宇宙学

Astronomy and Cosmology

The following notes are from the course Greatest Unsolved Mysteries of the Universe on edx

The Big Bang

Spectra and the Doppler Effect

A spectrum is a graph of flux per unit wavelength plotted against wavelength. It will often show emission or absorption lines due to particular elements.

If an object is moving towards or away from you, these spectral lines will be moved in wavelength away from their normal wavelength λo. If you observe a line at wavelength λ, you can define a redshift z as:

$$ z = \frac{\lambda-\lambda_0}{\lambda_0} $$

If this shift is due to the doppler effect, and the velocity $v\ll c$ (velocity much less than the speed of light - nearly always true), then:

$$ z = \frac{v}{c} $$

Hubble Law

Assuming that the brightest star in every galaxy had about the same luminosity (not a good assumption), Edwin Hubble calculated their relative distances. He found that the distances correlated with redshift. Everything was moving away from us and the speed correlated with how far things were from us.

The standard explanation is that space itself is expanding. Objects are not moving - they are just being carried apart by the expansion of space.

This means that unless more matter is created, the density of the universe must continuously go down (same amount of matter spread over more space). The alternative is that more matter is appearing out of nowhere - this is called the “steady state theory”.

The “Steady State Theory” predicts that the universe should always look the same. We actually observe, however, that the universe was different in the past (we can see the past by looking a distant objects). Quasars were more common and the microwave background emission comes from a time when space was opaque.

Peculiar Motions

If galaxies did not have any mass, and space was expanding fast, then every galaxy would move away from us, and their redshift would be proportional to their distance. But in reality, galaxies have mass, and their gravity pulls them towards each other. These motions due to gravity attracting neighbouring galaxies are called peculiar motions (peculiar because they do not fit in with the Hubble Law). The observed motion of galaxies is the combination of the Hubble Law (due to the expansion of the universe) and the peculiar motions.

When galaxies are close together, their mutual gravity is strong, and the amount of space to expand between them is weak. In this case, gravity can overcome the expansion and suck the galaxies together. Our own Milky Way galaxy and the nearby Andromeda Galaxy (M31) are being sucked together by their mutual gravity and will ultimately collide. But when galaxies are further apart, their gravitational attraction is weaker and there is more space between them to expand: in this case the expansion will win.

Scale Factor

Redshit $z$ is given by: $$ z = (1-a(t))/a(t) $$

or alternatively: $$ a(t) = 1/(1+z) $$

Density and energy evolution

The present-day mean energy of photons (the microwave background) is around $10^{-3}eV$ (an eV is an electron volt = $1.6×10^{−19}J$, defined as the energy an electroc acquires when accelerated across a one volt potential difference).

At some point in the past, the energy would be:

Microwave Background

At redshifts before around 1000, a fraction of photons had enough energy (13.6 eV) to knock the electrons off hydrogen atoms. Ionised hydrogen (hydrogen with its electrons knocked off) scatters light very strongly - so it would look like a glowing fog. Back then photons would be trapped bouncing between the atoms. Once the universe cooled down enough for the electrons to “recombine” with their nuclei, the universe became transparent and the photons have been flying freely in all directions ever since. This sea of photons is the observed microwave background.

Primordial nucleosynthesis

Neutrons have a half-life of 887 seconds when not locked securely away inside a nucleus. So if there had not been nuclear fusion in the first few minutes of the universe, the neutrons would have gone away.

Nuclear fusion didn’t start earlier because most reactions start with deuterium, and in the first 90 seconds, photons have enough energy to blow deuterium apart (the “Deuterium Bottleneck”).

The primordial nuclear reactions produced a universe of 25% Helium and 75% Hydrogen, plus trace amounts of Lithium and Deuterium.

Structure FOrmation and antimatter

The microwave background is extremely uniform - fluctuations of only one part in 100,000. Gravitational instability turned that very smooth universe into the lumpy one we see today. These tiny fluctuations probably came from quantum fluctuations when the universe was VERY young. These were somehow amplified up to the sizes of galaxy super-clusters. We don’t know why the fluctations are the size we observe.

We also don’t know why there is such a large imbalance between matter and antimatter in our universe.

Age of Universe

If you assume that space is expanding at a uniform pace, you can estimate the age of the universe.

One way is to look at an object at distance D, with redshift z. Its motion away from us is cz (speed of light times the redshift). So if you extrapolate backwards, it must have been right here on Earth a time 𝑡=𝐷𝑐𝑧 ago.

An alternative way to think about it is that when light set out from an object at redshift z, the universe had a scale factor $𝑎(𝑡)=\frac{1}/{1+𝑧}$ . The time t is just $\frac{D}{c}$. So if (for example) the scale factor 1 billion years ago was 0.9, then if the expansion was steady, the scale factor would have been 0 at a point in time ten billion years ago.

Some objects were discovered that seemed, according to this theory, to be older than the universe. A universe that started off expanding fast then decelerated would make this worse.

红移为0说明两个互相观测的天体在宇宙中的位置几乎是一样的,因此,红移越大,说明两个天体距离越远。因为 距离越远,膨胀速率越快。

Dark Energy

这一部分主要是讲了通过观测,发现宇宙的膨胀速率越来越慢,然而已知的可观测宇宙物质是不会引起宇宙膨胀减速的,所以提出了暗能量的假设,这个能量促使了宇宙膨胀减速。

Background

In the 1990s, it was generally believed that the expansion rate of the universe was slowing down. The big question was how rapidly it was slowing down - rapidly enough to come to a halt and then start shrinking?

Type 1a supernovae

his type of supernovae are probably caused when a white dwarf star becomes too massive and causes a run-away thermonuclear explosion.

It turns out that the peak luminosity correlates with how long they stay bright for. So if you measure the duration of staying bright, you can calculate the luminosity. If you measure the flux, you can use the inverse square law to work out the distance to the supernova, much more accurately than by previous methods.

这种超新星的亮度表明不太可能是核聚变。

Acceleration

By finding supernovae at a range of distances and measuring their redshifts, you can plot the expansion rate of the universe against time. This showed that the universe started off expanding slowly, and has been getting faster and faster.

Some form of long-range repulsive force must be responsible. One possibility is Dark Energy - a form of vacuum energy which has negative pressure.

Quasars

What is a quasar?

It stands for ‘Quasi Stellar Radio Source" - it means a something out in space which is emitting powerful radio waves, but which if you look at it with an optical telescope, looks like a star (i.e. a dot).

When you take a spectrum of one, you see broad emission lines, indicating gas that is swirling around at 10000 km/s, and often very high redshifts, indicating huge distances.

Because of the huge distances, the inferred luminosities of quasars are collossal. For things this far away to have the fluxes we observe, they must have luminosities of around 1040 W.

To calculate luminosities for things this far away, you have to allow for the redshifting of the photons as they travel. The inverse-square law relation between flux and luminosity is modified to become:

$$ f = \frac{L}{4\pi D^2(1+z)^2} $$

Note that even this isn’t quite correct - you have to worry about what exactly is meant by distance D in an expanding and curved universe. But it will do for purposes of this course.

A quasar 40 Astronomical units away would vaporise the Earth’s oceans in a fraction of a second.

To vaporise water (or anything else), you first need to bring it up to its boiling point. Energy needed to raise the temperature of something by $\delta T$K is $mc\delta T$ , where m is its mass and c its specific heat capacity ( 4200J kg−1K−1 for water). You then need to convert it from liquid to gas by supplying the latent heat of vaporisation ( 2260kJ kg−1 for water).

Quasars can vary on timescales of hours to days. In general, if an object varies on a timescale t, it cannot be larger than ct (where c is the speed of light). So quasars are the size of a solar system or less. They also squirt out jets than can be millions of light-years long, which means they must be capable of staying active for a millions of years.

类星体能量非常巨大,并且离我们非常遥远(红移系数很大),但是我们能观测到它的亮度能在数天之内变化,说明他的半径不是很大。

Energy source

Two possible energy sources were considered - nuclear fusion and gravity.

To get the most possible energy from nuclear fusion you would convert 56 Hydrogen atoms into 1 Iron atom. Each hydrogen atom has a mass of 1.00794 u (atomic mass units) while an iron atom has a mass of 55.845 u ( u=1.6605×10−27kg ). You will see that the 56 hydrogen nucei weigh about 1% more than the Iron nucleus. This 1% is converted into energy using the equation E=mc2 . About a solar mass of Hydrogen would need to be converted into Iron per day - which would rapidly produce a black hole.

如果能量来源于核聚变,根据它的光强测算,它会在很短时间内生成一个巨大的铁核而变成黑洞。

To get the maximum possible energy from gravity, you would drop something from infinity down to the event horizon (the radius of no return at which you need to travel at the speed of light to escape). The event horizon radius for a non-rotating black hole is called the Schwarschild Radius and is

$$ r_s = \frac{2GM}{c^2} $$

Gravitational potential energy is given by the equation

$$ E = \frac{-2GM}{c^2} $$

Combine these and you find that the kinetic energy of a particle dropped from infinity as it reaches the event horizon is 12mc2 , i.e. 50% of the rest mass can be converted to energy. This is much better than nuclear fusion can produce.

Most likely, infalling gas forms a disk, and friction within this disk causes it to glow, liberating energy.

另外一种能量来源就是来源于无限远的天体的重力势能。

The problem

You can orbit a black hole just like anything else. So how can you make black holes so massive when the universe is so young, when most of the matter you need to feed it with is in perfectly safe orbits a long way away?

First Light in the Universe

Formation of the elements

The Big Bang produced 75% Hydrogen and 25% Helium. Other elements are made in stars. There is a complicated network of nuclear reactions, with a bottle-neck producing carbon from helium (the triple-alpha reaction which required very high densities to proceed).

n-body codes

Quasar absorption

Hydrogen gas will absorb photons with an energy of 10.2 eV (enough energy to excite an electron from its ground state to the first excited energy level). Because of the expansion of space, photons emitted at higher energies (shorter wavelengths) are also absorbed, if there is neutral hydrogen around by the time they have been redshifted down to 10.2 eV.

This should produce a sudden cut-off in the spectra of all distant objects. Instead, a series of narrow absorption lines are seen (the Lyman-alpha forest) because most of the hydrogen has been ionised - only a few clouds are left neutral. This ionisation must have been done by hard ultra-violet photons, presumably from the first stars.

When did reionIsation happen?

There are two conflicting clues. The most distant quasars known, those at redshifts above 6, show the absorption-cut-off expected if the universe still had neutral hydrogen everywhere.

On the other hand, the haze in the microwave background suggests that reionisation took place earlier - before redshift 10.

A new generation of telescopes is looking for the 21cm radio emission from this neutral gas.

Before reionisation, neutral hydrogen is mixed in with the microwave background radiation. Normally it will emit and absorb equally, having no net effect on the spectrum. But when the first ultraviolet from the first stars emerges, it causes the absorption to increase, creating a brief dip in the spectrum. Later on, once all the hydrogen is ionised, the spectrum should be back to normal. But there should be this dip corresponding to first light in the universe. The EDGES telescope claims to have seen this dip, but it’s twice as large as predicted, and some are sceptical.

What caused reionisation?

There don’t seem to be enough galaxies or quasars at this redshift to do it. But maybe smaller galaxies or individual stars did it. The James Webb Space Telescope may be able to see these first stars.

An alternative approach is to look for low mass stars from this first generation that might be around. They would show a total lack of heavy element lines in their spectra. None have been seen, though some with very little heavy elements are being found.

Gamma-Ray Bursts

Introduction

Gamma ray bursts are pulses of gamma-rays coming from out in space. They come from all directions, and are divided into two classes - the short-hard ones and the long-soft ones.

flux and fluence

Flux is energy per unit area per unit time. Fluence is just energy per unit area.

TotalPer unit time
Physics TerminologyEnergyPower
What an stronomical object emitsEnergyLuminosity
What we dectec on EarthFluenceFlux

You can describe the radiation put out by some astronomical object either by its energy or its luminosity (which is energy per unit time). For an object which puts out energy at a constant rate, its total energy output is just its luminosity multiplied by how long it lasts. More generally, the energy is the integral with respect to time of the luminosity, and the fluence is the integral with respect to time of the flux.

Luminosity (and flux) are typically used for objects which put out power for long periods, like stars and quasars. Energy (and fluence) are typically used for objects which put out a lot of power but only for a small time, like supernovae and gamma-ray bursts.

The inverse square law works for both.

Posiible sources

To begin with, nobody knew how far away these things were. Three models were investigated:

  • Colliding comets in the Oort cloud. If the kinetic energy of two comets was converted into gamma-rays (unlikely) you could just about produce the required fluences if the comets were large and relatively close in.

  • Neutron star rearrangement. If 1 part in 10000 of a neutron star moved in radius by 1 part in 10000, and all the change in potential energy was converted into gamma rays, then you could get the necessary fluence from anywhere in our galaxy.

  • If a star died, and half the mass crashed down on the centre, the released potential energy would be enough to give the required fluences right across the observable universe.

Number counts

If you distribute identical objects uniformly through space, the closer ones will show larger fluxes or fluences because of the inverse square law from brightness. $f∝1/r^2$ , so the distance out to which you can see these things $r∝f^{−1/2}$ .

The number of these objects within range of your telescope will be proportional to the volume you survey, which is proportional to $r^3$ . Thus the number seen brighter than f is $∝f^{−3/2}$ . The number of Gamma ray bursts rises slower than this, meaning that they cannot be uniformly spread at all distances.

这里我也不是很明白

Counterpart

New satellites gave accurate enough positions so that optical counterparts could be tracked down. This showed that the gamma-ray bursts were coming from cosmological distances, and hence had stupendous luminosities: up to $10^{48}J$.

These luminosities were too great to be explained in any sensible model, unless we believe that they are beaming the flux in our direction. If an object is moving at speed v, then its radiation is focussed into a cone of angle θ along the direction of motion, where

$$ \theta=\sqrt{1-(\frac{v}{c})^2} $$

Because all the radiation is beamed into this small angle, the observed flux or fluence is enhanced by a factor of ~$\frac{4}{\theta^2}$ , where $\theta$ is measured in radians.

Long soft bursts seem to come from Type-2 supernova - when dying massive stars run out of fuel and collapse.

Short-hard bursts are more mysterious but seem to come from merging compact objects. The recent discovery of gravity waves from a short-hard burst GW170817 showed that this one, at least, came from a pair of merging neutron stars.

通过更先进的卫星定位,我们能够发现伽马射线暴的具体位置,我们能够得到射线源的辐射能量,这个能量非常巨大。如果这个能量是均匀的往空间辐射,超出了我们的认知。所以可能是只超某个方向辐射具体大的能量。这一次的能量发射相当于太阳30亿年的能量总和。

Fast Radio Bursts

More recently, a lot of interest has focussed on a newly discovered and different phenomenon - fast radio bursts. These are really fast: milli-seconds or less. On Earth we see a radio pulse that sweeps downwards in frequency, but this is due to the lower frequency signals being delayed as they pass through ionised inter-galactic gas. These bursts are very common, and at least some of them come repeatedly from the same objects. Most likely they come from magnetic neutron stars of some type, though the details are obscure.

Dark Matter

Measuring the mass of a galaxy

One way is by adding up the starlight. Use the inverse square law to work out the luminosity from the observed flux. Multiply this by a mass-to-light ratio (typically around 4 solar masses per solar luminosity) to estimate the mass in stars. The mass-to-light ratio differs for different types of star - a value of around 4 is an average of the sort of stars in our own galaxy (massive stars put out a lot of light per unit mass, while smaller stars have lower masses but their luminosities are lower still).

A second way is to find something orbiting around a galaxy and work out how strong the centripetal force needs to be to hold it in that orbit. Gravity must supply this centripetal force so you can work out how much mass needs to be present.

Centripetal force is given by

$$ f = \frac{mv^2}{r} $$

Set thie equal to the gravity…

$$ \frac{GMm}{r^2} = \frac{mv^2}{r} $$

测量一个星系的质量有两种方法,第一个是通过已知的每单位质量的亮度的统计数据,来估计其他星系的质量,当然我们需要预先假设星系的组成,因为不同的行星每单位质量的光强不同。另一种是观察向心加速度来确定一个星系的质量。这里涉及到一个旋转速率分布曲线。内部的可以用光学望远镜测量红移,外部的气体可以用射电望远镜测量多普勒红移来测量速度。

Thus if you know how far out the orbiting thing is, and how fast it is moving, you can work out the mass.

Note that this technique tells you (approximately) only the mass at radii less than r. By combining measurements at different radii you can estimate how the mass is distributed.

The gravitational technique finds much higher masses than counting starlight - so there must be something extra - dark matter. The discrepancy is greatest on larger scales - most of the dark matter is on the outskirts of galaxies.

Lucida avatar
Lucida
劳力人士