Cosmology (3.9.3)
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The Doppler effect refers to the change in frequency or wavelength of waves due to the motion of the source relative to an observer.
The Doppler effect is a very useful tool used in astrophysics for measuring the speed and position of stars and galaxies relative to Earth.
Redshift: When a light source moves away from the observer, the observed wavelength increases and the observed frequency decreases, shifting towards the red end of the spectrum.
Blueshift: When a light source moves towards the observer, the observed wavelength decreases and the observed frequency increases, shifting towards the blue end of the spectrum.
The extent to which the wavelength of light emitted from a distant star is shifted is given by the Doppler equation:
Where:
- is the source wavelength ,
- is the change in wavelength i.e. ,
- is the relative velocity of the source and the observer , and
- is the speed of light, .
By convention, the relative velocity, is taken to be positive when the source is moving towards the observer and negative when it is moving away from the observer.
The Doppler equation may also be expressed in terms of the frequencies of the emitted and observed light:
In this equation:
- is the source frequency (Hz)
- is the change in frequency, i.e. (Hz).
The two Doppler equations only apply when due to the derivation ignoring the effects of special relativity.
The quantity in the Doppler equation, is known as the redshift, can be represented using the symbol, Therefore, the Doppler equation may be written as:
Where the redshift may be represented more simply as . For sources moving away from the observer, is taken to be negative, so is positive.
Additionally, the Doppler equation in terms of frequency may be written in terms of :
It is useful to note that the Doppler equation can be derived by considering a wave source moving towards an observer.
- A source emits light of wavelength and time period , moving towards the observer at speed , where .
- In one period, the light travels a distance , while the source moves a distance towards the observer.
- The observer therefore measures a wavelength , so the change in wavelength is:
Since the time period, is equal to and then:
Therefore, the Doppler equation may be derived as follows:
Question walkthrough
Finding galaxy velocity from redshift
Uses Δλ/λ to find the redshift of a galaxy’s emission line from its emitted and observed wavelengths, then relates this to the galaxy’s recession velocity via the Doppler redshift approximation.
The spectra from distant galaxies are redshifted, demonstrating that they are all moving away from one another.
In current cosmological theory, galaxies are not physically moving apart. Instead, the space between galaxies is expanding, thereby stretching the light waves propagating through it. This is known as the cosmological redshift, named to distinguish it from the redshift of objects that are actually moving through space.
The red shift equations can be used to calculate both red shift and cosmological redshift as long as the velocity is much less than the speed of light . If the velocity is close to the speed of light, relativistic effects come into play, and the equations no longer work.
Astronomer Edwin Hubble determined that the more distant galaxies are moving away from us the faster. This suggests that the universe began as very hot and dense point, and is expanding at an increasing rate: this is known as the Big Bang theory.
Many stars in the universe are binary star systems, i.e. two stars orbiting one another. Most of these systems are too far away for telescopes to resolve. However, two stars can be distinguished by analysing their line spectra.
Observing the absorption lines in the system’s line spectra over time shows a shift in their positions. Therefore, the light coming from the system must be simultaneously red and blue shifted, providing evidence for two stars present in the system.
- When the two stars move sideways with respect to the observer’s line of sight, the light does not change, and the spectra show only one line.
- When the stars move towards or away from the observer, their light is blue-shifted and red-shifted, respectively. Therefore, two lines are formed in the spectra representing the blue-shifted and red-shifted light from each star.
In one half-period of orbit, the spectral lines go from a maximum to zero and back to a maximum again. This allows for the system’s orbital period to be calculated.
Question walkthrough
Finding radial velocity from redshift
Uses z=Δλ/λ to find the redshift of a binary star’s hydrogen spectral line, then z=v/c to find its maximum radial velocity along the line of sight.
The universe looks the same no matter the direction of observation: this is known as the cosmological principle.
The cosmological principle states that, on a large scale, the universe is both homogeneous and isotropic.
- Homogeneous: A universe where the matter is distributed uniformly such that the average density remains consistent.
- Isotropic: A universe that appears the same in all directions, no matter the point of observation.
Before the 1930s, it was believed that the universe was infinite in space and time and static: neither expanding nor contracting. This model seemed to be the only way the universe could be stable according to Newton’s law of gravitation.
In the late 1920s, Edwin Hubble examined the Doppler shift in the absorption spectra of various distant galaxies. His results confirmed that most galaxies exhibit redshift, indicating they are moving away from Earth.
Hubble also observed a correlation between distance and redshift. Specifically, more distant galaxies tend to have higher redshifts, suggesting they are moving away faster.
Hubble’s law states that the recessional velocity of a galaxy is directly proportional to its distance from Earth:
Where:
- is the recessional velocity of the galaxy in ,
- is Hubble’s constant in , and
- is the distance to the galaxy in megaparsecs,
The Hubble constant describes the rate of expansion of the Universe. It is approximately equal to:
The Hubble constant quantifies how quickly galaxies are receding from us, based on their distance. It is the rate of expansion per unit distance. For every megaparsec of distance from Earth, a galaxy’s recessional speed increases by kilometres per second.
The value of the Hubble constant is debated due to the difficulty of measuring distances to galaxies. This gives a range of values for typically in the range of:
Question walkthrough
Finding distance using Hubble’s law
Converts a galaxy’s recession velocity from metres per hour to km/s, then rearranges Hubble’s law d=v/H₀ to find its distance in Mpc.
Question walkthrough
Finding the age of the universe
Converts the Hubble constant from km/s/Mpc to SI units (s⁻¹), then uses t=1/H₀ to estimate the age of the universe, converting the result from seconds to years.
The model of the expanding Universe supports the Big Bang theory, indicating that the Universe began from a singular point approximately 13.8 billion years ago and has been expanding ever since.
The age of the Universe can be estimated using the inverse of the Hubble constant . The Hubble constant describes the rate of the Universe’s expansion, and the inverse of the equation gives us a rough estimate of the Universe’s age:
However, due to the uncertainty in the exact value of the Hubble constant, there is uncertainty in the exact age of the universe.
Based on modern measurements of the Hubble constant, the estimated age of the Universe is approximately 13.8 billion years. This is a crucial number in cosmology, reflecting the time that has elapsed since the Big Bang.
All matter in the universe is attracted to one another via gravity. This would suggest, therefore, that the rate of expansion of the universe would decrease.
However, cosmologists discovered that not only is the universe expanding, but it is also accelerating. The reason for this accelerating expansion is still unknown, but cosmologists state it is caused by a mysterious energy known as dark energy. They are trying to develop new theories and mathematical models to explain the accelerated expansion of the universe.
It is important to note that dark energy has yet to be directly detected; hence, the name dark, and it fills the entirety of space.
As the universe expands, it cools because it is a closed system. Therefore, if time is reversed, the universe gets smaller and hotter. Eventually, one arrives at the origin of the universe: Big Bang theory.
The Big Bang theory predicts that a large amount of electromagnetic radiation was produced at the very early stages of the universe’s life. As a result, this radiation should still be present and observable to this day, since it cannot escape. This radiation is known as the cosmic microwave background (CMB) and was detected by Penzias and Wilson in the 1960s.
In the 1980s, a satellite known as the Cosmic Background Explorer (COBE) was launched to investigate the properties of the CMB. It found that:
- the CMB has a perfect blackbody spectrum that corresponds to a temperature of 2.73 K.
- the CMB is largely homogenous and isotropic, in agreement with the cosmological principle.
Cosmic microwave background radiation (CMBR) serves as another critical piece of evidence for the Big Bang theory. CMBR is the afterglow of the early Universe; high-energy gamma photons from the early Universe have stretched into microwaves over time due to the expansion of space.
The picture of the cosmic microwave background above is a snapshot of the oldest light in our Universe, when the Universe was only years old. It shows small temperature fluctuations, and warmer areas are represented in red. These correspond to regions of slightly higher density, which are the areas where star clusters and galaxies eventually formed.
When the Universe was very young and extremely hot, space was filled with high-energy gamma photons. As the Universe expanded, the fabric of space stretched, which lengthened the wavelength of these photons to what we now observe as microwaves.
The Universe’s expansion over billions of years has cooled the initially hot, dense Universe to about At this temperature, the Universe acts as a black-body radiator, peaking at a wavelength around 1 in the microwave region of the spectrum.
The Big Bang model accounts for both the CMB and the high abundance of hydrogen and helium in the universe today, which together account for roughly 74% and 24% by mass. In the universe’s earliest, extremely hot stages, hydrogen could fuse to form helium outside of stars.
Comparing this early fusion to today’s observed hydrogen-helium ratio lets cosmologists estimate a timeframe for it, explaining the abundances we see. The theory of nucleosynthesis within stars similarly accounts for the abundances of all heavier elements.
Quasars are the highly luminous centres of distant galaxies. They were discovered in 1950 and were at first thought to be ordinary stars. However, they had a spectrum different to ordinary stars, and would occasionally emit jets of material.
They produced a continuous spectrum that was different to the normal blackbody radiation curve. Where ordinary stars produce absorption spectra, these ‘stars’ produce emission spectra of unknown elements.
In 1963, Maarten Schmidt discovered that the lines in emission spectra from quasars were redshifts of the Balmer series of hydrogen.
Quasars are also known to be extremely strong sources of radio waves: their name ‘quasar’ is derived from ‘quasi-stellar radio sources’.
Surrounding the supermassive black hole is an orbiting swirling mass of gas that is slowly falling into it, emitting light in the process. Similar to pulsars, magnetic fields are responsible for the emission of jets of radiation at the poles of the black hole. The measured energy emitted by the black holes suggests they consume the equivalent mass of ten Suns every year.
The size and luminosity of quasars initially caused controversy, as the values initially seemed far too large.
With the development of sensitive charged couple devices (CCDs), it was found that the intense glow observed in some extremely luminous galaxies was produced by quasars, which are approximately the size of the Solar System and have a luminosity larger than the entire Milky Way galaxy
It is now accepted that quasars are the highly luminous and powerful cores of young galaxies, where an active black hole drives their extreme energy output. These types of black holes at the centres of galaxies are known as supermassive black holes and have masses a million times greater than the Sun.
The large amount of red shift in the emission spectrum of quasars is evidence that they are incredibly far away. Quasars are known to be some of the most distant objects observed in the universe. They do not exist in the modern universe.
Measuring the red shift indicates that all quasars are billions of light years away. Because the light from quasars takes billions of years to reach Earth, observations of quasars reveal how they appeared in the early universe – they are viewed from the past.
The inverse square law for intensity gives an estimate of the brightness of quasars:
Where:
- is the power output ,
- is the intensity , and
- is the distance .
Question walkthrough
Finding quasar power output
Uses the inverse-square law P∝Id² to compare a quasar and a star of equal apparent intensity but different known distances, finding the quasar’s power output.
An exoplanet (or extrasolar planet) is defined as any planet that does not belong to the Solar System.
Exoplanets are difficult to discover due to:
- The star they are orbiting is much brighter than the exoplanet, diminishing any light coming from the exoplanet.
- The exoplanet is too small to distinguish from neighbouring stars. Most telescopes do not have enough power to resolve it.
Some powerful telescopes can directly resolve exoplanets. However, these planets are usually very large, very hot and far away from their orbiting star.
Exoplanets can be found using the Doppler shift (also known as the radial velocity method).
- Exoplanets orbiting a star cause small fluctuations (wobbles) in the star’s orbit due to both orbiting the same centre of mass that is not in the Star’s geometric centre.
- The small fluctuations in the star’s orbit cause subtle red and blue shifts in the light emitted from the star. These red and blue shifts can be detected on Earth and give evidence for the presence of an exoplanet.
- From the red and blue shifts, the minimum mass of the exoplanet can be calculated.
However, in order for the red and blue-shifted light from the star to be detected, the movement of the star needs to be aligned with the telescope’s line of sight. If the exoplanet is orbiting the star perpendicular to the line of sight, then no light will be shifted and, therefore, no exoplanet will be detected.
Exoplanets can also be found using the transit method.
- The transit method involves an exoplanet moving in front of a star and blocking some of the light observed on Earth.
- This results in a dip in the measured light over time on Earth. The time over which the dip occurs tells us about the orbital period of the exoplanet.
- The depth of the brightness dip allows the exoplanet’s radius to be calculated.
However, for the transit method to work, the path of the exoplanet must perfectly line up to cross the line of sight between the star and the observer. The chances of this occurring are very low.