Binary Stars

1
Binary Stars Stellar Masses
2
Why Binaries?

• We need to know about stellar masses to develop
  working models and theories
• The only way to measure mass directly is through
  gravitational interaction
• Half the stars in our galaxy are in binary
  systems!

3
Classes of Binaries

• Optical Double two stars along same line of
  sight
• Visual Binary both stars directly visible
• Composite Spectrum Binary two stars in one
  spectrum
• Eclipsing Binary light from system grows fainter
  brighter
• Astrometric Binary star wobbles through the
  sky
• Spectroscopic Binary oscillating spectral lines

4
Examples of Binaries
5
Era of Photographic Astronomy

• Background density of stars increases with
  magnitude
• Thus the number of (non-physical) optical pairs
  increases with magnitude

6
Doppler Shift

• Wavelength changes as a result of motion
• Decreased l means increased n, increased l means
  decreased n
• Movement perpendicular to line of sight has no
  effect

7
Doppler Shift ctd.

• Both the source observer can move
• Now change in l depends on their combined motion
• Shift only depends on changes in line of sight
  connecting the two radial velocity vr dr/dt
  (5.1)
• Moving apart ? vr gt 0, r increasing
• Moving closer ? vr lt 0, r decreasing

8
Moving Source Observer

• Source vs, angle ?
• Observer vo, angle f
• Relative radial velocity
• vr vscos? vocosf (5.2)

9
Measuring the Shift

• Wavelength shift Dl l-lo (5.3)
• lo is the rest wavelength
• l is the wavelength received
• For vr ltlt c Dl/l vr/c (5.4)
• vr gt 0 means Dl gt 0
• vr lt 0 means Dl lt 0
• shifts to longer ls are called redshifts
• shifts to shorter ls are called blueshifts

10
Frequency Shifts

• We know l c/n2
• dl/dn -c/n2 l/n
• Dn (-n/l)Dl
• Dn/no -vr/c (for vr ltlt c) (5.5)
• Also Dn/no -Dl/lo
• Shifts will be different between spectral lines
  because it is l-dependent spacing between lines
  will change

11
Circular Orbits

• Orbital speed v
• Radius r
• v v/r
• Instantaneous radial velocity vr
• Velocity along line of sight is vcos?
• ? vt
• vr vcost(vt) (5.6)

12
Period, Inclination

• Period P 2p/v
• Spectral line shift
• Dl/lo vcos(vt)
• Orbit is not always in our plane
• If i angle orbit is inclined, projection of
  orbit is vsin(i)
• Radial velocity vr vsin(i)cos(vt) (5.7)

13
Binary Stars Circular Orbits

• Circular orbits are not the most general case,
  but a good starting point
• Both stars orbit around center of mass
• m1, m2 r1, r2
• m1r1 m2r2 (5.8)

14
Center of Mass

• Center of mass unaffected by forces between the
  stars, only external forces apply
• Viewing from the center of mass
• Stars must always be on opposite sides of COM
• Stars have same period P, P 2pr/v (5.9)
• Gives v 2pr/P (5.10)
• Equating periods gives r1/v1 r2/v2 (5.11)
• v1/v2 r1/r2 m1/m2 (5.12)

15
Gravitational Forces

• Distance between stars is r1 r2
• F Gm1m2/(r1 r2) ? must balance acceleration
  of circular motion v2/r
• F m1v12/r1 (5.14)
• Balance and get
• m1v12/r1 Gm1m2/(r1 r2) (5.15)

16
Relate to P

• Divide both sides by m1 and relate v1 to P
• 4p2r1/P2 Gm2/(r1 r2)2 (5.16)
• Total distance (r1 r2) R r1(1 r1/ r2)
• 
  
  
  (5.18 5.19)
• Now have 4p2R3/G (m1 m2)P2 (5.20)

17
Determining P

• Watch
• Doppler shifts go through full cycle
• Wobble go through a full cycle
• Light curve dim and brighten
• If you can see both stars, measure R
• Once you know P, use
• 4p2R3/G (m1 m2)P2
• to determine m1 m2

18
Determining Individual Masses

• Obtain m1/m2 from
• r1/r2 if both stars can be seen
• v1/v2 if both Doppler shifts can be observed
• The sum of masses and ratio of masses can be used
  to determine m1 and m2
• But we dont always have all the required
  information!

19
Mass of the Sun

• Consider the sun earth as a binary system
• Msun 4p2R3/GP2 2 x 1033g
• Note The mass of the earth, when compared to the
  sun, is so small that m1 m2 ? m1
• This is defined as one solar mass

20
Keplers Third Law

• Now we know that for 2 objects with
• R 1AU
• P 1 year
• m1 m2 1 solar mass
• We obtain
• R/1AU3 (m1 m2)/1MsunP/1yr2 (5.21)
• Note that R3 P2 ? Keplers Third Law
• Originally found observationally, later used to
  show that gravity is an inverse square law

21
Measuring R On the Sky

• R is not directly measured instead we measure
  ?, the angular separation on the sky
• If d is the distance to the system
• R(AU) ?(rad)d
• R d will come out in the same units

22
R in Arcseconds

• But values of ? are very small so we convert to
  arcseconds
• R(AU) d(AU) ?()(2.06 x 105)
• 2.06 x 105 is used to convert rad ? arcsec
• Also used to convert AU to pc
• Note R(AU) d(pc) ?()
• Can also relate to parallax in arcsec
• R(AU) ?()/p() (5.22)

23
Behavior of Doppler Shifts

• Now look at Doppler Shifts
• P is the same for both stars
• v 2pr/P
• r1 r2 (P/2p)(v1 v2)
• Eliminate R in 5.20
• (P/2pG)(v1r v2r)3 m1 m2 (5.23)
• If orbit is inclined, vr vsin(i)
• (P/2pG)(v1r v2r)3/sin3(i) m1 m2 (5.24)

24
What Does This Mean?

• If binary is eclipsing, i 90º
• Otherwise we dont know i
• Circular orbits that are projected onto the sky
  appear elliptical
• If we dont know i, if we solve assuming i 90º
  we will get a lower limit on m1 m2
• The true value is this limit divided by sin3(i)

25
Using One Doppler Shift

• Sometimes only one Doppler shift can be seen
• Eliminate v2 ? v2 v1(m1/m2)
• v1 v2 v1(1 v2/v1)
• v1(1 m1/m2)
• (v1/m2)(m1 m2)
• Substitute into 5.24
• (P/2pG)(v1r)3 m23sin3(i)/(m1 m2)2 (5.25)
• The RHS is called the mass function
• Measuring one Doppler shift gives us only the
  mass function, not individual masses

26
Energy of the System

• Total energy
• E (1/2)m1v12 (1/2)m2v22 -Gm1m2/R (5.26)
• We know m1v12 Gr1(m1m2/R2), substitute back in
  
• E -(1/2)Gm1m2/R (5.27)
• Negative E means the system is bound
• Need to add at least (1/2)Gm1m2/R to break it up
• Analogous to an H atom
• As the orbits get smaller, the energy becomes
  more negative

27
Elliptical Orbits

• Semi-major axis a
• Semi-minor axis b
• Sum of distances from any point to two fixed
  point (foci) is constant
• r, r are distances r r constant 2a

28
Eccentricity

• Ellipses have eccentricity the distance between
  the foci / 2a
• Circle has eccentricity of zero
• Eccentricity lt 0
• r r a
• b2 a2 (ae)2
• a2(1-e2) (5.29)

29
Apastron Periastron

• In a binary system, the COM is at one focus
• Farthest point apastron
• r(apastron) a(1 e)
• Closest point periastron
• r(periastron) a(1 e)
• Average of these is a, replaces radius in a
  circular orbit

30
Relate to r and ?

• Using the law of cosines
• r2 r2 (2ae)2 2r(2ae)cos? (5.31)
• Using 5.28
• r a(1 e2)/(1 ecos?) (5.32)