Electromagnetic radiation from stars (5.5.2)
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A continuous spectrum is a type of light spectrum where you can observe all possible frequencies of light, spread smoothly over a wide range. It’s like seeing the full range of colours in a rainbow without any gaps.
For example: If you were to look at the spectrum of light produced by a white-hot filament, you would see a continuous blend of colours from red to violet without any missing sections.
Even though the Sun’s light appears white, its spectrum is not continuous.
When we examine it closely, we see dark lines in the Sun’s absorption spectrum called absorption lines where some frequencies are missing. These gaps are caused by elements in the Sun’s outer layers absorbing certain specific wavelengths of light.
Bohr’s atomic model:
- Electrons orbit the nucleus: Similar to how planets orbit the Sun, electrons circle around the nucleus, but they can only exist in specific orbits.
- Electrons have specific, quantized energy levels: Electrons cannot just orbit anywhere. They are confined to certain paths or energy levels that correspond to particular energies. These paths are called electron shells.
- Energy transitions: Electrons can move between these orbits, but to do so, they must either absorb energy to move to a higher shell or release energy to transition to a lower shell.
Picture an electron moving up or down steps in a building. Each step represents a specific, discrete energy level – the electron cannot stop in between the steps, only on one or the other. This means their energy is quantized, meaning they are limited to specific values.
Bohr used the idea of photons (particles of light) to explain the phenomenon of spectral lines. He explained that atoms emit or absorb light at specific, discrete frequencies, producing spectral lines instead of a continuous range of colours.
These lines correspond to specific energies associated with the energy levels electrons may occupy in an atom. This is because photons are either emitted or absorbed when electrons in the atom move between energy levels.
Excitation occurs when an electron gains energy and jumps from a lower energy level (closer to the nucleus) to a higher energy level (farther from the nucleus). This requires the electron to absorb a specific amount of energy.
The energy needed for this can come from various sources.
- Photon absorption: The electron can absorb a photon (a packet of light energy) with exactly the right amount of energy corresponding to the difference between two energy levels.
- Heat energy: Energy from the surroundings can also excite electrons, such as heating a gas.
- Electric field: Applying an electric field can provide energy to excite electrons.
De-excitation occurs when an electron loses energy and falls from a higher energy level to a lower energy level. When this happens, the electron releases the energy it no longer needs. This energy is emitted as electromagnetic radiation (usually visible light or other forms of radiation, depending on the atom).
The frequency of the emitted radiation is directly related to the energy difference between the higher and lower energy levels. This is why atoms emit light at specific frequencies, which we can observe as spectral lines.
Absorption when an electron absorbs energy (such as from a photon), it moves to a higher energy level. This is also called excitation.
Only photons with exactly the right amount of energy – the difference between two energy levels – can be absorbed. If the photon doesn’t match, the electron won’t move. Thus, only specific frequencies of light are absorbed.
Emission occurs when the electron drops back to a lower energy level, it emits a photon with energy exactly equal to the difference between the two levels.
This process is the basis for the emission spectra of elements.
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Energy, as a physical quantity, only has meaning when quoted relative to a defined zero point.
For electrons orbiting a nucleus, the energy is defined to be zero when it is infinitely far from the nucleus. At this point, the electron is said to be free from the atom, and the forces of attraction between the electron and the nucleus are practically zero.
This does not mean that the electron has absolutely zero energy, as it may still be moving. The zero point defines which direction represents positive and negative energy, remembering that the vector nature of energy includes both magnitude and direction.
Energy levels within an atom are given as negative values. This is because external energy must be supplied to transition an electron from one energy level to another, or to move it to a point far from the nucleus where its energy is zero.
The negative energy represents how much energy an electron is ‘missing’ compared to being free. The more negative the value, the more tightly the electron is bound to the nucleus.
The value of a given energy level tells you the amount of energy required to remove the electron from that specific energy level and move it to infinity, where it is free of the atom.
Think of the negative energy as the depth of a well. To pull the electron out of the well, you need to add enough energy to overcome its negative value.
The energy level with the most negative value is the ground state, This is the lowest energy level an electron can occupy in an atom.
It is the most stable position for the electron and requires the most energy to remove the electron from the atom compared to any other energy level.
In a hydrogen atom, the energy of the ground state is This means you would need to supply of energy to completely remove the electron from a hydrogen atom.
This complete removal of an electron from an atom is called ionisation.
The way electron energy levels are defined is similar to gravitational potential energy.
When two masses are far apart (like a planet and a distant asteroid), the gravitational force between them becomes negligible, and we consider the potential energy to be zero at an infinite distance.
Just like with gravitational potential energy, where energy is required to bring a mass from infinity to a closer position, energy is needed to move an electron from its energy level in the atom to a point far from the nucleus.
When gases are heated, they emit light at specific wavelengths, resulting in a series of bright lines on a dark background. This occurs because electrons absorb energy when the gas is heated, becoming excited to a higher energy level.
Electrons cannot remain in this excited state indefinitely; they will eventually return to a lower energy level through a process called de-excitation. During de-excitation, energy must be conserved. The energy lost by the electron is emitted as a photon with a specific frequency.
In summary, this absorption/emission process occurs when an electron absorbs energy and is promoted from a lower energy level to a higher energy level. When it then transitions from a higher energy level back to a lower energy level, a photon is released.
Each time an electron transitions between energy levels, it emits photons with discrete frequencies or wavelengths. Since each atom has multiple possible electron transitions, a variety of wavelengths are emitted.
The emitted photons produce a line spectrum, which is a series of bright lines against a dark background. Each line corresponds to a specific wavelength of light emitted during electron transitions.
An emission-line spectrum serves as a unique fingerprint for each element. Since no two elements have the same energy level structure, the pattern of emission lines is unique to each element, allowing for precise identification.
By analysing these spectral lines, scientists can identify the composition of distant stars and other celestial bodies.
For example, a portion (known as the Balmer series) of the emission spectrum of hydrogen is shown above. The different coloured lines indicated the wavelengths of light that correspond to different energy changes that an electron in a hydrogen atom may experience.
The energy of the emitted photon corresponds to the difference in energy between the two levels and can be calculated as follows:
Where:
- is planck’s constant,
- is the frequncy of the emitted photon,
- is the energy of the emitted photon,
- is the energy of the higher energy level, and
- is the energy of the lower energy level.
It is important to note that in exam questions, energy levels may be quoted in or
Question walkthrough
Photon Energy from Hydrogen Transition
Uses the hydrogen energy level formula to find the energy, in joules, of the photon released when an electron drops from n=3 to n=2.
Electromagnetic radiation can be described using both wave and particle models. The photon model treats electromagnetic radiation as a stream of particles called photons, which are packets of energy. Each photon has a discrete amount of energy, making radiation ‘quantized’. Photons have no mass, but they carry energy and momentum.
The electron volt ( is a unit of energy commonly used when discussing photons and subatomic particles.
is the amount of energy gained or lost by an electron when it moves through a potential difference of one volt.
The energy of a photon is directly related to its frequency:
Where:
- is the photon energy in joules,
- is Planck’s constant, which is equal to and
- is the frequency in hertz ().
This equation may alternatively be expressed as:
Where:
- is the speed of light in a vacuum, equal to approximately , and
- is the wavelength in metres (m). This is because the frequency, wavelength and speed of a photon are related by:
The wavelength of the emitted photon is inversely proportional to the energy of the transition.
- Larger energy transitions result in photons with shorter wavelengths (higher frequency).
- Smaller energy transitions produce longer wavelength photons (lower frequency).
For example, transitions to different energy levels in the hydrogen atom produce photons with different characteristics:
- Transition to (ground state): Photons emitted are in the ultraviolet range (short wavelength, high energy, high frequency).
- Transition to Photons emitted are in the visible light range.
- Violet light corresponds to the highest energy (shorter wavelength).
- Red light corresponds to the lowest energy (longer wavelength).
- Transition to Photons emitted are in the infrared range (long wavelength, lower energy, lower frequency).
In some questions, you may be asked to calculate the frequency or wavelength of a photon. The question may provide you with the energy levels and the corresponding energy difference between them.
The energy of the photon is calculated from the difference between the final and initial energy levels:
Once you have in electron volts (eV), you’ll need to convert it to joules ( by using the conversion:
After converting the energy into joules, you can then use either formulae:
to calculate either the frequency or the wavelength of the emitted photon.
Question walkthrough
Photon Wavelength from Hydrogen Energy Levels
Calculates the wavelength of a photon emitted when a hydrogen electron drops from n=3 to n=2, converting the energy difference from eV to joules and identifying the line as red light.
Photons generated by nuclear fusion in a star’s core move outward over thousands of years, passing through the various layers of plasma and gas in the star as they do.
These photons encompass all frequencies of the electromagnetic spectrum, forming what is called a continuous spectrum.
As the photons travel through the gas layers, they are absorbed by ions or atoms in the gas, exciting the electrons. These excited atoms then re-emit photons, but in random directions and often at different frequencies than which they were absorbed.
Spectroscopy is used to analyse the light emitted by stars. The outer atmospheres of stars are not hot enough to produce an emission line spectrum. Instead, stars emit an absorption line spectrum. An absorption line appears when light from a source passes through a cooler gas and is observed by a detector.
An absorption line spectrum is essentially the opposite of an emission spectrum, dark lines are superimposed on a continuous background. These dark lines correspond to specific wavelengths of light that are absorbed by the gas as photons excite the atoms.
Each element has a unique set of energy levels, meaning the pattern of spectral lines is unique for each gas.
An example of the absorption spectrum for hydrogen is shown above. The different dark lines indicate the wavelengths of light that are absorbed when electrons are excited within a hydrogen atom.
The absorption lines (unique pattern of dark lines) in a star’s spectrum acts like a fingerprint for the elements present in the star. By analysing the absorption spectrum, scientists can determine the chemical composition of a star, even if it is located far away.
If a particular element is present in a star, its characteristic absorption lines will appear in the star’s spectrum. By comparing the emission line spectra of elements confirmed in a laboratory, such as hydrogen and helium with the absorption line spectrum of the Sun, we can verify the Sun’s chemical makeup.
It is important to note that when provided with an absorption spectrum of a star, you can be asked to identify a star with a similar chemical composition. Pay close attention to the spacing and pattern of the dark lines to match the spectrum to the corresponding element.
Continuous spectra
- A continuous emission spectrum is one that contains light across all wavelengths of the electromagnetic spectrum.
- This type of spectrum is produced by hot, dense objects, such as the cores of stars.
- Photons emitted from these sources include all possible wavelengths and frequencies, creating a seamless spectrum without gaps.
Emission line spectra
- An emission line spectrum occurs when electrons transition from higher to lower energy levels, releasing photons.
- Each transition corresponds to a specific wavelength, producing coloured lines on a black background.
- This type of spectrum is characteristic of hot, low-pressure gases.
Absorption spectra
- Absorption spectra arise when an atom absorbs specific wavelengths of light, resulting in missing lines.
- When a continuous spectrum passes through a cool, low-pressure gas, specific wavelengths of light are absorbed, leading to a spectrum with missing wavelengths.
- This spectrum consists of a continuous background with dark lines where certain wavelengths have been absorbed.
The missing wavelengths in an absorption spectrum correspond exactly to the wavelengths emitted in the emission spectrum of the same element. When electrons return to lower energy levels, they emit photons in all directions, which is why some wavelengths appear absent.
The three kinds of spectra you should be familiar with:
- Continuous spectra: Contains all possible wavelengths and frequencies
- Emission line spectra: Discrete coloured lines on a dark background
- Absorption line spectra: Discrete dark lines on a continuous background
The key differences between how these spectra are produced and what they look like are shown in the image below:
Question walkthrough
Formation of Absorption Line Spectra
Explains why passing light from a hot, dense source through a cooler, low-pressure gas produces dark absorption lines in an otherwise continuous spectrum.
Dispersion refers to the process of splitting visible white light into its constituent colours, forming a spectrum. This can be achieved using either a glass prism or a diffraction grating.
A transmission diffraction grating is a tool often used in spectrometers for high-resolution separation of light by wavelength. It consists of a glass or plastic slide with many closely spaced, parallel slits or lines. When light passes through these slits, it diffracts and spreads out, allowing for detailed analysis of the light’s components.
For example, a continuous spectrum from a halogen light may be passed through a sample and then through a diffraction grating to produce an absorption spectrum. This absorption spectrum can then be used to identify the sample’s chemical composition.
Diffraction gratings are particularly useful for analysing light from stars. By separating the light into its individual wavelengths, scientists can determine the composition of stars based on their emission or absorption spectra.
It is important to note that:
- Diffraction is the bending or spreading out of waves as they pass through a gap or move around an obstacle. The extent of diffraction depends on the wavelength of the wave and the size of the gap or obstacle. The effect is most pronounced when the gap size is comparable to the wavelength.
- Interference occurs when two or more waves overlap and combine, resulting in a new wave pattern. There are two types of interference:
- Constructive interference: When the crests of two waves align, their amplitudes add together, resulting in a wave with greater amplitude.
- Destructive interference: When the crest of one wave aligns with the trough of another, their amplitudes subtract, resulting in a reduction in amplitude or cancellation.
- Phase difference is the difference in phase between two points on a wave or between two waves. It measures how ‘in sync’ or ‘out of sync’ two waves are. It is typically measured in degrees or radians, where 360∘ (or 2π radians) corresponds to one full cycle.
A diffraction grating separates light into its component wavelengths through the process of diffraction and interference.
When light hits the grating, it passes through the narrow slits, causing the light waves to spread out, or diffract. The amount of diffraction depends on the wavelength of the light. Longer wavelengths (such as red) diffract more than shorter wavelengths (such as blue).
Once the light has been diffracted, the waves from each slit interact with each other through interference. There are two types of interference:
- Constructive interference occurs when waves from adjacent slits are in phase (their crests and troughs align), resulting in a bright fringe.
- Destructive interference happens when the waves are out of phase, cancelling each other out and resulting in a dark region.
Each wavelength of light will produce constructive interference at a specific angle, which depends on the spacing between the slits and the wavelength of the light.
This leads to angular dispersion, where the different wavelengths of light are spread out into a spectrum at different angles. Longer wavelengths (like red) are diffracted at a greater angle than shorter wavelengths (like blue).
When light passes through the grating, it creates a series of bright lines (spectral orders) at various angles. Each line corresponds to a specific wavelength of light.
Compared to prisms or double-slit setups, diffraction gratings produce sharper fringes and offer higher resolution because the large number of slits increases the precision of constructive and destructive interference, allowing fine details of the spectrum to be observed.
Advantages of diffraction gratings
- Greater angular dispersion: Compared to optical prisms, diffraction gratings provide a much higher angular dispersion, meaning the colours are separated more distinctly.
- Sharper fringes: the fringes (or bands of light) produced by diffraction gratings are sharper than those created by a double-slit experiment, making them ideal for precise spectral analysis.
- Accuracy: Because diffraction gratings rely on the interference of light, they provide more accurate wavelength measurements compared to prisms, which are subject to imperfections in glass and material dispersion.
- Efficiency: Transmission diffraction gratings can transmit more light than a prism, making them more efficient for capturing fainter light sources, such as distant stars.
- Customization: Gratings can be manufactured with a variety of line spacings (grating density), allowing them to be optimised for specific wavelength ranges or applications. This flexibility makes them versatile for different types of spectroscopy.
- Compact Design: Diffraction gratings are often smaller and lighter than prisms, making them easier to integrate into compact and portable instruments, such as modern spectrometers.
Diffraction grating works by diffracting light through many slits, causing the light waves to interfere. Different wavelengths constructively interfere at different angles, producing a separated spectrum of colours.
The angles at which maxima of intensity (constructive interference) occur can be found using the diffraction grating equation:
Where:
- is the spacing between slits, typically in metres or millimetres,
- is the diffraction angle in radians or degrees depending on calculator settings,
- is the order of the maxima (first, second, etc.) and
- is the wavelength of the light, typically in metres or millimetres.
This equation may use and in either metres or millimetres as long as the same unit is used for both, as the units cancel each other out.
In exam questions, you may be given the number of lines per metre (or millimetre, nanometre, etc.) on the grating, called This can be used to calculate the spacing between slits, using the following equation:
This equation converts the number of lines per unit length, into the distance between adjacent slits,
To calculate the angular separation of each maxima, you can rearrange the diffraction grating equation to solve for
In this equation, is the angle measured from the centre (zero order) to the maxima.
Higher-order maxima (larger values of ) will occur at greater angles from the centre.
The angular separation between two maxima is simply the difference between their angles. For example, the separation between the first-order maxima, and second-order maxima is calculated as:
The highest order of maxima is observed when a beam of light is incident at a right angle to the diffraction grating.
This happens when the angle reaches 90°:
In this case, the highest order of maxima is found by rearranging the grating equation:
Remember,
However, must always be an integer. If the calculated value of is not an integer, you must round down to the nearest whole number.
For example, if the highest visible order is
Why must the order of maxima be an integer?
In diffraction, constructive interference occurs when the path difference between light waves from adjacent slits is an exact multiple of the wavelength This ensures that the light waves arrive in phase, reinforcing each other to create a bright spot at the order
Since the path difference must be a whole number of wavelengths for constructive interference, can only take integer values (first order, second order, etc.).
For non-integer values of the light waves are not perfectly in phase, resulting in partial interference that does not produce a distinct bright fringe.
Why must non-integer values of the order of maxima be rounded down?
When you calculate the maximum possible value of you sometimes get a non-integer value. Since must be an integer, and only integer orders produce visible maxima, you must round down to the nearest whole number.
For example, if rounding down gives meaning the highest visible order is the second order.
Rounding down ensures that you do not count an order that cannot physically exist, as an order like would not satisfy the condition for constructive interference if is less than three. This guarantees you stay within the physical limits of the diffraction grating, only counting the orders that produce visible maxima.
Question walkthrough
Diffraction Grating Spacing and Maximum Order
Calculates the line spacing of a diffraction grating from lines per mm, then uses the grating equation to find the highest observable order of maxima.
A black body radiator is a theoretical object that absorbs and emits radiation at all wavelengths. While true black bodies are ideal and do not exist in reality, stars provide the closest real-world approximation.
The spectrum of radiation emitted by a black body is determined solely by its temperature.
The intensity–wavelength graph for black bodies shows the relationship between the temperature and the peak wavelength of emitted radiation for different objects. As the temperature in kelvin rises, the peak wavelength reduces, and the intensity increases.
Wien’s displacement law relates the peak wavelength of radiation emitted by an object to its surface temperature. It states that the wavelength at which the radiation curve peaks is inversely proportional to the object’s temperature:
Where:
- is the peak wavelength (),
- is the surface temperature (), and
- is Wien’s constant (; metres kelvin).
Based on Wein’s displacement law:
- Hotter objects emit radiation with shorter peak wavelengths, meaning they appear white or blue.
- Cooler objects have longer peak wavelengths, giving them a red or yellow appearance.
- Hotter objects also emit greater intensity at each wavelength compared to cooler ones.
Recall how wavelength varies along the electromagnetic spectrum, so reducing wavelength means the radiation moves from the radio end of the spectrum towards the gamma end of the spectrum. Within the visible light region, a lower wavelength means light moves from red to violet.
Do not forget to convert all temperatures given to kelvin instead of Celsius. This is done by adding 273 to the temperature in Celsius.
Question walkthrough
Wien's Law, Temperature, and Star Colour
Uses Wien's displacement law to find a star's surface temperature in Celsius from its peak emission wavelength, then links that wavelength to the star's apparent colour.
Intensity is the power per unit area carried by a wave and is proportional to the square of the amplitude. This means that if the amplitude of a wave doubles, its intensity increases by a factor of four.
Intensity represents the amount of energy transmitted by the wave per second over a given area. In a progressive wave, intensity decreases as the wave spreads:
Where:
- is intensity (in watts per square metre,
- is the power carried by the wave (in watts, W), and
- is area over which the wave is spread (in square metres, ).
Luminosity is the total amount of energy that a star (or any radiating object) emits per second in the form of electromagnetic radiation.
The luminosity of an object is determined by two main factors:
- Its surface temperature
- Its surface area
The relationship between these factors is described by the Stefan-Boltzmann law (or Stefan’s law). This states that 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:
Where:
- is the luminosity of the star ,
- is the radius of the star ,
- is the Stefan-Boltzmann constant , and
- is the surface temperature of the star .
From the Stefan-Boltzmann law:
We can see that the luminosity of a star is proportional to:
- Its radius:
- Its surface area:
- Its absolute surface temperature:
Remember that the surface area of a star (or any spherical object) can be calculated using the following formula:
Question walkthrough
Star Luminosity from Radius and Temperature
Uses the Stefan-Boltzmann law to calculate a star's luminosity from its radius and surface temperature, rounding the answer to an appropriate number of significant figures.
Radiant flux is the amount of energy per unit area detected from a star. The inverse square law of flux relates the observed flux of radiation from a star to its luminosity and distance:
Where:
- is the radiant flux
- is the luminosity of the star , and
- is the distance of the star from Earth .
If the radiant flux and distance to the star are known, this equation can be rearranged to calculate the luminosity of the star:
Astronomers use the combination of Wien’s displacement law, the Stefan-Boltzmann law, and the inverse square law of flux to estimate properties of stars that we cannot measure directly, such as their radius or luminosity. These laws link observable quantities on Earth, such as a star’s brightness and colour, to its fundamental properties.
Wien’s displacement law allows us to estimate the surface temperature of a star based on the peak wavelength of the light it emits. The peak wavelength corresponds to the colour of the star and can be observed using telescopes that measure the star’s spectrum.
Luminosity is the total amount of energy a star emits per second. Although we cannot measure the luminosity directly, we can estimate it using the inverse square law of flux, which links the observed brightness (flux) from Earth to the star’s luminosity and distance. If we measure the flux (brightness per unit area) of a star and know its distance (from techniques such as parallax), we can calculate its luminosity.
Finally, once we know the star’s luminosity and surface temperature (from previous steps), we can use the Stefan-Boltzmann law to calculate the star’s radius. This is important because the radius of a star is not something we cannot measure directly from Earth due to the star’s vast distance from us.
By combining Wien’s displacement law, the Stefan-Boltzmann law, and the inverse square law of flux, we can use the relationships between temperature, luminosity, and the distance to the star to estimate its radius.
This is the process:
Step 1: use Wien’s displacement law to find surface temperature
Wien’s displacement law relates the peak wavelength of the radiation emitted by the star to its surface temperature:
Where:
- is the peak wavelength (),
- is the surface temperature (), and
- is Wien’s constant ().
Use this equation to calculate the temperature of the star.
Step 2: use the inverse square law of flux to find luminosity
The inverse square law of flux relates the observed flux of radiation from a star to its luminosity and distance:
Where:
- is the radiant flux
- is the luminosity of the star (), and
- is the distance of the star from Earth ().
Rearrange this equation for and use known values of flux and distance from earth to calculate luminosity.
Step 3: use the Stefan-Boltzmann law to find the stellar radius
Once the luminosity, and surface temperature, have been determined, the radius of the star can be calculated using the Stefan-Boltzmann law:
Where:
- is the luminosity of the star (),
- is the radius of the star (),
- is the Stefan-Boltzmann constant and
- is the surface temperature of the star ().
Rearranging for the stellar radius can be calculated as:
Question walkthrough
Determining Stellar Radius via Blackbody Laws
Combines Wien's displacement law, the inverse square law, and the Stefan-Boltzmann law to find a star's surface temperature, luminosity, and radius from its peak wavelength and flux.