PX158 - G2 - spectroscopic binaries

• most binaries cannot be spatially resolved
• however, in many cases, they can be resolved spectroscopically
• the two stars will have different emission spectra
• the velocity of the stars can be measured by the doppler shift of the lines
• assuming circular orbit:

• the doppler shift is given by:

where, $\lambda =$ observed wavelength, ${\lambda }_0=$ rest wavelength, $v_{RV}=$ radial velocity
- $v_{RV}<<c$

• the inclination of the binary star orbit is given by $i$

  • $i=90\mathrm{°}:$ edge on $⟹v_{RV}=v$
  • $i=0\mathrm{°}:$ face on $⟹v_{RV}=0$

• consider an observer is edge a binary system, ie: $i=90\mathrm{°}$, and star 1 is revolving about the centre of mass in the anticlockwise direction

  • $(a):\theta =0:$ star 1 is closest to the observer and the velocity is perpendicular to the line of sight
  • $(b):\theta =\frac{\pi }{2}:$ star 1 is moving away from the observer with maximum (positive) radial velocity
  • $(c):\theta =\pi$ star 1 is furthest from the observer and the velocity is perpendicular to the line of sight
  • $(d):\theta =\frac{3\pi }{4}:$ star 1 is moving towards the observer with minimum (negative) radial velocity

• for eclipsing binaries, $i\sim 90$
• as $i\to 0$, the amplitude will decrease given by $\sin i$
• the motion of a binary star through the galaxy can be seen as a constant velocity ($\gamma$)

• if $i$ is unknown, $m=m_1\sin i$
• only a minimum mass can be obtained