Chapter 14 Astronomy & Cosmology Flashcards
Luminosity L is defined as
The total power output of radiation emitted by a star
- It is measured in units of Watts (W)
Radiant flux intensity F is defined as
The observed amount of intensity, or the radiant power transmitted normally through a surface per unit of area, of radiation measured on Earth
The best way to picture this is:
- The luminosity is the total radiation that leaves the star
- The radiant flux intensity is the amount of radiation measured on Earth
- By the time the radiation reaches the Earth, it will have spread out a great deal, therefore, it will only be a fraction of the value of the luminosity
Light sources which are further away appear …
- fainter because the light it emits is spread out over a greater area
- The moment the light leaves the surface of the star, it begins to spread out uniformly through a spherical shell
- The surface area of a sphere is equal to 4πr2
- The radius r of this sphere is equal to the distance d between the star and the Earth
- By the time the radiation reaches the Earth, it has been spread over an area of 4πd2
- The inverse square law of flux can therefore be calculated using:
- Where:
- F = radiant flux intensity, or observed intensity on Earth (W m-2)
- L = luminosity of the source (W)
- d = distance between the star and the Earth (m)
- This equation assumes:
- The power from the star radiates uniformly through space
- No radiation is absorbed between the star and the Earth
The inverse square law of flux equation tells us
- For a given star, the luminosity is constant
- The radiant flux follows an inverse square law
- The greater the radiant flux (larger F) measured, the closer the star is to the Earth (smaller d)
- A standard candle is defined as:
An astronomical object which has a known luminosity due to a characteristic quality possessed by that class of object
- Examples of standard candles are:
-
Cepheid variable stars
- A type of pulsating star which increases and decreases in brightness over a set time period
- This variation has a well defined relationship to the luminosity
-
Type 1a supernovae
- A supernova explosion involving a white dwarf
- The luminosity at the time of the explosion is always the same
Using Standard Candles as a Distance Indicator
- A direct distance measurement is only possible if the object is close enough to the Earth
- For more distant objects, indirect methods must be used – this is where standard candles come in useful
- If the luminosity of a source is known, then the distance can be estimated based on how bright it appears from Earth
- Astronomers measure the radiant flux intensity, of the electromagnetic radiation arriving at the Earth
- Since the luminosity is known (as the object is a standard candle), the distance can be calculated using the inverse square law of flux
- Each standard candle method can measure distances within a certain range
- Collating the data and measurements from each method allows astronomers to build up a larger picture of the scale of the universe
Hubbles Law
Wien’s displacement law relates the
- observed wavelength of light from a star to its surface temperature, it states:
The black body radiation curve for different temperatures peaks at a wavelength which is inversely proportional to the temperature
- This relation can be written as:
- λmax is the maximum wavelength emitted by the star at the peak intensity
A black-body is an object which:
- Absorbs all the radiation that falls on it, and is also a good emitter
- Does not reflect or transmit any radiation
- A black-body is a theoretical object, however, stars are the best approximation there is
- The radiation emitted from a black-body has a characteristic spectrum that is determined by the temperature alone
The intensity-wavelength graph shows how thermodynamic temperature links to the peak wavelength for four different stars
- The full equation for Wien’s Law is given by
λmaxT = 2.9 × 10-3 m K
- Where:
- λmax = peak wavelength of the star (m)
- T = thermodynamic temperature at the surface of the star (K)
Wien’s Law tells us the higher the temperature of a body
λmaxT = 2.9 × 10-3 m K
- The shorter the wavelength at the peak intensity, so hotter stars tend to be white or blue and cooler stars tend to be red or yellow
- The greater the intensity of the radiation at each wavelength
Table to compare surface temperature and star colour
A star’s luminosity depends on two factors:
- Its surface temperature
- Its radius
- The relationship between these is known as the Stefan-Boltzmann Law, which states:
The total energy emitted by a black body per unit area per second is proportional to the fourth power of the absolute temperature of the body
Stefan-Boltzmann Law Equation
L = 4πr2σT4
- Where:
- L = luminosity of the star (W)
- r = radius of the star (m)
- σ = the Stefan-Boltzmann constant
- T = surface temperature of the star (K)
Estimating the Radius of Stars
- The radius of a star can be estimated by combining Wien’s displacement law and the Stefan–Boltzmann law
Estimating the Radius of Stars: The procedure for this is as follows:
- Using Wien’s displacement law to find the surface temperature of the star
- Using the inverse square law of flux equation to find the luminosity of the star (if given the radiant flux and stellar distance)
- Then, using the Stefan-Boltzmann law, the stellar radius can be obtained
Summary of Equations
-Stefan-Boltzmann law= L = 4πr2σT4
Emission Spectra
- Astronomers are very limited in how they can investigate objects in the space
- All of the techniques used involve analysing the light emitted from the star, or galaxy
- One of these techniques involves analysing the emission and absorption spectra of stars
- More details on this can be found in the revision notes “Line Spectra” in the Quantisation of Energy topic
- Elements in the star, predominantly hydrogen and helium, absorb some of the emitted wavelengths
- The top emission spectra shows spectral lines of hydrogen measured on Earth*
- The bottom emission spectra shows the shifted spectral lines of hydrogen measured from a distant galaxy*
When astronomers observe light from distant galaxies, they observe differences in the
- spectral lines to the light from the Sun
- The lines have the same characteristic pattern, meaning the element can still be easily identified, they just appear to be shifted sightly
- The lines in the spectra from distant galaxies show an increase in wavelength
- The lines are moved, or shifted, towards the red end of the spectrum
the Doppler effect is defined as:
The apparent change in wavelength or frequency of the radiation from a source due to its relative motion away from or toward the observer
- On Earth, the Doppler effect of sound can be easily observed when sound waves moves past an observer at a notable speed
In space, the Doppler effect of light can observed when spectra of distant stars and galaxies are observed, this is known as:
- Redshift if the object is moving away from the Earth, or
- Blueshift if the object is moving towards the Earth
- Redshift is defined as:
The fractional increase in wavelength (or decrease in frequency) due to the source and observer receding from each other
- For non-relativistic galaxies, Doppler redshift can be calculated using:
- Where:
- Δλ = shift in wavelength (m)
- λ = wavelength emitted from the source (m)
- Δf = shift in frequency (Hz)
- f = frequency emitted from the source (Hz)
- v = speed of recession (m s-1)
- c = speed of light in a vacuum (m s-1)
An Expanding Universe
- After the discovery of Doppler redshift, astronomers began to realise that almost all the galaxies in the universe are receding
- This lead to the idea that the space between the Earth and the galaxies must be expanding
- This expansion stretches out the light waves as they travel through space, shifting them towards the red end of the spectrum
- The more red-shifted the light from a galaxy is, the faster the galaxy is moving away from Earth
The expansion of the universe can be compared to dots on an inflating balloon
- As the balloon is inflated, the dots all move away from each other
- In the same way as the rubber stretches when the balloon is inflated, space itself is stretching out between galaxies
- Just like the dots, the galaxies move away from each other, however, they themselves do not move
Edwin Hubble investigated the
- light spectra emitted from a large number of galaxies
- He used redshift data to determine the recession velocities of these galaxies, and standard candles to determine the distances
- From these measurements, he formulated a relationship, now known as Hubble’s Law
- Hubble’s Law states:
The recession speed of galaxies moving away from Earth is proportional to their distance from the Earth
Hubble’s Law can be calculated using:
v = H0d
- Where:
- v = the galaxy’s recessional velocity (m s-1)
- d = distance between the galaxy and Earth (m)
- H0 = Hubble’s constant, or the rate of expansion of the universe (s-1)
- This equation tells us:
- The further away a galaxy, the faster it’s recession velocity
- The gradient of a graph of recession velocity against distance is equal to the Hubble constant
A key aspect of Hubble’s law is that the furthest galaxies appear to move away the fastest
Age of the Universe
- If the galaxies are moving away from each other, then they must’ve started from the same point at some time in the past
- If this is true, the universe likely began in an extremely hot, dense singular point which subsequently began to expand very quickly
- This idea is known as the Big Bang theory
- Redshift of galaxies and the expansion of the universe are now some of the most prominent pieces of evidence to suggest this theory is true
- The data from Hubble’s law can be extrapolated back to the point that the universe started expanding ie. the beginning of the universe
the age of the universe T0 is equal to:
- Current estimates of the age of the universe range from 13 – 14 billion years
- There is still some discussion about the exact age of the universe, therefore, obtaining accurate measurements for the Hubble constant is a top priority for cosmologists
