Turning Stars Inside-Out: Chapter 30

1. Turning Stars Inside-Out:Chapter 30 1915: We have learned much about the surfaces of stars, but burning questions remain about what is below the surface. How were the stars made? How long do they live? Where does their energy come from? Do they die? But first, how heavy is a star?

2. Binary Stars Are the Key to Weighing a Star • Roughly half the stars we see have a seen or unseen companion gravitationally bound to it. • As we saw for our solar system, companions and their orbital parameters provide means for weighing stars First we need to remember some simple mechanics…

3. Center of Mass: like a see-saw m2 m1 r2 r1 Center of Mass: An imaginary point of balance of forces. The torques around this fulcrum cancel: m1gr1=m2gr2 so m1r1=m2r2 . For Sun-Earth, r1(m1/m2)=r2=1AU*3x10-6=4.5x105 m from center of Sun. Since Rsun=7x108m, center of mass just 0.1% from center of Sun!

4. unequal but comparable If m1 > m2 then r1 < r2 m2 F = m1 R=r1+r2 Gm1m2 R2 equal masses If m1 = m2 then r1 = r2 3 cases forrelative masses 2 1 (CM = center of mass) In all cases: m1r1 = m2r2 2 M1 dominant m1 >> m2 then r1 ~0 e.g., Sun and Earth 2

5. Newton's Law and Finding Stellar Masses • Recall from Newton’s law of gravity when m1>>m2: As an exercise you can use the requirement of matching periods of each star and definition of center of mass to show that: and with So 2 eqs and 2 unknowns (m1,m2). For visual binary we measure P, r1,r2 to get m1,m2. Note: need parallax distance for true radii (i.e., r=θ*D). Also works for ellipses (R is Semi- major axis) but need to determine orbit inclination, i, to get true R.

6. Visual Binary

7. Castor, a visual binary observed through a telescope (it was the first binary discovered, by Herschel in 1790). So if we measure the period and physical radii (I.e., angular position and distance) we can measure both of these masses. By 1925 this had been done for a few hundred stars

8. The Sun’s mass Centripetal force= Inward force needed to Travel in a circle • For circular motion, mv2/R=GMm/R2 and using v=2R/P it follows that • Explains Kepler’s Law. Can now measure P and R, then derive M ! Example: What is the Sun's () mass? • Assume mEarth << M. Then M e.g., transits of Venus

9. Enormously important tool Determine radial velocity (speed toward or away from us) Stationary emitter of waves, l0 Wave crests 4 3 2 1 Another Way to Find Binaries: from Motion: TheDopplerEffect

10. Moving emitter of waves l < l0 l > l0 5 4 3 2 1 4 3 2 1 l = l0 TheDopplerEffect Christian Doppler 1803-1853 (see Chap 26) “line of sight” effect red shift blue shift no shift

11. Doppler Shift: Another Way to Detect Binary Stars http://astro.ph.unimelb.edu.au/software/binary/binary.htm

12. Doppler shift of spectral lines Hg Hb Ha Hydrogen NO SHIFT (v=0) b b l l(Å) Blue Red R B 4861 6563 Redshift (v away from us) Blueshift (v toward us) b l R B Dl l0

13. Example: v = 90 km/s; • HaÞ v/c = 3 x 104 • Dl = 2 ÅÞ l = 6565Å, • since l0= 6563Å Note that stretching not just shifting, red moves farther than blue Doppler shift of spectral lines V C Doppler Shift (v<<c)

14. Spectroscopic Binaries Visual binaries are rare. More common are spectroscopic binaries where you measure radial velocity from Doppler shift of absorption lines (single or double set). Gives velocities and periods. -observer

15. Eclipsing Binaries Binary with orbit plane along line Of sight. Results in periodic variation of brightness. Can also yield sizes of stars and masses if System is also a double lined spectroscopic binary

16. Astrometric Binary When you can only see one star but it appears to Wobble around a fixed point (center of mass) , better if face-on

17. Frequently both members of binary cannot be resolved (i.e., not a visual binary) in which case other information is needed such as velocities (spectroscopic binary), or eclipses (eclipsing binaries) to determine inclination, eccentricity, PA, period, and masses. Such data can be compared to computer codes. Local Binaries Binaries in Solar Neighborhood

18. Example 1: Earth-Sun (M1=1, M2=0.01, a=1.0, e=0.0, I=0.0, w=0.0) Notice that the period is 1 year (change M1 and period changes) Change a=2.0 and the period changes, so need P,a to get M1) Go back to start, change e=0.5 Example 2: Star-Star: M1=2.0, M2=1.0, ratio of radii gives ratio of masses (try M1=3.0, M2=1.0 to see ratio r1/r2 change). Period gives sum of the masses (e.g., try M1=4.0, M2=2.0, same ratio, shorter period). This is a visual binary. Example 3: star-star, real life! M1=2.0, M2=1.0, e=0.5 (same period), turn off trail (hard to find CM!). Now, incline orbit I=45 degrees. Notice difference between Earth view and priveledged view. Also notice that we need Doppler shifts to get inclination. This requires system to be a spectroscopic binary. Now try changing w. Finally, try w=90, I=90….need a computer to solve this! Slow down, remove trail, add bad weather, now you are an astronomer! Lets Play with Binaries http://astro.ph.unimelb.edu.au/software/binary/binary.htm

19. Stellar masses along the Main Sequence Table of main sequence stellar parameters SpectralRadius Mass Luminosity Temp. local Examples Type R/R☉ M/M☉ L/L☉K O5 18 40 500,000 38,000 Sanduleak −66° 41, Zeta Puppis B0 7.4 18 20,000 30,000 Phi1 Orionis B5 3.8 6.5 800 16,400 Pi Andromedae A A0 2.5 3.2 80 10,800 Alpha Coronae Borealis A A5 1.7 2.1 20 8,620 Beta Pictoris F0 1.4 1.7 6 7,240 Gamma Virginis F5 1.2 1.29 2.5 6,540 Eta Arietis G0 1.05 1.10 1.26 6,000 Beta Comae Berenices G2 1.00 1.00 1.00 5,920 Sun, Alpha Centauri A G5 0.93 0.93 0.79 5,610 Alpha Mensae K0 0.85 0.78 0.40 5,150 70 Ophiuchi A K5 0.74 0.69 0.16 — 61 Cygni A M0 0.63 0.47 0.063 3,920 Gliese 185 M5 0.32 0.21 0.0079 3,120 EZ Aquarii A M8 0.13 0.10 0.0008 — Van Biesbroeck's star

20. Sir Arthur Eddington CompilesLargest Possible list of Star Masses One of the greatest astrophysicists…knighted in 1930 This was a Hard won dataset Luminosities And distances… 1882-1944 Eclipse expedition in 1919, Eddington limit, etc

21. The Mass-Luminosity Relationshipon Main Sequence L  M4 Masses range from 0.1 to 50 Msun Eddington’s Original data Absolute Magnitude Sun Log Mass Luminosity is (roughly) proportional to the 4th power of the mass. (Twice the mass means sixteen times the luminosity) (M = 2 x 1033 g) This is a terribly important discovery! Why does star need mass? Fuel! But, ever hear of burning a candles at both ends…?

22. Question A star with half the mass of the Sun would have a luminosity of approximately: a) 6% L b) 50% L c) 100% L d) 1600% L e) none of the above answer, a) L= L x(M/M)4= L (1/2)4= L (1/16) ~ 6% L

23. Mass, Luminosity, Lifetimeon the Main Sequence Energy supply = Mass x (energy/kilogram) Energy use rate = Luminosity [energy/time] • Lifetime =energy supply energy use rate • So lifetime, t~M/L • From binaries, we saw there is a relation between Mass and Luminosity: L~M4 • So, t~M/L = M/M4=M-3= 1/M3, bigger stars-shorter lives • Massive stars: like Bonfires, live fast, die young • Small stars: like candles, misers, live a long time

24. Compare to the Sun In 1924 the Earth was known to be a few billion years old…(assume the Sun’s age near this) So stars 10 times more massive would have lives ~103 times shorter or only a few million years! What about smallest stars? So stars have a large range of lifetimes!

25. Is this the Lifecycle of Stars? Big Spenders ?? Misers Recall we found that stars come in a wide variety of sizes, luminosities and temperatures. Are these stages of their lives? Where are newborns and how do they age?

26. How Does Sun Get Energy? Chemical energy? • Say the Sun was made out of coal. • 1 kg of coal releases 2x107 Joules • If Sun was made of coal, burning 21030 kg of coal would release 41037 J. At Solar Luminosity (4x1026 Joules/sec) just 3800 yrs of life! • (original argument by Lord Kelvin, 19th century) • Most chemical energy sources are within a factor of 100 of this.

27. Fuel Efficiency Heat of combustion for common fuels Fuel MJ/kg Mcal/kg BTU/lb Hydrogen 141.9 33.9 61,000 Gasoline 47 11.3 20,400 Diesel 45 10.7 19,300 Ethanol 29.8 7.1 12,800 Propane 49.9 11.9 21,500 Butane 49.2 11.8 21,200 Wood 15 3.6 6,500 Coal 15–27 4.4–7.8 8,000–14,000 Natural Gas ~54 ~13 ~23,000

28. Question: If the Sun was burning regular gas (“fill er up!”) to power it how long would it last? a) Since the Dinosaurs roamed Earth b) less than recorded human history c) less than the Obama Administration d) more than a billion years answer, b) twice as long as coal, about 7,000 years

29. Announcements • Turn in lab Monday • Hw due Monday • Midterm March 13 • Review session Mon, March 11, 8pm • Watch video lecture, short quiz Monday • Last time…Sun around for billions of years but Sun’s mass of coal would only last 3800 years. Another source?

30. Way you did it in Calculus class, volume integral Potential energy of two bodies, m1, m2 Add up contribution of each shell, m2=(4r2dr) central, m1=4/3  r3 R Easy Way: Pretend half mass in outer shell, half in center: GM2/4R A Better Energy Source: Gravity! • Say the sun was slowly contracting – how much energy could this liberate? Best guess in 19th century. Let’s add up potential energy of shells falling to center

31. Conclusion of all this Age of the Earth: 4.5 Gyr4.5Gyr >> 0.03 Gyr Need a much more productive energy source ! A Great mystery in the 1920’s !

32. thermal (heat) pressure Another Mystery:If stars are gasbags… Star gravity Eddington wondered, if stars are mostly gas, why don’t stars 1) just waft away or 2) collapse into a small solid? Sun does appear to be in equilibrium Can demo here. What happens and why? http://www.youtube.com/watch?v=j3lQUHOPCgI

33. In any given layer of a star, there is a balance between the thermal pressure (outward) and the weight of the material above pressing downward (inward). This balance is called hydrostatic equilibrium. Pressure must be intense near center! Eddington Figures Out What Holds up a Star If a star became too compact, the increase in pressure (heat) would make it expand again. Too big and the drop in Pressure (heat) would make it contract. self-regulating bags of hot air So stars are self-regulating bags of hot air…neither waft away Nor shrink into oblivion….

34. A Dangerous Existence A star is constantly putting out energy just to keep its equilibrium No energy source lasts forever. What happens when the “fire” goes out? What holds the star (gas bag) up? (A star is like a shark that will die if it stops swimming!)