earth is 4.31 light years, which is equivalent to 1.3pc. This means that it would take 4.31 years to send or receive a message to/from Proxima Centauri by electromagnetic wave transmission.

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

To recognize the scale of the universe compared to common objects, see the link below.

http://htwins.net/scale2/

• Stellar parallax and its limitations

Stellar 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.

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.

However, if a star is too far away from Earth, its parallax will be too small to be measured with accuracy.

• Luminosity and apparent brightness

The total power radiated by a star in all directions is known as its luminosity and the SI unit for luminosity is watts ( W ). When you compare this to the power received by an observer on the Earth, you can see that the two quantities are quite different. The power received per unit is known as the star’s brightness and this is measured in watts per metre squared (W/m^2).

If two stars were at the same distance from Earth, the one that had the greatest luminosity would also have the greatest brightness. However, because stars are at different distances from the Earth, their brightness will depend on the luminosity as well as the distance from Earth. The luminosity of a star will decrease with distance according to the inverse square law.

D.2 – Stellar characteristics and stellar evolution

• Stellar spectra

The stellar spectra can be used to identify elements in stars.

Most stellar spectra use the absorption spectrum which is a continuous spectrum that passes through a cool gas and has specific spectral lines removed (inverse of an emission spectrum). The missing wavelengths in a star’s absorption spectra correspond to the absorption spectrum of a number of elements in the star.

There are 7 basic spectral classes: O, B, A, F, G, K and M.

Class	Surface temperature (K)	Color
O	28000-50000	Blue
B	9900-28000	Blue-white
A	7400-9900	White
F	6000-7400	Yellow-white
G	4900-6000	Yellow
K	3500-4900	Orange
M	2000-3500	Orange-red

As temperature increases, electrons are kicked up to higher levels by collisions with other atoms. Large atoms have more kinetic energy, and their electrons are excited first, followed by lower mass atoms.

If the collision is strong enough (high temperatures) then the electron is knocked off the atom and we say that the atom is ionized. So as we go from low temperatures in stars (a few thousand Kelvins), we see heavy atoms, like calcium and magnesium, in the stellar spectra. For stars with higher temperatures, we see lines from lighter atoms, such as hydrogen. The heavier atoms are all ionized by this point and have no electrons to produce absorption lines.

• Hertzsprung–Russell (HR) diagram

The Hertzsprung-Russell (HR) diagram shows the relationship between absolute magnitude, luminosity, classification, and surface temperature of stars.

Most of the stars occupy the region along the line called the main sequence. During this stage, stars are burning hydrogen.

The H-R diagram is also used by scientists to help the figure out roughly how far away the stars are from Earth. This can be done because if we know the apparent magnitude, we can plot the star onto the graph using its spectral class and the type of star it is. We can then use the graph to deduce the absolute magnitude of the star.

• Mass–luminosity relation for main sequence stars

For main sequence stars, the luminosity increases with the mass with the approximate power law

where L⊙ and M⊙ are the luminosity and mass of the Sun. The value a = 3.5 is commonly used for main-sequence stars and does not apply to red giants or white dwarfs.

• Cepheid variables

Cepheid variables are stars in which its luminosity increases sharply and falls gently in a period of time. Thus, the period is correlated to the luminosity of the star and the Cepheid variable can be used to estimate the distance of the star.

Cepheid variables on a luminosity-period graph, due to their brightness increase and gradual fade offs, curves on the graph, giving a sine graph picture. The outer layers of the star go through contractions and expansions periodically. When it expands outward, the star becomes brighter because of high velocity, and when it contracts, the star becomes dimmer as the surface it moves inward.

Cepheid variables are thousands of times more luminous than the Sun and provide us with such a benchmark which is known in astronomy as a “standard candle”.

• Stellar evolution on HR diagrams

The nebulae in space from which stars are created are actually the remains of a previous star that has reached the end of its lifecycle and died. Generally speaking, they consist of hydrogen and helium and small amount of the other heavier elements. The nebula, under the influence of gravity, begins to condense, and eventually, a protostar is formed. Such protostars can be observed in nebulas such as the horsehead nebula and the crab nebula. It is in this stage that the process of nucleosynthesis begins. Nucleosynthesis, in contrast to the nuclear processes that we are used to on Earth, is fusion, not fission. That is, instead of splitting a heavy nucleus, light nuclei are smashed together and fuse to produce a heavier nucleus, and gamma rays. It is called the proton-proton cycle. The star will continue to react its core of hydrogen into helium for all of its main-sequence lifetime (see previous section: the nature of stars).

Once the star runs out of hydrogen, the core collapses, and, under the additional gravitational pressure, the hydrogen in the core will start to undergo fusion. This causes the outer layers of the star to expand, however, the outer layers also cool, and the star becomes a red giant. The core continues to react and elements such as carbon, neon, oxygen, silicon and iron are produced. It is here that the elements that compose our world are created. Without the stars then universe would be composed of hydrogen and little else.

When the star finally runs out of fuel completely; usually when the core becomes iron, the red giant star collapses. The next stage of the star is determined by the mass of that star and the Chandrasekhar limit.

If a star is below 1.4 solar masses (Type G), it is less that the Chandrasekhar limit and when it collapses, its forms a white dwarf of 1.4 solar masses or less, along with a planetary nebula. The white dwarf star continues to cool and eventually becomes invisible.

If a star is above 1.4 solar masses (Type A, B, O), it is above the Chandrasekhar limit and instead of becoming a regular red giant, it becomes a super red giant. In this case, when the star dies, it takes a rather more spectacular path than the star below the Chandrasekhar limit, becoming a supernova. Depending on the mass of the star, it will either go on to become a black hole or a neutron star.

For stellar masses less than about 1.4 solar masses, the energy from the gravitational collapse is not sufficient to produce the neutrons of a neutron star so the collapse is halted by electron degeneracy to form white dwarfs. Electron degeneracy is a stellar application of the Pauli Exclusion Principle, as is neutron degeneracy. No two electrons can occupy identical states, even under the pressure of a collapsing star of several solar masses.

H-R diagrams can also be used to plot the evolution of a star from its birth as a protostar until its death as a white dwarf.

• Red giants, white dwarfs, neutron stars and black holes

See previous section.

• Chandrasekhar and Oppenheimer–Volkoff limits

Chandrasekhar limit

The largest mass a white dwarf can have is about 1.4 solar masses.

Oppenheimer-Volkoff limit

Oppenheimer-Volkoff limits the largest mass a neutron star can have to approximately 2-3 solar masses. The uncertainty in this limit comes from the fact that the equation of state of the matter inside a neutron star is not precisely known.

D.3 – Cosmology

• The Big Bang model

The Big Bang theory states that both space and time originated with the expansion from a singularity.

The evidence that supported the Big Bang theory was observed through the redshift (Doppler effect) of almost all the galaxies. This indicates that all of the galaxies are moving away from us.

Although that observation would seem to indicate that we, or rather, the Earth, are at the centre of the universe, this is not the case. It only appears to be this way as we are observing from the Earth. If we were on a different galaxy, we would see our own galaxy moving away in the same manner as we are observing that galaxy moving away. This can be related to the idea of painted dots on the surface of a balloon; as the balloon is inflated, all of the dots move away from each other equally.

Ultimately, however, what gave the Big Bang theory weight above all others was the discovery of the Cosmic Microwave Background radiation.

This discovery supports the Big Bang theory in two major ways:

1. The early universe was in thermal equilibrium and the radiation from then had a black body spectrum, which has traveled through space, becoming increasingly redshifted up to this point in time. This reduces the temperature of the black body spectrum and the radiation should be visible from every point in space.
2. As the radiation travels throughout the universe, space has expanded, causing the wavelength to increase and its energy to decrease.

All these observations are in accordance with the Big Bang theory.

• Cosmic microwave background (CMB) radiation

See previous section.

• Hubble’s law

Hubble’s law states 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 where the amount of shift was approximately proportional to the distance of the galaxy from us. Thus, the velocity is proportional to the distance.

We can use Hubble’s law to estimate the age of the universe.

However, Hubble’s law really describes the speed at which celestial bodies move away from each other at the present time and changes because the expansion of the universe if accelerating.

• The accelerating universe and redshift (z)

The evidence for an accelerating expansion comes from observations of the brightness of distant supernovae. We observe the redshift of a supernova which tells us by what the factor the Universe has expanded since the supernova exploded. This factor is (1+z), where z is the redshift. However, in order to determine the expected brightness of the supernova, we need to know its distance now. If the expansion of the Universe is accelerating due to a cosmological constant, then the expansion was slower in the past, and thus the time required to expand by a given factor is longer, and the distance now is larger. But if the expansion is decelerating, it was faster in the past and the distance now is smaller. Thus for an accelerating expansion, the supernovae at high redshifts will appear to be fainter than they would for a decelerating expansion because their current distances are larger.

• The cosmic scale factor (R)

The cosmic scale factor is a function of time which represents the relative expansion of the universe.

This may be represented by

where d(t) is the proper distance at time t, d0 is the distance at time t0, and a(t) is the cosmic scale factor.

Astrophysicists would out the cosmic scale factor using Einstein’s theory of general relativity laws.

D.4 – Stellar processes

• The Jeans criterion

Stars form when a portion of interstellar cloud collapses gravitationally.

The Jeans mass criterion is determined by asking when the magnitude of the gravitational potential energy exceeds the magnitude of the gas’s kinetic energy.

The collapse of an interstellar cloud may begin if M>Mj where Mj is the Jeans criterion.

• Nuclear fusion

Nucleosynthesis is the process of combining light elements into heavier elements, also known as fusion. Nucleosynthesis requires high speed collisions and high temperatures. Temperatures in the core are much higher than on the surface of a star. The main process for energy production in a star is nuclear fusion of hydrogen to helium. This is the process that occurs during most of a star’s lifetime. After the hydrogen in the star’s core is exhausted, the star can burn helium to form progressively heavier elements, carbon and oxygen and so on, until iron and nickel are formed. Up to this point the process releases energy. The formation of elements heavier than iron and nickel requires the input of energy. Supernova explosions result when the cores of massive stars have exhausted their fuel supplies and burned everything into iron and nickel. The nuclei with mass heavier than nickel are thought to be formed during these explosions.

All stars follow a simple proton-proton cycle in order to maintain equilibrium between gravity and pressure. When the star is expanding, it rises in temperature and therefore rises in pressure. This is required in order to keep a balance between the 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 98% made of hydrogen. There are three basic stages of the proton-proton cycle:

• Two hydrogens fuse to form deuterium plus a positron and a neutrino. Each positron is annihilated to create 2 gamma particles which are in turn absorbed and re-emitted as 200,000 photons of light per gamma particle.
• A deuterium and a hydrogen fuse to create helium and a gamma particle. Another deuterium and a hydrogen fuse to create helium and a gamma particle. Thus far, there have been 4 gamma particles created with 800,000 photons of light. That’s one bright star!
• 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 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 (see later section).

• Nucleosynthesis off the main sequence

Clouds of hydrogen and helium form into main sequence stars, where nuclear fusion takes place, fusing hydrogen to form helium. Further fusion only takes place in heavier stars, otherwise the pull of gravity forces the star to contract and cool to a red dwarf. If further fusion takes place, the star becomes a red giant.

Red giants are formed when the hydrogen in the core of the star has fused into heavier helium and helium fusions occur to create beryllium. Gravity causes the star to contract and heat up. The hydrogen around the core burns more fiercely and causes the outer part of the star to expand and cool down. Small red giants (1.4 solar masses – Chandrasekhar limit) cannot withstand the pull of gravity, so it shrinks, becomes extremely hot, until it finally cools into a white dwarf. Larger red giants fuses until iron is formed, however, further fusion cannot take place without energy input. Therefore, the star contracts and heat up because of the large kinetic energy in the particles and explode as a supernova, spilling its rich elements into space to form future stars and planets

• Type Ia and II supernovae

Type Ia supernova

Type Ia supernovae occur in a binary system — two stars orbiting one another. One of the stars in the system must be a white dwarf star, the dense, carbon remains of a star that was about the size of our Sun. The other can be a giant star or even a smaller white dwarf.

White dwarf stars are one of the densest forms of matter, second only to neutron stars and black holes. Just a teaspoon of matter from a white dwarf would weigh five tons. Because white dwarf stars are so dense, their gravity is particularly intense. The white dwarf will begin to pull material off its companion star, adding that matter to itself.

When the white dwarf reaches 1.4 solar masses, or about 40 percent more massive than our Sun, a nuclear chain reaction occurs, causing the white dwarf to explode. The resulting light is 5 billion times brighter than the Sun.

Because the chain reaction always happens in the same way, and at the same mass, the brightness of these Type Ia supernovae are also always the same. To find the distance to the galaxy that contains the supernova, scientists just have to compare how bright they know the explosion should be with how bright the explosion appears. Using the inverse square law, they can compute the distance to the supernova and thus to the supernova’s home galaxy. Thus, Type Ia supernovae are also known as “standard candles”.

Type II supernova

Most stars that are eight or more times the mass of our sun die as a Type II Supernova. A Type II Supernova is a supernova that is classified as having hydrogen lines in its spectra that are made by the explosion of a very large star. The hydrogen lines come from the hydrogen-rich outer layers of the star as the star explodes.

The top shows the evolution of a Type Ia supernova while the bottom shows the evolution of a Type II supernova.

D.5 – Further cosmology

• The cosmological principle

The idea of a uniform universe is called the cosmological principle. There are two aspects of the cosmological principle:

1. The universe is homogeneous. This means there is no preferred observing position in the universe (everywhere looks the same). However, homogenous does not mean that all regions of space should appear identical or be smoothly filled with particles. It only means that the same types of structures — stars, galaxies, clusters, and superclusters — are seen everywhere.
2. The universe is also isotropic. This means you see no difference in the structure of the universe as you look in different directions. In other words, no observation can be made that will identify an edge or a center. The concept of isotropy is supported by the fact that galaxies do not bunch up in any direction in the sky and by the fact that we observe the same Hubble relation in different directions in the sky. Large telescopes have been used to count faint and distant galaxies in different direction and the numbers are always statistically the same.

• Rotation curves and the mass of galaxies

How do we measure the amount of mass in the universe? We measure gravity, indirectly by measuring motion and applying Newton’s law of gravity.

The orbital period of the Sun around the galaxy gives us a mean mass for the amount of material inside the Sun’s orbit. A detailed plot of the orbital speed of the galaxy as a function of radius reveals the distribution of mass within the galaxy. The simplest type of rotation is wheel rotation shown below.

Rotation following Kepler’s 3rd law is shown above as planet-like or differential rotation. Notice that the orbital speeds falls off as you go to greater radii within the galaxy. This is called a Keplerian rotation curve.

However, plotting the curves of observed data points, the rotation curve of the galaxy stays flat out to large distances, instead of falling off as predicted in the figure above (planet-like rotation). This means that the mass of the galaxy increases with increasing distance from the center (radius).

The surprising thing as there is very little visible matter far beyond the center of the galaxy. The rotation curve of the galaxy indicates a great deal of mass but these masses cannot be observed. In other words, the halo of our galaxy is filled with a mysterious matter of unknown composition and type known as dark matter.

• Dark matter

Roughly 80 percent of the mass of the universe is made up of material that scientists cannot directly observe. Known as dark matter, this bizarre ingredient does not emit light or energy. Most scientists think that dark matter is composed of non-baryonic matter. The lead candidate, WIMPS (weakly interacting massive particles), have ten to a hundred times the mass of a proton, but their weak interactions with “normal” matter make them difficult to detect.

• Fluctuations in the CMB

Although the temperature of the CMB is almost completely uniform at 2.7K, there are very tiny variations, or anisotropies, in the temperature on the order of 10^-5K. The anisotropies appear on the map as cooler blue and warmer red patches. But what do these minute fluctuations mean?

These anisotropies in the temperature map correspond to areas of varying density fluctuations in the early universe. Eventually, gravity would draw the high-density fluctuations into even denser and more pronounced ones. After billions of years, these little ripples in the early universe evolved, through gravitational attraction, into the planets, stars, galaxies, and clusters of galaxies that we see today.

• The cosmological origin of redshift

During the first 380000 years after the Big Bang, the universe was so hot that all matter existed as plasma. During this time, photons could not travel undisturbed through the plasma because they interacted constantly with the charged electrons and baryons, in a phenomenon known as Thompson Scattering. As a result, the universe was opaque.

As the universe expanded and cooled, electrons began to bind to nuclei, forming atoms. The introduction of neutral matter allowed light to pass freely without scattering. This separation of light and matter is known as decoupling. The light first radiated from this process is what we now see as the Cosmic Microwave Background.

The CMB is a perfect example of redshift. Originally, CMB photons had much shorter wavelengths with high associated energy, corresponding to a temperature of about 3000K. As the universe expanded, the light was stretched into longer and less energetic wavelengths. By the time the light reaches us, 14 billion years later, we observe it as low-energy microwaves at a frigid 2.7K. This is why CMB is so cold now.

• Critical density

The more mass there is, the more gravity there is to slow down the expansion. Is there enough gravity to halt the expansion and recollapse the universe or not? If there is enough matter (gravity) to recollapse the universe, the universe is “closed’”. A closed universe would be shaped like a four-dimensional sphere (finite, but unbounded). Space curves back on itself and time has a beginning and an end. If there is not enough matter, the universe will keep expanding forever. Such a universe is “open”‘. An open universe would be shaped like a four-dimensional saddle (infinite and unbounded). Space curves away from itself and time has no end.

Instead of trying to add up all of the mass in the universe, a more reasonable thing to do is to find the density of a representative region of the universe where density=(mass in the region)/(volume of the region). If the region is truly representative, then the total mass of the universe=(the density)*(the total volume of the universe). If the density is great enough, then the universe is closed. If the density is low enough, then the universe is open. In the popular astronomy magazines, you will probably see the mass density of the universe specified by the symbol “W”, also known as the ratio of the current density to the “critical density” described in the next paragraph. If W < 1, the universe is open; if W > 1, the universe is closed.

Critical Density

The boundary density between the case where the universe has enough mass/volume to close universe and too little mass/volume to stop the expansion is called the critical density, which may be represented as

where H is the Hubble constant for a given cosmological time (such as the present). The current critical density is approximately 1.06*10^-29g/cm^3. This amounts to six hydrogen atoms per cubic meter on average overall.

A critical density universe has “flat” curvature. The W density parameter equals to exactly 1 in a flat universe. It is to be noted that the Hubble “constant’” is not really a constant – it is different at different cosmological times. The greater the value of the Hubble constant at a given cosmological time, the faster the universe is expanding at that time.

• Dark energy

Although dark matter makes up most of the matter of the universe, it only makes up about a quarter of the composition of the universe. The universe is dominated by dark energy.

After the Big Bang, the universe began expanding outwards. Scientists once thought that it would eventually run out of the energy, slowing down as gravity pulled the objects inside it together (Big Crunch). But studies of distant supernovae revealed that the universe today is expanding faster than it was in the past, not slower, indicating that the expansion is accelerating (Big Rip). This would only be possible if the universe contained enough energy to overcome gravity – dark energy.

To understand the Big Crunch and the Big Rip as possible hypotheses of the end of the universe, see the link below.