Physics | Astrophysics and Cosmology

Welcome to Astrophysics and Cosmology!

Hello future astrophysicist! This chapter is where we apply all the Physics you've learned—especially Thermodynamics, Radiation, and Waves—to understand the vastness of the Universe, the life cycles of stars, and the history of everything!

Don't worry if this seems intimidating. We will break down these gigantic concepts into small, manageable chunks, using analogies and simple maths. By the end of this section, you'll be able to discuss the evidence for the Big Bang and calculate the age of the Universe!

1. Measuring the Stars: Luminosity, Intensity, and Distance

1.1 Luminosity (L) vs. Intensity (I)

When we look up at the night sky, some stars look brighter than others. This apparent brightness depends on two things: how powerful the star actually is, and how far away it is.

• Luminosity (\(L\)): This is the total power radiated by a star in all directions. It is an intrinsic property of the star (it doesn't change regardless of where you are). Luminosity is measured in Watts (W).
• Intensity (\(I\)): This is the power received per unit area at a specific distance from the star. It is what we actually measure on Earth. Sometimes referred to as apparent brightness. Intensity is measured in \(W\,m^{-2}\).

Think of it like a light bulb: a 100W bulb has a fixed Luminosity (L). But the light Intensity (I) is much greater if you hold it right next to your eye than if it's 10 meters away.

1.2 The Inverse Square Law

As light travels out from a star, it spreads out over an increasingly large spherical area. The surface area of a sphere is \(4\pi d^2\), where \(d\) is the distance from the star.

Since the total power (\(L\)) must be distributed over this entire area, the Intensity (\(I\)) falls off quickly with distance:

$$I = \frac{L}{4\pi d^2}$$

Key Point: Intensity is inversely proportional to the square of the distance (\(I \propto 1/d^2\)). If you double the distance, the intensity drops to a quarter of its original value.

Common Mistake to Avoid: Confusing Luminosity (\(L\)) and Intensity (\(I\)). \(L\) is fixed for the star; \(I\) changes based on your position.

Quick Review: Luminosity is the total power output; Intensity is the power per unit area we receive, governed by the Inverse Square Law.

2. Stellar Radiation and Temperature (Black Body Physics)

Stars are excellent approximations of Black Bodies—idealized objects that absorb all incident radiation and emit a characteristic spectrum purely based on their temperature.

2.1 Wien’s Displacement Law

The spectrum of light emitted by a star is not uniform; it peaks at a specific wavelength, \(\lambda_{max}\). This peak wavelength tells us the star's surface temperature.

Wien’s Displacement Law states that the peak emission wavelength (\(\lambda_{max}\)) is inversely proportional to the absolute temperature (\(T\)) of the black body:

$$\lambda_{max} T = b$$

Where \(b\) is Wien's constant (\(2.90 \times 10^{-3}\,m\,K\)).

• If a star is Hotter (larger \(T\)), \(\lambda_{max}\) is Shorter (shifts towards blue/UV light).
• If a star is Cooler (smaller \(T\)), \(\lambda_{max}\) is Longer (shifts towards red/IR light).

Memory Aid: Think of colours. Red is cool (long wavelength); Blue/White is hot (short wavelength).

2.2 The Stefan-Boltzmann Law

While Wien's Law relates temperature to the colour of the star, the Stefan-Boltzmann Law relates the temperature to the star’s total Luminosity (\(L\)).

For a perfect black body of surface area \(A\) and absolute temperature \(T\), the power emitted is:

$$L = A \sigma T^4$$

Since stars are roughly spherical, the surface area \(A = 4\pi R^2\), where \(R\) is the star's radius.

$$L = 4\pi R^2 \sigma T^4$$

Where \(\sigma\) is the Stefan constant (\(5.67 \times 10^{-8}\,W\,m^{-2}\,K^{-4}\)).

The Power of T to the Fourth: Notice the \(T^4\) term. This is incredibly important. It means temperature is the dominant factor determining luminosity. If two stars have the same size, but Star A is twice as hot as Star B, Star A is \(2^4 = 16\) times more luminous!

3. The Doppler Effect and Redshift

Astrophysics depends heavily on observing light, but light waves change if the source is moving relative to the observer. This is the Doppler Effect.

3.1 Understanding the Doppler Effect

You are likely familiar with sound waves: an ambulance siren sounds higher pitched (higher frequency) as it approaches and lower pitched (lower frequency) as it moves away.

The same effect happens with light waves:

• Blueshift: If a star or galaxy is moving TOWARDS us, the observed wavelength decreases (shifts to the blue end of the spectrum). Frequency increases.
• Redshift: If a star or galaxy is moving AWAY from us, the observed wavelength increases (shifts to the red end of the spectrum). Frequency decreases.

3.2 Quantifying Redshift (\(z\))

We calculate redshift (\(z\)) using the difference between the observed wavelength (\(\lambda_{obs}\)) and the wavelength measured in the lab (\(\lambda_{rest}\)):

$$z = \frac{\Delta \lambda}{\lambda_{rest}} = \frac{\lambda_{obs} - \lambda_{rest}}{\lambda_{rest}}$$

For speeds (\(v\)) much less than the speed of light (\(c\)), the redshift is also directly related to the velocity of the source:

$$z \approx \frac{v}{c}$$

If we measure the redshift (\(z\)), we can determine the recession velocity (\(v\)) of the distant object, provided it is not moving too close to the speed of light.

Key Takeaway: Almost every distant galaxy we observe shows a significant Redshift. This tells us that the galaxies are nearly all moving away from us.

4. The Expanding Universe and Hubble’s Law

In the 1920s, Edwin Hubble analyzed the redshift of dozens of galaxies. He made a revolutionary discovery: the further away a galaxy is, the faster it is moving away from us.

4.1 Hubble’s Law

Hubble's observations established a direct, linear relationship between the recession velocity (\(v\)) of a galaxy and its distance (\(d\)) from Earth:

$$v = H_0 d$$

Where \(H_0\) is the Hubble Constant.

The units of \(H_0\) are typically given as \(km\,s^{-1}\,Mpc^{-1}\) (kilometers per second per megaparsec), but the crucial underlying unit is simply \(s^{-1}\) (inverse seconds).

Implications of Hubble's Law:

1. The Universe is expanding.
2. The expansion is uniform across the Universe.

Analogy: Imagine dots painted on an inflating balloon. As the balloon expands, all dots move away from each other, and the dots that are further apart appear to move away fastest. There is no central point of expansion.

4.2 Estimating the Age of the Universe

Hubble’s Law gives us a simple way to estimate the time since the expansion began (the age of the Universe, \(T\)).

If velocity \(v = d/T\), and we know \(v = H_0 d\), we can substitute and solve for \(T\):

$$\frac{d}{T} = H_0 d$$ $$T = \frac{1}{H_0}$$

This means that the age of the Universe is approximately the reciprocal of the Hubble constant, provided the expansion rate has been constant throughout history.

Did you know? Current measurements of \(H_0\) give the age of the Universe as approximately 13.8 billion years.

5. Evidence for the Big Bang

The Big Bang Theory is the leading model for the history of the Universe. It posits that the Universe started from an extremely hot, dense point and has been expanding and cooling ever since. Hubble’s Law proves expansion, but two other pieces of evidence are essential:

5.1 Cosmic Microwave Background Radiation (CMBR)

This is arguably the most powerful evidence for the Big Bang.

Step-by-Step Explanation:

1. In the very early Universe (for about 380,000 years), the temperature was so high that all matter existed as plasma (ions and electrons). Photons could not travel far before scattering off charged particles; the Universe was opaque.
2. As the Universe expanded, it cooled. When the temperature dropped below about 3000 K, electrons and nuclei could finally combine to form stable atoms (mostly hydrogen and helium). This event is called decoupling.
3. Once atoms formed, the Universe became transparent. The photons that existed at that moment were suddenly free to travel through space.
4. Since that time, the Universe has continued to expand dramatically. This expansion has caused the wavelength of those original photons to stretch out significantly (extreme redshift).
5. Today, we detect these ancient photons as low-energy microwave radiation coming uniformly from every direction in space. They correspond to the black body emission of an object at a temperature of just 2.7 K.

The Significance: The existence of the CMBR is the 'fossil record' of the early hot Universe, perfectly matching the prediction of a hot starting point that has expanded and cooled.

5.2 Abundance of Elements

The Big Bang model predicts the ratio of the lightest elements formed in the first few minutes (Big Bang Nucleosynthesis). The observed ratio in the Universe—about 75% Hydrogen and 25% Helium (by mass), with trace amounts of Lithium—matches the theoretical predictions extremely accurately.

Summary: The combination of Hubble's Law (Expansion), the existence of CMBR (Cooling Evidence), and the correct Elemental Abundance provides overwhelming support for the Big Bang model.

Final Thought

You have just explored concepts ranging from the energy output of a single star to the birth of the entire cosmos. Keep practicing the formula applications, especially the links between temperature (Wien's, Stefan-Boltzmann) and velocity (Hubble's Law, Redshift). You've got this!