1. The Family of Stars Lecture 12

3. Homework assignment 6 is due today. Homework 7 – Due Monday, March 19 Unit 52: TY4 Unit 54: P3, TY3 Unit 56: P1 Unit 58: RQ2, TY1, TY2 Announcements

4. Exam 2 The test is this Wednesday, March 7 th. Required Pencil/pen Equation sheet Recommended Calculator Scratch paper Covers Units 22-25,28-30,49-52,54,56, and 58 (more or less)

5. Exam 2 Details 85 minutes Wednesday (5-6:25 pm) Approximate Test Format: –20 multiple-choice questions (2 points each) –10 True/False questions (2 points each) –6 Short answer/problem questions (5 points each) –Equation sheet (10 points) Mostly conceptual

6. How to prepare? –Focus on the lecture material: These units contain a LOT more material than what we actually went over in class. You ARE responsible for understanding the topics covered in class (including details in the book that I may not have mentioned). You are NOT responsible for other stuff in these chapters not covered at all in lecture. Test Prep

7. Forbidden –Cell phone (not even for use as calculator) –Communication with anyone other than me –Textbook or any other reference material not written by you. Equation Sheet –Single page (front and back) HAND WRITTEN notes, equations, or any information you want to bring to the test. –Max size 8.5 x 11 inches. –Will be turned in – counts 10% of your test grade. Test Rules

8. Properties of Stars We already know how to determine a star’s surface temperature chemical composition surface density In this lecture, 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.

9. Distances to Stars Trigonometric Parallax: 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 1 pc = 3.26 LY

10. 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.

11. 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.

12. Intrinsic Brightness/ Absolute Magnitude The more distant a light source is, the fainter it appears.

13. Intrinsic Brightness / Absolute Magnitude 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): F ~ L __ d2d2 Star A Star B Earth Both stars may appear equally bright, although star A is intrinsically much brighter than star B.

14. Distance and Intrinsic Brightness Betelgeuse Rigel Example: App. Magn. m V = 0.41 Recall that: Magn. Diff. Intensity Ratio 12.512 22.512*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

15. Distance and Intrinsic Brightness 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

16. 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.

17. Absolute Magnitude Betelgeuse Rigel BetelgeuseRigel mVmV 0.410.14 MVMV -5.5-6.8 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

18. The Distance Modulus 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 = -5 + 5 log 10 (d [pc]) Distance in units of parsec Equivalent: d = 10 (m V – M V + 5)/5 pc

19. 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

20. 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 100 2 = 10,000 times more than our sun’s.

21. 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 or Absolute mag.

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

23. The Hertzsprung-Russell Diagram Stars spend most of their active life time on the Main Sequence (MS). Same temperature, but much brighter than MS stars  Must be much larger  Giant Stars Same temp., but fainter → Dwarfs

24. 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 100 times smaller than the sun RigelBetelgeuse Sun Polaris

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

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

27. 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)

28. 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.

29. 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.

30. Estimating Stellar Masses Recall Kepler’s 3 rd 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)

31. 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 rd Law must have a total mass of 1 solar mass.

32. Visual Binaries The ideal case: Both stars can be seen directly, and their separation and relative motion can be followed directly.

33. 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

34. Spectroscopic Binaries 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

35. Spectroscopic Binaries Time Typical sequence of spectra from a spectroscopic binary system

36. 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!

37. Eclipsing Binaries Peculiar “double-dip” light curve Example: VW Cephei

38. Eclipsing Binaries 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

39. The Light Curve of Algol

40. Masses of Stars in the Hertzsprung- Russell Diagram The higher a star’s mass, the more luminous (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. 0.5 18 6 3 1.7 1.0 0.8 40 Masses in units of solar masses Low masses High masses Mass L ~ M 3.5 t life ~ M -2.5

41. Maximum Masses of Main-Sequence Stars  Carinae M max ~ 50 - 100 solar masses a) More massive clouds fragment into smaller pieces during star formation. b) Very massive stars lose mass in strong stellar winds Example:  Carinae: Binary system of a 60 M sun and 70 M sun star. Dramatic mass loss; major eruption in 1843 created double lobes.

42. Minimum Mass of Main-Sequence Stars M min = 0.08 M sun At masses below 0.08 M sun, stellar progenitors do not get hot enough to ignite thermonuclear fusion.  Brown Dwarfs Gliese 229B

43. Surveys of Stars Ideal situation: Determine properties of all stars within a certain volume. Problem: Fainter stars are hard to observe; we might be biased towards the more luminous stars.

44. 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

45. Work on your equation sheet I have reversed Labs 7 and 9 so that we can do another “pencil & paper” lab after the test. For Next Time