🌌 Welcome to Astronomy and Cosmology (9702 Topic 25) 🔭
Hello future astrophysicist! This final topic of A-Level Physics is perhaps the most exciting, covering how we measure the vastness of the Universe, determine the properties of stars, and understand the origins of space and time itself.
Don't worry if the concepts seem huge; we will use fundamental physics laws (like power, energy, and waves) to measure things trillions of kilometers away. By the end of this chapter, you'll understand how simple observations on Earth can reveal the scale of the cosmos!
25.1 Standard Candles: Measuring Stellar Distances
What is Luminosity? \(L\)
The first step in astronomy is understanding how bright a star truly is.
- Luminosity (\(L\)) is defined as the total power of radiation emitted by a star.
- It is an intrinsic property of the star itself, measured in Watts (W).
- Analogy: Think of the wattage printed on a lightbulb packaging (e.g., 60 W). This is its Luminosity.
What is Radiant Flux Intensity? \(F\)
When we look at a star, we don't measure its total power output ($L$); we measure how much of that power reaches us.
- Radiant Flux Intensity (\(F\)) is the power received per unit area at a certain distance from the source.
- It is measured in \(\text{W} \text{m}^{-2}\).
- Analogy: This is the brightness you actually see when the lightbulb is placed far away.
The Inverse Square Law for Intensity
As light energy travels away from a star, it spreads out over an increasingly large spherical surface area. The surface area of a sphere of radius \(d\) is \(4\pi d^2\).
Since the total power \(L\) is constant, the flux intensity \(F\) at a distance \(d\) must be:
$$F = \frac{L}{4\pi d^2}$$
This is the Inverse Square Law. It tells us that if you double the distance (\(d\)), the intensity (\(F\)) drops by a factor of four (\(2^2\)).
Quick Review Box: The Key Difference
Luminosity (L): Intrinsic property (W). Does not change.
Flux Intensity (F): Measured property (\(\text{W} \text{m}^{-2}\)). Decreases with distance.
What are Standard Candles?
We can easily measure \(F\) (the brightness we see), but to find the distance \(d\), we need to know the star's actual luminosity \(L\).
A Standard Candle is an object (like a certain type of star or supernova) that has a known or predictable luminosity (\(L\)).
If we know \(L\), and we measure \(F\), we can use the rearranged Inverse Square Law to calculate the distance \(d\):
$$d = \sqrt{\frac{L}{4\pi F}}$$
Why they are important: Standard candles are crucial for determining distances to very distant galaxies, allowing us to map the Universe. They act like cosmic mile markers!
Key Takeaway (25.1)
We use the relationship \(F = L/(4\pi d^2)\) to find astronomical distances. A standard candle is required because it provides a reliable, known value for the star's total power output, \(L\).
25.2 Stellar Properties: Temperature and Radius
Stars are treated as black body radiators—objects that emit electromagnetic radiation across a range of wavelengths depending only on their temperature. This allows us to determine both their surface temperature and their physical size.
1. Determining Surface Temperature using Wien's Displacement Law
A hotter object emits light at shorter wavelengths (bluer light), while a cooler object emits light at longer wavelengths (redder light).
Wien's Displacement Law relates the peak wavelength of emitted radiation (\(\lambda_{max}\)) to the absolute temperature (\(T\)) of the star:
$$\lambda_{max} \propto \frac{1}{T}$$
In equation form, this is often written as:
$$\lambda_{max} T = b$$
where \(b\) is Wien's constant (which is given in the data booklet).
To estimate the peak surface temperature (\(T\)) of a star:
- Measure the star's spectrum (the intensity of light at different wavelengths).
- Identify the wavelength at which the intensity is maximum, \(\lambda_{max}\).
- Use Wien's Law to calculate \(T\).
Did you know? Our Sun's peak wavelength (\(\lambda_{max}\)) is in the visible spectrum, making it appear yellowish-white. If it were hotter, it would peak in the UV and appear blue!
2. Determining Radius using the Stefan-Boltzmann Law
Once we know the temperature \(T\) and the luminosity \(L\) (from Section 25.1), we can determine the star’s radius \(r\).
The Stefan-Boltzmann Law links a star's luminosity to its radius and temperature:
$$L = 4\pi\sigma r^2 T^4$$
Where:
- \(L\) is Luminosity (W)
- \(r\) is the radius of the star (m)
- \(\sigma\) is the Stefan-Boltzmann constant (given in the data booklet)
- \(T\) is the absolute surface temperature (K)
Why \(T^4\) is crucial:
Notice the temperature term is raised to the power of four (\(T^4\)). This means a small change in temperature results in a massive change in luminosity. A star that is twice as hot is \(2^4 = 16\) times more luminous!
Step-by-Step: Estimating Stellar Radius
To estimate the physical size (radius \(r\)) of a star, you need to combine both laws:
- Find \(T\): Use Wien's Law and the observed peak wavelength (\(\lambda_{max}\)) to calculate \(T\).
- Find \(L\): Use the Inverse Square Law \(L = 4\pi d^2 F\). You need the distance \(d\) (often found using standard candles) and the measured flux \(F\).
- Find \(r\): Rearrange the Stefan-Boltzmann law to solve for the radius: $$r = \sqrt{\frac{L}{4\pi\sigma T^4}}$$
Key Takeaway (25.2)
Wien's Law determines a star’s surface temperature \(T\). The Stefan-Boltzmann Law (\(L \propto r^2 T^4\)) then uses this temperature, combined with luminosity \(L\), to calculate the star's physical radius \(r\).
25.3 Hubble's Law and the Big Bang Theory
Observing Redshift
When astronomers look at the light spectrum emitted by distant galaxies, they observe that the characteristic emission and absorption lines (spectral lines) are shifted compared to where they should be if the object were stationary.
- This shift is always towards the longer wavelength (red) end of the spectrum, known as redshift.
- Redshift indicates that the objects are moving away from the observer (receding).
Redshift is a direct application of the Doppler Effect (which you studied in the Waves chapter).
The Redshift Equation (Approximate for \(v \ll c\))
For sources moving at velocities \(v\) much smaller than the speed of light \(c\), the fractional change in wavelength or frequency can be related to the recession velocity:
$$\frac{\Delta\lambda}{\lambda} \approx \frac{\Delta f}{f} \approx \frac{v}{c}$$
Where:
- \(\Delta\lambda\) is the change in wavelength (redshift).
- \(\lambda\) is the original (rest) wavelength.
- \(v\) is the recession velocity of the galaxy.
- \(c\) is the speed of light (\(3.00 \times 10^8 \text{m}\text{s}^{-1}\)).
Note for calculations: You must use SI units (metres and seconds) for all terms.
Hubble's Law: The Expanding Universe
In 1929, Edwin Hubble analyzed the recession velocities (\(v\)) and distances (\(d\)) of many galaxies. He made a revolutionary discovery: the speed at which a galaxy moves away from us is proportional to its distance from us.
Hubble's Law:
$$v \approx H_0 d$$
Where:
- \(v\) is the recession velocity (\(\text{m}\text{s}^{-1}\))
- \(d\) is the distance to the galaxy (m)
- \(H_0\) is the Hubble Constant (\(\text{s}^{-1}\)).
The fact that galaxies farther away are moving away faster (as shown by a larger redshift) leads directly to the idea that the Universe is expanding.
Analogy: Imagine baking raisin bread. As the bread rises (expands), every raisin moves away from every other raisin. A raisin further away from you appears to move faster because the space between you and it is increasing everywhere.
The Big Bang Theory
Hubble's Law provides strong evidence for the Big Bang theory.
If every galaxy is moving away from every other galaxy (expansion), then running time backward must lead to a point where all matter was compressed into an infinitesimally small, hot singularity.
How Hubble's Law leads to the Big Bang Theory:
1. Expansion: Hubble's Law (\(v \propto d\)) confirms the universal expansion observed via redshift.
2. Age of the Universe: Since \(v = d/T\) (where \(T\) is time/age) and \(v = H_0 d\), we can equate these:
$$\frac{d}{T} \approx H_0 d$$ $$T \approx \frac{1}{H_0}$$
By calculating the inverse of the Hubble constant, we can estimate the age of the Universe. This constant expansion tracing back to a single moment is the core evidence for the Big Bang.
Common Mistake Alert!
Redshift is *not* caused by galaxies slowing down or stopping. It is caused by the expansion of the space itself between the galaxy and the observer. The light waves are stretched as they travel through the expanding space.
Key Takeaway (25.3)
Redshift ($\Delta\lambda/\lambda \approx v/c$) shows distant galaxies are receding. Hubble's Law ($v \approx H_0 d$) quantifies this recession, proving the Universe is expanding and leading to the Big Bang model, where the age of the universe is estimated by $T \approx 1/H_0$.