⭐ Star Properties

Sun ≈ 5778 K
Sun ≈ 6.957e8 m
Peak Wavelength (λ_max): 0 nm
Luminosity (L): 0 W

🌌 Hubble's Law & Redshift

Distance (d): 0 Mpc
Redshift (z): 0
Age of Universe: 0 Billion Years

The Dynamics of the Cosmos

Cosmology is the study of the universe as a whole—its origin, evolution, and eventual fate. In A Level Physics, we utilize the properties of light emitted from stars and distant galaxies to measure unimaginably large distances and speeds.

1. Wien's Displacement Law

All objects with a temperature above absolute zero emit electromagnetic radiation. Wien's Law relates the temperature of a "black body" star to the wavelength at which its emission is most intense.

λmax T = 2.90 × 10⁻³ m·K

2. Stefan-Boltzmann Law

The Luminosity (L) of a star—its total power output—depends on both its size and its temperature. Because temperature is raised to the fourth power, even small changes in T lead to massive changes in L.

L = 4πr²σT⁴

3. Hubble's Law and Redshift

Edwin Hubble discovered that the light from distant galaxies is 'redshifted'—their spectral lines are moved toward longer wavelengths. This occurs because the galaxies are moving away from us as space itself expands.

v = H₀d

The Hubble Constant (H₀) represents the rate of expansion. By taking the reciprocal of H₀ (after unit conversion), we can estimate the time since the Big Bang began.

Exam Tip: Always be mindful of units. Wavelength is often in nanometers (10⁻⁹ m), while distances might be in light-years or Megaparsecs (Mpc). SI conversion is vital before plugging values into equations.

Deep Dive: Worked Examples

✅ Example 1: Wien's Law (Star Temperature)

A red supergiant star, Betelgeuse, has a peak emission wavelength of approximately 850 nm. Calculate its surface temperature.

Step 1: Identify Formula and Constants Wien's Law: λ_max × T = 2.90 × 10⁻³ m·K
Step 2: Convert Wavelength to Meters 850 nm = 850 × 10⁻⁹ m
Step 3: Solve for T T = (2.90 × 10⁻³) / (850 × 10⁻⁹) ≈ 3412 K

✅ Example 2: Stefan-Boltzmann (Radius Calculation)

A white dwarf star has a luminosity 10⁻³ times that of the Sun (L☉ ≈ 3.83 × 10²⁶ W) and a temperature of 25,000 K. Calculate its radius.

Step 1: Values L = 3.83 × 10²³ W, T = 25,000 K, σ = 5.67 × 10⁻⁸
Step 2: Rearrange Stefan's Law for r r = √(L / (4πσT⁴))
Step 3: Solve r = √(3.83×10²³ / (4π × 5.67×10⁻⁸ × 25000⁴)) ≈ 1.17 × 10⁷ m
(This is roughly the same size as Earth!)

✅ Example 3: Redshift and Velocity

The H-alpha line (656.3 nm) in a galaxy's spectrum is observed at 670.0 nm. Calculate the recession velocity of the galaxy.

Step 1: Calculate Redshift (z) z = Δλ / λ₀ = (670.0 - 656.3) / 656.3 = 13.7 / 656.3 ≈ 0.02087
Step 2: Relate to Velocity v = z × c = 0.02087 × (3.00 × 10⁸ m/s)
Step 3: Solve v ≈ 6.26 × 10⁶ m/s (or 6260 km/s)

✅ Example 4: Hubble's Law (Distance)

A galaxy is moving away from Earth at 15,000 km/s. Assuming H_0 = 70 km/s/Mpc, how far away is the galaxy?

Step 1: Formula v = H₀d → d = v / H₀
Step 2: Plug in values d = 15000 / 70
Step 3: Solve d ≈ 214 Mpc

✅ Example 5: Estimating the Age of the Universe

Using a Hubble Constant of 70 km/s/Mpc, estimate the age of the universe in years.

Step 1: Convert H₀ to SI Units (s⁻¹) H₀ = (70 × 1000) / (3.086 × 10²²) = 2.27 × 10⁻¹⁸ s⁻¹
Step 2: Age t = 1/H₀ t = 1 / (2.27 × 10⁻¹⁸) = 4.41 × 10¹⁷ seconds
Step 3: Convert to Years 4.41 × 10¹⁷ / (31,557,600 s/year) ≈ 14 Billion Years