Define the Hubble constant H_0 as the ratio of the speed at which the galaxy is moving away from the Earth to its distance from the Earth; recall and use the equation H_0 = v / d

6.2.3 The Universe – The Hubble Constant

Learning Objective

Define the Hubble constant \$H0\$ as the ratio of the speed at which a galaxy is moving away from the Earth to its distance from the Earth; recall and use the equation \$H0 = \frac{v}{d}\$.

Key Concepts

• Red‑shift and recession velocity.
• Linear relationship between recession velocity and distance (Hubble’s Law).
• Units of \$H_0\$ (km s⁻¹ Mpc⁻¹).
• Typical measured value of \$H_0\$ (≈ \$70\ \text{km s}^{-1}\text{Mpc}^{-1}\$).

Derivation of Hubble’s Law

Observations of distant galaxies show that their spectral lines are shifted towards longer wavelengths. The shift is quantified by the red‑shift \$z\$, where

\$\$

z = \frac{\Delta \lambda}{\lambda_0}

\$\$

For speeds much less than the speed of light, the recession velocity \$v\$ can be approximated by

\$\$

v \approx cz

\$\$

where \$c = 3.00 \times 10^5\ \text{km s}^{-1}\$ is the speed of light. Hubble’s empirical law states that \$v\$ is directly proportional to the distance \$d\$ of the galaxy from the observer:

\$\$

v = H_0 d

\$\$

Re‑arranging gives the definition of the Hubble constant:

\$\$

H_0 = \frac{v}{d}

\$\$

Using the Equation \$H_0 = v/d\$

1. Measure the red‑shift \$z\$ of the galaxy’s spectral lines.
2. Calculate the recession velocity \$v = cz\$.
3. Determine the distance \$d\$ (e.g., using standard candles).
4. Compute \$H_0 = v/d\$.

Worked Example

Galaxy A has a measured red‑shift \$z = 0.005\$. Its distance, determined from Cepheid variables, is \$d = 20\ \text{Mpc}\$. Find the recession velocity and the value of \$H_0\$ implied by this galaxy.

1. Calculate \$v\$:

   \$\$

   v = cz = (3.00 \times 10^5\ \text{km s}^{-1})(0.005) = 1.5 \times 10^3\ \text{km s}^{-1}

   \$\$

2. Use \$H_0 = v/d\$:

   \$\$

   H_0 = \frac{1.5 \times 10^3\ \text{km s}^{-1}}{20\ \text{Mpc}} = 75\ \text{km s}^{-1}\text{Mpc}^{-1}

   \$\$

The result is close to the accepted range of \$70 \pm 5\ \text{km s}^{-1}\text{Mpc}^{-1}\$, showing good agreement.

Typical \cdot alues from Observations

Common Misconceptions

• Thinking \$H_0\$ is a speed rather than a rate; it has units of speed per distance.
• Assuming the same \$H_0\$ applies to all galaxies regardless of scale; local motions (peculiar velocities) can cause deviations.
• Confusing red‑shift caused by Doppler effect with cosmological expansion; at large distances the latter dominates.

Practice Questions

1. A galaxy is observed to have a recession velocity of \$9.0 \times 10^3\ \text{km s}^{-1}\$. Using \$H_0 = 70\ \text{km s}^{-1}\text{Mpc}^{-1}\$, estimate its distance.
2. If a galaxy 50 Mpc away shows a red‑shift \$z = 0.012\$, calculate the implied \$H_0\$ and comment on whether it agrees with the accepted value.
3. Explain why galaxies that are very close to the Milky Way may not follow Hubble’s law precisely.

Summary

The Hubble constant \$H0\$ quantifies the rate of expansion of the Universe. It is defined by the simple ratio \$H0 = v/d\$, where \$v\$ is the recession speed of a galaxy and \$d\$ its distance from Earth. Accurate determination of \$H_0\$ is central to cosmology, linking observations of distant objects to the age and size of the Universe.