Outline - Feb. 25, 2010 Black Hole Binary System Observational evidence for Black Holes (pgs. 600-601) Properties of Stars (Ch. 16) Luminosities (pgs. 519-523) Temperatures (pg. 524) Optical image of Cygnus X-1 A star is observed to be in orbit (by Doppler shift of spectrum) around an invisible object. If star is sufficiently large and is sufficiently close to its unseen companion, matter from the star may transfer over and build up in an accretion disk around the black hole. Radii (pgs. 534-535) Masses (pgs. 528-529) Spectral Type (pgs. 525-527) Hertzsprung-Russel Diagram (pgs. 530-533) Artist s conception of black hole binary system Cygnus X-1 Gas spirals toward BH, is accelerated up to high speeds by gravity, suffers violent collisions and heats up (millions of degrees = X-ray emission). Note: stars are not strong X-ray sources Constraining the Size of the Region that Contains the Invisible Mass World s Longest, Loudest Marching Band All band members play one short, staccato note. If the X-ray light flickers (on/off) very rapidly, this places a direct constraint on the size of the accretion disk (just outside the event horizon). Time scale over which you observe the light to be flickering must be smaller than the time it takes for light to travel across the accretion disk, or you won t notice the flickering - it will be smeared out!! Example: sound waves (time delay of arrival of sound due to its distance; e.g. thunder vs. lightning) What do you hear? Speed of sound = 343 m/s, so you don t hear the back row of the band until 10 seconds after the single note is played If band plays 1 staccato note every half second you would hear continuous sound (no quiet or off time) If band plays staccato notes more than 10 seconds apart, then you will notice breaks in the sound Band is 37.5 times the length of a football field Time to traverse the length of the band has to be shorter than the time between which the notes occur in order for you to experience off time (same goes for light) 1
Constraining the Size, II Diameter of the emitting region has to be less than the distance light could travel over a time equal to the time scale for flickering (Δt) Back to Cygnus X-1 Cygnus X-1 consists of a bright star with mass = 18 M sun and an unseen companion with mass = 10 M sun D < c Δt Rapidly flickering of X-rays says companion is much too small to be a star If Δt < 1 second, D < 300,000 km (i.e., 20% of the diameter of the sun) So, if you see flickering on a time scale less than about 4 or 5 seconds, the size of the emitting region (the accretion disk) is smaller than a star, so the companion cannot possibly be a star! Most theoretically conservative conclusion: companion is a black hole Many such X-ray binary systems exist in our Galaxy, with black holes that have masses between 4 M sun and 10 M sun These black holes were formed when an extremely massive star died in a supernova explosion M BH > 10 6 M sun M BH > 10 6 M sun If something (star, disk of gas) is orbiting about a black hole, the speed of rotation should decrease with distance from the black hole: V = (G M BH / R) 1/2 If you can measure V and R, you can deduce M BH Look for rapidly rotating disks at the very centers of big galaxies, motions of stars near the very center of our own galaxy (Milky Way). What do you find? Rapidly rotating disk within only 16 light years of the center of giant elliptical galaxy M87 gives M BH = 3 x 10 9 M sun 2
MBH > 106 M sun MBH > 106 M sun Rapidly rotating disk within only 0.64 light years of the center of spiral galaxy NGC 4258 gives MBH = 4 x 107 Msun Last word on Black Holes Over course of about 20 years astronomers have followed the motions of stars at the very center of the Milky Way, and have determined their orbits with very high accuracy. From orbital speeds of stars within 0.03 light years of the center of the Milky Way, MBH = 106 Msun Properties of Stars (Ch. 15) Black holes really do exit Black holes with mass MBH < 10 M sun probably result from death of massive star in a supernova explosion What colors can you see? Probably all large galaxies (galaxies at least as big as our own) harbor supermassive black holes at their centers (formation mechanism not yet understood) You have nothing to fear from black holes, you just want to stay far enough away that the maximum speed of your space ship exceeds the local escape speed. What does the color tell you? Are bigger stars on the image intrinsically more luminous that the smaller stars? It s time to put all of our tools to use!!!! 3
What is a star? (Two catch phrases) Luminosity vs. Brightness How much light do stars emit? The sun is a mass of incandescent gas. A star is a self-gravitating nuclear reactor. Incandescent (think standard light bulb ) gas = extremely dense, opaque gas, emits Black Body radiation Self-gravitating = star holds itself together by gravity (has to balance the pressure that pushes outward) Nuclear reactor = power source is nuclear fusion (E=mc 2 ); for 90% of a star s lifetime it is hydrogen that is fused, like a controlled H-bomb Luminosity (L) = intrinsic brightness of a source of light (amount of radiative energy emitted per second) Units of Luminosity: Watts (W), 1 W = 1 J/s Apparent Brightness (b) = amount of radiative energy passing through a given area per second Units of Brightness: Watts per square meter (W/m 2 ) Luminosity is INDEPENDENT of distance (d) to source Brightness DEPENDS on distance (d) to source Luminosity vs. Brightness Direct measurement of distance: stellar parallax b = L / (4π d 2 ) Flip the equation around and you get L = 4π b d 2 If you measure b and d, you can determine L. How would you do this for your favorite star in the sky? The farther is a star, the smaller is its parallax. If p is measured in arcseconds, the distance to the star is d = 1/p where d is in units of parsecs 1 parsec = 1 pc = 3.26 ly Most accurate measurements done from space (Hipparchos satellite); stars with distances < 1,000 pc. New satellite (GAIA) to be launched in 2012 will be able to measure distances of > 10,000 pc using parallax. 4
Example: What is the luminosity of Betelgeuse? L = 4π b d 2 We need b in units of Watts/meter 2 and d in units of meters, then L will be in units of Watts (W) How luminous are stars intrinsically? Brightness of Betelgeuse is 5.19x10-11 times the brightness of the sun and the brightness of the sun is 1.30x10 3 W/m 2. The brightness of Betelgeuse is b = (5.19x10-11 )(1.30x10 3 ) = 6.75x10-8 W/m 2 Huge range (a factor of 10 billion) in stellar luminosities: 10-4 L sun to 10 6 L sun Parallax of Betelgeuse is p = 0.0076 arcseconds, so distance to Betelgeuse is d = 1/p = 1/0.0076 = 131 pc = (131 pc)(3.09x10 16 m/pc) = 4.05x10 18 m So, the luminosity of Betelgeuse is L = 4π (6.75x10-8 )(4.05x10 18 ) 2 = 1.39x10 31 W Interestingly, the most intrinsically luminous stars are very rare, while the most intrinsically dim stars are very numerous Luminosity of the sun is L sun = 3.84x10 26 W, so Betelgeuse is intrinsically (1.39x10 31 / 3.84x10 26 ) = 36,000 times more luminous than the sun. Surface Temperatures of Stars Radii of Stars For λ max measured in cm and T measured in Kelvin, λ max = 0.29 / T T = 0.29 / λ max Since stars are a good approximation to being black bodies, we know that the relationship between Luminosity, Temperature, and Radius is L = 4π R 2 σ T 4 For Betelgeuse, λ max = 8.53x10-5 cm, so T = 0.29 / 8.53x10-5 = 3,400 K where σ = 5.67x10-8 W / (m 2 K 4 ) is the Stefan-Boltzmann constant Rearrange the equation and you get: R = (L / 4π σ T 4 ) 1/2 Range of stellar surface temperatures is small: about 3,000 K to about 30,000 K. Surface temperature of the sun is 5,800 K. Range of radii is about a factor of 50,000: 0.01 R sun ( white dwarf ) to 500 R sun ( supergiant star = Betelgeuse) 5
Stellar Masses Binary Stars Stellar Masses Binary Stars When the object is moving tangentially with respect to your line of sight, there is no Doppler shift and you see the zero velocity line pattern. The curved magenta line above shows you how one particular black absorption line sweeps up and down the spectrum due to orbital motion. As one object orbits around another, the lines in its spectrum will be shifted back and forth. When the object is coming towards you, the lines will be blueshifted. When the object is going away from you, the lines will be redshifted. To measure the masses of the stars in a binary system, we need to see the lines of both stars. When one star is moving away from us, the other will be coming towards us. The star with the smaller mass will have the larger, faster orbit (like the planets orbiting the sun). Stellar Masses Binary Stars Both stars orbit on ellipses, and they share a common focus (the center of mass ). The more massive star has the smaller orbit. The relative amount of the Doppler shift of the two sets of lines tells us the ratio of the stellar masses. If both sets of lines shift by the same amount, the stars have the same mass. If one set of lines shifts twice as far as the other set, the big star is twice as massive as the small star. If one set of lines shifts three times as far as the other set, the big star is three times as massive as the small star, etc. Get (M 1 + M 2 ) from Newton s form of Kepler s 3rd law and M 1 / M 2 from the relative Doppler shift. After a little algebra you get M 1 and M 2 separately (see Mathematical Insight 15.4 on pg. 529) Stellar Masses Masses of stars range from about 0.08 M sun to 150 M sun (about a factor of 1,800). Lower limit set by how much mass you need to start H-fusion going, and upper limit set by pressure-gravity balance. Just because a star has a radius that is bigger than the sun doesn t necessarily mean that it is more massive than the sun! Just because a star is more luminous than the sun doesn t necessarily mean that it is more massive than the sun! Just because a star is hotter than the sun, doesn t necessarily mean that it is more massive than the sun! It turns out that this is due to the fact that the radius, temperature and luminosity of stars evolve over time 6
Patterns to the Stars Stellar Spectra Spectral Type (Astronomers like to classify things / put them in bins) The letters O, B, A, F, G, K, M are called the spectral type of the star and describe the appearance of the spectrum (i.e., strong helium lines but weak hydrogen lines, strong hydrogen lines but no helium lines). Depending upon the surface temperature of the star, you see different absorption lines. The hottest stars show strong Helium lines, stars with T = 10,000 K show the strongest Hydrogen lines, and the very coolest stars show strong lines due to molecules (like titanium oxide). This is really is a temperature effect, it is not reflective of different chemical composition for the different stars! The spectral type classifications are historical and come from a time when we didn t know that the different spectra were due to different stellar temperatures. The notation persists today, though! Time-honored mnemonic: Oh Be A Fine Girl/Guy, Kiss Me Hertzprung-Russel (H-R) Diagram for Stars Take a huge random sample of stars and plot up their luminosity (vertical) and their temperature / spectral type (horizontal, with T increasing to the LEFT). Remarkably, you don t get a random plot at all! Roughly 90% of all stars fall on the Main Sequence. These are stars that produce energy by fusion of hydrogen (E = mc 2 ). Any star that is not on the Main Sequence is getting close to the end of its life. 7