T. , where σ = m 2 K. σ is called the Stefan - Boltzmann constant. The luminosity

Transcription

1 The Sun and Stars The Sun is a typical star with a mass of about kilograms or about earth masses and a radius R km or about 109 earth radii. It is a huge ball of gas held together by its own selfgravity. The spectrum of a star The Sun and other stars emits a typical stellar continuous spectrum with dark absorption lines (Fraunhoffer lines.) The continuous spectrum follows closely the laws of ideal blackbody radiation, or thermal radiation. The predominat color in the continuous spectrum of the sun is yellow with a wavelength λ max 500 nm. According to Wien s law λ max = T this gives a temperature of about 5,800 K for the photosphere. Other stars may be cooler (reddish) or hotter (bluish.) The absorption lines are produced by the gases in the outer atmospheres of the star. Luminosity of a star The total energy output of a star is called its luminosity, L. The luminosity of a star with absolute temperature T and radius R obeys the law of a blackbody radiator L = σt 4 4πR 2 8 Watt, where σ = m 2 K. σ is called the Stefan - Boltzmann constant. The luminosity increases rapidly with temperature; increasing the temperature

2 by a factor of 2 increases the luminosity by a factor of 16. Clearly a luminous star is hot and/or large, while a faint one is cool and/or small. For the Sun L = Watts. Apparent brightness of a star The apparent brightness of a star is measured by the intensity I of the light we receive from it. I is defined as the energy received by a detector of area 1m 2. Since the energy emitted by a star gets spread over a larger sphere as we get farther away from the star, we see that the intensity at a distance d from a star of luminosity L is I = L 4πd 2. I is measured in Watt/m 2. So, a star that is 10 times farther away appears 100 time less bright. The brightness is sometimes expressed not in Watt/m 2 but in magnitudes. The magnitude of Sirius (the brightest star in our sky) is -1.46, of Canopus -0.72, of Vega 0.04, of Deneb 1.26,... more or less according to an ancient scale developed by Hipparchus. Each step in magnitude amounts to a factor of about 2.5 in brightness. The faintest stars visible to the naked eye are of about 6th magnitude. The light year and the parsec The Astronomical Unit (AU) is the distance from the Earth to the Sun. It is 1 AU = km. Stars are much farther away than 1 AU Their distances are often given in light-years (lty). One light year is the distance that light travels in one year. The speed of light is c = km/s. One year has about seconds, so 1 lty = km/s s = 9.6

3 10 12 km. 1 AU is only 8.3 light-minutes. Since the apparent brightness or magnitude of a star depends on its distance from us, we define absolute brightness (or absolute magnitude) as the apparent brightness it would have if it were viewed from a distance of 10 parsecs. A parsec is approximately 3.26 lightyears and it is the distance from which the average radius of the earth s orbit around the sun (1 AU) would be seen to subtend an angle of 1 arcsecond. The parallax p of a star is the angle that the AU would subtend as viewed from the star. distant stars 1 AU = 150 million km Sun Earth p Star apparent position 1 1 AU apparent position 2 orbit of Earth As the earth goes around the sun in a year a nearby star shifts its position against the background of more distant stars by an angle p to each side of its average position. The angle p, is always very small so it is measured in arcseconds (arcs). If we can measure the angle p, we can deduce that the star is at a distance d given by d = 1 parsec arcs, so a star with p = 1 arcs has d = parsec. p The nearest star, Proxima Centauri, is 1.3 parsecs (4.2 lty) away

4 Nuclear Fusion The source of a star s energy is the process of nuclear fusion which takes small nuclei as a fuel and joins them to form larger nuclei whose mass is less than the mass of the fuel. The mass difference m is transformed into energy according to Einstein s formula E = mc 2. In a normal (main sequence) star the fuel is hydrogen that is converted into helium. The net nuclear reaction to produce one helium-4 nucleus is 4 1 1H e 4 2 He + mc 2. For the fusion reaction to occur very high temperatures are needed, on the order of 10 7 K, which are achieved in the core of the star. The sun s luminosity of L = Watts is produced by transforming kilograms of mass into energy each second due to the fusion of 600 million metric tons of hydrogen into helium each second. As a star gets older it begins to run short of hydrogen in its core and begins to fuse helium into carbon and so on with increasingly heavier elements. The life of a star, from youth, to maturity, to old age is the struggle between gravity that tends to contract the star and the gas pressure that opposes the contraction. The pressure is maintained by the fusion energy releases at the core of the star. As the star begins to run out of hydrogen at the core it begins to evolve. The Hertzsprung-Russell Diagram The spectrum of a star yields a lot of information about the star. When examined closely, the discrete absorption lines in the spectrum of a star allow us to classify the stars into spectral classes A, B, C,... Here class A stars have the strongest lines in the visible part (Balmer series) of the hydrogen (H) spectrum, stars with somewhat weaker H lines are in class B, and so on.

5 The order O, B, A F, G, K, M (remembered as Oh be a fine girl kiss me ) is arranged from hottest (bluest), (O about 30,000 K) to coolest (reddest) (M about 3,000 K). The Hertzsprung- Russell diagram is a plot of individual stars where the vertical axis is the absolute luminosity and the horizontal axis is the temperature. It reveals that stars that are mature lie on a diagonal line

6 called the Main Sequence. These are stars like the Sun, that are burning hydrogen in the core. As they start to run out of hydrogen in the core, gravity wins and the core contracts. Then they start to burn helium to produce carbon at the core. Hydrogen is still available outside the core so hydrogen burning begins in a shell. This causes the star to expand to become a red giant (bright, cool, and large) as it evolves toward the upper right in the H-R diagram. This will happen to the Sun in some 5 billion years, when it will swallow the Earth. Then they start to burn oxygen at the core, and helium in a shell, and hydrogen in another shell. An onion like pattern of layers forms with heavier and heavier elements fusing and the star may begin to expand and contract becoming a variable star, specially a Cepheid variable. These variable stars have a period-luminosity relation that makes them useful for measuring far away distances,

7 Death of Stars Eventually, the core becomes mostly iron. Beyond iron fusion does not produce energy and the star enters its final stages and collapses under its gravity. White Dwarfs If the star is not to massive, M star < 1.3M sun the star collapses to a white dwarf, where the iron core is supported by electron gas pressure. The density is very large, a solar mass compressed to the size of the earth. A teaspoon of white dwarf material has the mass of some 6 tons. The collapse of a white dwarf produces an explosion, a nova, that ejects gas that goes to form a planetary nebula, so called because it appears as a fuzzy ball in small telescope. Neutron Stars If 1.3M sun < M star the collapse is a supernova. This is one of the most violent events known in the universe. The crab nebula below is the remnant of a supernova explosion observed by the chinese in the year 1054 AD.

8 If M star < 4 or 5M sun the final stage after the supernova explosion is a neutron star. Every electron is packed into the nucleus and the star is a ball of neutrons with a few kilometers diameter and a tremendous density It is a solar mass compressed to a few kilometers diameter. One cc would have a mass of some kg, or about tons. Neutron stars spin very rapidly because of conservation of angular momentum and have a fantastically strong magnetic field. Their radiation can only escape in a beam along the magnetic axis. As the star spins it acts like the rotating beam of a lighthouse and the star appears as a pulsating star, called a pulsar. If the beam does not pass by the Earth, then we do not receive the pulses. At the center of the carb nebula there is a neutron star pulsating so that it seems to blink on and off as the light beam sweeps around, as shown in the figure below.

9 Black Holes If M star > 5M sun the final stage may be a black hole. If a mass M is so concentrated that its size is less than the Schwarzschild radius R s = 2GM/c 2, where G is Newton s gravitational constant and c is the sped of light, a black hole is formed. This is a solution of Einstein s equations of general relativity R s is the radius of the event horizon. Anything that gets closer than the event horizon will never escape back out of the event horizon, and will eventually be crushed at the center singularity. Astronomers have found at the core of many galaxies, including

10 ours, a rapidly spinning accretion disk with radiation coming out perpendicular to the disk, as shown below. Calculations show that there is an enormous mass at the center, millions of solar masses, in a fairly small area. Is it a black hole? Strange things happen to space and time as one approaches the event horizon. Imagine a mother ship in orbit far away from the black hole. A volunteer is sent in a spaceship to approach the black hole. As the volunteer approaches the event horizon his time slows down. As he speaks by radio to the mother ship his speech sslllooooowwwwsssss dddooooowwwwwnnnnnn more and more, although the volunteer does not particularly notice this. When he nears the event horizon he might say IIII ammmmm nearrrrrrrrrrrr... And we would never hear the end of it. He might survive into the black hole, but to us he would seem to be slowing down more and more as he approaches the event horizon, and it would take an infinite time by the mother ship clock for him to reach the event horizon. However the volunteer enters the blackhole in a finite time by his measure, but we would never receive any message from him. THIS IS WEIRD. Actually, he would probably be spaghettized by gravitational forces as he nears the event horizon.