Binary Stars

1
Binary Stars

• Physical Astronomy
• Professor Lee Carkner
• Lecture 6

2
Questions

• 1) Consider a planet of mass m in a circular
  orbit of radius R around a star of mass M. What
  is the gravitational potential energy?
• U -(GMm/R)
• 2) What is the kinetic energy of the planet?
• K ½mv2
• 3) If you equate gravitational and centripetal
  force, what is the velocity?
• GMm/R2 mv2/R
• v2 GM/R
• 4) Does the virial theorem hold?
• K ½m(GM/R) ½(GMm/R) -½U
• (yes)

3
Visual Binaries

• Consider a binary star composed of two masses in
  two elliptical orbits around the center of mass
  (cm)
• m1/m2 a2/a1
• a1 a1/d
• m1/m2 a2/a1
• even if we dont know the distance, if we can
  measure as we can find the mass ratio

cm
a
4
Finding Mass

• Since a a1 a2, Keplers 3rd law gives
• P2 (4p2a3)/G(m1m2)
• Have to use m1m2 and m1/m2 to pull out ms
  separately

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Inclination

• The true orbit will be inclined to our line of
  sight by an inclination angle, i
• ? a cos i
• so Keplers third law becomes
• m1m2 (4p2/G)(d/cos i)3(?3/P2)

6
Spectroscopic Binaries

• Instead we measure velocity
• vr v sin i
• If we observe over time, velocity is periodic
• For circular, edge-on orbits, vmax is the true
  orbital velocity

7
Mass and Velocity

• We can get the ratio of masses even if we dont
  know i,
• If we do know i,
• m1m2 (P/2pG)(v1rv2r)3/sin3 i
• Where we can measure both v1r and v2r

8
Inclination and Statistics

• Often we cant find i
• We cant find mass for one star, but we can find
  an average mass for a class of stars
• Gives mass-luminosity relationship
• How does mass produce luminosity?

9
Eclipsing Binaries

• Eclipsing binaries show changes in brightness
• From Doppler shift we can get the velocity of
  each star
• Smaller vs
• Larger vl
• Relative velocity v vs vl

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Eclipsing Binaries and Radius

• rs (v/2)(tb-ta)
• Time for smaller to emerge from behind larger is
  just tc-ta, so radius is
• rl (v/2)(tc-ta)

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Eclipse Brightness

• Maximum light B0
• Primary minimum Bp
• Secondary minimum Bs
• Cooler star completely behind hotter

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Temperature

• B0-Bp
• B0-Bs
• Since flux is proportional to temperature,
• (B0-Bp)/(B0-Bs) 1-(Bp/B0)/1-(Bs/B0)
  (Ts/Tl)4
• Note that B1/B2100(m2-m1)/5

13
Next Time

• Read 8.1
• Homework 7.5, 7.6, 8.1, 8.1, 8.6a