Download AS 3004 Stellar Dynamics

Determining orbits from radial
velocities
• Observe system over n times (observations)
– n observations
– V = F(K,e,w,g,T,P)
n (Vr, t)
– 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)
2
E  esin E 
(t  T)
P
– Thus:
1
2
q  1  e
E 
tan    
 tan  
2  1  e
2 
• what are values of (K,e,w,g,T,P) that best represent
observations
• requires least squares procedure
AS 3004
Stellar Dynamics
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
AS 3004
Stellar Dynamics
Pulsar Timing
• In binary system, time between pulses affected by
orbital motions
– due to light travel time (distance) changing along orbit
AS 3004
Stellar Dynamics
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
z 
c
a p sin i
c
sin(q  w )
(1  e )
1  e cos q
2
for a circular orbit w =0, e=0
a
sin
i
2

p
z 
sin (t  T0 )
c
 P

c
AS 3004
Stellar Dynamics
light travel time
AS 3004
Stellar Dynamics
Pulsar timing III
• Hence, time difference between predicted , jn t, and
actual (binary) pulse arrival times, tn is
2 (t  T0 ) 
t  t n  jn t  at  b sin 



P
– 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
AS 3004
Stellar Dynamics
Mass determinations
• If optically visible companion
– OB star in massive X-ray binaries
– A-F star in low-mass X-ray binaries
2 12
(1  e )
ac sin i 
Kc P
2
hence, the mass function is
m2 sin 3 i
f (m) 
(m1  m2 ) 2
and the mass ratio, q,
mp
ac sin i
q

mc a p sin i
– If inclination, i, can be found, then mass follow directly
AS 3004
Stellar Dynamics
Frequency shifts
• Binary orbit also affects pulsar frequency
– radio pulsars , very narrow pulse widths
– pulse frequency affected by orbital velocity
– Doppler shift:
Vrad
z
f  f
 f
c
c
– gives a phase lag of:
t
z
   f dt  f 0 t  T0 
c
T0
z
 f0
c
AS 3004
z

c

Stellar Dynamics
Pulsar Phase lag
• Combined phase lag is
– from light travel time due to orbit
z
 L   f
c
– and from Doppler shift
z  z 
 D  f 0  
c c
– hence
–
 z   z

   D   L     f 0  f 0  f0 (t  T0 )
 c  c

generally
 D  0.001 L
– but measurable in radio pulsars
AS 3004
Stellar Dynamics
Eclipsing Binaries
• Determine binary properties from eclipsing
systems
– sizes, shapes of stars, temps, apsidal motion
AS 3004
Stellar Dynamics
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
2 R2








 2 1  4 3 
2a
3  1    4   2  
AS 3004
2 R1
2a
Stellar Dynamics
Partial Eclipse
AS 3004
Stellar Dynamics
Light Curves
AS 3004
Stellar Dynamics