Chapter 1 Cosmic Journey
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- 考生常見錯誤
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Size
planet < star < star cluster < nebula < galaxy(Milky Way) < galaxy cluster < supercluster < filament
Astronomical unit (AU) 天文單位
天文單位（Astronomical unit，縮寫為AU）是一種用於衡量天體距離的單位。它是以地球和太陽之間的平均距離作為基準，被定義為約 1.5 x 1011m Formula Sheet 有這個number
AU最初是用來衡量太陽系中行星之間的距離，特別是地球和其他行星之間的距離。它也被用於測量太陽系外行星（外行星）與恆星之間的距離。
AU的定義是基於地球和太陽之間的平均距離，因為地球繞太陽運轉的軌道是橢圓形的，並且距離太陽的距離會有些變化。因此，AU的值是基於這個平均距離而不是具體的瞬時距離。
AU在天文學中是一個非常有用的單位，因為它提供了一個方便的標準來比較行星之間的距離，而不必依賴於特定的度量單位。例如，地球到太陽的距離約為1 AU，而火星到太陽的平均距離約為1.5 AU。
Light year (ly) 光年
這就是光走一年的距離
1 ly = (3.0 × 108) × (365 × 24 × 60 × 60) = 9.46 × 1015 m Formula Sheet 有這個number
Terrestrial planets 類地行星
- Small
- Relatively close to the star they orbit around
- Made of mostly rocks and metals
- e.g. Mercury, Venus, Earth and Mars
Jovian planets 類木行星
- Large
- relatively farther from the star they orbit around
- Made of liquid hydrogen and helium with no solid surface
- Thick atmosphere
- e.g. Jupiter, Saturn, Uranus and Neptune
Satellites 衛星
- Orbit around planet
- Except Mercury and Venus, satellites orbit around each planet
- The Moon is the only satellite of the Earth
Dwarf planets 白矮星
Similar to planets except they have not ‘cleared’ the neighborhood celestial bodies around their orbits
e.g. Pluto
Asteroids 小行星
- Small rocky bodies that revolve around the Sun
- Irregular in shape
- Lie mainly in the asteroid belt between the orbits of Mars and Jupiter
- e.g. 4 Vesta
Comets 彗星
- Small icy bodies revolve around the Sun in very elliptical orbits
- Solid part only about the size of Hong Kong Island
- When close to the Sun, the materials on its surface vaporize and forming a cloud of gas with a long tail
- e.g. Comet ISON
Star 恆星
- Luminous celestial bodies that generates energy by nuclear fusion
- e.g. the Sun
- The life cycle of stars:
- A star may shine for millions to billions of yearsAfter the fuel is used up, it may end its life in a violent explosion
- Matter is then shattered to
- space as nebulae (e.g. The Ring Nebula)
- Which may later form another generation of stars
Star clusters 星團
- Groups of stars that bound together by gravity (e.g. M3 cluster)
- Stars that belong to the same cluster are formed from the same nebula
Nebulae 星雲
- Cloud of gas and dust between stars in space
- Birth place of stars
- e.g. The Eagle Nebula
The formation of stars 恆星的形成
Galaxy and galaxy cluster 星系團
- A great cloud of stars, gas and dust
- There are several kinds of galaxies
Superclusters and filaments 超星系團
- Galaxy clusters join together to form superclusters
- Superclusters further join up in the form of filaments
- The spaces between filaments are called voids, they contain relatively few galaxies
Chapter 2 Modelling the Universe
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- 考生常見錯誤
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這一個章節基本上都是背爲主，因此大家要準備好紙和筆！
而且這一課有些奇怪，它是融合了歷史，所以大家可以當聼故事。而且同學要學習一些現在看來是錯，但過去的人認爲正確的理論。
Apparent motion of celestial bodies
The term ‘apparent motion’ describe how a celestial body appears to move in the sky
Daily motion
- The Earth completes one rotation about its axis every 24 hours
- We see celestial bodies e.g. the Sun, the Moon, planets,
and stars, move around us once a day
Yearly motion of the Sun
- The Earth revolves around the Sun once a year
- We see the Sun move across a background of distant stars once a year
The apparent path of the Sun is called the ecliptic
Constellations
The ecliptic passes through 12 constellations
- Ancient civilizations divided groups of stars in the sky and associated them with legends
- In modern astronomy, there are 88 constellations with well-defined boundaries
Motion of planets
- There are eight planets moving around the Sun. Their orbits lie nearly on the same plane
- As seen from the Earth, the planets appear to move close to the ecliptic
Retrograde motion 逆行運動
- Planets usually moves from west to east (eastwards) relative to the background stars
- They move from east to west (westward) only when undergo retrograde motion
Motion of Mercury and Venus
- Mercury and Venus are never far away from the Sun
- They can only be seen shortly before sunrise or shortly after sunset
- This is why Mercury and Venus are sometimes called the ‘morning stars’ or ‘evening stars’ (they are not stars)
Celestial sphere
- A model that very useful for astronomical observation
- In this model, all celestial bodies are attached to the inner surface of a very large sphere centred at the Earth
Astronomy of ancient Greeks
- Our ancestors tried to explain how the celestial bodies move with different models
- It takes humans almost two thousand years to understand our position in the universe
Early geocentric models 早期的地心説
- The ancient Greeks were the first civilization that believed humans could understand the universe in a rational way
- Most of them thought that the heavens were a universe with the Earth at the centre
- Plato was one of the most influential philosophers
- He thought that the heavens which should be perfect, must be made up of spheres
簡單而言就是“老屈”
Ptolemaic model 勒密模型
Evolution of geocentric models and explanation of retrograde motion
因爲這個現象所以要對模型做出修正
- They proposed that a planet moved on a small circle called an epicycle, which in turn moved around the Earth on a larger circle called a deferent
這樣就能解釋逆行運動。
- Greek astronomer Ptolemy improved the model by adjusting the radii and the rates
of the epicycles and deferents - He also made the Earth slightly off centre from the deferent
- By assuming the centres of the epicycles of Mercury and Venus were fixed on a line joining the Sun and the Earth
- Why Mercury and Venus always appear close to the Sun
was explained
簡單而言，就是“阿茂整餅無嗰樣整嗰樣”，以及“死雞撐飯蓋”，堅持地心説是正確的！
Copernican revolution 哥白尼
- The Polish astronomer Copernicus put forward a heliocentric model: Sun was the centre of the universe
真相近了！
- The daily motions of the celestial bodies were explained by the self-rotation of the Earth
- Orbital speed decreases from Mercury (fastest) to Saturn (slowest)
Explanation of retrograde motion
- By placing the Sun at the centre, Copernicus explained the retrograde motion of planets without using epicycles
Explanation of morning stars and evening stars
- Copernicus also explained why Mercury or Venus appears as a morning star and an evening star
- Simply because the orbit of an inner planet lies within that of the Earth
- Thus the planet appears close to the Sun
超伏位：Copernican model vs Ptolemaic model
論accuracy，Ptolemaic model是比Copernican model更準確。
因爲Ptolemaic model是根據觀察數據而製作的，Copernican model不是根據觀察數據而草擬的。
Copernican model的可惜之處就是Copernican依然困住自己的想法，認爲orbit一定是circular。
因此，Ptolemaic model是比Copernican model更準確。
必背：Galileo’s discoveries 伽利略的發現
Galileo made the following discoveries:
1.There are hilly terrains on the Moon
2.There are some black spots (sunspots) on the Sun’s surface
3.There are four satellites orbiting around Jupiter
4.A complete cycle of phase change of Venus
5.The Milky Way is made up of numerous stars too faint to be seen with the naked eye
Phases of Venus
- Galileo found that Venus passed through phases like the Moon
- The phase changes of Venus can be explained using the Copernican model
Kepler’s laws of planetary motion
經過多年歷史的資料，終於有人建構出一個類似現在宇宙的物理定律！
- Kepler’s 1st law: All planets move in elliptical orbits, with the Sun at one focus
- Kepler’s 2nd law: An imaginary line joining the Sun and the planet sweeps out equal areas in equal time intervals
- Kepler’s 3rd law: For any planet, the square of its orbital period T is proportional to the cube of the semi-major axis a of its orbit
Chapter 3 Orbital Motions under Gravity
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Kepler’s first law: Elliptical orbits
All planets move in elliptical orbits, with the Sun at one focus
semi-major axis
An ellipse has two axes about which it is symmetric:
- The longer one is the major axis. The semi-major axis a is half of the major axis
- The shorter one is the minor axis. The semi-minor axis b is half of the minor axis
perihelion and aphelion
The semi-major axis
$$a=\frac{r_1+r_2}2$$
Kepler’s second law: Equal area in equal time
An imaginary line, called the radius vector, joining the Sun and the planet sweeps out equal areas in equal time intervals
The planet moves faster when it is closer to the Sun, and slower when it is farther from the Sun.
Kepler’s third law: Period and semi-major axis
For any planet, the square of its orbital period T is proportional to the cube of the semi-major axis a of its orbit, i.e. T2 ∝ a3
For two celestial bodies orbiting the same object, having periods T1 and T2, and orbits of semi-major axes a1 and a2 :
$$\left(\frac{T_2}{T_1}\right)^2=\left(\frac{a_2}{a_1}\right)^3$$
$$T^2=\frac{4\pi^2}{GM}a^3$$
Energy in orbital motion
When an object is in orbital motion due to gravity, its mechanical energy is conserved
New Formula for Gravitational Potential Energy U
Gravitational Potential Energy U
$$U=-\frac{GMm}r$$
我們定義物件在無限遠時的 PE=0
然後約接近我們，PE就會越細。
因此，Gravitational Potential Energy U是永遠 negative
同學可能會覺得很奇怪，讓小編再解釋多一點
原本的 PE=mgh
即是越接近地面，PE越細；即是越遠離地面，PE越大
此時您看回U，其實是一樣的。當物件越接近我們時，U是一個負數；當物件越接近我們時，U=0；
可見都是越遠PE越大，令同學們感到奇怪的是，原本的PE=mgh是把0設在地面，所以越高就會越大。而U是把設在無限遠，所以與接近我們時數值會越細。
Work Done
- m and M are always attractive
- Work has to be done to pull m and M apart
- Suppose mass m is brought from A to B away from the Earth (mass M), the work done:
$$W=\Delta U=-GMm\left(\frac1{r_B}-\frac1{r_A}\right)$$
Escape speed of a celestial body
逃逸速度（Escape speed）是指在一個天體上，克服該天體引力的最低速度，使得一個物體能夠完全離開該天體，並永遠遠離它。換句話說，逃逸速度是指物體需要具備的最小速度，以便克服天體的引力束縛，使其能夠逃逸到無限遠處。
因此，這個速度應該要剛剛好讓物件逃逸到無限遠處而沒有多餘的能量，剛好停在無限遠。
$$\mathrm{KE}_1+\mathrm{PE}_1=\mathrm{KE}_2+\mathrm{PE}_2\\\frac12m\nu_1^2-\frac{GMm}{r_1}=\frac12m\nu_2^2-\frac{GMm}{r_2}$$
如果遠處速度要是0，一方會全部 = 0
因此，要克服天體引力，您需要具備Mechanical energy = 0
無能量就可以走？？？？
直覺上的確如此，但大家要想想：
U<0，KE>0
您要讓物件逃到無限處而又剛好無能量，即是KE要剛好覆蓋U所製造的負數，所以爲什麽Mechanical energy = 0是用來計算逃逸速度
當然，如果物件有比這個速度更快的速度，這件物件會逃到無限遠而還有能量，即是在無限遠依然有KE，依然在移動。
Newton’s cannonball
- Let M and R be the mass and the radius of the Earth
- Suppose the cannonball is projected with a velocity v and air resistance is neglected:
$$\begin{aligned}
&1.\text{ If }\nu<\sqrt{\frac{GM}R}\text{ , the ball will fall to the ground} \\
&\begin{aligned}2.&\mathrm{~If~}\nu=\sqrt{\frac{GM}R}&\text{, the ball will move in a circular orbit around}\\&\text{the Earth}\end{aligned} \\
&3.\text{ If }\sqrt{\frac{GM}R}<\nu<\sqrt{\frac{2GM}R}\text{ , the ball will move in an elliptical orbit}\\\text{around the Earth.} \\
&4.\mathrm{~If~}\nu\geq\sqrt{\frac{2GM}R}\text{ , the ball will escape from the Earth}
\end{aligned}$$
Black hole
In the most extreme situations, uesc could be larger than or equal to the speed of light in a vacuum:
$$u_{\mathrm{esc}}=\sqrt{\frac{2GM}R}\geq c=3\times10^8\text{ m s}^{-1}$$
就是連光也不能逃脫，因此黑洞才這麽”黑“
Chapter 4 Starlight: Messengers from the Stars
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Distance and brightness of stars
Angular measurement
The apparent distance (or the angular distance/separation) between two stars is the angle that separates them in the sky
The apparent diameter (or the angular diameter) of an object (e.g. the Moon) is the angle that its diameter subtends in the sky
arc degrees (°), arc minutes (′), and arc seconds (″)
1′ = 60″ and 1° = 60′ = 3600″
Radian
$$1\mathrm{~rad}=\frac{180^\circ}\pi\quad\\\ \quad1^\circ=\frac\pi{180}\mathrm{~rad}$$
Small-angle approximation
- For a celestial body, the apparent diameter θ, actual distance d and actual diameter D form a trigonometric relation
- When θ is small, i.e. d >> D:
$$\theta\approx\frac Dd\quad\text{(in radians)}$$
Method of parallax
An effect known as parallax , happens when a nearby object is viewed from two different positions
Stellar parallax
The stellar parallax p of the nearby star is defined as half of the apparent shift in the star’s position over the six months
一會兒你就會知道爲甚麽是“ half of the apparent shift”
Distance measured using Stellar parallax
The Sun-to-star distance d, the Sun-to-Earth distance D (1 AU), and the parallax p formed a trigonometric relation
$$\theta=\frac Dd\quad\text{(in radians)}\\p=\frac{1\text{ AU}}d\quad\text{(in radians)}$$
One Parsec 長度
A stellar parallax p is usually measured in arc seconds (″)
正常的，所有星都離我們很遠，因此角度變化不會很大
$$p\text{(in arc seconds)}$$
$$\begin{aligned}
&=\frac{(60\times60)\cdot(180)}{\pi}\times p\text{ (in radians)} \\
&=206265\cdot p\text{ (in radians)} \\
&=\frac{206\text{ 265 AU}}d
\end{aligned}$$
因爲要簡化206265這個數字
我們就會定義1 parsec = 206265 AU= 3.56 光年（ly）
$$p\text{ (in arc seconds)}=\frac{1 pc} {d\text{ (in pc)}}$$
因爲兩個單位是pc才能互相抵消！
最終方程：
$$d\text{ (in pc)}=\frac1{p\text{ (in arc seconds)}}$$
Apparent and absolute magnitudes
- Astronomers use magnitude as a scale to measure the brightness of celestial bodies
- The smaller the value, the brighter the body appears
沒錯！數字越細，亮度越光。很反直覺！、
由於以前的天文學家認爲星體有六個等級的光度，1等星最光，、6等星最暗（好似講緊龍珠）
他們定義“1等星比6等星光100倍”
因此，若要平均分出六個等級，每個等級的倍數差距是：
$$f=100^{1/5}\approx2.512$$
這個意思就是n等星比n+1等星光2.512倍
小編也不明白爲何要弄得如此複雜？？
Apparent magnitude
- Measures the apparent brightness of a celestial body as seen from the Earth
- Depends on how much light the body emits and its distance
| Situation | Apparent Magnitude |
|---|---|
| Sun | -26.7 |
| Moon (full) | -12.9 |
| Venus (max.) | -4.9 |
| Vega | 0.0 |
| Limit of a naked eye | 6.5 |
| Limit of a 5 m telescope | 20 |
| Limit of Hubble Space Telescope | 30 |
太陽apparent magnitude = -26.7 XD
早知有今日，何必當初定義六個等級的星
Absolute magnitude
- The apparent magnitude that a celestial body would have if it were at a distance of 10 pc away from the Earth
- Depends on how much light the body emits only
- Note: most celestial bodies appear very dim only because they are very far away
簡單而言就是公平競賽
Brightness and distance
- The brightness of a celestial body decreases with distance
- At a distance d from a star, the energy it radiates is spread evenly onto a sphere whose surface area is A = 4πd2
Intensity
$$I=\frac{\text{total power emitted}}{4\pi d^2}$$
- Measure the power per unit area
- Unit: W m-2
Blackbody
- A perfect absorber of radiation in theory
- No radiation is reflected or passes through a blackbody when it is illuminated by light
∴ A blackbody appears black when it is cold
- A blackbody is also a perfect emitter of radiation
- The radiation it emits is called blackbody radiation
- Blackbody radiation consists of a continuous spectrum of wavelengths
blackbody radiation curve
- A blackbody radiation curve tells us how the intensity of radiation changes with the wavelength
- A blackbody emits most radiation of wavelengths around the peak of its curve
Surface temperature
Ideally, the radiation emitted by a blackbody depends only on the temperature but not its chemical composition
When it gets hotter, its radiation curve:
- Becomes higher as a whole, gives out more radiation at all wavelengths
- The peak shifts to a shorter wavelength, gives out more radiation of shorter wavelengths
A hot star appears bluer because its spectrum peaks at a shorter wavelength
A cool star appears redder because its spectrum peaks at a longer wavelength
Spectral classification O B A F G K M
首先講講如何記，大部分書應該會寫這句：
‘Oh, Be A Fine Girl, Kiss Me!’
小編有同學改為：
‘Oh, Be A Fine Girl, Kill Me!’
| Surface Temperature (K) | Class | Color |
| 30,000 | O | Blue |
| 10,000 – 30,000 | B | Blue White |
| 7,500 – 10,000 | A | White |
| 6,000 – 7,500 | F | Yellow White |
| 5,000 – 6,000 | G | Yellow |
| 3,500 – 5,000 | K | Orange |
| 2,000 – 3,500 | M | Red |
absorption spectrum
A continuous spectrum with absorption lines is called an absorption spectrum
From the width and patterns of the absorption lines, astronomers deduce the abundances of various chemical elements on the surface of the stars
Stellar luminosity
The total radiation power that a celestial body gives out
The total energy emitted per unit time
- Luminosity = total radiation power given out by the star, does not drop with distance
- Intensity = radiation power per unit area,
drops with distance
因此
- luminosity有點像absolute magnitude
- intensity有點像apparent magnitude
Stefan–Boltzmann law
The radiation power given out per unit area J by a blackbody α the fourth power of its absolute temperature T :
$$J=\sigma\cdot T^4$$
$$\sigma=5.67\times10^{-8}\mathrm{~W~m^{-2}~K^{-4}}$$
$$L=4\pi R^2\cdot J$$
$$\begin{aligned}L&=4\pi R^2\cdot J\\&=4\pi R^2\cdot\sigma T^4\end{aligned}$$
Hertzsprung–Russell diagram
- The horizontal axis is the spectral classes, correspond to surface temperatures
- The vertical axis is the luminosity which is expressed in absolute magnitude or in solar luminosity Lʘ
Doppler effect
簡單而言，就是一個會移動的wave source
- The Doppler effect of light can be observed when light is emitted from a moving star:
The spectrum of a star may change as we observe it from the Earth
When a star is approaching
the observed wavelength appears shorter and the spectral lines shift to the
blue end ⇒ blue shift
When a star is receding
the observed wavelength appears longer and the spectral
lines shift to the red end ⇒ red shift
Doppler effect calculation
For the Doppler effect of light, the emitted wavelength λ and observed wavelength λ′ of a spectral line are related by:
$$\frac{\Delta\lambda}\lambda=\frac{\lambda^{\prime}-\lambda}\lambda\approx\frac{\nu_r}c$$
where Δλ is the change in wavelength, c is the speed of light in a vacuum and vr is the radial velocity
Expansion of the universe
Astronomer Edwin Hubble measured the spectra of some distant galaxies and found that they are all red shifted,
i.e. all moving away from us at high speeds
Hubble also discovered that a galaxy which is farther away has a higher recession velocity
Hubble’s law
\nu=H\cdot d
The velocity v is usually measured in km s-1, and the distance d in Mpc
H = 73.2 ± 1.7 km s-1 Mpc-1
Physics 必修課題
- Temperature and Thermometers
- Heat Capacity
- Change of State
- Gas laws and Kinetic Theory
- Motion
- Force
- More about Force
- Work Energy and Power
- Momentum
- Projectile Motion
- Uniform Circular Motion
- Gravitation
- Wave Motion
- Reflection Refraction and Diffraction
- Interference and Stationary Waves
- Light and Sound
- Reflection of Light
- Refraction of Light
- Lenses
- Electrostatics
- Circuit and Power
- AC and Domestic Electricity
- Electromagnetism
- Electromagnetic Induction
- Radiation and Radioactivity
- Rate of Decay and Uses of Radionuclides
- Nuclear Energy