The Family of Stars Chapter 9

If you want to study anything scientifically, the first thing you have to do is find a way to measure it. But measurement in astronomy is very difficult. Astronomers must devise ingenious methods to find the most basic properties of stars. As you will see in this chapter, combining those basic properties reveals important relationships among the family of stars. Your study of stars will reveal answers to five basic questions: How far away are the stars? How much energy do stars make? How big are stars? What is the typical star like? Guidepost

Making measurements is the heart of science, and this chapter will answer two important questions about how scientists go about their work: How can scientists measure properties that can’t be directly observed? How do scientists accumulate and use data? With this chapter you leave our sun behind and begin your study of the billions of stars that dot the sky. In a sense, the star is the basic building block of the universe. If you hope to understand what the universe is and how it works, you must understand the stars. Guidepost (continued)

I. Measuring the Distances to Stars A. The Surveyor's Method B. The Astronomer's Method C. Proper Motion ( 自行 ) II. Intrinsic Brightness A. Brightness and Distance B. Absolute Visual Magnitude ( 絕對星等 ) 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 Outline

D. Interferometric Observations of Diameter E. Luminosity Classification F. Luminosity Classes G. Spectroscopic Parallax ( 視差 ) 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 Outline

The Properties of Stars 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 Trigonometric Parallax ( 三角視差 ): Star appears slightly shifted from different positions of the Earth on its orbit The farther away the star is (larger d), the smaller the parallax angle p. d = __ p 1 d in parsec (pc) p in arc seconds (arcsec) 1 pc = 3.26 LY

The Trigonometric Parallax Example: Nearest star,  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.

Intrinsic Brightness / Absolute Magnitude (2) More quantitatively: The flux received from the light is proportional to its intrinsic brightness or luminosity (L, unit: erg/s or J/s) and inversely proportional to the square of the distance (d): F ~ L __ d2d2 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 Betelgeuse Rigel Example: App. Magn. m V = 0.41 Recall that: Magn. Diff.Intensity Ratio *2.512 = (2.512) 2 = 6.31 …… 5(2.512) 5 = 100 App. Magn. m V = 0.14 For a magnitude difference of 0.41 – 0.14 = 0.27, we find an intensity ratio of (2.512) 0.27 = 1.28

Distance and Intrinsic Brightness (2) Betelgeuse Rigel Rigel is appears 1.28 times brighter than Betelgeuse, Thus, Rigel is actually (intrinsically) 1.28*(1.6) 2 = 3.3 times brighter than Betelgeuse. but Rigel is 1.6 times further away than Betelgeuse.

Absolute Magnitude To characterize a star’s intrinsic brightness, define Absolute Magnitude (M V ): Absolute Magnitude = Magnitude that a star would have if it were at a distance of 10 pc.

Absolute Magnitude (2) Betelgeuse Rigel BetelgeuseRigel mVmV MVMV d152 pc244 pc Back to our example of Betelgeuse and Rigel: Difference in absolute magnitudes: 6.8 – 5.5 = 1.3 => Luminosity ratio = (2.512) 1.3 = 3.3

The Distance Modulus ( 距離模數 ) If we know a star’s absolute magnitude, we can infer its distance by comparing absolute and apparent magnitudes: Distance Modulus = m V – M V = log 10 (d [pc]) Distance in units of parsec Equivalent: d = 10 (m V – M V + 5)/5 pc

The Size (Radius) of a Star We already know: flux increases with surface temperature (~ T 4 ); hotter stars are brighter. But brightness also increases with size: A B Star B will be brighter than star A. Absolute brightness is proportional to radius squared, L ~ R 2 Quantitatively: L = 4  R 2  T 4 Surface area of the star Surface flux due to a blackbody spectrum

Example: Star Radii Polaris ( 北極星 ) 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 = 10,000 times more than our sun’s.

Organizing the Family of Stars: The Hertzsprung-Russell Diagram We know: Stars have different temperatures, different luminosities, and different sizes. To bring some order into that zoo of different types of stars: organize them in a diagram of Luminosity versus Temperature (or spectral type) Luminosity Temperature Spectral type: O B A F G K M Hertzsprung-Russell Diagram ( 赫羅圖 ) or Absolute mag.

The Hertzsprung-Russell Diagram Most stars are found along the Main Sequence

The Hertzsprung-Russell Diagram (2) Stars spend most of their active life time on the Main Sequence (MS). Same temperature, but much brighter than MS stars

The Brightest Stars The open star cluster M39 The brightest stars are either blue (=> unusually hot) or red (=> unusually cold).

The Radii of Stars in the Hertzsprung-Russell Diagram 10,000 times the sun’s radius 100 times the sun’s radius As large as the sun Rigel Betelgeuse Sun Polaris

Eyeball: Balloon = Sun :Supergiants The Relative Sizes of Stars in the HR Diagram

Luminosity Classes Ia Bright Supergiants Ib Supergiants II Bright Giants III Giants IV Subgiants V Main-Sequence Stars Ia Ib II III IV V

Example: Luminosity Classes Our Sun: G2 star on the Main Sequence: G2V Polaris: G2 star with Supergiant luminosity: G2Ib

Spectral Lines of Giants => Absorption lines in spectra of giants and supergiants are narrower than in main sequence stars Pressure and density in the atmospheres of giants are lower 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, r A = r B The more unequal the masses are, the more it shifts toward the more massive star.

Estimating Stellar Masses Recall Kepler’s 3rd Law: P y 2 = a AU 3 Valid for the Solar system: star with 1 solar mass in the center We find almost the same law for binary stars with masses M A and M B different from 1 solar mass: M A + M B = a AU 3 ____ Py2Py2 (M A and M B in units of solar masses)

Examples: Estimating Mass a) Binary system with period of P = 32 years and separation of a = 16 AU: M A + M B = = 4 solar masses 16 3 ____ 32 2 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

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

Spectroscopic Binaries (3) Time Typical sequence of spectra from a spectroscopic binary system

Eclipsing Binaries Usually, the 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

Eclipsing Binaries (3) 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. Example: Algol in the constellation of Perseus

The Light Curve of Algol

Masses of Stars in the Hertzsprung- Russell Diagram The higher a star’s mass, the brighter it is: High-mass stars have much shorter lives than low-mass stars: Sun: ~ 10 billion yr. 10 M sun : ~ 30 million yr. 0.1 M sun : ~ 3 trillion yr. Low masses High masses Mass L ~ M 3.5 t life ~ M -2.5

Surveys of Stars Ideal situation for creating a census of the stars: Determine properties of all stars within a certain volume

Surveys of Stars Main Problem for creating such a survey: Fainter stars are hard to observe; we might be biased towards the more luminous stars.

A Census of the Stars Faint, red dwarfs (low mass) are the most common stars. Giants and supergiants are extremely rare. Bright, hot, blue main-sequence stars (high-mass) are very rare.