Secular and Statistical Parallax

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Secular and Statistical Parallax

• Rachel Matson
• Extragalactic Distance Presentation
• 11/29/07

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Trigonometric Parallax

• Limited by size of angle
• Extend range with larger baseline

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Space Velocity

• Stars velocity with respect to the LSR

• U - radially outward from the galactic center
• V -in the direction of galactic rotation
• W -in the direction of the galactic north pole

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Secular Parallax

• v? 20 km/s ? Sun moves 4 AU/yr disk
• v? 200 km/s ? Sun moves 40 AU/yr halo
• After a time t of several decades, accumulated
  baseline vt is several orders of magnitude larger
  than the Earth-Sun baseline.

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Secular Parallax

• If the speed of the individual star was known to
  be v, then it is better by a factor of vt/AU over
  trigonometric parallax (same astrometric
  accuracy).
• The drawback is that the stars have a dispersion
  s, so the precision of the distance determination
  is inversely proportional to the Mach number, ?
  v/s (Popowski 1998).
• Apply method to a sample of N stars, drive down
  noise and achieve a precision aN-1/2 ?-1

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Secular Parallax

• Use point of convergence to find space velocity 
• Deduce distance by comparing actual speed with
  its apparent speed

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Secular Parallax

• Green stars-small mean distance
• Red stars -large mean distance
• Average proper motion of stars away from the
  point of the sky the Solar System is moving
  towards (apex)
• ? angle to apex
• d?/dt proper motion
• ? proportional to sin ?
• µ slope of line
• Mean distance
• Dcm V? cm/sec / µ radians/sec
• Dpc 4.16 / (µ "/yr )

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Secular Parallax
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Hipparcos and Secular Parallax

• Hipparcos proper motions (larger relative
  precision than trig. parallax) to derive 3
  times more precise distance estimates (de Bruijne
  et al. 2000)
• Secular parallax
• A 4.740 km yr s-1
• µ proper motion vectors
• v common space motion
• ? angular dist between a
• star and cluster apex

Vx -5.84km/s Vy 45.68km/s Vz 5.54km/s s
0.30 km/s
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Hipparcos and Secular Parallax

• Color-absolute magnitude diagrams of Hyades

Hipparcos trigonometric parallaxes (L), Hipparcos
secular, and Tycho-2 secular parallaxes (R)
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Statistical Parallax

• When the stars have measured radial velocities,
  then the scatter in their proper motions can be
  used to determine the mean distance.
• Assume velocities isptropically distributed

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Statistical Parallax

• Observed radial velocity
• Since v? is xiµi
• Statistical parallax

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Secular and Statistical Parallax

• Secular parallax is based on forcing equality
  between the three first moments of the velocity
  distribution
• from radial velocity and proper-motion
  measurements
• Classical statistical parallax is based on
  forcing equality of the six second moments
• six independent components of the velocity
  covariance
• Modern statistical parallax determines ten
  parameters simultaneously by applying maximum
  likelihood

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Statistical Parallax

• Ideally, the statistical parallax method should
  be applied to a group of stars that are
• (1) dynamically homogeneous, i.e., all the stars
  are drawn from a single velocity distribution
  (not constrained to be Gaussian), regardless of
  their locations
• (2) standard candles, i.e., all have the same
  absolute magnitude.

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RR Lyrae and Statistical Parallax

• One of the main luminosity calibration methods
  for RR Lyrae stars
• All systematic errors in the statistical parallax
  measurement of the absolute magnitude of RR Lyrae
  stars will have been removed or quantified, and
  the statistical errors will become the true
  uncertainty (Popowski and Gould 1999).

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LMC Distance Comparison

• Recent distant moduli to the LMC from statistical
  parallax of RR Lyrae stars (L) and trigonometric
  parallax of cepheids (R)

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Conclusions

• Secular and statistical parallax
• Statistical depends on stars random velocities
• Secular exploits solar motion
• Clusters of stars
• Range d500 pc