Note that the following lectures include animations and PowerPoint effects such as fly ins and transitions that require you to be in PowerPoint's Slide Show mode (presentation mode). Chapter 9 The Family of Stars Outline I. Measuring the Distances to Stars A. The Surveyor's Method B. The Astronomer's Method C. Proper Motion II. Intrinsic Brightness A. Brightness and Distance B. Absolute Visual Magnitude C. Calculating Absolute Visual Magnitude D. Luminosity III. The Diameters of Stars A. Luminosity, Radius, and Temperature B. The H-R Diagram C. Giants, Supergiants, and Dwarfs 1 Outline D. Luminosity Classification E. Spectroscopic Parallax IV. The Masses of Stars A. Binary Stars in General B. Calculating the Masses of Binary Stars C. Visual Binary Systems D. Spectroscopic Binary Systems E. Eclipsing Binary Systems V. A Survey of the Stars A. Mass, Luminosity, and Density B. Surveying the Stars Light as a Wave (1) We already know how to determine a star’s • surface temperature • chemical composition • surface density In this chapter, we will learn how we can determine its • distance • luminosity • radius • mass and how all the different types of stars make up the big family of stars. Distances to Stars d in parsec (pc) p in arc seconds _1_ d = p Trigonometric Parallax: Star appears slightly shifted from different positions of the Earth on its orbit 1 pc = 3.26 LY The farther away the star is (larger d), the smaller the parallax angle p. 2 The Trigonometric Parallax Example: Nearest star, α Centauri, has a parallax of p = 0.76 arc seconds d = 1/p = 1.3 pc = 4.3 LY With ground-based telescopes, we can measure parallaxes p ≥ 0.02 arc sec => d ≤ 50 pc This method does not work for stars farther away than 50 pc. Proper Motion In addition to the periodic back-and- forth motion related to the trigonometric parallax, nearby stars also show continuous motions across the sky. These are related to the actual motion of the stars throughout the Milky Way, and are called proper motion. Intrinsic Brightness/ Absolute Magnitude The more distant a light source is, the fainter it appears. 3 Brightness and Distance (SLIDESHOW MODE ONLY) Intrinsic Brightness / Absolute Magnitude (2) More quantitatively: The flux received from the light is proportional to its intrinsic brightness or luminosity (L) and inversely proportional to the square of the distance (d): L F ~ __ d2 Star A Star B Earth Both stars may appear equally bright, although star A is intrinsically much brighter than star B. Distance and Intrinsic Brightness Example: Recall that: M Intensityagn. Ratio Betelgeuse Diff. App. Magn. m = 0.41 1 2.512 V 2 2.512*2.512 = (2.512)2 = 6.31 … … 5 (2.512)5 = 100 Rigel For a magnitude difference of 0.41 App. Magn. mV = 0.14 – 0.14 = 0.27, we find an intensity ratio of (2.512)0.27 = 1.28 4 Distance and Intrinsic Brightness (2) Rigel is appears 1.28 times brighter than Betelgeuse, But Rigel is 1.6 times further Betelgeuse away than Betelgeuse Thus, Rigel is actually (intrinsically) 1.28*(1.6)2 = 3.3 times brighter than Betelgeuse. Rigel Absolute Magnitude To characterize a star’s intrinsic brightness, define Absolute Magnitude (MV): Absolute Magnitude = Magnitude that a star would have if it were at a distance of 10 pc. Absolute Magnitude (2) Back to our example of Betelgeuse and Rigel: Betelgeuse B etelgeuseRigel V 0.41m 0.14 V -5.5M -6.8 d 152 pc 244 pc Rigel Difference in absolute magnitudes: 6.8 – 5.5 = 1.3 => Luminosity ratio = (2.512)1.3 = 3.3 5 The Distance Modulus If we know a star’s absolute magnitude, we can infer its distance by comparing absolute and apparent magnitudes: Distance Modulus = mV –MV = -5 + 5 log10(d [pc]) Distance in units of parsec Equivalent: d = 10(mM V – V + 5)/5 pc The Size (Radius) of a Star We already know: flux increases with surface temperature (~ T4); hotter stars are brighter. But brightness also increases with size: A Star B will be B brighter than star A. Absolute brightness is proportional to radius squared, L ~ R2. Quantitatively: L = 4 π R2 σ T4 Surface flux due to a Surface area of the star blackbody spectrum Example: Star Radii Polaris has just about the same spectral type (and thus surface temperature) as our sun, but it is 10,000 times brighter than our sun. Thus, Polaris is 100 times larger than the sun. This causes its luminosity to be 1002 = 10,000 times more than our sun’s. 6 , . sizes temperatures We know: , and different Organizing the Family of Stars: The Hertzsprung-Russell Diagram luminosities Stars have different different Temperature (or spectral type) versus To bring some order into that zoo of different types of stars: organize them in a diagram of Luminosity Hertzsprung-RussellTemperature Diagram or Luminosity Absolute mag. Absolute Spectral type: O B A F G K M The Hertzsprung-Russell Diagram Most stars are found along the Main Sequence 7 Same temperature, but much brighter than MS stars Must be → much larger Giant → Stars The Hertzsprung-Russell Diagram (2) Stars spend most of their active life time on the Main Sequence (MS). ut p., b e tem fs Sam war r → D fainte 10,000 times the sun’s radius The Radii of Stars in the Hertzsprung-Russell Diagram Rigel Betelgeuse 100 times the sun’s radius Polaris As large as the sun Sun 100 times smaller than the sun Ib Supergiants II Bright Giants Luminosity ClassesIb Ia BrightIa Supergiants II III Giants IV Subgiants III IV V Main-Sequence Stars V 8 Example Luminosity Classes G2V luminosity: • Our Sun: G2 star on the Main Sequence: G2Ib • Polaris: G2 star with Supergiant Spectral Lines of Giants Pressure and density in the atmospheres of giants are lower than in main sequence stars. => Absorption lines in spectra of giants and supergiants are narrower than in main sequence stars => From the line widths, we can estimate the size and luminosity of a star. → Distance estimate (spectroscopic parallax) Binary Stars More than 50 % of all stars in our Milky Way are not single stars, but belong to binaries: Pairs or multiple systems of stars which orbit their common center of mass. If we can measure and understand their orbital motion, we can estimate the stellar masses. The Center of Mass center of mass = balance point of the system. Both masses equal => center of mass is in the middle, rA = rB. The more unequal the masses are, the more it shifts toward the more massive star. 9 Center of Mass (SLIDESHOW MODE ONLY) Estimating Stellar Masses Recall Kepler’s 3rd Law: 2 3 Py = aAU Valid for the Solar system: star with 1 solar mass in the center. We find almost the same law for binary stars with masses MA and MB different from 1 solar mass: 3 ____aAU M A + MB = 2 Py (MA and MB in units of solar masses) Examples: Estimating Mass a) Binary system with period of P = 32 years and separation of a = 16 AU: ____163 M + M = = 4 solar masses. A B 322 b) Any binary system with a combination of period P and separation a that obeys Kepler’s 3. Law must have a total mass of 1 solar mass. 10 Visual Binaries The ideal case: Both stars can be seen directly, and their separation and relative motion can be followed directly. Spectroscopic Binaries Usually, binary separation a can not be measured directly because the stars are too close to each other. A limit on the separation and thus the masses can be inferred in the most common case: Spectroscopic Binaries Spectroscopic Binaries (2) The approaching star produces blue shifted lines; the receding star produces red shifted lines in the spectrum. Doppler shift → Measurement of radial velocities → Estimate of separation a → Estimate of masses 11 Spectroscopic Binaries (3) Typical sequence of spectra from a spectroscopic binary system Time Eclipsing Binaries Usually, inclination angle of binary systems is unknown → uncertainty in mass estimates. Special case: Eclipsing Binaries Here, we know that we are looking at the system edge-on! Eclipsing Binaries (2) Peculiar “double-dip” light curve Example: VW Cephei 12 Eclipsing Binaries (3) Example: Algol in the constellation of Perseus From the light curve of Algol, we can infer that the system contains two stars of very different surface temperature, orbiting in a slightly inclined plane. The Light Curve of Algol Masses of Stars in the Hertzsprung- Russell Diagram The higher a star’s mass, Masses in units of the more luminous solar masses (brighter) it is: 40 3.5 L ~ M H 18 ig h High-mass stars have m a s 6 much shorter lives than se s low-mass stars: 3 1.7 t ~ M-2.5 M life 1.0 as s 0.8 0.5 L m o w a s Sun: ~ 10 billion yr. s e s 10 Msun: ~ 30 million yr. 0.1 Msun: ~ 3 trillion yr. 13 Maximum Masses of Main-Sequence Stars Mmax ~ 50 - 100 solar masses a) More massive clouds fragment into b) Very massive stars lose smaller pieces during star formation. mass in strong stellar winds η Carinae Example: η Carinae: Binary system of a 60 Msun and 70 Msun star. Dramatic mass loss; major eruption in 1843 created double lobes. Minimum Mass of Main-Sequence Stars Mmin = 0.08 Msun At masses below 0.08 Msun, stellar Gliese 229B progenitors do not get hot enough to ignite thermonuclear fusion. → Brown Dwarfs Surveys of Stars Ideal situation: Determine properties of all stars within a certain volume.