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Chapter 4- Timing
Contents
4.0 Introduction
4.1 Pulsars
4.2 Pulsating stars
4.2.1 White dwarfs
4.2.2 Hot subdwarfs
4.3 Eclipsing binaries
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4.0 Introduction (1)
 Principle
 The motion of planets in orbit around a star causes the star to
undergo a reflex motion around the barycenter which can be
measured either through a change in the radial velocity of the
star or a change in its position in the sky (astrometry).
 The reflex motion of the star can also be inferred if the star
features a periodic signal that will vary due to the Doppler effect.
 Three types of periodic signal
 Radio pulsars
 Pulsating stars (white dwarfs, subdwarfs)
 Eclipsing binaries
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4.0 Introduction (2)
 The timing method refers to the timing delay τp
associated with the light travel time associated with the
reflex motion of the star around the barycenter, defined
as
For a circular orbit, the amplitude of the time delay is
Orbital plane
observer
(4.1)
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4.0 Introduction (3)
Discovery status (end of 2010)
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4.0 Introduction (3)
Discovery status (past 2010; not complete)
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Contents
4.0 Introduction
4.1 Pulsars
4.2 Pulsating stars
4.2.1 White dwarfs
4.2.2 Hot subdwarfs
4.3 Eclipsing binaries
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4.1 Pulsars (1)
 Rapidly rotating highly-magnetized neutron stars with
magnetic axis beamed towards the Earth
 Violent formation through core collapse of massive (~8-40
M) stars in a supernova explosion.
 Emit two narrow beams of radio emissions aligned with
magnetic axis.
 Two classses:
 Normal pulsars with P~1s
 msec pulsars, i.e., ‘recycled’ old neutron stars
spun-up to vey short periods during mass and
angular momentum transfer from a binary
companion. Most msec pulsars still have
binary companions, either white dwarfs or
neutron stars.
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4.1 Pulsars (2)
 msec pulsars are extremely accurate clocks
 Periods changing through tiny spin-down rate (10-19 ss-1).
 Pulse arrival time residuals measured with μs accuracy.
 Known pulsar population: 1700
 80 msec pulsar in the Galaxy
 130 in Galactic globular clusters
 11 with distance less than 300 pc
 For a circular edge-on orbit of period P, and assuming a
canonimal neutron star mass of 1.34 M,
(4.2)
 Possible to detect moon-size planets with μs accuracy.
• Ex: Mp~0.01 M, P~30 days, τp~2 μs
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4.1 Pulsars (3)
Factors affecting the timimg delay
 The general formulation of the barycentric pulse arrival time, tB, is
(Wolszczan & Kuchner 2011; Eqn 11)
tclk :
r:
n:
clock correction that accounts for differences between the
observatory clocks and terrestrial time standards.
net vector of the observatory to the barycenter. Sum of three
vectors pojnting Earth’s center, from there to the center of the
Sun, and then to the source.
unit vector in the direction of the pulsar.
with the ecliptic latitude. This is the Roemer delay, the travel
time within the Solar system. This is the most uncertain correction
since it requires a very accurate knowledge of the pulsar’s sky
position.
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4.1 Pulsars (4)
Factors affecting the timimg delay
D’ : a constant related to the column density of interstellar
electrons along the line of sight.
f : the frequency of the observations.
: ‘Shapiro’s delay’ acquired by light propagating through curved
space,
is the pulsar-Sun-Earth angle computed from the solar system
ephemeris.
 ~120 μs for the Sun, ~200 ns for Jupiter
:
tR :
Einstein’s delay. The combined effect of time dilation and
gravitational redshift of the signal due to the annual variation of a
terrestrial atomic clock as Earth moves around the Sun on its
eccentric orbits and to the presence of other masses in the solar
system. Maximum correction of this term is ~1.66 ms.
additional Roemer delay due to the Keplerian orbital motion of the
planet.
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4.1 Pulsars (5)
PSR B1257+12
 First exoplanets ever found (Wolszczan & Frail 1992)
 Distance: d~300 pc
 Initially discovered as a two planet system
 M sin i = 2.8 and 3.4 ME, a=0.47 and 0.36 AU.
 Orbital periods (98.22 and 66.54 d) close to a 3:2 resonnance
 Long term monitoring show evidence of planet-planet
interaction (see next slide)
 Enable mass estimate without a priori knowledge of the inclination
 Provide evidence for a third close-in planet: P=25.34 d, Mp=0.02 ME
• One the smallest planet ever discovered.
 Dynamical studies suggest the system to be stable over a
timescale
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4.1 Pulsars (6)
PSR B1257+12 – Planet-planet interaction
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4.1 Pulsars (7)
PSR B1257+12- Formation mechanisms
1. Planet formed around a normal (massive) star, the pulsar progenitor.
Its present existence implying that it must have survived the
supernova explosion.
2. Planet formed around another star before being captured by the
pulsar through dynamical interaction.
3. ‘’Fallback accretion’’. Planet formed after the supernova explosion
which created the neutron star.

Supernova need to retain some residual material that could fall back to form a
debris disk around the young neutron star.
 Difficulties in modelling planets which survive the supernova
explosion may favour the ‘fallback’ accretion disk model.
 Fallback model imply the existence of a dust disk.

Attempt to detect it by SPITZER failed so far.
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4.1 Pulsars (8)
PSR B1620-26
 11-ms pulsar discovered in globular cluster M4.
 Has a binary companion, a 0.3 M white dwarf in a 191-d low eccentricity
orbit.
 10 MJ planet orbiting pulsar-WD binary with a~35 AU, P~100 yr.
 Such a wide companion can survive the dense environment of the cluster
(Woolson 2004).
 Signatures of Newtonian interactionb between planet and WD observed.
Formation scenarios
1.
Planet forms around a main sequence star, then



Migration towards the cluster core where it encounters a neutron star binary.
One neutron star captures the star and planet, and ejects its original neutron
star companion.
The main sequence star evolves into a red giant (and eventually a WD),
transfering mass and so spinning up the neutron star to its final ms pulsar
status.
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4.1 Pulsars (9)
PSR B1620-26 – Fomation mechanisms
2. Planetary system encounters a pre-existing binary millisecond pulsar
or a pulsar/WD binary.

Planetary system is disrupted, with the main sequence star being ejected and the
planet captured.
 Both scenarios require the formation of a planetary system is a lowmetallicity globular cluster environment, which appears to occur with
low probability. Hence a third scenario,
3. Planet formed through gravitational instability through a passing
star perturbing the common-envelope of a main sequence/giant
binary.


Most massive component becomes a supernova
Main sequence star then transfers mass, spining up the neutron star to its msec
pulsar status befie evolving to a white dwarf.
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4.1 Pulsars (10)
Pulsar planetary formation mechanisms
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4.1 Pulsars (10)
Pulsar planetary formation through gravitational instability
Beer et al. (2004)
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Contents
4.0 Introduction
4.1 Pulsars
4.2 Pulsating stars
4.2.1 White dwarfs
4.2.2 Hot subdwarfs
4.3 Eclipsing binaries
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4.2 Pulsating stars
 Scientific interests of probing evolved stars for
planetary systems
 Provides insight into the future of the solar system in general and
Earth in particular.
 In survivability is robust, detecting such planets is another route
to characterizing their frequency and distribution.
4.2.1 White dwarfs
 End point of most stars up to ~8 M.
 Common in the solar neighborhood.
 Planetary system survical to red giant phase depends on several
parameters:




Initial orbit separation.
Stellar mass-loss rate
Tidal forces
Details of dynamical interaction with ejected material.
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4.2.1 Pulsating stars – white dwarfs
 Fate of planetary systems during the red giant phase.
 All planets within the final extent of the red giant envelope will
be engulfed and migrate inwards.
 Planets further out will have greater chance of survival,
migrating outwards as mass is lost from central star.
 In mass is loss instantaneously, planet could escape the system.
 Planetary orbits should expand adiabatically (constant energy)
• Ex: for a 1 M projenitor evolving into a 0.5 M white dwarf leads to
orbits expanding by a factor of two.
• In the process, some stable orbits might become unstable.
 Orbits with initial a > 0.7 AU remains larger than the primary
star radius at all stages of its evolution.
• WD could potentially host surviving planets with P > 2.4 yr.
 Two observing approachs for finding planets around WD:
 Pulsation timing
 Direct imaging (later)
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4.2.1 Pulsating stars – white dwarfs
Pulsating white dwarf
 As WD cools through certain temperature ranges,
C/O (~ 105 K; GW Vir), He (2.5x104 K; DBV) and H (104
K; DAV) in its photospheres progressively become
partially ionized, driving multi-periodic non-radial
g-mode (gravity driven) pulsations.
 Pulsation periods: 100-1000s.
 Include some of the most stable know, both in amplitude
and phase.
 Ideal targets for the timing method.
 e.g., G117-B15A
(4.3)
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4.2.1 Pulsating stars – white dwarfs
Pulsating white dwarf
 A planet with an orbital period much longer than the observational
baseline gives rise to an apparently linear change in pulsation
period,
(4.4)
where P is the white dwarf pulsation period.
 Ex: Mp=1 MJ, a=10 AU, P=100 s,
 Three possibilities for period variation not associated with a planet.
 Inherent to the white dwarf cooling:
 Known wide-separation proper motion companion not necessarily gravitationally
bound
 Proper motion. Important for nearby system (Pajdosz 1995)
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4.2.1 Pulsating stars – white dwarfs
Pulsating white dwarf
 GD 66 b, best candidate planet aound a WD.
 Monitoring since 2003 (2.1, McDonald Observatory)
 Variations consistent with a ~2 MJ in a 4.5 yr orbit
 Under the list of ‘unconfirmed’ planets in www.exoplanet.eu
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4.2.2 Pulsating stars – hot subdwarfs
 Hot subdwarfs, of spectral types O and B, also termed
"extreme horizontal-branch stars, represent a late stage
in the evolution of solar-mass stars caused when a red
giant star loses its outer hydrogen layers before the core
begins to fuse helium.
 Like WD, some subdwarfs are pulsating
 Première découverte: V391 Peg b (Silvotti et al, 2007)
 Découvertes récentes par la mission Kepler
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4.2.2 Pulsating stars – hot subdwarfs
V391 Peg b
 Progenitor mass: ~1 M
 Planet properties: Mp sin i = 3.2 MJ, a=1.7 AU and P=3.2 yr
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Contents
4.0 Introduction
4.1 Pulsars
4.2 Pulsating stars
4.2.1 White dwarfs
4.2.2 Hot subdwarfs
4.3 Eclipsing binaries
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4.3 Eclipsing binaries
HR Vir
 Short period: Porb=2.8 hrs
 sdB + M star, i=81°
 Studied since 1980
 Secular variation of the period due to
magnetic breaking.
 Now confirmed with two planets
 Mp sin i=19.2 MJ; P=15.8 yrs, a=5.3 AU, e=0.46
 Mp sin i=8.5 MJ; P=9.1 yrs, a=3.6 AU, e=0.31
 Periods suggest 5:3 or 2:1 resonnance
 High e in line with planet-planet
interactionn
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4.4 Summary
 Dynamical method
 Like astrometry, sensitive to long period planets.
 Amplitude of the time delay (circular orbit)
 Three types of objects used for timing
 Pulsars
 Pulsating stars (white dwarfs, hot subdwarfs)
 Eclipsing binaries
 Current sensus
 A dozen detection so far.
 Similar technque used with transit (TTV; TDV)
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