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					   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
                            Limitations imposed by oscillations,
   astrometry
                                 granulations and activity –
Urban Eriksson


Introduction
                                    µ–arcsec astrometry
Optical
Astrometry
Astrometric detection
of exoplanets
                                                Urban Eriksson1
Astrophysical
limitations
                        1
                            Dept. of Mathematics and Science, Kristianstad University College
Astrometric
effects of
stellar surface
structures
                                        Towards Other Earths
Simulating                                    urban.eriksson@hkr.se
stellar surface
structures

Results                                           October 21, 2009
Conclusions
                        Contents

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
                          Introduction
    µ–arcsec
   astrometry
                          Astrometry
Urban Eriksson


Introduction
                          Astrometric effects of
Optical                   stellar surface
Astrometry
Astrometric detection
of exoplanets             structures
Astrophysical
limitations               Simulating stellar
Astrometric
effects of                surface structures
stellar surface
structures

Simulating
                          Results
stellar surface
structures                Conclusions              Figure: The Sun, a spotted star with
Results                                            planets (Hα )
Conclusions
                        Introduction

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson


Introduction

Optical
Astrometry
Astrometric detection
of exoplanets


Astrophysical
limitations

Astrometric
effects of
stellar surface
structures

Simulating
stellar surface
structures
                        Figure: A view from our place in the Universe, but where are the stars
Results
                        and planets?
Conclusions
                        Introduction

   Limitations
  imposed by
  oscillations,
                        We do not know the locations of very many stars in the Galaxy.
 granulations
 and activity –             Position A few million stars to an accuracy level of about
    µ–arcsec
   astrometry                        0.01 arcsecond.
Urban Eriksson             Distance Only some 20 000 stars to 10% or better...
Introduction            Basic stellar data obtained with the astrometric method are:
Optical
Astrometry                  parallax
Astrometric detection
of exoplanets
                            position
Astrophysical
limitations                 proper motion
Astrometric
effects of
                        The Hipparcos project (∼ 1990), accuracy of about
stellar surface
structures
                        0.001 arcsec (1 milliarcsec = 1 mas).
Simulating
stellar surface
structures

Results

Conclusions
                        Introduction

   Limitations
  imposed by                New instruments with about 100 times higher astrometric
                            accuracy, or about 10 µas:
  oscillations,
 granulations
 and activity –
    µ–arcsec                    Gaia
   astrometry
                                SIM
Urban Eriksson                  Ground-based interferometers:
Introduction                         Radio interferometry (VLBA already at 10 µas)
Optical
                                     Optical interferometry (VLTI PRIMA, ...)
Astrometry
Astrometric detection       We now approach the fundamental limits for how accurate
of exoplanets


Astrophysical
                            we can measure stellar positions!
limitations
                            The stars themselves set one of these limits, depending
Astrometric
effects of                  on stellar surface structures such as spots, plages,
stellar surface
structures                  faculae, granulation and non-radial oscillations.
Simulating
stellar surface
                            This affects astrometric measurements at µas level!
structures

Results

Conclusions
                        Introduction

   Limitations
  imposed by
  oscillations,
                        In this presentation I present stellar models for (the astrometric
 granulations
 and activity –
                        effect of) a spotted surface. I have investigated the influence of
    µ–arcsec            stellar surface structures on
   astrometry

Urban Eriksson
                          1   the total flux
Introduction
                          2   the photocentric position
Optical
Astrometry
                          3   radial velocity
Astrometric detection
of exoplanets
                          4   the third central moment
Astrophysical                 (of interest for interferometry)
limitations

Astrometric             using both numerical simulations and an analytical model.
effects of
stellar surface
structures              Finally, I evaluate the expected astrometric effects for different
Simulating
stellar surface
                        types of stars, and draw some conclusions concerning the
structures
                        possibility of (Earth-like) exoplanet detection around these
Results
                        stars.
Conclusions
                        Detection of exoplanets

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson


Introduction

Optical
Astrometry
Astrometric detection
of exoplanets


Astrophysical
limitations

Astrometric
effects of
stellar surface
structures

Simulating
stellar surface
structures

Results
                        Figure:      Perryman (2000) created a diagram, giving an overview of the present and future methods of detecting
                        exoplanets. This diagram is an updated version from October 3, 2007, taken from the Jean Schneider’s Extrasolar
                        Planets Encyclopaedia (http://exoplanet.eu). c M.A.C. Perryman
Conclusions
                        Astrometric detection of exoplanets

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson


Introduction

Optical
Astrometry
Astrometric detection
                            The apparent path of a star on the sky can be described
of exoplanets
                            by five astrometric parameters (α, δ, , µα∗ , µδ ).
Astrophysical
limitations                 Any deviation from this model indicates some perturbation
Astrometric
effects of
                            of the star, e.g. from an exoplanet.
stellar surface
structures                  The size of this deviation is called the astrometric
Simulating                  signature of the exoplanet.
stellar surface
structures                  The size of the signature is conveniently expressed in µAU
Results
                            (1 µAU at 1 pc = 1 µas).
Conclusions
                         Expected astrometric effect of exoplanets

   Limitations
  imposed by            The astrometric signature, α                       The RMS value of the photocentric
  oscillations,
 granulations                                                              displacement is
 and activity –                           Mp                 Mp
    µ–arcsec                       α=             a             a                                       α
   astrometry
                                        M∗ + Mp              M∗                                  σpos = √
                                                                                                               3
Urban Eriksson          since Mp     M∗ and where a is the
Introduction
                        semi-major axis. For a main sequence
                        star (luminosity L∗ ∝ M∗ .5 ), the mean
                                               4
Optical
Astrometry              distance of the habitable zone is
Astrometric detection
of exoplanets
                        approximately (Kasting et al. 1993)
Astrophysical                                                2.25
limitations                           L∗        M∗
                            a=           [AU] =                     [AU]
Astrometric                           L         M
effects of
stellar surface
structures              which gives
Simulating                                            1.25
                                         Mp M∗
stellar surface
structures                     α    3×                       [µAU]
                                         M⊕ M
Results
                        where M⊕      3 × 10−6 M
                                                                           Figure:       Schematic diagrams describing the motion of a
Conclusions                                                                star in an inclined orbit around the barycentre.
                        Expected astrometric effect of exoplanets

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson


Introduction

Optical
Astrometry
Astrometric detection
of exoplanets


Astrophysical
limitations

Astrometric
effects of
stellar surface
structures

Simulating
stellar surface
structures
                        Figure:      Graph of the expected astrometric RMS dispersion for different main sequence stars, in the mass
                        range 0.2–2 M , corresponding to spectral classes A – M, caused by an Earth-like (in mass) exoplanet in the
Results                 habitable zone. Note that 1µAU ∼ 150 km.

Conclusions
                        Astrophysical limitations to ultra–high–precision
                        astrometry
   Limitations
  imposed by
  oscillations,
                        Any deviation from a simple, symmetric object is a potential
 granulations
 and activity –
                        limitation to the accuracy, e.g.:
    µ–arcsec
   astrometry

Urban Eriksson              Circumstellar disks                                            Stellar surface structures
                            Multiplicity                                                   (spots, plages, flares, faculae,
Introduction
                                                                                           granulation, non-radial oscillations, ...)
Optical
Astrometry
                            Weak microlensing
Astrometric detection
of exoplanets
                            (distortion by gravitational
Astrophysical               fields)
limitations

Astrometric
effects of
stellar surface
structures

Simulating
stellar surface
structures              Figure:        The position of a star is altered by a lensing object close to the lines of sight of different epochs. The
                        parallax will appear larger than without the lensing object, and thereby making the star appear closer to the
Results                 observer. The positional change over time can erroneously be interpreted as an extra proper motion.
Conclusions
                        Astrometric effects of stellar surface structures
                        Introducing the integrated properties
   Limitations
  imposed by            Stellar surface structures affect the brightness distribution of the star
  oscillations,
 granulations           by blocking/emitting light in different areas on the star. These
 and activity –
    µ–arcsec
                        brightness fluctuations influence integrated properties such as
   astrometry

Urban Eriksson
                          1   total flux

Introduction
                          2   photocentric displacement
Optical                   3   radial velocity
Astrometry
Astrometric detection
of exoplanets             4   third central moment (skewness, closure phase)
Astrophysical
limitations             These effects have been investigated
Astrometric             – numerically, using a ‘spotted star’ model
effects of
stellar surface
                        – analytically, using a statistical model
structures
                        We seek for distinct relations between the dispersions of the different
Simulating
stellar surface         properties.
structures

Results

Conclusions
                        The moments of the brightness distribution

   Limitations
  imposed by            The following theoretical relations are used in the model
  oscillations,
 granulations
 and activity –                            F (t )   =       I(r, t )µ dS
    µ–arcsec
   astrometry                                           S

Urban Eriksson                                            1
                                         ∆ x (t )   =                I(r, t )x µ dS
                                                        F (t )
Introduction                                                     S
Optical                                                   1
Astrometry                               ∆ y (t )   =                I(r, t )y µ dS
Astrometric detection                                   F (t )
of exoplanets                                                    S

Astrophysical                                             1
limitations                               µ3 (t )   =                I(r, t )(x − ∆x (t ))3 µ dS
                                                        F (t )
Astrometric                                                      S
effects of
                                                          1
stellar surface                        ∆vR (t )     =                I(r, t ) (ω × r ) · ˆ µ dS
                                                                                         z
structures                                              F (t )
                                                                 S
Simulating
stellar surface
structures

Results

Conclusions
                        The moments of the brightness distribution

   Limitations
  imposed by            Generally, the normalised moments are
  oscillations,
 granulations                                  Mmn   1
 and activity –                                    =               I(x , y )x m−n y n µ dS
    µ–arcsec                                   M00   F
   astrometry                                                S

Urban Eriksson          and the central moments
                                        Mmn   1
Introduction                                =          I(x , y )(x − ∆x )m−n (y − ∆y )n µ dS
                                        M00   F
Optical                                           S
Astrometry
Astrometric detection   In conclusion, we find that
of exoplanets

                                                                                M10
Astrophysical                                         F = M00         ∆x =      M00
limitations

Astrometric                                                  M30
effects of                                            µ3 =   M00
                                                                      ∆y =      M01
                                                                                M00
stellar surface
structures
                        For the radial velocity we have
Simulating
stellar surface                                   ∆ vR    =        ωx ∆y − ωy ∆x
structures
                                                                      M01      M10
Results                                                   =        ωx     − ωy
                                                                      M00      M00
Conclusions
                        The Equivalent ARea Spot (EARS) model

   Limitations
  imposed by
  oscillations,
                        We have created a simulation model where the stellar surface
 granulations
 and activity –
                        is featureless except for a number of localized spots. We
    µ–arcsec            assume N spots that are
   astrometry

Urban Eriksson               absolutely black,
Introduction                 small in comparison to the stellar radius,
Optical
Astrometry
                             of equal area A expressed as a fraction of the total
Astrometric detection
of exoplanets
                             surface,
Astrophysical                randomly spread over the entire surface of the star, and
limitations

Astrometric                  fixed in position on the surface, while the star rotates.
effects of
stellar surface         The star itself is treated as a solid, spherical body with a
structures
                        rotation period P about a tilted axis.
Simulating
stellar surface
structures

Results

Conclusions
                        An example

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson


Introduction

Optical
Astrometry
Astrometric detection
of exoplanets


Astrophysical
limitations

Astrometric
effects of
stellar surface
structures

Simulating
stellar surface
structures              Figure:      This figure shows the effect of one dark spot on flux (mag), photocentric displacement, third central
                        moment and radial velocity as the star rotates about its axis. Here a spot covering 1% of the visible stellar surface is
Results                 located at 30◦ latitude on the surface of a star with its rotation axis perpendicular to the line of sight.

Conclusions
                        Some statistical considerations

   Limitations
  imposed by            What can be expected from a statistical point of view?
  oscillations,
 granulations
 and activity –              Spot filling factor, f ∝ A · N .
    µ–arcsec
   astrometry                Some previous studies (Saar & Donahue 1997; Hatzes 2002)
Urban Eriksson               suggest σpos , σvR ∝ f 0.9 or f 0.92 .
Introduction                 If A · N     1 all effects are proportional to A.
Optical
Astrometry                   N spots are randomly spread over the surface but only k visible
Astrometric detection
of exoplanets                at any given time. The number of visible spots follows a
Astrophysical                binomial distribution function (p = 0.5, N). The RMS dispersion
                                                      √
                             of such a distribution is N /2.
limitations

Astrometric
effects of
stellar surface
                             Expected RMS dispersion for the integrated properties:
structures
                                              √
Simulating                              σj ∝ A N       where      j = F , m, pos, vR and µ3
stellar surface
structures

Results
                             rather than depending on the spot filling factor, f, or some power
                             of f.
Conclusions
                          Results of Monte Carlo simulations

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson


Introduction

Optical
Astrometry
Astrometric detection
of exoplanets


Astrophysical
limitations

Astrometric
effects of
stellar surface
structures              Figure:        Results of Monte Carlo simulations of the effect   Figure:        Results of Monte Carlo simulations of a rotating
                        of varying spot size clearly show that the RMS dispersion of      star with different number (N) of spots of size A = 0.0025.
Simulating              the magnitude, σm , is proportional to the spot size, σm ∝ A ,    The different graphs represent from top to bottom
stellar surface         as predicted. The upper graph represents the case σm with         σm , σpos , σµ3 and, σvR , expressed on an arbitrary scale.
structures              ten spots. The lower graph represents the case for only one       Dots and error bars represent mean value and dispersion of
                        spot. In both cases the spot size varies from 0.0025 to 0.08 of   the σ values for a set of simulations with a given N. The
                                                                                                                                                 √
Results                 the total surface area. The slopes of the dashed lines are 1,     dashed lines have slopes 0.5, corresponding to σ ∝ N.
                        corresponding to a linear relationship. The same linear
Conclusions             relation is found for the other dispersions.
                        Results from the Monte Carlo simulations

   Limitations
  imposed by            The following mean relations were found for a large number of spotted model
  oscillations,
 granulations
                        star with only dark spots and a random i:
 and activity –
    µ–arcsec
   astrometry                                           √
                             σm        (1.17 ± 0.60) · A N                    σpos      0.49 R σm
Urban Eriksson                                          √
                            σpos       (0.57 ± 0.25) · A N · R                σµ3       0.19 R 3 σm
                                                        √
Introduction
                             σµ3       (0.22 ± 0.09) · A N · R 3              σvR       0.43 R ω σm
Optical                                                 √
Astrometry
Astrometric detection
                             σvR       (0.51 ± 0.26) · A N · R ω
of exoplanets


Astrophysical
limitations                  The simulations show that there are statistical relations between the
Astrometric                  integrated properties.
effects of                                                                    √
stellar surface              All RMS dispersions follow the predicted σ ∝ A    N.
structures
                             The relationships are supported by the analytical model (Eriksson &
Simulating
stellar surface              Lindegren, 2007).
structures

Results

Conclusions
                        Results from the Monte Carlo simulations and the
                        analytical model
   Limitations
  imposed by
  oscillations,
 granulations
 and activity –         Table: Results from simulations for different types of models. Case
    µ–arcsec
   astrometry
                        1–4 are from the rotating model, Case 5–6 from the static model and
Urban Eriksson
                        Case 7 is the predicted result from the analytical model in Appendix
                        of Eriksson & Lindegren (2007). In Case 5–7 we do not have any
Introduction
                        rotation and therefore no radial velocity dispersion.
Optical
Astrometry
Astrometric detection            D[M10 ]     D[M30 ]      σpos     σ µ3    σvR
of exoplanets            Case    RD[M00 ]   R 3 D[M00 ]   R σm    R 3 σm   R ωσm
                                                                                   Comments
Astrophysical              1       0.53         0.20       0.49    0.19     0.43   Rotating, dark spots, random i
limitations                2       0.47         0.18       0.42    0.16     0.47   Rotating, dark spots, i = π/2
                           3       0.54         0.20       0.49    0.19     0.43   Rotating, dark and bright spots, random
Astrometric                4       0.40         0.15       0.37    0.14     0.46   Rotating, dark and bright spots, i = π/2
effects of                 5       0.49         0.15       0.45    0.14       –    Static, dark spots
stellar surface            6       0.41         0.15       0.38    0.14       –    Static, dark and bright spots
structures                 7      0.409        0.151      0.376   0.139    0.307   Analytical model

Simulating
stellar surface
structures

Results

Conclusions
                        Astrometric jitter estimated from photometric and
                        radial velocity variations
   Limitations
  imposed by
  oscillations,
                        Our models give the following mean relations for the
 granulations
 and activity –
                        astrometric jitter
    µ–arcsec
   astrometry
                                                σpos   0.376R σm                    (12)
Urban Eriksson
                                                σpos   0.195P σvR                   (18)
Introduction

Optical                 where R is the radius and P the rotation period. These may be
Astrometry
Astrometric detection
of exoplanets
                        upper limits if other effects (e.g. pulsations) contribute to σm
Astrophysical
                        and σvR .
limitations

Astrometric             Svensson & Ludwig (2005) used hydrodynamical model
effects of
stellar surface         atmospheres to estimate the astrometric jitter due to
structures

Simulating
                        granulation, from which a lower limit is derived:
stellar surface
structures
                                          σpos = (300 µAU) × 101−log g              (27)
Results

Conclusions             where g the surface gravity of the star (in cm s−2 ).
                        Variability across the HR diagram

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson


Introduction

Optical
Astrometry
Astrometric detection
of exoplanets


Astrophysical
limitations

Astrometric
effects of
stellar surface
structures

Simulating
stellar surface
structures

Results

Conclusions
                        Figure:     Stellar variability in the HR diagram. Stars in different bins are labelled with the mean intrinsic scatter
                        (Eyer & Grenon 1997).
                        Estimated astrometric jitter for different stellar
                        types
   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson


Introduction

Optical
Astrometry
Astrometric detection
of exoplanets


Astrophysical
limitations

Astrometric
effects of
stellar surface
structures

Simulating
stellar surface
structures

Results

Conclusions
                        Estimated astrometric jitter for different stellar
                        types
   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson


Introduction

Optical
Astrometry
Astrometric detection
of exoplanets


Astrophysical
limitations

Astrometric
effects of
stellar surface
structures

Simulating
stellar surface
structures

Results
                         Figure:       HR diagram visualising the astrometric jitter from the previous table. The diameters of the circles are
Conclusions              proportional to log σpos (in µAU).
                        Comparison with the effects of exoplanets

   Limitations
  imposed by                The astrometric effect of Earth-like exoplanets in the habitable
  oscillations,
 granulations               zone around late type main-sequence stars is σplanet 4 µAU
 and activity –
    µ–arcsec                Astrometric detection is only possible if
   astrometry

Urban Eriksson
                                                       σpos   σplanet                       (†)
Introduction
                            or
Optical
Astrometry                                             σpos   0.5α                          (‡)
Astrometric detection
of exoplanets
                            and then only if other noise sources (photon noise, instrument
Astrophysical
limitations
                            effects, etc.) are even smaller (Sozzetti 2005).
Astrometric                 Therefore, Earth-like exoplanets can only be detected
effects of
stellar surface             astrometrically around unusually stable main-sequence stars,
structures
                            like the Sun.
Simulating
stellar surface             For most old, solar-type stars the expected astrometric jitter is
structures
                            just about the limit of (†), making detection difficult if not
Results
                            impossible.
Conclusions
                        Conclusions

   Limitations
  imposed by
  oscillations,
                        Numerical simulations and an analytical model have been used
 granulations
 and activity –
                        to study the effects of stellar surface structures on integrated
    µ–arcsec            stellar properties, such as position, photometry and radial
   astrometry

Urban Eriksson
                        velocity.

Introduction
                          1   There are statistical relationships between the different
Optical
                              integrated properties (σm , σpos , σvR , ...).
Astrometry                                     √
Astrometric detection
                          2   We find σ ∝ A · N. Thus, the spot filling factor (∝ A · N)
of exoplanets


Astrophysical
                              is not the most relevant parameter for spottiness.
limitations
                          3   It will be difficult to detect Earth-like exoplanets by
Astrometric
effects of                    astrometric techniques. It should however be possible for
stellar surface
structures                    the most stable stars, similar to the Sun.
Simulating
stellar surface
                          4   Detection of larger exoplanets ( 10M⊕ ) should be
structures
                              possible for most MS stars.
Results

Conclusions
                        Thanks...

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson


Introduction

Optical
Astrometry
Astrometric detection
of exoplanets
                                    Thank you for your attention!
Astrophysical
limitations

Astrometric
effects of
stellar surface
structures

Simulating
stellar surface
structures

Results

Conclusions
                  Graph1

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph2

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph3

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph4

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph5

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph6

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph7

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph8

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph9

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph10

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph11

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph12

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph13

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph14

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph15

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph16

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph17

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson
                  Graph18

   Limitations
  imposed by
  oscillations,
 granulations
 and activity –
    µ–arcsec
   astrometry

Urban Eriksson

				
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