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Unformatted text preview: _6 Will be assigned next class h is on Tues. April 5, 2011 : ill be the same as the first exam _' entrate on the material since the last exam (including . Ures on the Sun). I here will be an extra credit assignment associate with this exam. NASA, NOAO, ESAand The Hubble Heritage Team (STScl/ Stellar Evolution Continued... (Kulner Ch. 10; also Shu, Ch. 4, 9) AST 203 (Spring 2011) Cepheid Variables Mass and composition completely I specify the structure of a star. .10 Supergiants The 4;» If we perturbed the star, it would _5 adjust back to this stable equilibrium state. (diJis Kiiiiqaisui/EipadMi/vi) _p o “X if; +5 Main Sequence“. Dwarfs Evolved stars on the instability 5'5 S m fl strip on the H-R diagram are not “‘1" m D H stable. +15 Bm +2 0 B A F G K M L T Spectral Class AST 203 (Spring 2011) Cepheid Variables Variable stars: Brightness oscillations with periods from seconds to years. apparent brightness Due to changes in the star's structure. Types are named after the prototype _L I I i L L__ i t 2 0 4 6 8 10 12 14 time (days) Figure 9.11. The light curve of a Cepheid variable. The period here is about a week (interval between, say, peak light to peak light). Long period (P ~ 3 mo. to 2 yr). Can change in brightness by 6 mag—a factor of 250 in brightness. Cepheids variables: Named after 6 Cephei. Periods range from 1 to 100 days. More than 1000 in our galaxy (including Polaris). AST 203 (Spring 2011) Cepheid Variables Doppler shifts in the spectra of Cepheids vary throughout the period of oscillation—the surface of the star is moving. The change in luminosity is associated with the change in radius. For Cepheid variables, ionization of He+ to He++ affects the opacity, causing radiation to be trapped when the star is compressed and radiation to escape when the star expands—drives oscillations. AST 203 (Spring 2011) Cepheid Variables Cepheids have a period and PERIOD-LUMINOSlTY RELATlONSHIP luminosity relationship. m4 From observing Cepheids in 1 9 . a 8 —f a the Magellanic Clouds. E m d TYDE‘HWWS) g age Cepheids .5 E 10 W RRtyrae 1 ' i t i i i i y 0.5 2 3 5 10 30 50 100 Period (03515} http://outreach.atnf.csim.au/education/seniodastrophysicslvariable_oepheids.htm| Australia Telescope Outreach and Education Measure Cepheid period in distant galaxy —> get L from P-L relation you can compute the distance. When used in this fashion, Cepheids are called standard candles. AST 203 (Spring 2011) Cepheid Variables Cepheid Variable in M100 HST-WFPC2 (NASA, HST, W. Freedman (CIW), R. Kennicull (U. Arizona), J. Mould (ANU)) AST 203 (Spring 2011) Cepheid Variables Two populations of Cepheids (Jeulnx) Separate P-L relationships. Not known originally—led to errors in distances. Relative number l 10 Classncal Cephe Ids _,_ijf:i?§jié?§)_M,WAA Distribution of periods for Cepheids, Note that .. 1 e are two distinct groupings. l Shorter periods. Type that were originally observed in the Magellanic Clouds. Type II Cepheids Sometimes called WVirginis stars Found in globular clusters in our galaxy. 1.5 magnitudes fainter than a classical Cepheid with same period. AST 203 (Spring 2011) Cepheid Example Original calibration done with type II Cephids Easier to study in our galaxy In other galaxies, easier to see the brighter classical Cephids. Not knowing the difference, the distance was computed using an absolute magnitude estimate 1.5 magnitudes too faint. Galaxies were really 2x further away from the original estimate AST 203 (Spring 2011) Degeneracy Pressure Recall that degeneracy pressure plays an important role in stellar evolution. Degeneracy pressure is a “quantum mechanical pressure” that arises from: The Pauli exclusion principle: no two electrons can have the same quantum mechanical state. The Heisenberg uncertainty principle: we cannot measure both the position and momentum of a particle to a precision greater than 21. Mathematically, this is: AxAp > h AsT 203 (Spring 2011) Degeneracy Pressure Thermal pressure in an ordinary gas arises from atomic collisions For an ideal gas,P oc pT, so as T —> 0 the pressure vanishes. In a degenerate electron gas, all the electrons cannot occupy the same state (position, momentum, and spin). As we compress them, the number density of electrons, n6, increases, and Ax = ng1/3 decreases. The Heisenberg uncertainty principle tells us that Ap ~ h/Ar If Age is small, then the momentum is large. Even if the temperature were 0, there is still a pressure. Note, Eq. 10.2 in your book is a little misleading. Kutner writes Ap 2 h/27r Ar but he really means Ap 2 h/(27rAas) - AST 203 (Spring 2011) Pressure (Shu Ch. 4) Consider a volume of gas with electron density neand particle velocity 22,, Pressure = force / unit area. Force = momentum / unit time. ‘90 /O O\ \ When the particles hit the wall, they \0 /O rebound, so Ap 2 2p /0 / The number of particles moving toward the wall is A '7vat % (1/2)n.,v : (1 /2)n€(UmAtA) H 1/2 the particles in this wall Over At, the total momentum volume are moving I transferred to the wall is toward the Wa”’ carry” positive momentum. (1/2)n6(vatA)2p Now pressure is momentum / unit time / unit area, so P = He,va AST 203 (Spring 2011) Degeneracy Pressure Now we can compute the pressure of a degenerate electron gas. Starting with P = nevmp , we have p ~ h/Am = hug/3 The velocity iS just U = p/me = fini/3/me Together, we have P N 71271? 3 /me A more detailed calculation finds a pressure ~ 2x higher. AST 203 (Spring 2011) Degeneracy Pressure In terms of mass density: Each of the positive ions has a mass Amp , so the total density is p = Arnan + mane ~ Ampnz The star is ionized by overall electrically neutral, so n6 2 an The number density of electrons is then / Z p n6 2 —— Number Number A mp denSIty of density of electrons nuclei and the electron degeneracy pressure becomes PNh—g g 5/3 A 5/3 me A mp Notice that there is no temperature dependence here! AST 203 (Spring 2011) Degeneracy Pressure A more detailed calculation would find P = ll?) <11“ Z 5/3 5/3 P = 9.9 x 1012 dyn cm—2 (—> or A AST 203 (Spring 2011) White Dwarfs Using the H-R diagram, we inferred a WD radius to be ~ 0.01 Re. A typical mass of a white dwarf is 1 MO. We can compute the average density: M 2.0 x 1033 g 6 3 _ Z — Z —A = 1.4 10 — p (4/3)7TR3 (4/3)7r(0.01 - 7 x 1010 any X g cm Recall that the average density of the Sun is only fie : 1.4 g (urn—3 A white dwarf is much more dense than the Sun. AST 203 (Spring 2011) White Dwarfs Take the temperature of this white dwarf to be 107 K. The electron degeneracy pressure is given by: Z 5/3 5/3 Pdeg = 9.9 x 1012 dyn arm—2 <—) \ / A 1gcm— K We will take (Z/A) = 1/2 (equal numbers of protons and neutrons). This gives Pdeg = 3.1 x 1022 dyn arm—2 AST 203 (Spring 2011) White Dwarfs For an ideal gas, the pressure is Ptherm : (me, + Note: your text is AS before, 77/6 I ZTLZ , SO assuming thatZ >> 1 here. This is not Ptherm I (Z + “Tlsz as easily satisfied as saying mp >> me Now, we already know that p 2 A11sz , so Z 1 Z = (—+ > (A) H (1)” A mp A mp Putting in our density and temperature, we find Ptherm : X 1020 dyn CIIli2 Ideal gas pressure can provide only 1/100th of that provided by electron degeneracy pressure. AsT 203 (Spring 2011) White Dwarfs Hydrostatic equilibrium tells us that d_P _ _GM(r) d7“ — 7‘2 p0“) As we did for the Sun, we can approximate this as: P GM (M) _> GM2 E”? E If we take the pressure to be electron degeneracy pressure, then 5/3 5/3 H45) (—p A) A 1gcm—3 Z 5/3 M 5/3 “((2) (e) AST 203 (Spring 2011) White Dwarfs Equating these two, we have GMg—K g 5/3 M 5/3 R4 — A R3 . K Z 5/3 Ml/S : _ _ R G (A) or Notice that the righthand side is a constant. Mass-radius relation for a white dwarf, R (X M ‘1/3 . The more massive the white dwarf, the smaller its radius. We saw this when we considered stellar evolution. AST 203 (Spring 2011) White Dwarfs As electrons pack close together, their momentum increases Special relativity: no particle can travel faster than the speed of light When v ~ 0, we must revise our estimate of the pressure. Recall that P = Hevmp . v cannot be faster than c, so we can rewrite this as P = necp The momentum is again p N h/Am 2 Eng/3 Relativistic degeneracy pressure is P = hang/3 AsT 203 (spring 2011) White Dwarfs Using the electron density: Zp rte :: ——-——— 14in? 4/3 4/3 (5) (A) 14 rnp A more detailed calculation would give: 2 1/3 4/3 4/3 P43“ 16(5) (A) 4 14 Hip Note the difference in the density dependence between non- relativistic and relativistic degeneracy pressure. 4 3 4 3 P = 1.2 x 1015 dyn cm—2 (5) / (L) / 14 1 g Chi—3 \\_______‘\/,_______,/ Kl we have AST 203 (Spring 2011) Chandrasekhar Mass We need to use the relativistic degenerate pressure expression when the density becomes high (high mass white dwarf). When relativistic effects are important, the M-R relation is GMg—K/ Z 4/3 M 4/3 R4 _ X E Notice that the radius drops out. Solving for the mass, we find K, 3/2 Z 2 Z 2 For Z/A = 1/2, we have M = 0.3M9 AST 203 (Spring 2011) Chandrasekhar Mass A more detailed calculation gives M : 1.4MG This is called the Chandrasekhar mass. Maximum mass supported by electron degeneracy pressure. Largest mass white dwarf possible—anything higher will collapse. AST 203 (Spring 2011) White Dwarfs A degenerate gas is a good 105 . . conductor 2101;. 08 105 x T is uniform in the WD us u Eng 10“ 7 White dwarfs are the end 103* state of low mass stars No other energy sources. 4. Can't compress any more. Star radiates away thermal energy, stays the same 101’ size (P independent of T) white dwarfs 10’3 a ‘ ,-.I.’ii.I1;. WD will just cool and dim ‘ »_ w and eventually fade awaY' 104 40000 20000 10000 5000 2500‘ T00 AST 203 (Spring 2011) White Dwarfs Ground" I - I White Dwarf Stars in'M4 HST - WFPC2 PRC95-32 - ST Scl 0P0 - August 28, 1995 - H. Bond (ST Sci), NASA Cooling white dwarfs are observed—they can tell us about the age of the Universe (they take a long time to cool—homework problem, Kutner 10.13) AST 203 (Spring 2011) http://antwrp.gsfc.nasa.gov/apod/apOOOQ10.htm| Planetary Nebula The outer layers of the AGB star are illuminated from the inside by the white dwarf core—a planetary nebula. Photons absorbed—momentum pushes gas shell outward. M57 (NASA/ESA) AST 203 (Spring 2011) NGC 6543 (NASA/ESA) Planetary Nebula Planetary nebula are spherical shells. < g 7 At the edges, we are looking through more material. —-—e—-—> / \ Spectra/Doppler shift show that the nebular is expanding outward. Different lines indicate the density and temperature of the nebula. Observations show typical mass of a planetary nebula is ~ 0.1 Me. This material is enriched in heavy elements—returned to the interstellar medium and the next stars can form out of it. AST 203 (Spring 2011) ...
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