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Determining orbits from radial velocities

Determining orbits from radial velocities. Observe system over n times (observations) n observations n (V r , t) V = F(K,e, w,g ,T,P) since T,P enter expression for V via q , from time of observation t, estimates or known values of P,e,T, hence E (ecc. anomoly) Thus:

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Determining orbits from radial velocities

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  1. Determining orbits from radial velocities • Observe system over n times (observations) • n observations n (Vr, t) • V = F(K,e,w,g,T,P) • since T,P enter expression for V via q, • from time of observation t, estimates or known values of P,e,T, hence E (ecc. anomoly) • Thus: • what are values of (K,e,w,g,T,P) that best represent observations • requires least squares procedure

  2. Application to Binary Pulsars • binary system where one star is a pulsar • emits ‘pulses’ of radiation • accurate timing possible (accurate clocks) • need narrow pulses • radio signals from neutron stars • solitary pulsar • if at 0 velocity relative to us • time between pulses, dt = constant (unless being spun up/down) • if at Vrel relative velocity • dt = constant x pulse number • if pulsar spins up, dt decreases with pulse number • concave curve • if pulsar spins down, dt increases with pulse number • convex curve

  3. Pulsar Timing • In binary system, time between pulses affected by orbital motions • due to light travel time (distance) changing along orbit

  4. Pulsar timing II • Projected distance across orbit at any time t is • z = r sin (q+w) sin i • For the pulsar orbit • rp = ap (1-e2) / (1+ e cosq) • So the light travel time is for a circular orbit w =0, e=0

  5. light travel time

  6. Pulsar timing III • Hence, time difference between predicted , jnt, and actual (binary) pulse arrival times, tn is • where P is the orbital period, T0 is a reference time and a,b are determined by the velocity of the pulsar • a: from systematic velocity • b: from Keplerian velocity • for circular orbits: b = ap/c sini

  7. Mass determinations • If optically visible companion • OB star in massive X-ray binaries • A-F star in low-mass X-ray binaries hence, the mass function is and the mass ratio, q, • If inclination, i, can be found, then mass follow directly

  8. Frequency shifts • Binary orbit also affects pulsar frequency • radio pulsars , very narrow pulse widths • pulse frequency affected by orbital velocity • Doppler shift: • gives a phase lag of:

  9. Pulsar Phase lag • Combined phase lag is • from light travel time due to orbit • and from Doppler shift • hence • generally • but measurable in radio pulsars

  10. Eclipsing Binaries • Determine binary properties from eclipsing systems • sizes, shapes of stars, temps, apsidal motion

  11. Eclipsing systems • Four contact points • obscuring star touching background star entering eclipse • obscuring star fully in front of background star • touching rim on entering • touching rim on exiting • obscuring star touching background star exiting eclipse • Relative sizes

  12. Partial Eclipse

  13. Light Curves

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