哈工大天文學(xué)概論——銀河

1、Stellar Mass,Gravity and Orbits(2),Newtonian Gravity an explanation for planetary motions,Newton’s 3 Laws of Motion starting-pointNecessary ingredient to calculate motionsExact description of forces involvedHow to

2、quantify gravity ?Galileo’s Law of Falling BodiesNewton, the Apple and the Moon !,Galileo’s Mechanics Experiments,Prior to work with telescope, Galileo performed fundamental research on motion.Explored the rate of fal

3、ling bodies by dropping different weights, or sliding them down inclined planes.Law of Falling Bodies:In the absence of air, heavy objects and light objects fall at the same, constant rate of acceleration.,Astronaut Da

4、vid R. Scott,Apollo 15 (Falcon) near Hadley Rille, August 1971,The Idea of Universal Mutual Gravitation,Newton, in his Principia, formulated the Law of Universal Mutual Gravitation:Gravity is an Attractive force:Work

5、s to bring massive objects closer together.Gravity is a Universal force:Works everywhere in the Universe. Gravity is a Mutual force:Works between pairs of massive objects.,Towards an exact description,Force of gravit

6、y between any two objects depends only upon:The masses of the two objects:More massive objects feel a stronger force.The distance between them:Objects closer together feel a stronger force.It does not depend at all

7、on the shapes, colors, or compositions of the two objects.,Towards an exact description,Newton’s Idea :Force depends linearly on masses of objectsTwice as much mass ? twice the forceHalf as much mass ? half the force

8、Newton’s Laws then guarantee Galileo’s observation of Falling BodiesWhat is the dependence on distance ?,Comparing an apple to the moon,Newton’s fascinating thought process :Assumptions : Fall of an apple and path of

9、moon governed by the same principle ? Gravitational attraction (only !)Gravity independent of shape of EarthCareful numerical analysis leads to deeper understanding of gravity,What Newton knew about apples,Falling appl

10、es on Earth :Constant acceleration near surface of Earth : a=9.8 meters/second2Radius of Earth : 6378 km = 6378,000 meters (Eratosthenes !),What Newton knew about the Moon,Distance to the Moon ~380,000 km = ~ 60 x Ear

11、th radiusSidereal Orbital Period28.3 days? Speed of Moon on its path : ~ 1000 m/sec,,A Curved Path,But, of course the Moon really moves along a curved path:According to the first law, it is deflected from a straight-

12、line path by the force of gravity.This causes the moon to fall a little bit towards the Earth, deflecting its path into an arc.,The curved path of the Moon,How much does the moon have to fall in 1 second to ‘close the l

13、oop’ ?Simple geometry : 0.00136 meters (about 1.4 mm!)Newton’s clever idea : compare to apple’s fall !,Comparing apple and Moon,Apple on Earth falls in 1 second :dApple = 4.8 m Moon falls in 1 second :dMoon = 0.001

14、36 meters Ratio of these distances = ratio of accelerations = ratio of forcesdApple/dMoon = 4.8m/0.00136m ~ 3600,Comparing apple and Moon,Ratio of forces : FApple/FMoon ~ 3600Ratio of distances from center of Earth :

15、Dapple-Earth/DMoon-Earth ~ 60 Price winning question :What is the relation between these two ratios ?,Newton’s exact formula,The force of gravitational attraction between any two massive bodies is proportional to thei

16、r masses and inversely proportional to the square of the distance between their centers.,Gravitational Force Law,F = force due to gravity.M1 = mass of the first objectM2 = mass of the second objectd = distance between

17、 their centers.G = “Gravitational Force Constant”,Why is this such a powerful concept ?,The Law of Gravity is Universal:Governs the fall of apples on the EarthGoverns the fall of the Moon around the EarthGoverns the

18、fall of the Earth/Moon system around the SunGoverns the fall of the Sun around the center of the Milky Way Galaxy.Governs the fall of the Milky Way and Andromeda Galaxies in their mutual orbit...,The Gravitational Forc

19、e Constant,The force constant, G, is a number which gives the size of the gravitational coupling between two massive objects.G is very small, in metric units:G=6.7?10–11 Newtons meter2 / kilogram2The Newton is the met

20、ric unit of force:4.41 Newtons = 1 poundG has to be measured experimentally.,Example 1 : Weighing the Earth,Measure the acceleration of gravity by dropping weights (Galileo):a = 9.8 meters/second2Measure the radius o

21、f the Earth using geometry (Eratosthenes):RE=6378 kilometers = 6,378,000 metersEarth’s Mass is:,Example 2 : The Concept of Mutuality,What is the force of the Earth on the appleF = GME MA/RE2What is the apple’s accele

22、ration (2nd Law):a = F/MA = GME/RE2 = 9.8 meters/sec2The acceleration due to gravity is independent of the mass of the apple!,Example 2 : The Concept of Mutuality,The third law says that all forces come in equal yet op

23、posite pairs.What is the force of the apple on the EarthF = GME MA/RE2How much does the Earth accelerate towards the apple?a = F/ME = GMA/RE2a = 9.8 m/sec2 ? (MA/ME) = very small amount,Second Law of Orbital Motion,

24、Orbital motions conserve angular momentum.This doesn’t sound much like “equal areas in equal times”, but in fact it is the same thing.Angular Momentum:L = mvr = constantm=mass, v=velocity, r = distance from the cente

25、r of mass.,Angular Momentum & Equal Areas,L is a constant. If the distance changes, the velocity must change to compensate:Near Perihelion:Planet is closer to the sun, hence smaller rSpeed increases proportionall

26、y to compensate.Near Aphelion:Planet is farther from the sun, hence larger rSpeed decreases proportionally to compensate.,Third Law of Orbital Motion,Newton’s Generalization of Kepler’s 3rd Law:,P = period of the orbi

27、ta = semi-major axis of the orbitM1 = mass of the first bodyM2 = mass of the second body,A Third Law for Every Body,The proportionality now depends on the masses of the two bodies.For planets orbiting the Sun, Msun i

28、s so much bigger than any planet (even Jupiter, at 1/1000th Msun), we recover Kepler’s version:,Measuring Masses,Newton’s form of Kepler’s 3rd law is a way to measure masses from orbital motions!Mass of the Sun from the

29、 Earth’s orbit:Pearth = 1 year = 3.156?107 secondsaearth = 1 AU = 1.496?1011 meters,Universal Method for Masses,Measure mass of Jupiter from the orbits of the Galilean moons, since MJupiter>>MmoonsFind MJupiter

30、? 300 MearthMeasure the mass of the Earth and Moon by measuring their orbital parameters.Earth is only ~81x more massive than the Moon, so you have to use the full formula. Measure the masses of binary stars using the

31、 full formula.,The Predictive Power of Gravity,Newton’s description of planetary positions is only a start.It also allows quantitative new predictions.Halley’s Comet:Using Newtonian Gravity, Edmund Halley found that t

32、he orbit of the great comet of 1682 was similar to comets seen in 1607 and 1537.Predicted it would return in 1758/59.It did, dramatically confirming Newton’s laws.,The Discrepant Orbit of Uranus,In 1781, William Hersch

33、el discovered the planet Uranus orbiting beyond Saturn.By 1840, the discrepancies between the predicted and actual positions of Uranus grew larger than 1 arcminute.Two theorists, Adams in the UK and Leverrier in France

34、, predicted that the deviations were due to the gravitational influence of another, unknown massive planet beyond Uranus.,The Discovery of Neptune,Using the deviant motions of Uranus, they independently calculated where

35、this unknown 8th planet should be.Adams was ignored by English astronomers.Leverrier convinced Galle in Berlin to search.On Sept 23, 1845, Galle found Neptune only 52 arcminutes from where Adams and Leverrier predicte

36、d it would be!,The Why of Planetary Motions,Kepler’s Laws are descriptions of the motion:Arrived at by trial and error, and some vague notions about celestial harmoniesOnly describe the motions, without explaining why

37、they move that way.Newton provides the explanation:Kepler’s Laws are a natural consequence of Newton’s 3 Laws of Motion and gravitation.Gives the laws predictive power.,雙星和恒星的質(zhì)量(17.9),1. 雙星 (binary stars)由在彼此引力作用下以橢

38、圓軌道互相繞轉(zhuǎn)的兩顆恒星組成的雙星系統(tǒng)。 大部分的恒星位于雙星和聚星系統(tǒng)中。,研究雙星的意義,→驗(yàn)證萬(wàn)有引力定律→測(cè)量恒星質(zhì)量→研究恒星結(jié)構(gòu)（形狀、大小、大氣）→研究恒星演化→研究物質(zhì)交流和吸積過程,2.目視雙星和恒星質(zhì)量的測(cè)定,(1)目視雙星 (visual binaries)在望遠(yuǎn)鏡內(nèi)能夠分辨出兩顆子星的雙星系統(tǒng)。,Krueger 60,,,A Binary Star System,雙星的軌道運(yùn)動(dòng),兩顆子星圍繞公共質(zhì)心作

39、橢圓運(yùn)動(dòng)，半長(zhǎng)徑分別為a1和a2. 公共質(zhì)心位于橢圓的焦點(diǎn)上，子星在運(yùn)動(dòng)時(shí)與公共質(zhì)心始終位于一條直線上。橢圓軌道的大小與子星的質(zhì)量有關(guān)，M1a1＝M2a2如果以一顆子星以參照點(diǎn)，另一顆子星的相對(duì)運(yùn)動(dòng)也是一個(gè)橢圓，其半長(zhǎng)徑為a＝a1 + a2,,,,,,,,,,,,a,a1,a2,CM,Removing the tilt from a visual binary's orbit,目視雙星質(zhì)量的測(cè)定 利用Kepler第

40、三定律和Newton萬(wàn)有引力定律 得到：以太陽(yáng)－地球系統(tǒng)為參照其中a, P為雙星的軌道半長(zhǎng)徑和周期。,(2)天體測(cè)量雙星 (astrometric binaries),某些雙星的一顆子星較暗，很難觀測(cè)到，但通過較亮子星的自行軌跡的變化推測(cè)其伴星的存在。 雙星系統(tǒng)的質(zhì)心以直線運(yùn)動(dòng)，但每一顆子星的運(yùn)動(dòng)軌跡是波浪形的， 如天狼星（Sirius）。,,The Motion of Sirius A and B,3. 分光雙星 (

41、spectroscopic binaries),通過子星軌道運(yùn)動(dòng)引起的譜線的Doppler位移確定其雙星性質(zhì)。雙線、單線分光雙星。,譜線位移量也與雙星軌道傾角的大小有關(guān)。,,A,,A,,B,,B,,,,,,,,,,,,,,,,,B,A,,,,,,,A,B,視向速度曲線 (radial velocity curve)由子星譜線的Doppler位移得到的子星的視向速度隨時(shí)間的變化曲線。如子星1的軌道運(yùn)動(dòng)速度為V1,0，雙星軌道平面的

42、法線與視線的夾角為i, 它的視向速度為 由于 得到 且,4. 食雙星 (eclipsing binaries),子星相互交食造成亮度變化的雙星

43、。 光變曲線 (light curve)：子星間的相互交食造成雙星亮度的變化曲線。,由光變曲線可以得到： 兩顆子星的溫度比、軌道傾角（→恒星質(zhì)量）和恒星的大小。,,Eclipsing Binary,,,,Time,3,,,1,Brightness,,,,4,2,,,1,3,,,2,4,,,,,,,,,,,Algol (Beta Persei), the Prototype Eclipsing Binary,Dips from ma

44、gnitude 2.1 to 3.4 every 2.87 days. Each eclipse, including the partial phases, takes nearly 10 hours.,Measure Mass of a Single Star,Astronomers have directly measured the mass of a single star — the first time for any

45、solitary star other than our own Sun. The measurement has been done on a small red star located some 1,800 light-years from Earth, by use of the effect of microlensing.,5. 主序星的質(zhì)光關(guān)系和質(zhì)量－半徑關(guān)系,恒星的質(zhì)量和密度分布：,,~100,,10-6

46、 1.4 106,褐矮星,超巨星 太陽(yáng) 白矮星,M (M⊙),r (gcm-3 ),The Most and Least Massive Star,LBV 1806–20 with a diameter of at least 200 times the Sun's and a mass of roughly

47、150 solar masses.,Luminosity 2x10-6 L⊙, Temperature 700 K Mass 20 - 50 MJ, Separation 44 AU.,,主序星的質(zhì)光關(guān)系(mass-luminosity relation)：L ~ M 2-4主序星的質(zhì)量－半徑關(guān)系(mass-radius relation)：R ~ M 0.5-1,不同質(zhì)量的恒星在H-R圖上的分布,恒星的質(zhì)量決定了恒星

48、在H-R圖上的位置。高質(zhì)量的恒星明亮且高溫，位于主序帶的上部。低質(zhì)量的恒星黯淡且低溫，位于主序帶的下部。,Main Sequence,Mass matters !,The most important physical quantity for a Main Sequence star is the mass High mass stars are blue and very luminous.Low mass stars ar

49、e red and very dim.The Main Sequence is a mass sequence !Mass determines the history of a star How long it existsHow luminous it is during its lifeHow it will ‘die’ What remains of it,Sizes of Stars,Really hard to