# Spectroscopy, the Doppler Shift and Masses of Binary Stars

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1 Doppler Shift At each point the emitter is at the center of a circular wavefront extending out from its present location. Spectroscopy, the Doppler Shift and Masses of Binary Stars The Doppler Shift since ν = c λ v λ obs = λ emit + λ emit c

2 λ obs =λ emit 1 + v c λ obs = λ emit 1 v c The Doppler Shift since ν = c λ if moving away from you with speed v if moving toward you with speed v v λ obs = λ emit + λ emit c Astronomical Examples of Doppler Shift A star or (nearby) galaxy moves towards you or away from you (can t measure transverse motion) Motion of stars in a binary system Δλ =λ obs λ emit = λ emit (1 ± v c 1) Δλ = ± v c λ emit This formula can only be used when v << c Otherwise, without proof, 1/ 1+ v/c λ obs = λ emit 1 v/c if v is away from you Δλ >0 if v is toward you Δλ < 0 red shift blue shift Thermal motion in a hot gas Rotation of a star Doppler Shift: 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? λ obs = ( ) = A Note that the Doppler shift only measures that part of the velocity that is directed towards or away from you. Note different from a cosmological red shift!

3 A binary star pair Again, only see a shift due to motion along our line of sight Thermal Line Broadening 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, T, is 1 m atom v = 3 k T towards v average = 0 speed 3kT m atom away where k = 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. The full range of wavelengths, hence the width of the spectral line will be Δλ λ = v average c = c 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". ( ) T ( )( A) ( ) Δλ λ = Thermal Line Broadening Δλ λ = T ( ) A 1/ A = 1 for hydrogen 4 for helium 1 for carbon 16 for oxygen etc. where T is in K; nb depends on A

4 Thermal Line Broadening Full width = Δλ = T(in K) λ A ROTATION (as viewed in the plane of the equator) Eg. Hα at 5800 K (roughly the photospheric temperature of the sun) A=1 λ = 6563 A T= Δλ = 1.05 x / (6563) = 0.53 A This is (another) way of measuring a star's temperature Wien's law (wavelength where most emission comes out) Spectral class (O,B, F, G, K, M and subsets thereof) L L= 4 π R σ T4 T = 4π R σ 1/4 Thermal line broadening Note: Potential complications: 1) Star may have both thermal and rotational broadening ) May see the star at some other angle than in its equatorial plane. Example: Hα in a star with equatorial rotational speed 100 km/s = 10 7 cm/s v Full width = Δλ = λ c 10 7 = ()(6563) = 4.4 A

5 Average rotational velocities (main sequence stars) Stellar v equator Class (km/s) O5 190 B0 00 B5 10 A0 190 A5 160 F0 95 F5 5 G0 1" 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, a G star rotates at km/s at its equator Red giant stars rotate very slowly. Single white dwarfs in hours to days. Neutron stars may rotate in milliseconds 3 sources of spectral line broadening 1) Pressure or Stark broadening (Pressure) ) Thermal broadening (Temperature) 3) Rotational broadening (w, rotation rate) One can distinguish rotational broadening from thermal broadening because rotation affects all ions equally, but in a hot gas the heavier ions move slower. Thermal width depends on A -1/. SUMMARY planets?

6 Hyperfine Splitting The 1 cm Line 8) Magnetic fields From Zeeman splitting 9) Expansion speeds in stellar winds and explosions Less tightly bound Supernovae, novae, planetary nebulae 10) From 1 cm - rotation rates of galaxies. Distribution of neutral hydrogen in galaxies. Sun s motion in the Milky Way. More tightly bound 1 cm (radio) = 1 cm λ =cν = 1.4 x 10 9 Hz h = (6.63 x 10-7 )(1.4 x 10 9 ) = 9.5 x erg = 5.6 x 10-6 ev Must have neutral H I 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 1 cm Emission collisionally excited Lifetime of atom in excited state about 10 7 yr Galaxy is transparent to 1 cm Aerecibo m radio telescope - Puerto Rico

7 Binary and Multiple Stars (about half of all stars are found in multiple systems) Getting Masses in Binary Systems 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.0 and 9.0. Now known to be a triple. Separation ~000 AU for distant pair. When the star system was born it apparently had too much angular momentum to end up as a single star. Polaris 1. Msun Polaris Ab Type F6 - V 4.5 Msun Polaris A Cepheid Period 30 yr Polaris B is F3 - V

8 585 years > 10 5 years 1165 years Epsilon Lyra a double double. The stars on the left are separated by.3 about 140 AU; those on the right by.6. The two pairs are separated by about 08 (13,000 AU separation, 0.16 ly between the two pairs, all about 16 ly distant). Each pair would be about as bright as the quarter moon viewed from the other. 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 An optical double Stars not really bound. Center of Mass m 1 =m r 5000 years for Mizar A and B to complete an orbit 6 months 0.5 d For constant total separation, 0 AU, vary the masses

9 Circular Orbit Unequal masses Two stars of similar mass but eccentric orbits Two stars of unequal mass and an eccentric orbit E.g. A binary consisting of a F0v and M0v star Aside The actual separation between the stars is obviously not constant in the general case shown of non-circular orbits. The separation at closest approach is the sum of the semi-major axes of the two elliptical orbits, a = a 1 +a, times (1-e) where e is the eccentricity. At the most distant point the separation is a times (1+e). So if one measures the separation at closest approach and farthest separation, adds and divides by, one has a 1 + a - visbin For circular orbits e = 0 and the separation is constant, +r.

10 (a 1 +a ) (1+ e) (a 1 +a ) (1 e) + farthest separation closest separation (a 1 +a ) / = (a 1 + a ) Period = 50.1 years semi-major axis of A is 6.4 AU and B is 13.4 AU (In Kepler s equation use the sum of the semimajor axes) Some things to note: A binary star 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 For the case of circular orbits, the distance from the center of mass to the star times the mass of each star is a constant. (next page) 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. ASSUME CIRCULAR ORBITS m 1 v 1 x r = Gm 1m ( + r ) π v 1 = πr v = CM m 1 v m v r = m v r v 1 = v r r m 1 = m r m = Period r = m m 1

11 Circular Orbit Unequal masses r = v 1 v So since r = m m 1 m m 1 = v 1 v E.g.. Motion of the sun because of Jupiter; Roughly the same as two stars in circular orbits m 1 = m r 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 = 5.0 AU = 7.80 x cm M J = 1.90 x gm = 9.55 x 10-4 M Note: wobble of the star is bigger if the planet is bigger or closer to the star (hence has a shorter period). 1.5 m/s years Can ignore the influence of the other planets. P = years

12 Doppler shift As of today 05 extra solar planets in 1300 stellar systems and the number is growing rapidly. About 40 mph Many were detected by their Doppler shifts. Many more by the transits they produce as they cross the stellar disk. KEPLER S THIRD LAW FOR BINARIES Circular Orbit Unequal masses + GM 1 M ( ) = M 1 v 1 +r GM 1 M ( +r ) = M v r M r x M 1 G(M 1 +M ) ( +r ) = v 1 + v = 4π r P + 4π r P r = 4π P ( +r ) P = K ( +r ) 3 K = i.e., just like before but 4π G(M 1 +M ) M M 1 +M R +r In the general case ( + r ) (a 1 + a ) where a 1 and a are the semimajor axes of the two stars in ellipical orbits around each other v = πr P

13 ( M 1 + M ) = 4π M = 4π GP ( + r )3 ( AU )3 G(1 yr) for the earth M 1 + M M = 3 ( + r ) AU P yr M 1 M = r or M 1 M = v v 1 Divide the two equations If you know, r in AU, or v 1, v, and P in years you can solve for the two masses. Measure: Getting Stellar Masses Visual binaries E.g., Sirius A and B; distance.67 pc) Period Separation (if circular orbit; sum of semi-major axes if elliptical; average of max and min separations) Ratio of speeds or separations from center of mass if circular (or ratio of semi-major axes if elliptical) Period = 50.1 years semi-major axis of A is 6.4 AU and B is 13.4 AU (In Kepler s equation use the sum of the semimajor axes)

14 Calculation M A + M B P (yr) = A 3 (AU) M A= A A + A B = =19.8 AU P =51 yr M A + M B M = =.99 So the sum of the masses is.98 solar masses. The ratio of the masses is A A / A B = 6.4 / 13.4=0.478 =M B / M A M A M A =.99 M A =.0 M M B = M A = 0.97M and since we see the orbits face on these are the actual masses. Sirius B only has a radius times that of the sun. What is it? In the case of circular orbits seen face on the separation is a constant. As shown on the next page the separation in AU is just the distance in pc times the angular separation in arc seconds. This is a consequence of the way the pc is defined. One can also measure, in the same way the separation between sta and the center of mass,, and star and the center of mass, r. The total mass is then given by the usual equation and M 1 + M P (yr) = R 3 (AU) = ( r M 1 + r ) 3 M 1 M = r s (in pc) =r (in pc) θ (in radians) s (in AU) = r (in AU) θ (in radians) r (in AU) = r (in pc) θ in radians = θ (in arc sec) s in AU = r (in pc) number AU 1 pc number AU 1 pc 1 radian number arc sec 1 θ (in arc sec) radian number arc sec e.g. The separtion of Sirius A and B varies from 3.5 to 11.5 arc sec. The average is therefore 7.5 arc sec. Sirius is.67 pc away so a 1 + a = (.67)(7.5)= 0 AU % r s Spectroscopic Binaries

15 (away) Star Sta (toward) Complication: The viewing angle Getting Stellar Masses For Sprectroscopic Binaries 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 P= πr v r = Pv π Assume circular orbits CALCULATION π r P = v First get and r from v 1 and v r i = v i P π Example: v 1 = 75 km s 1 v = 5 km s 1 P= 17.5 days Note - the bigger the speeds measured for a given P the bigger the masses

16 Complication The Inclination Angle Eclipsing Binary Let i be the angle of the observer relative to the rotation axis, i.e., i = 0 if we re along the axis. Only if i = 90 degrees do we measure the full velocity. Measure v Sin i which is a lower bound to v. 4π 3 P = ( r1 + r ) G( M + M ) 1 P vi ri = π but measure v = vsin i, so we end up measuring r = r Sin i and calculate + = GP π when the actual mass is 3 4 v1 v 3 M 1 M π + P M M 3 4π v + v 3 P 1 + = actual GP π hence the measurement gives a low bound on the actual mass 3 ( M + M ) = ( M + M )Sin i 1 1 measured Since Sin i < 1, the measurement is a lower bound. To Observer i v 3 Sin i =0.59 But we tend to discover more edge on binaries so /3 is perhaps better For an eclipsing binary you know you are viewing the system in the plane of the orbit. I.e., Sin i = 1 Limits of stellar mass: Observed stars end up having masses between 0.08 M and about 150 M. The upper number is uncertain (130? 00?). The lower number will be derived later in class (minimum mass to ignite H burning before becoming degenerate).

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