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Unformatted text preview: Announcements was grade ' W ' Closed book, closed notes. You are expected to know the general formulae A sheet of constants vyill be provided. . ' There. will be an extra credit assignment dealing with this exam —that will be discussed next class.. T 5‘3: NASA, NOAO, ESAand The Hubble Heritage Team (STScl/ Hertzprung-Russell Diagram Horizontal axis: spectral class, B — VI or T 1|:n:u:u:i (increasing to left). E-IIIIIiEil-. r: '. mm. 300le :Em lEIIII DEM] Vertical axis: Luminosity or absolute magnitude. main sequence: diagonal line running through all the spectral classes. Some T-L combinations not realized in nature. [I Lilli Wide range in L for stars of the same [mum Low L population: white [Minimum dwarfs. AST 203 (Spring 2011) if: E [biog-KEEN: Binary Stars (Kulner, Ch. 5) AST 203 (Spring 2011) Binary Stars ~ 1/2 of all stars are in binary systems. A :3 (I) 5 .1 (I) E Period + separation tell us about the gravitational force. Binary systems can tell us about stellar masses. visual binaries: both stars seen independently—details of the orbits of both stars in the system can be measured. optical doubles: two stars that appear close in the sky, but do not orbit one another, and may in fact be quite a distance apart. AST 203 (Spring 2011) Binary Stars (HewholooksMnkipedia) Albireo: a double star. It is not known whether these stars are gravitationally bound—if so, the period is > 100,000 years. AST 203 (Spring 2011) Binary Stars 7 iii-:7 ii "if -' Ci ‘5': =1" iiifi‘f} mass retro: 2.00 path of primary star ECCEHtFICItYT 0-00 path of unseen companion W astrometric binary: we only see one of the stars in the system— wobble in its proper motion indicates an unseen companion. AST 203 (Spring 2011) Binary Stars “ rpf eclipsing binary: magnitude __ variation with a regular period. Nearly edge-on orbit showing eclipses. n e 9: . . . . J . fl.‘ ‘0.“ '0.“ us [LIB 0.2 v.5 $6 “.5 0.5 Ms UIBITEFI‘ESE a ( d Small hot star and larger cool star eslipsing: V HHGNI TIDE 10.5 “1.4 is NJ MM 8 1a.. Flux drops when either star blocked. Stellar radii can be found from eclipse ft: m * shapes. :1. ll. :1 i. l l, 4;. HP flux Binary Stars composite spectrum binary: spectrum does not correspond to known spectral type but rather is a blend of two—one from each star in the system. spectroscopic binary: shows shifts in the position of the spectral lines about some average wavelength (Doppler shift). AST 203 (Spring 2011) Doppler Shift Source moving toward us: wavelength of emission is compressed—blueshift. Source moving away from us: wavelength stretched out—redshift. Only the relative speed along the line of sight matters. As long as this velocity is not relativistic, shift in wavelength is: d°PP'er-a“ AA i A — A0 12—7. A—O A0 C Here, A0 , is the rest wavelength of the emission. AsT 203 (Spring 2011) Doppler Shift We can also write an expression for the Doppler shift relating to frequency. Recall that A : C/l/ SO dA : —ch//I/2 Therefore, AsT 203 (Spring 2011) Doppler Shift Example Ha is one of the strongest lines in stellar spectra. If we have a star with a radial velocity of 10 km/s, what is the observed wavelength shift? Recall that Ha has a wavelength of 656.28 nm. 106 cm 8—1 U A Z _: ,2 —= .22 A A00 656 8nm3xlolocm871 00 nm This is a small, but measurable shift. AST 203 (Spring 2011) Spectroscopic Binary AST 203 (Spring 2011) Orbital Motion Our knowledge of orbital motion started with the planets. Kepler described three laws of planetary motion that were later derived by Newton. ’lSt law: The orbit of each planet is an ellipse with the Sun at 1 foci. 2nd law: The radius vector to a planet sweeps out equal areas in equal time 3rd law: The square of the period of the planet is proportional to the cube of the semi-major axis of the orbit. In reality the planets and the Sun orbit the common center of mass of the system. AST 203 (Spring 2011) Circular Motion Starting point: circular orbits Consider two stars of masses m1 and m2 orbiting the center of mass atdistances 7‘1 and 7‘2. The center of mass lies on a line connecting the two masses, so mlrl : m2r2 Note: 71 + T2 is the distance between the centers of the stars, not their surfaces. Newton showed that for spherical objects, the force of gravity is equivalent is all the mass of the object were placed at the center. AST 203 (Spring 2011) Circular Motion The stars are always opposite one another—period is the same: 27W" P : — : const. 1) Each star experiences a gravitational force from the other: 77217722 F:G— (7”1+7“2)2 Gravitational force balanced by centripetal acceleration. For star 1, we have: 77111]? AST 203 (Spring 2011) Circular Motion Rewriting, vi : G m2 r1 (M + r2)2 we know that 271'7‘1 ’U : 1 P so 47T2T1 Gmg P2 (M + r2)2 Defining R = 71 + T2 , we have 7" m 7' R=r1+r2=r1<1+—2) :r1(1+—1)=—1(m1+m2) 7'1 m2 m2 AST 203 (Spring 2011) Circular Motion Note: your text Putting it all together, we have misses the power of 2 3 2 on Tr—be sure to 477 R 2 (m1 + m2)P2 look at the errata. G Solar system: Kepler's third law, also known as the harmonic law. P is an observable. distance to the system + resolve components —> separation. Eclipsing binary: get the period by studying dips in lightcurve. This allows us to determine the total mass of the system. Can we do any better? AST 203 (Spring 2011) Circular Motion Ratio of the masses: m1 /m2 : rz/r1 Together, we can find the mass of each component! Nature isn't this simple however, the orbital plane may be inclined to our line of sight the orbits may be elliptical, instead of circular AST 203 (Spring 2011) Mass of the Sun The easiest star that we can find the mass of is our Sun. Let's write down Kepler's law for the Sun and the earth: 4 2 gift 2 (MG + [email protected])P2 What are R and P ? R is just 1 astronomical unit, 1.5 X 1013 cm P is just 1 year We take [email protected] + M69 % MG SO 4W2R3 471'2 1.5 x 1013 cm 3 A. MG: 2 Z is ( 2 i2 ) 7 2:2X1033g GP (6.67 X 10 dyn cm g )(3.16 x 10 s) AsT 203 (Spring 2011) Mass of the Sun Solar system: MG + Mg m MG P2 = 1:553 for P in years and a in AU, k = 1 Kepler's Third Law Nauru? 102 — Uranus+1 - Kepler discovered this Jup.:r::n’+ . . l _ observationally In 1609. 10 ,v T: Mars+’l I I 3‘ 100 _ Earthfl; Newton used this to derive M “9””? I ercury the Inverse square Q _] . F" . 10 ' / / dependence of graVIty. {Calm ‘ Ganymede 10'2 — ’ Europa I. ID 1073 3 ‘ 2 I-l I0 ‘1 10' 10' 10 10 10 AST 203 (Spring 2011) semiemajor axis (AU) Doppler Shift Eliminate the separation in favor of velocity. First, we use 271'?" v : — P now, the total separation is P RZTl-l-T'Q : gfifl +112) so we can completely eliminate the separation: P ‘ —(’01 + 1J2)d : m1 + m2 27rG AST 203 (Spring 2011) Inclined Orbits Complication: inclined orbits. Doppler shifts only measure the velocity along the line of sight, UT. . If the orbit is inclined by some angle 1’, then we measure 1),. = v sini and our period/velocity relation becomes P (“m + 02,1)3 . . 2 m1 + m2 2770 $1113 2 AST 203 (Spring 2011) Orbital wane Plane of sky Inclined Orbits Eclipsing binary: inclination close to 90°. With enough observations, we can trace out the orbit in the sky, we can determine the inclination from the orbital characteristics. When we don't know i, the best we can do is assume that it is 90°, and get a lower limit to the sum of the masses. AsT 203 (Spring 2011 :1 Example Observations of a binary system show that it has a period of 10 years. The radial velocities of the two stars are determined to be 10 km/s and 20 km/s. Assuming circular orbits, with an inclination angle of 90°, what are the masses? We begin by finding the sum of the masses via: P (1)177“ + U2,r>3 m1 + m2 = — . . 2770 Slug 2 _ 10 yr - 3.16 x 107 s/yr (106 cm/s+ 2 x 106 cm/s)3 _ 27r - 6.67 X 10—8 dyn cm2/g2 singz' : 2.04 X 1034 g/sin3z' = 10.2 MQ/sin3i AST 203 (Spring 2011) Example Since the periods of the stars are the same, then r1/v1 : 72/222 and from the center of mass relation, we have m2/m1 : rl/r2 : v1/v2 (Note this is also just conservation of momentum: mlvl : 777,202 ) Thus the ratio of masses is just ml/mg : U2/U1 : 2 If we take 1' = 90° , then we have m1 + m2 = MG ml/m2:2 AsT 203 (Spring 2011) Example SO m1 = MG and m2 = [email protected] More massive star has lower velocity (momentum conservation) If we instead take 1' = 45° , then the sum of the masses is m1 + m2 = 10.2 MQ/sin3z‘ = 28.8 Me and we have or m1 2 19-2 and 7712 = 9.6 does-not change. AsT 203 (Spring 2011) ...
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