## **23.5 Hubble's Law and the Age of the Universe** How do we know the universe is getting bigger, and how can we possibly know how old it is? The answers come from an amazing discovery called **Hubble's Law**. It shows that distant galaxies are speeding away from us, and the farther they are, the faster they go. This section explains this crucial law, how it provides the best evidence for the Big Bang theory, and how it allows scientists to calculate the age of the universe itself. We'll also see how space telescopes like Hubble and James Webb help us gather this cosmic evidence. ### **Hubble's Law** - **Definition:** States that the recession speed ($v$) of a galaxy is directly proportional to its distance ($d$) from the Earth. - **Formula:** $$ v=H_{0} d $$ Where: - $v$ is the recessional velocity of the galaxy. - $d$ is the distance of the galaxy from Earth. - $H_{0}$ is Hubble's constant. - **Interpretation:** The further away a galaxy is from Earth, the faster its recessional velocity. - **Graphical Representation:** The gradient of a graph plotting recession velocity against distance ($v/d$) is equal to Hubble's constant ($H_0$). ### **Hubble's Constant ($H_0$)** - **Definition:** Represents the rate of expansion of the universe. - **Units:** Measured in $\mathrm{s}^{-1}$ (SI unit), though often expressed in $\mathrm{km\ s^{-1}\ Mpc^{-1}}$. - **Significance:** Crucial for determining the age and expansion rate of the universe. ### **The Big Bang Theory and the Age of the Universe** - **Premise:** If galaxies are moving away from each other, they must have originated from a single point in the past. - **Big Bang Theory:** Proposes that approximately 13.8 billion years ago, the entire universe originated from an extremely hot, dense singular point and has been expanding outwards ever since. - **Evidence:** Redshift of galaxies and the observed expansion of the universe are primary pieces of evidence supporting this theory. - **Age Calculation:** By extrapolating Hubble's Law backward, the age of the universe ($T_0$) can be estimated. $$ T_{o}=\frac{1}{H_{o}} $$ - **Current Estimate:** The estimated age of the universe ranges from 13 to 14 billion years, with cosmologists actively seeking more accurate measurements of $H_0$. ### **Space Telescopes** - **Hubble Space Telescope (HST)** - **Launch:** 1990 - **Orbit:** Approximately 540 km above Earth. - **Observations:** Ultraviolet, visible, and near-infrared light. - **Contributions:** Significant understanding of galaxy evolution, star formation, cosmology, and determining the universe's expansion rate. - **James Webb Space Telescope (JWST)** - **Launch:** 2021 - **Orbit:** Approximately 1.5 million km from Earth (L2 point). - **Observations:** Near-infrared and mid-infrared light. - **Objectives:** Studying the universe's early stages, understanding galaxy formation and evolution, observing star formation and planetary system development. ### **Possible Questions/Answer** - **Q:** How is a galaxy's recessional velocity determined? **A:** By analyzing the redshift data of the light spectra emitted from the galaxy. This relates to concepts in . - **Q:** What is used to determine the distance of a galaxy from Earth? **A:** Standard candles (objects of known luminosity). - **Q:** A specific line in the laboratory spectrum has a frequency of $4.570 \times 10^{14} \mathrm{~Hz}$. The same line from a distant galaxy has a frequency of $4.547 \times 10^{14} \mathrm{~Hz}$. What speed is the distant galaxy moving in relation to the Earth, and is it moving towards or away? **A:** Shift in frequency $\Delta f = (4.570 - 4.547) \times 10^{14} = 2.3 \times 10^{12} \mathrm{~Hz}$. Using the equation $\frac{\Delta f}{f}=\frac{v}{c}$: $v=\frac{c \Delta f}{f}=\frac{\left(3 \times 10^{8}\right)\left(2.3 \times 10^{12}\right)}{4.57 \times 10^{14}}=1.5 \times 10^{6} \mathrm{~m} \mathrm{~s}^{-1}$. Since the observed frequency ($4.547 \times 10^{14} \mathrm{~Hz}$) is less than the emitted frequency ($4.570 \times 10^{14} \mathrm{~Hz}$), the light is redshifted, meaning the galaxy is moving away from Earth at $1.5 \times 10^{6} \mathrm{~m} \mathrm{~s}^{-1}$. - **Q:** A galaxy is moving away with a speed of $2.1 \times 10^{7} \mathrm{~m} \mathrm{~s}^{-1}$ at a distance of $9.5 \times 10^{24} \mathrm{~m}$ from Earth. Assuming constant speed, what is the age of the universe in years? **A:** 1. Calculate Hubble's constant $H_0 = v/d = (2.1 \times 10^7 \mathrm{~m/s}) / (9.5 \times 10^{24} \mathrm{~m}) \approx 2.21 \times 10^{-18} \mathrm{~s}^{-1}$. 2. Calculate the age of the universe $T_0 = 1/H_0 = 1 / (2.21 \times 10^{-18} \mathrm{~s}^{-1}) \approx 4.52 \times 10^{17} \mathrm{~s}$. 3. Convert seconds to years: $1 \text{ year} \approx 3.1536 \times 10^7 \text{ seconds}$. $T_0 \approx (4.52 \times 10^{17} \mathrm{~s}) / (3.1536 \times 10^7 \mathrm{~s/year}) \approx 1.43 \times 10^{10} \text{ years}$ or **14.3 billion years**. ### **Summary** - **Hubble's Law** states that a galaxy's recession speed is directly proportional to its distance from Earth ($v=H_0 d$). - **Hubble's constant ($H_0$)** represents the universe's expansion rate and is the gradient of a velocity-distance graph. - The **Big Bang Theory** posits the universe began from an extremely hot, dense point about 13.8 billion years ago, a theory supported by galactic redshift and observed expansion. - The **age of the universe ($T_0$)** can be estimated as the inverse of Hubble's constant ($T_0 = 1/H_0$). - **Hubble (HST)** and **James Webb (JWST)** Space Telescopes are crucial observatories providing data on cosmology, galaxy evolution, and star formation. **Key Formulas:** | Concept | Formula | Description | | :--- | :--- | :--- | | Hubble's Law | $v=H_{0} d$ | Recessional velocity proportional to distance | | Age of the Universe | $T_{o}=\frac{1}{H_{o}}$ | Age calculated from Hubble's constant | | Redshift Velocity | $\frac{\Delta f}{f}=\frac{v}{c}$ | Relates frequency shift to recessional velocity |