1	Astr 1050 Wed. Feb. 16, 2005 Today: Finish Chapter 6 Chapter 7, the Sun for Friday
2	Planck and other Formulae Planck formula gives intensity of light at each wavelength It is complicated. We’ll use two simpler formulae which can be derived from it. Wien’s law tells us what wavelength has maximum intensity Stefan-Boltzmann law tells us total radiated energy per unit area
3	Examples of Wien’s law What is wavelength at which you glow? Room T = 300 K so This wavelength is about 20 times longer than what your eye can see. Camera in class operated at 7-14 μm. What is temperature of the sun – which has maximum intensity at roughly 0.5 mm?
4	Example of the Stefan-Boltzmann law Suppose a brown-out causes the temperature of a lamp filament to drop to 0.9 of its original value. By what factor does the light output of the lamp drop? Using the Stefan-Boltzmann law (with the numerical value of s) we could have calculated how big (in m²) a light filament would have to be to emit 100 W of light, at any given temperature. We could also use it to find the size of a star, if we know how much light energy that star emitted
5	Kirchoff’s laws Hot solids emit continuous spectra Hot gasses try to do this, but can only emit discrete wavelengths Cold gasses try to absorb these same discrete wavelengths In stars we see absorption lines – what does that tell us? Stars have “atmospheres” of gasses Stars must be colder on the outside, hotter on the inside
6	Hydrogen Lines Energy absorbed/emitted depends on upper and lower levels Higher energy levels are close together Above a certain energy, electron can escape (ionization) Series of lines named for bottom level To get absorption, lower level must be occupied Depends upon temperature of atoms To get emission, upper level must be occupied Can get down-ward cascade through many levels
7	Which levels will be occupied? The higher the temperature, the higher the typical level Collisions can knock electrons to higher levels, if moving atoms have enough kinetic energy At T ~ 300 K (room T) almost all H in ground state (n=1) At T ~ 10,000 K many H are in first excited state (n=2) At T ~ 15,000 K many H are ionized Because you have highest n=2 population at ~10,000K you also have highest Balmer line strength there. This gives us another way to estimate temperatures of stars
8	Sense larger T range using many atoms Different atoms hold on to electrons with different force Use weakly held electrons to sense low temperatures (Fe, Ca, TiO) TiO molecule is destroyed above 4000K Ca has lost 1 electron by ~5000K, but still has others to give lines Use moderately held electrons to sense middle temperatures (H) Below 6000 K most H electrons in lowest state – can’t cause Balmer lines Above 15,000K most H electrons completely lost (ionized) Use tightly held electrons to sense high temperatures (He, ionized He) Below 10,000K most He electrons in ground state – just like H, no visible absorption lines Above 15,000K most H has lost one electron, but still has a second one to cause absorptions
9	Classification of stars O B A F G K M scheme Originally in order of H strength – A,B,etc Above order is for decreasing temperature Standard mnemonic: Oh, Be A Fine Girl (Guy), Kiss Me Use numbers for finer divisions: A0, A1, ... A9, F0, F1, ... F9, G0, G1, ...
10	Composition of Stars Somewhat complicated – we must correct for temperature effects Regular pattern: More of the simplest atoms: H, then He, ... Subtle patterns later – related to nuclear fusion in stars
11	Doppler effect Effect similar in light and sound Waves compressed with source moving toward you Sound pitch is higher, light wavelength is smaller (bluer) Waves stretched with source moving away from you Sound pitch is lower, light wavelength is longer (redder) v = velocity of source c = velocity of light (or sound) l = apparent wavelength of light l_o = original wavelength of light
12	Doppler effect examples Car with horn blowing, moving away from you at 70 MPH. Speed of sound is ~700 MPH = 1000 ft/sec Original horn pitch is 200 cycles/sec Þ l_o ~ 5 ft Star moving toward you at 200 km/sec = 2.0´10⁵ m/s Speed of light c = 3.00 ´ 10⁸ m/s Original Ha l_o= 0.65647 mm
13	Chapter 7: The Sun Basic Properties of the Sun Radius: R = 6.96 ´ 10⁵ km (q=diameter/distance) Mass: M = 1.99 ´ 10³⁰ kg
14	Basic Properties of the Sun Surface Temperature 5800 K (from l_max or spectral type) Not all that hot by laboratory standards Central Temperature 15 ´ 10⁶ K (explained later) Central temperature IS very high Luminosity (L) 3.8 ´ 10²⁶ J/s ( L = sT⁴_surface ´ 4 p R²_sun) ( L = F_{at Earth} ´ 4 p R²_{1 AU}) Will be important for understanding energy generation in Sun
15	Physical State of material in Sun At these T’s, r’s, hydrogen will be a gas At high enough T, as pressure (P) increases and r increases, you never really get a “liquid”, just a dense gas. H ionization? On outside, H mostly neutral (a small fraction is ionized) remember H ionized and Balmer lines gone only above 10,000 K Over most of interior, H completely ionized separate electrons (e^-) and protons (p⁺) Ionized gas called a “plasma” No discrete “surface” – just increasing r, T, P, and “opacity”
16	How we determine T, r, P vs depth I. From theory: Pressure (P) and density (r) must increase with depth Weight of overlying gas compresses lower material -- “Hydrostatic equilibrium” Temperature (T) must increase with depth Energy is flowing out of the sun – and it flows from hot to cold -- so hot inside Numerical modeling of details let us calculate T(r), r(r), P(r)
17	How we determine T, r, P vs depth II From observations of “oscillations” or “solar seismology” The sun oscillates like a bell (or the air in an organ pipe) The frequency depends upon sound speed, which depends upon T(r), r(r), P(r) Observations from “Global Oscillation Network Group (GONG) telescopes. From using Kirchoff’s laws The Sun looks like continuous emission: Solid or hot dense gas Absorption lines in the spectrum: Cooler gas between us and the dense gas
18	The “surface” of the Sun No discrete “surface” – just increase r, T, P, and “opacity” “Surface” or photosphere defined by depth from which visible photons can escape. Opacity depends on wavelength, so apparent “surface” will be at different depths for different wavelengths High opacity in absorption lines because these photons easily absorbed/emitted Won’t see very far in at these wavelengths. Low opacity in between absorption lines Can see in deeper at these wavelengths. Eventually r so high gas opaque at all wavelengths (just as in solid) “surface” high = cool = dark in lines; deep = hot = bright between lines
19	T,r dependence upon depth inside the sun
20	Complicated T dependence at the very edge We see emission lines at some wavelengths: Implies very THIN HOT overlying gas at top of atmosphere Gas is so thin it has trouble radiating heat away Sound waves or magnetic fields heat thin gas Chromosphere (“colored region” glows at a few wavelengths) Corona (“crown” seen during solar eclipses) Solar Wind ( escaping wind of tenuous gas)
21	Detailed structure of the outer photosphere CONVECTION: Granulation and Supergranulation Heat carried by actual motion of gas Different than radiative transport energy carried by photons dominates deeper in sun SUNSPOTS Darker (and cooler) regions of sun Strong magnetic fields limit convection Come and go in 11 (really 22) year cycle Magnetic energy releases cause “flares” Material ejected causes aurora
22	Nuclear forces and nuclear energy What holds protons in the nuclei of atoms? Coulomb (electric) repulsion should make protons fly apart They are packed so close together – must have very strong force to hold them Nuclear “Strong force” attracts nucleons (protons, neutrons) Why doesn’t strong force collapse all atoms into a giant nucleus? Nuclear Strong force is very short range falls off quickly after a few proton radii Coulomb force is long range falling off only as 1/r² At large distances only coulomb force is important Þ repulsion Nuclear Strong Force important only close together Requires very high speed (high temperature) collision for fusion
23	The Curve of Binding Energy If you keep adding protons to a nucleus? Coulomb repulsion continues to increase new proton feels repulsion from all other protons Strong force attraction reaches limit new proton can’t feel attraction from protons on far side of a big nucleus Gain energy only up to point where Coulomb repulsion outweighs strong force attraction. Most “stable” nucleus is ⁵⁶Fe (26 protons, 30 neutrons, 56 total) Release energy by fusion of light nuclei to make heaver ones– up to ⁵⁶Fe Release energy by fission of heavy nuclei to make lighter ones – down to ⁵⁶Fe
24	The Role of the Nuclear “Weak Force” Why can’t we keep adding neutrons rather than protons? They feel strong force attraction – with no Coulomb repulsion You should be able to get lots of energy by adding neutrons Nuclear Weak Force can (slowly) convert protons to neutrons and back The reactions involve an electron or “positron” to keep charges balanced The reactions produce a new almost massless particle called a neutrino p⁺ + e^- Û n + n p⁺ is proton e^- is electron n is neutron n is neutrino p⁺ Û n + e⁺+ n e⁺ is positron = antiparticle of electron If this second reaction happens then the positron annihilates the next electron it encounters, thereby producing the equivalent of the first reaction. The weak force likes to keep ~equal protons and neutrons in a nucleus The proton repulsion tips the balance towards slightly more neutrons in big nuclei
25	The four fundamental forces Gravity Dominates on astronomical scales Electromagnetic Holds atoms together: Chemistry Strong force Holds nuclei together: Nuclear energy Weak force n Ûp⁺, e^-Radioactive decay (will also play critical role in solar fusion)
26	Fusion in the sun I. Have lots of hydrogen (p⁺ and e^-) – what can we make from it? If ²He (2 protons, 0 neutrons) were stable, fusion would be “easy” Run two protons into each other at fast enough to overcome Coulomb repulsion Once they get close enough strong force takes over, and holds them as nucleus
27	Fusion in the sun II. “Unfortunately” ²He isn’t stable To get stable He need to add one or two neutrons to: Increase Strong Force, without increasing Coulomb force Not really “unfortunate” – If ²He were stable: Sun would burn energy way too fast – and would have gone out by now Weak force converts proton to neutron–fusion will be slow In solar fusion no excess neutrons lying around Hydrogen bombs use deuterium: ²H = (p⁺n) or tritium: ³H = (p⁺n n) to provide it
28	The Proton-Proton chain The first step is slow because it relies on two rare events happening simultaneously Two protons collide with enough energy to overcome the Coulomb barrier While they are close the weak force turns one proton into a neutron The resulting combination of a proton and a neutron IS a stable nucleus ¹H + ¹H ® ²H + e⁺+ n p⁺ + p⁺ ® (p⁺ n) + e⁺ + n The next two steps go quickly because they rely only on the strong force ²H + ¹H ® ³H (p⁺ n) + p⁺ ® (p⁺ n n ) ³H + ³H ® ⁴He + ¹H + ¹H (p⁺ n n) + (p⁺ n n) ® (p⁺ p⁺ n n) + p⁺ + p⁺ The net effect is 4 ¹H ® ⁴He
29	Energy Released? Could work it out “classically” by strength of forces Classical mechanics doesn’t work at this scale – Need quantum mechanics Strength of nuclear forces not originally known Use E=mc² to do accounting Mass is a measure of the energy stored in a system Loss of mass from a system means release of energy from that system Compare mass of four ¹H to mass of one ⁴He 6.693 ´ 10^-27 kg - 6.645 ´ 10^-27kg = 0.048 ´ 10^-27 kg drop in mass E = mc² = 0.048 ´ 10^-27 kg ´ (3 ´ 10⁸ m/s)^{2 =} 0.43 ´ 10^-11 kg m²/s² = 0.43 ´ 10^-11 J (note == a Joule is just shorthand for kg m²/s²) So 4.3 ´ 10^-12 J of energy released This is huge compared to chemical energy: 2.2 ´10^-18 J to ionize hydrogen
30	About how long will Sun’s fuel last? Luminosity of sun: 3.8 ´ 10²⁶ J/s H burned rate: H atoms available: Lifetime: In reality not all the atoms we start with are H, and only those near the center are available for fusion. The structure of the sun will change when about 10% of the above total have been used, so after about 10 billion years.
31	Testing solar fusion model Does lifetime of sun make sense? Oldest rocks on earth ~4 billion years old Oldest rocks in meteorites ~4.5 billion years old Other stars with higher/lower luminosity Causes for different luminosity Lifetimes of those stars Look for neutrinos from fusion Complicated story – due to neutrino properties Example of how astronomy presents “extreme” conditions
32	Neutrinos Generated by “weak” force during p⁺® n + e⁺ + n “Massless” particles which interact poorly with matter In that first respect, similar to photons Can pass through sun without being absorbed Same property makes them very hard to detect Davis experiment at Homestake Mine in Black Hills 100,000 gallon tank of C₂Cl₄ dry cleaning fluid in Cl nuclei n + n ® p⁺ + e^- so Cl (Z=17) becomes Ar (Z=18) Physically separate out the Ar, then wait for it to radioactively decay Saw only 1/3 the neutrinos predicted
33	Missing Neutrino Problem Lack of solar neutrinos confirmed by Kamiokande II detector in Japan. (Using different detection method) Apparent explanation in terms of Neutrino physics 3 different types of Neutrinos: electron, muon, and tau neutrinos Sun generates and Cl detectors see only electron neutrinos Can electron neutrinos can change to another type on way here? These “neutrino oscillations” are seen and imply neutrino has non-zero mass Kamiokande II evidence of muon neutrinos becoming electron ones Read “Window on Science 7-2” on “scientific faith” Neutrino mass may have implications for “cosmology” Neutrinos also used to study supernova 1987A