# Spectroscopy, the Doppler Shift and Masses of Binary Stars.

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1 Spectroscopy, the Doppler Shift and Masses of Binary Stars

2 Doppler Shift At each point the emitter is at the center of a circular wavefront extending out from its present location.

3

4 The Doppler Shift since! = c " " v %! obs =! emit +! emit # \$ c& '

5 The Doppler Shift since! = c " " v %! obs =! emit +! emit # \$ c& ' "! obs =! emit 1 + v % # \$ c & ' "! obs =! emit 1 ( v % # \$ c & ' if moving away from you with speed v if moving toward you with speed v )! =! obs (! emit =! emit (1 ± v c ( 1) )!! emit = ± v c This formula can only be used when v << c Otherwise, without proof, " 1+ v/c%! obs =! emit # \$ 1( v/c& ' if v is away from you )! >0 if v is toward you )! < 0 1/2

6 Doppler Shift: Note different from a cosmological red shift!

7 Astronomical Examples of Doppler Shift A star or galaxy moves towards you or away from you (can t measure transverse motion) Motion of stars in a binary system Thermal motion in a hot gas Rotation of a star

8 E.g. A H atom in a star is moving away from you at 3.0 x 10 7 cm s -1 = times c. At what wavelength will you see H?! obs = ( ) = A Note that the Doppler shift only measures that part of the velocity that is directed towards or away from you.

9 A binary star pair

10 Thermal Line Width In a gas with some temperature T atoms will be moving around in random directions. Their average speed will depend upon the temperature. Recall that the definition of temperature is 1 2 m atom!v 2 " = 3 2 k T towards v average = 0 speed 3kT m atom away where k =1.38 # 10 \$16 erg K \$1 Here!" means "average". Some atoms will be moving faster than the average, others hardly at all. Some will be moving towards you, others away, still others across your line of sight.

11 The full range of wavelengths, hence the width of the spectral line will be!" " = 2 v average c = 2 3kT m atom The mass of an atom is the mass of a neutron or proton (they are about the same) times the total number of both in the nucleus, this is an integer "A". ( ) 1.38#10 \$16 ( ) T ( )( A)!" " = 2 % 3 ' & 1.66#10 \$24 Thermal Line Broadening ( )!" " = 1.05# 10\$6 T A ( * ) 1/2 % & ' #10 10 ( ) * where T is in K A = 1 for hydrogen 4 for helium 12 for carbon 16 for oxygen etc.

12 Full width =!" = 1.05 # 10 \$6 T(in K) A "

13 ROTATION

14

15 Note: Potential complications: 1) Star may have both thermal and rotational broadening 2) May see the star at some other angle than in its equatorial plane. Example: H! in a star with equatorial rotational speed 100 km/s = 10 7 cm/s Full width = "# = 2 \$ % & v c ' ( ) # \$ =(2)(6563) & % 10 7 ' 3 * ) ( = 4.4 A

16 Average rotational velocities (main sequence stars) Stellar v equator Class (km/s) O5 190 B0 200 B5 210 A0 190 A5 160 F0 95 F5 25 G0 12 Stellar winds and magnetic torques are thought to be involved in slowing the rotation of stars of class G, K, and M. Stars hotter than F5 have stable surfaces and perhaps low magnetic fields. The sun rotates at 2 km/s Red giant stars rotate very slowly. Single white dwarfs in hours to days. Neutron stars may rotate in milliseconds

17 3 sources of spectral line broadening 1) Pressure or Stark broadening (Pressure) 2) Thermal broadening (Temperature) 3) Rotational broadening (, rotation rate)

18

19 planets?

20 8) Magnetic fields From Zeeman splitting 9) Expansion speeds in stellar winds and explosions Supernovae, novae, planetary nebulae 10) From 21 cm - rotation rates of galaxies. Distribution of neutral hydrogen in galaxies. Sun s motion in the Milky Way.

21 Hyperfine Splitting The 21 cm Line Less tightly bound More tightly bound

22 21 cm (radio) = 21 cm = 1.4 x 10 9 Hz h = (6.63 x )(1.4 x 10 9 ) = 9.5 x erg = 5.6 x 10-6 ev Must have neutral H I Emission collisionally excited Lifetime of atom in excited state about 10 7 yr Galaxy is transparent to 21 cm

23 Merits: Hydrogen is the most abundant element in the universe and a lot of it is in neutral atoms - H I It is not so difficult to build big radio telescopes The earth s atmosphere is transparent at 21 cm Aerecibo m radio telescope - Puerto Rico

24 Getting Masses in Binary Systems

25 Binary and Multiple Stars (about half of all stars) Beta-Cygnus (also known as Alberio) Separation Magnitudes 3.0 and 5.3. Yellow and blue. 380 ly away. Bound? P > y. The brighter yellow component is also a (close) binary. P ~ 100 yr. Alpha Ursa Minoris (Polaris) Separation Magnitudes 2.0 and 9.0. Now known to be a triple. Separation ~2000 AU for distant pair. When the star system was born it apparently had too much angular momentum to end up as a single star.

26 Polaris 1.2 Msun Polaris Ab Type F6 - V 4.5 Msun Polaris A Cepheid Period 30 yr Polaris B is F3 - V

27

28 Epsilon Lyra a double double. The stars on the left are separated by 2.3 about 140 AU; those on the right by 2.6. The two pairs are separated by about 208 (13,000 AU separation, 0.16 ly between the two pairs, all about 162 ly distant). Each pair would be about as bright as the quarter moon viewed from the other. 585 years > 10 5 years 1165 years

29 Castor A and B complete an orbit every 400 years. In their elliptical orbits their separation varies from 1.8 to 6.5. The mean separation is 8 billion miles. Each star is actually a double with period only a few days (not resolvable with a telescope). There is actually a C component that orbits A+B with a period of of about 10,000 years (distance 11,000 AU). Castor C is also a binary. 6 stars in total

30 An optical double Stars not really bound years for Mizar A and B to complete an orbit 6 months 20.5 d

31

32 Circular Orbit Unequal masses

33 Two stars of similar mass but eccentric orbits

34 Two stars of unequal mass and an eccentric orbit E.g. A binary consisting of a F0v and M0v star - visbin

35 Aside The actual separation between the stars is obviously not constant in the general case shown. The separation at closest approach is the sum of the semi-major axes a times (1-e) where e is the eccentricity. At the most distant point the separation is a times (1+e). For circular orbits e = 0 and the separation is constant.

36 Period = 50.1 years 6.4 (A) and 13.4 (B) AU

37 Some things to note: The system has only one period. The time for star A to go round B is the same as for B to go round A A line connecting the centers of A and B always passes through the center of mass of the system The orbits of the two stars are similar ellipses with the center of mass at a focal point for both ellipses The distance from the center of mass to the star times the mass of each star is a constant. (will be shown) r B M B = r A M A

38 x r 1 CM r 2 m 2 m 1 both stars feel the same gravitational attraction and thus both have the same centrifugal force More massive star is closer to the center of mass and moves slower. m 1 v = m 2v 2 2!r 1 = 2!r 2 r 1 r 2 v 1 m 1 r 1 2 v 2 2 v 2 = Gm 1m 2 (r 1 + r 2 ) 2 " v 1 = r 1v 2 r 2 r 1 r 2 2 = m 2 v 2 r 2 2 m 1 r 1 = m 2 r 2 = Period r 1 r 2 = m 2 m 1

39 r 1 r 2 = v 1 v 2 So M 1 M 2 = v 2 v 1

40 Motion of the sun because of Jupiter m 1 r 1 = m 2 r 2 M! d! = M J d J d! = M J M! d J = (9.95 x 10-4 )(7.80 x ) = 7.45 x cm d! = radius of sun's orbit around center of mass d J = Jupiter's orbital radius = 5.20 AU = 7.80 x cm M J = 1.90 x gm = 9.55 x 10-4 M! Can ignore the influence of the other planets. P = years

41 Doppler shift

42 12.5 m/s years

43 KEPLER S THIRD LAW FOR BINARIES + GM 1 M 2 ( ) 2 = M 1 v 1 r 1 r 1 +r 2 GM 1 M 2 (r 1 +r 2 ) 2 = M 2 v 2 r M 2 r 2 x r 1 M 1 2 G(M 1 +M 2 ) (r 1 +r 2 ) 2 = v 1 + v 2 = 4!2 r 1 r 1 r 2 P 2 + 4!2 r 2 r 1 P 2 r 2 2 = 4!2 P 2 (r 1 +r 2 ) P 2 = K (r 1 +r 2 ) 3 K = i.e., just like before but 2 4! 2 G(M 1 +M 2 ) M! M 1 +M 2 R! r 1 +r 2 2

44 ( M 1 + M 2 ) = 4! 2 GP 2 (r 1 + r 2 )3 M! = 4! 2 ( AU )3 2 G(1 yr)

45 M 1 + M 2 M! =! # # " ( r + r ) AU P yr 2 \$ & & % M 1 M 2 = r 2 r 1 or M 1 M 2 = v 2 v 1 If you know r 1, r 2, or v 1, v 2, and P you can solve for the two masses.

46

47 P = 50 y 1 pc = AU 1 radian = arc sec! = radian! arc sec ! P = ( total separation) G( M + M ) ( Total M )( P )"( separation) 2 3 and since you can measure the angle of inclination of the orbit, you get the actual masses.

48 s (in pc) =r (in pc)! (in radians) s (in AU) = r (in AU)! (in radians) r (in AU) = r (in pc)! in radians =! (in arc sec) s in AU = r (in pc) " number AU % # \$ 1 pc & ' " # \$ 1 radian number arc sec " number AU % 1 # \$ 1 pc & '! (in arc sec) % & ' " # \$ & r radian s number arc sec % & '

49 Spectroscopic Binaries

50 (away) Star 2 Star 1 (toward) Complication: The viewing angle

51 GETTING STELLAR MASSES #2 For spectroscopic binaries measure: Period Velocity of each star Inclination will be unknown so mass measured will be a lower bound (TBD) CM r v CALCULATION P = 2! r v First get r 1 and r 2 from v 1 and v 2 r i = v i P 2! Example: v 1 = 75 km s "1 v 2 = 25 km s "1 P= 17.5 days

52 P= 2!r v " r = Pv 2! Note - the bigger the speeds measured for a given P the bigger the masses

53 Complication The Inclination Angle Let i be the angle of the observer relative to the rotation axis, i.e., i = 0 if we re along the axis. Measure v Sin i which is a lower bound to v. 4! P r r = ( 1 + 2) G( M1 + M 2) P vi ri = 2! but measure v! = vsin i, so we end up measuring r! = r Sin i and calculate v1 v2 3 M! 1 M!! "! +! # 2 P 2 + = \$ % GP & 2! ' when the actual mass is 2 3 4! " v2 + v2 # 3 M1 M 2 P 2 + = \$ % GP & 2! ' hence the measurement gives a low bound on the actual mass 3 ( M! + M! ) = ( M + M )Sin i Since Sin i < 1, the measurement is a lower bound. Only if i = 90 degrees do we measure the full velocity. To Observer i v 3 Sin i =0.59 But we tend to discover more edge on binaries so 2/3 is perhaps better

54 Eclipsing Binary For an eclipsing binary you know you are viewing the system in the plane of the orbit. I.e., Sin i = 1

55

56

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