Our place in the universe
| Module 5: Rise and the fall of the clockwork universe 5.1 Models and rules |
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| 5.1.3 |
Our place in the universe a) Describe and explain: I) The use of radar-type measurements to determine distances within the solar system; how distance is measured and defined in units of time, assuming the relativistic principle of the invariance of the speed of light II) Effect of relativistic time dilation using the relativistic factor [math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}[/math] III) The measurement of relative velocities by radar observation IV) Evidence for a ‘hot big bang’ origin of the universe from cosmological red-shifts (Hubble’s law); cosmological microwave background. b) Make appropriate use of, by sketching and interpreting: I) Logarithmic scales of magnitudes of quantities: distance, size, mass, energy, power, brightness. c) Make calculations and estimates involving: I) Distances and ages of astronomical objects II) Distances and relative velocities from radar-type measurements. |
a) Describe and explain:
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I) Measuring Distances Using Radar in the Solar System and Time Dilation in Relativity
- ⇒ Radar-Type Measurements in the Solar System
- Radar measurements involve sending electromagnetic signals (e.g., radio waves or light) to a target (like a planet or asteroid) and detecting the reflected signal. By measuring the time taken for the signal to travel to the target and back, the distance can be determined.
- 1. Radar Distance Measurement
- Basic Principle:
- – Electromagnetic waves travel at the speed of light ([math]3.00 × 10^8 m/s[/math]) in a vacuum.
- – The total travel time ([math]Δt[/math]) of the signal corresponds to a round trip (to and back from the target).
- Figure 1 Radar distance sensor
- Equation for Distance:
- [math]d = \frac{c∆t}{2}[/math]
- – d: Distance to the target.
- – c: Speed of light ([math]3.00 × 10^8 m/s[/math]).
- – [math]Δt[/math]: Time for the signal to make a round trip.
- Steps in Measurement:
- – A signal is transmitted from Earth.
- – The signal reflects off the target.
- – The reflected signal is received back on Earth.
- – The total round-trip time is measured.
- Assumptions:
- – The speed of light is constant and invariant in all inertial frames of reference (relativity principle).
- – The path of the signal is straight (ignoring gravitational bending for simplicity in most measurements).
- 2. Distances in Units of Time
- Why Measure Distance in Time?
- In relativistic physics, time and space are deeply interconnected. Distances can be expressed in terms of the time it takes for light to travel that distance.
- For example:
- – 1 light-second: Distance light travels in 1 second [math]3 × 10^8m/s[/math].
- – 1 light-year: Distance light travels in 1 year ([math]9.46 × 10^{15} m[/math]).
- Conversion from Time to Distance:
- [math]d = c . t[/math]
- – t: Time taken by light to travel the distance.
- ⇒ Applications of Radar Distance Measurement:
- Planetary Distances:
- – Radar has been used to measure the distance to Venus, Mars, and other planets with high precision.
- Tracking Spacecraft:
- – Signals sent to and received from spacecraft help determine their location and trajectory.
- Mapping Asteroids and Moons:
- – Radar is also used for detailed surface mapping and measuring asteroid orbits.
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II) Relativistic Time Dilation:
- Relativity introduces the idea that time does not pass at the same rate for all observers. The faster an object moves relative to an observer; the more time dilation
- ⇒ Relativistic Time Dilation Equation:
- Time Dilation Factor (γ):
- [math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
- – v: Velocity of the moving object relative to the observer.
- – c: Speed of light.
- Relation Between Proper Time and Dilated Time:
- [math]Δt’ = γΔt[/math]
- – [math]Δt[/math]: Time interval measured by a stationary observer (proper time).
- – [math]Δt'[/math]: Time interval measured by a moving observer.
- A moving clock appears to tick more slowly compared to a stationary clock.
- Time dilation becomes significant when v approaches c.
- ⇒ Effects of Time Dilation
- Practical Implications:
- GPS Satellites:
- – GPS satellites move at high speeds and experience time dilation. Their onboard clocks run slightly slower than clocks on Earth.
- Figure 2 GPS satellites
- – This relativistic effect is corrected to maintain accurate positioning.
- High-Speed Space Travel:
- – For astronauts traveling at relativistic speeds, time passes more slowly for them relative to people on Earth. This is often called the “twin paradox.”
- Particle Physics:
- – High-energy particles, such as muons produced in the atmosphere, decay more slowly when moving at relativistic speeds, allowing them to reach Earth’s surface.
- ⇒ Visualizing Time Dilation
- Small Velocities ([math]v≪c[/math]):
- – When v is much smaller than [math]c, γ ≈ 1[/math], so time dilation is negligible.
- Relativistic Velocities ([math]v ≈ c[/math]):
- – As v approaches c,[math]γ → ∞[/math] , meaning time dilation becomes extreme.
- – For example, if v=0.99c:
- [math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}} \\
\gamma = \frac{1}{\sqrt{1 – (0.99)^2}} \\
\gamma = 7.1[/math] - A clock on a spaceship moving at 99c ticks 7.1 times more slowly than a stationary clock.
- ⇒ Combining Radar and Relativity
- Relativistic Corrections in Radar Measurements:
- – At high velocities or in strong gravitational fields, relativistic time dilation and length contraction must be considered for precise measurements.
- – Light travel time is influenced by both the speed of the target and the curvature of spacetime.
- Example: Measuring Distance to a Fast-Moving Spacecraft:
- – If the spacecraft moves at a relativistic speed (v), the time interval ([math]Δt[/math]) recorded by the radar must account for relativistic time dilation to ensure accurate distance calculations.
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III) Measurement of Relative Velocities by Radar Observation
- 1. Radar Principle for Measuring Velocities
- Radar observation can determine the relative velocity between two objects (e.g., spacecraft, asteroids, or planets) using the Doppler effect.
- ⇒ Doppler Effect and Radar:
- Doppler Shift:
- – When a radar signal (e.g., electromagnetic wave) is reflected from a moving object, the frequency of the reflected wave is shifted relative to the emitted frequency. This shift is caused by the relative motion between the radar source and the object.
- Observed Frequency Shift ([math]f'[/math]):
- [math]f’ = f \left(\frac{c + v_r}{c}\right)[/math]
- – [math]f[/math]: Frequency of the transmitted wave.
- – [math]f'[/math]: Frequency of the received wave.
- – c: Speed of light ([math]3 × 10^3 m/s[/math]).
- – [math]v_r[/math]: Relative velocity of the object toward or away from the radar (positive when approaching, negative when receding).
- Figure 3 Doppler effect
- Relative Velocity Measurement:
- – By measuring the frequency difference ([math]Δf = f’ – f[/math]), the relative velocity ([math]v_r[/math]) can be calculated:
- [math]v_r = c \frac{\Delta f}{f}[/math]
- 2. Practical Application of Radar Velocity Measurement
- Astronomy and Space Exploration:
- – Radar is used to track spacecraft, measure asteroid velocities, and determine orbital properties of planets and moons.
- Earth-Based Examples:
- – Police speed radars work using the same Doppler principle to measure the velocity of moving vehicles.
- Relativistic Corrections:
- – At high velocities close to the speed of light, relativistic Doppler effects must be used:
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IV) Evidence for a ‘Hot Big Bang’ Origin of the Universe
- The “Hot Big Bang” model of cosmology is supported by multiple lines of evidence, primarily:
- i) Cosmological Redshifts (Hubble’s Law)
- ii) Cosmic Microwave Background Radiation (CMB)
- i) Cosmological Redshifts and Hubble’s Law
- ⇒ Cosmological Redshift
- What is Redshift?
- – Light from distant galaxies appears shifted toward the red end of the spectrum due to the expansion of the universe.
- – This is called cosmological redshift and is quantified by the redshift parameter (z):
- [math]z = \frac{\lambda_{\text{observed}} – \lambda_{\text{emitted}}}{\lambda_{\text{emitted}}}[/math]
- – [math]\lambda_{\text{observed}}[/math]: Wavelength of light received.
- – [math]\lambda_{\text{emitted}}[/math]: Wavelength of light at the source.
- Figure 4 Cosmological right shift
- Interpretation of Redshift:
- – A higher z value means the object is farther away and its light has been traveling longer through expanding space.
- ⇒ Hubble’s Law:
- Statement:
- – The velocity of recession of galaxies is proportional to their distance from us:
- [math]v = H_0 d[/math]
- – v: Recession velocity of the galaxy.
- - [math]H_0[/math]: Hubble constant (current estimate:[math]∼70 km/s/Mpc[/math] ).
- – d: Distance to the galaxy.
- Figure 5 Hubble Law
- Evidence for the Big Bang:
- – Hubble’s observation in the 1920s showed that most galaxies are moving away from us, implying the universe is expanding.
- – If we extrapolate backward in time, all matter and energy converge to a singularity, consistent with the “Big Bang” origin.
- ii) Cosmic Microwave Background Radiation (CMB)
- ⇒ Discovery and Nature of the CMB:
- In 1964, Arno Penzias and Robert Wilson discovered a faint microwave radiation coming uniformly from all directions in space.
- This radiation is the afterglow of the Big Bang, providing a snapshot of the universe when it was about 380,000 years old.
- Figure 6 Cosmic microwave radiation
- ⇒ Characteristics of the CMB
- 1. Blackbody Spectrum:
- The CMB has a perfect blackbody spectrum with a temperature of 725 K.
- This is consistent with predictions of the cooling universe after the Big Bang.
- Figure 7 Blackbody spectrum
- 2. Uniformity and Anisotropies:
- The CMB is nearly uniform, but tiny temperature fluctuations ([math]~10^{-5} K[/math]) exist.
- These fluctuations represent density variations that later grew into galaxies and large-scale structures.
- ⇒ CMB as Evidence for the Big Bang
- The Big Bang theory predicts that the early universe was hot and dense, filled with a plasma of photons and particles.
- As the universe expanded and cooled, photons decoupled from matter (known as recombination), forming the CMB we observe today.
- The uniformity and spectrum of the CMB match predictions of this model.
- ⇒ Combining Hubble’s Law and the CMB
- Expansion History:
- – Hubble’s Law shows that the universe is expanding.
- – The CMB provides evidence of the early hot and dense state of the universe.
- Big Bang Timeline:
- – Big Bang (~13.8 billion years ago): The universe begins as a hot, dense singularity.
- – Recombination (~380,000 years after): CMB photons are released as the universe cools.
- – Structure Formation (~1 billion years onward): Galaxies and stars form as gravitational forces act on matter.
b) Make appropriate use of, by sketching and interpreting:
- ⇒ Logarithmic Scales of Magnitudes of Quantities:
- Logarithmic scales are widely used in science to represent quantities that span a vast range of magnitudes.
- These scales are based on the logarithm of a quantity rather than the quantity itself. This allows large and small values to be represented compactly and compared effectively.
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I) Logarithmic Scale Definition:
- Definition:
- – A logarithmic scale is a nonlinear scale where each unit increase corresponds to a multiplication by a constant factor.
- – For example:
- [math]\text{Logarithmic Scale} = \log_{10} (\text{quantity})[/math]
- Figure 8 Logarithmic Scale
- Why Use Logarithmic Scales?
- – Large ranges of values (e.g., distances from atomic nuclei to galaxies).
- – Easier visualization of exponential growth or decay.
- – Commonly used in astronomy, physics, and biology.
- ⇒ Quantities Represented on Logarithmic Scales
- 1. Distance:
- From subatomic scales ([math]10^{-15}m[/math]) to cosmic scales ([math]10^{26}m[/math]).
- Examples:
- – Radius of a hydrogen atom ([math]10^{-10}m[/math]).
- – Diameter of the observable universe ([math]∼10^{26}m[/math]).
- 2. Size:
- – Objects range from elementary particles ([math]∼10^{-15}m[/math]) to stars and galaxies ([math]∼10^{21}m[/math]).
- 3. Mass:
- – Ranges from an electron’s mass ([math]∼10^{-31}kg[/math]) to the mass of a galaxy ([math]∼10^{42}kg[/math]).
- 4. Energy:
- – Covers energy scales from chemical reactions ([math]∼10^{-19}J[/math]) to supernova explosions ([math]∼10^{44}kg[/math]).
- 5. Power:
- – Ranges from the power output of a flashlight ([math]∼10^{-1}W[/math]) to the luminosity of stars ([math]∼10^{26}W[/math]).
- 6. Brightness (Apparent Magnitude):
- – Logarithmic due to human perception of light.
- – Apparent brightness of celestial objects is measured using the magnitude system, where a 5-magnitude difference corresponds to a factor of 100 in brightness.
- ⇒ Logarithmic Scale Equation
- General Equation:
- [math]\text{Logarithmic Value} = \log_{10} \left( \frac{\text{Quantity}}{\text{Reference Value}} \right)[/math]
- – For example,
- In decibels (dB) for sound intensity:
- [math]L = 10 \log_{10} \left( \frac{I}{I_0} \right)[/math]
- – I: Intensity of sound.
- - [math]I_0[/math]: Reference intensity.
- ⇒ How to Sketch Logarithmic Scales?
- 1. Axes:
- – Use a logarithmic scale on one or both axes. Each tick mark represents a power of 10 ([math]10^1, 10^2, 10^3,[/math] etc.).
- Spacing between ticks corresponds to logarithmic progression.
- 2. Range:
- – Clearly label the range of values, showing large and small quantities.
- 3. Linear Relationships on Logarithmic Scales:
- – Exponential relationships ([math]y = a.e^{bx}[/math]) appear linear when plotted on a logarithmic scale.
- Examples of Logarithmic Scales for Quantities
- 1. Distance
| Logarithmic Scale (Base 10) | Description |
|---|---|
| [math]10^{-15} m[/math] | Size of a Proton |
| [math]10^{-10} m[/math] | Size of an atom |
| [math]10^{0} m[/math] | Human scale (1 meter) |
| [math]10^{6} m[/math] | Size of earth (radius) |
| [math]10^{21} m[/math] | Size of a galaxy |
| [math]10^{26} m[/math] | Observable universe |
- 2. Mass
| Logarithmic Scale (Base 10) | Description |
|---|---|
| [math]10^{-31} kg[/math] | Mass of electron |
| [math]10^{-15} kg[/math] | Virus mass |
| [math]10^{0} kg[/math] | Human mass (average adult) |
| [math]10^{30} kg[/math] | Mass of Sun |
| [math]10^{42} kg[/math] | Mass of galaxy |
- 3. Energy
| Logarithmic Scale (Base 10) | Description |
|---|---|
| [math]10^{-19} J[/math] | Chemical reaction energy |
| [math]10^{0} J[/math] | Energy in lifting an apple (1m) |
| [math]10^{26} J[/math] | Total Sun’s energy per second |
| [math]10^{44} J[/math] | Energy in a supernova explosion |
- 4. Power
| Logarithmic Scale (Base 10) | Description |
|---|---|
| [math]10^{-1} W[/math] | Flashlight |
| [math]10^{0} W[/math] | Human energy output |
| [math]10^{26} W[/math] | Power output of a star (Sun) |
- 5. Apparent Brightness (Magnitude System)
- Brightness of stars is logarithmic due to human perception:
- – [math]m = -2.5 \log_{10} \left( \frac{I}{I_0} \right)[/math]
- – I: Observed intensity.
- – [math]I_0[/math]: Reference intensity.
- ⇒ Interpretation of Logarithmic Plots
- Compression of Data:
- – Large values are compressed, making it easier to compare objects across different scales.
- – For example, comparing the energy of atomic reactions ([math]10^{-19}[/math]J) with that of stars ([math]10^{44}[/math]).
- Linear Trends:
- – Exponential growth or decay appears as straight lines.
- ⇒ Example Sketch Ideas:
- 1. Logarithmic Distance Scale:
- – X-axis: Object size/distance ([math]10^{-15} \text{ to } 10^{26}m[/math]).
- – Y-axis: Representative objects (proton, atom, human, Earth, galaxy).
- 2. Logarithmic Mass Scale:
- – X-axis: Object mass ([math]10^{-31} \text{ to } 10^{42}m[/math]).
- – Y-axis: Examples (electron, virus, human, star, galaxy).
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c) Calculations and Estimates
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I) Distances and Ages of Astronomical Objects
- 1. Distances of Astronomical Objects
- Astronomical distances are measured using various techniques depending on the object’s distance and location.
- a) Parallax Method
- Principle:
- – The apparent position of a nearby star changes relative to background stars when observed from two opposite points in Earth’s orbit (six months apart).
- Formula:
- [math]d = \frac{1}{p}[/math]
- – d: Distance to the star in parsecs (pc).
- – p: Parallax angle in arcseconds (“).
- Example:
- – For a star with p=0.1′′:
- [math]d = \frac{1}{p} \\
d = \frac{1}{0.1} \\
d = 10 \text{ pc}[/math] - Converting parsecs to light-years (1pc[math]≈3.26 ly[/math])
- [math]d = 10 \times 3.26 \\
d = 32.6 \text{ ly}[/math] - b) Luminosity-Distance Relation
- Principle:
- – Measure the apparent brightness (B) of an object and compare it to its intrinsic luminosity (L).
- Formula:
- [math]d = \sqrt{\frac{L}{4\pi B}}[/math]
- – L: Luminosity in watts (W).
- – B: Observed brightness in [math]W/m^2[/math].
- Example:
- – A star has [math]L = 3.8 \times 10^{26} \text{ W} \quad \text{(like the Sun)}, \quad B = 1.0 \times 10^{-10} \text{ W/m}^2[/math]
- [math]d = \sqrt{\frac{L}{4\pi B}} \\
d = \sqrt{\frac{3.8 \times 10^{26}}{4\pi \times (1.0 \times 10^{-10})}} \\
d = 3.46 \times 10^{16} \text{ m}[/math] - Converting to light-year ([math]1 ly ≈ 9.46 × 10^{15} m[/math])
- [math]d \approx \frac{3.46 \times 10^{16}}{9.46 \times 10^{15}} \\
d = 3.66 \text{ ly}[/math] - c) Redshift (Cosmological Distance)
- Principle:
- – Light from distant galaxies is redshifted due to the expansion of the universe. The redshift parameter (z) provides a measure of distance.
- Formula:
- [math]d = \frac{c}{H_0} z[/math]
- – c: Speed of light ([math]3.00 × 10^8 m/s[/math]).
- – [math]H_0[/math]: Hubble constant ([math][/math]).
- – z: Redshift.
- Example:
- – A galaxy has z=0.1:
- [math]d = \frac{c}{H_0} z \\
d = \frac{3 \times 10^8}{70 \times 10^3} \times 0.1 \\
d = 428.6 \text{ Mpc}[/math] - Converting to light-years ([math]1 \text{ Mpc} \approx 3.26 \times 10^6 \text{ ly}[/math])
- [math]d = 428.6 \times 3.26 \times 10^6 \\
d = 1.4 \times 10^9 \text{ ly}[/math] - 2. Ages of Astronomical Objects
- a) Age of the Universe (Hubble Time)
- Formula:
- [math]t_{\text{universe}} = \frac{1}{H_0}[/math]
- – [math]H_0[/math]: Hubble constant (70 km/s/Mpc70).
- Calculation: Converting [math]H_0[/math]to [math]s^{-1}[/math]
- [math]H_0 = \frac{70 \times 10^3}{3.09 \times 10^{19}} \\
H_0 = 2.27 \times 10^{-18} \text{ s}^{-1} \\
t_{\text{universe}} = \frac{1}{H_0} \\
t_{\text{universe}} = \frac{1}{2.27 \times 10^{-18}}[/math] - Converting to year ([math]\left( 1 \text{ yr} \approx 3.15 \times 10^7 \text{ s} \right)[/math])
- [math]t_{\text{universe}} = \frac{4.4 \times 10^{17}}{3.15 \times 10^7} \\
t_{\text{universe}} = 13.9 \text{ billion years}[/math] - b) Stellar Ages
- Determined using stellar models and nuclear fusion rates, such as [math]t_{\text{sun}}[/math] 6 billion years.
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II) Distances and Relative Velocities from Radar-Type Measurements
- Radar observations involve sending electromagnetic pulses to an object and measuring the time delay and frequency shift of the reflected signal.
- 1. Distance Measurements
- Principle:
- – The time delay between transmitting and receiving the radar signal gives the distance.
- Formula:
- [math]d = \frac{c × t}{2}[/math]
- – c: Speed of light ([math]3 × 10^8 m/s[/math]).
- – t: Time delay for the radar pulse to return.
- Example:
- – A radar pulse takes 002 s to return:
- [math]d = \frac{c \times t}{2} \\
d = \frac{3 \times 10^8 \times 0.002}{2} \\
d = 3 \times 10^5 \text{ m} \\
d = 300 \text{ km}[/math] - 2. Relative Velocity Measurements
- Principle:
- – The frequency shift ([math]\Delta f[/math]) of the reflected radar signal is used to determine the object’s velocity using the Doppler effect.
- Formula:
- [math]v_r = c \frac{\Delta f}{f}[/math]
- – [math]v_r[/math]: Relative velocity of the object.
- – f: Transmitted frequency:
- – [math]\Delta f[/math]:Observed frequency shift.
- Example:
- – Transmitted frequency:[math]f = 10^{10} Hz[/math].
- – Observed frequency shift:[math]∆f = 10^3 Hz[/math]
- [math]v_r = c \frac{\Delta f}{f} \\
v_r = \frac{3 \times 10^8 \times 10^3}{10^{10}} \\
v_r = 30 \text{ m/s}[/math] - ⇒ Practical Applications
- Distances:
- – Radar is used to measure the distance to planets (e.g., Venus) and objects within the solar system.
- Velocities:
- – Radar Doppler shifts help determine the motion of asteroids, spacecraft, and planets.