Measuring and classifying stars (Topic 10A)
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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, ).
Amplitude is the maximum displacement from the wave’s equilibrium position. The energy carried by a wave is directly related to its amplitude. Specifically, intensity is proportional to the square of the amplitude:
This means that doubling the amplitude leads to a fourfold increase in intensity.
Therefore, a wave with double the amplitude carries four times the energy per unit area and time.
Frequency is the number of wave cycles that pass a point per second, measured in Hz.
Since a larger number of wave cycles passing a point per second leads to a greater energy transfer, intensity increases with frequency. In fact, intensity is directly proportional to frequency squared:
If the frequency of a wave doubles, its intensity quadruples, meaning the wave is transferring four times as much energy per unit area and time.
Spherical waves are waves originating from a point source that spread equally in all directions, forming spherical wavefronts.
As a spherical wave moves further from the source, it covers a larger area. This area expands according to the surface area of a sphere:
where:
- is the distance from the source in metres.
By substituting this formula for area into the equation for intensity, we can see that intensity is inversely proportional to the distance from the spherical wave source:
When the distance from a point source doubles, the intensity reduces to a quarter of its original value.
Question walkthrough
Star Intensity After Wavelength Shift
Combine the inverse square law with a given proportionality between intensity and frequency to find the intensity of light reaching Earth from a star whose peak emission wavelength has changed.
Astronomical unit (AU)
- Definition: An astronomical unit is the average distance between Earth and the sun.
- Value: Approximately
- Usage: Primarily used to describe distances within the Solar System, such as the distance between planets.
Light year (ly)
- Definition: The distance that light travels in one year in a vacuum.
- Value: Approximately
- Usage: Used to measure distances to stars and galaxies beyond the Solar System. For example, the Andromeda Galaxy is about 2.54 million light years away from Earth.
The Parsec (pc) is the distance at which one astronomical unit subtends an angle of one arcsecond:
- Value: Approximately
- Usage: Commonly used in astronomy to measure stellar and galactic distances. For instance, the nearest star system, Alpha Centauri, is about away from Earth.
Stellar parallax refers to the apparent shift in the position of a nearby star against the backdrop of distant stars as the Earth orbits the sun:
- This shift is due to observing the star from two different points in Earth’s orbit, six months apart.
- Stellar parallax is used to measure distance to the nearby star from Earth.
You can more intuitively understand stellar parallax by holding your finger in front of you and alternately closing one eye, then the other.
The apparent ‘shift’ in your finger’s position mimics how a nearby star appears to shift against distant stars as Earth orbits the sun. The closer the star, the larger the shift (i.e. the parallax angle).
There are limitations to the stellar parallax method. It is effective for stars within only . Beyond this, parallax angles become too small to measure accurately.
Even with advanced instrumentation, distant stars remain difficult to measure; alternative methods, such as standard candles or Doppler-shift analysis of absorption spectra, are used instead.
The parallax angle is half the total angular shift of the star against the backdrop of distant stars over a time period of six months as observed from Earth. The formula to calculate the parallax angle is:
Where:
- is the parallax angle in arcseconds, and
- is the distance to a star in parsecs.
A larger parallax angle means the star is closer to the observer.
Type 1 supernovae have a subtype known as type 1a supernovae. These occur when a white dwarf in a binary system accretes (draws in) matter from its companion and exceeds the Chandrasekhar limit (~1.4 solar masses), triggering a thermonuclear explosion.
Because this mass threshold is consistent, these supernovae always explode with nearly the same peak luminosity, around −19.3 in absolute magnitude.
This uniform brightness allows astronomers to treat them as standard candles: objects of known luminosity used to measure distances via the inverse square law.
Their immense brightness means they are visible up to 1000 mega parsecs (1000 MPc) away, making them essential for measuring vast cosmic distances and detecting the accelerating expansion of the universe.
A type 1a supernova has a distinctive light curve, which makes it useful as a standard candle.
After the initial explosion, the luminosity increases rapidly over a few days to a sharp peak, corresponding to an absolute magnitude of about –19.3. This rise is due to the sudden onset of thermonuclear fusion in the white dwarf.
- After peaking, the brightness declines in two stages: a rapid drop over a few weeks followed by a slower exponential decay lasting months.
- The decline is driven by the radioactive decay of nickel-56 to cobalt-56, which then decays to iron-56.
- The predictable shape and brightness of the curve allow astronomers to identify type 1a events and use them to determine vast cosmic distances.
Hertzsprung-Russell diagrams are graphs which plot the temperature of a star against its luminosity:
- X axis: surface temperature is measured in kelvin, with hotter stars on the left.
- Y axis: luminosity compared to our sun, with brighter stars towards the top.
When astronomers first plotted the stars, they had clustered them together in four groups:
- Main sequence
- White dwarfs
- Giants
- Supergiants
Most stars in the universe are on the main sequence. This is where stars spend most of their lives, in their stable core hydrogen-burning phase. When a star leaves the main sequence, it becomes either a white dwarf, a giant, or a supergiant, depending on its mass:
- Brighter stars have a proportionally higher surface temperature.
- The coolest stars are red, while the hottest stars are blue.
A star is considered low mass if it has between roughly 0.5 and 10 solar masses. Our Sun falls within this range.
When a low mass star exhausts the hydrogen in its core, hydrostatic equilibrium is disrupted. The core contracts while the outer layers expand and cool, forming a red giant. The star eventually sheds its outer layers as a planetary nebula (unrelated to planets, the name is historical), leaving behind a dense, hot remnant called a white dwarf.
A star is considered high mass if it has a mass greater than approximately 10 solar masses. Unlike low mass stars, high mass stars have cores hot enough to fuse elements far beyond carbon and oxygen, progressing through successive stages of fusion up to iron.
Stars with between 10 and 40 solar masses will expand into red supergiants before ending their lives in a violent explosion called a supernova. The remnant left behind is either a neutron star or a black hole, depending on the remaining core mass, these are not represented on an HR diagram.
It is useful to note these supernovae are responsible for distributing heavy elements throughout the interstellar medium, the same material that forms new stellar nebulae. Every element heavier than iron found on Earth was produced during a supernova explosion.