A&A 476, 1389-1400 (2007)
U. Eriksson1,2 - L. Lindegren1
1 - Lund Observatory, Lund University, Box 43, 221 00 Lund, Sweden
2 - Dept. of Mathematics and Science, Kristianstad University, 291 88 Kristianstad, Sweden
Received 7 June 2007 / Accepted 9 August 2007
Aims. To investigate the astrometric effects of stellar surface structures as a practical limitation to ultra-high-precision astrometry (e.g. in the context of exoplanet searches) and to quantify the expected effects in different regions of the HR-diagram.
Methods. Stellar surface structures (spots, plages, granulation, non-radial oscillations) are likely to produce fluctuations in the integrated flux and radial velocity of the star, as well as a variation of the observed photocentre, i.e. astrometric jitter. We use theoretical considerations supported by Monte Carlo simulations (using a starspot model) to derive statistical relations between the corresponding astrometric, photometric, and radial velocity effects. Based on these relations, the more easily observed photometric and radial velocity variations can be used to predict the expected size of the astrometric jitter. Also the third moment of the brightness distribution, interferometrically observable as closure phase, contains information about the astrometric jitter.
Results. For most stellar types the astrometric jitter due to stellar surface structures is expected to be of the order of 10 micro-AU or greater. This is more than the astrometric displacement typically caused by an Earth-size exoplanet in the habitable zone, which is about 1-4 micro-AU for long-lived main-sequence stars. Only for stars with extremely low photometric variability (<0.5 mmag) and low magnetic activity, comparable to that of the Sun, will the astrometric jitter be of the order of 1 micro-AU, sufficient to allow the astrometric detection of an Earth-sized planet in the habitable zone. While stellar surface structure may thus seriously impair the astrometric detection of small exoplanets, it has in general a negligible impact on the detection of large (Jupiter-size) planets and on the determination of stellar parallax and proper motion. From the starspot model we also conclude that the commonly used spot filling factor is not the most relevant parameter for quantifying the spottiness in terms of the resulting astrometric, photometric and radial velocity variations.
Key words: stars: general - starspots - planetary systems - techniques: interferometric - methods: statistical
The accuracy of astrometric measurements has improved tremendously over the past few decades as a result of new techniques being introduced, both on the ground and in space, and this is likely to continue in the next decade, e.g. Gaia (Perryman 2005) is expected to improve parallax accuracy by another two orders of magnitude compared to Hipparcos. As a result, trigonometric distances will be obtained for the Magellanic Clouds, and thousands of Jupiter-size exoplanets are likely to be found from the astrometric wobbles of their parent stars. Even before that, ground-based interferometric techniques are expected to reach similar precisions for relative measurements within a small field. How far should we expect this trend to continue? Will nanoarcsec astrometry soon be a reality, with parallaxes measured to cosmological distances and Earth-size planets found wherever we look? Or will the accuracy ultimately be limited by other factors such as variable optical structure in the targets and weak microlensing in the Galactic halo? We aim to assess the importance of such limitations for ultra-high-precision astrometry and we consider the effects of stellar surface structures found on ordinary stars.
Future high-precision astrometric observations will in many cases be able to detect the very small shifts in stellar positions caused by surface structures. In some cases, e.g. for a rotating spotted star, the shifts are periodic and could mimic the dynamical pull of a planetary companion, or even the star's parallax motion, if the period is close to one year. These shifts are currently of great interest as a possible limitation of the astrometric method in search for Earth-like exoplanets. We want to estimate the extent of these effects for different types of stars, especially in view of current and future astrometric exoplanet searches such as VLTI-PRIMA (Reffert et al. 2005), SIM PlanetQuest (Unwin 2005) and Gaia (Lattanzi et al. 2005).
Astrometric observations determine the position of the centre of gravity of the stellar light, or what we call the photocentre. This is an integrated property of the star (the first moment of the intensity distribution across the disc), in the same sense as the total flux (the zeroth moment of the intensity distribution) or stellar spectrum (the zeroth moment as function of wavelength). In stars other than the Sun, information about surface structures usually comes from integrated properties such as light curves and spectrum variations. For example, Doppler imaging (DI) has become an established technique to map the surfaces of rapidly rotating, cool stars. Unfortunately, it cannot be applied to most of the targets of interest for exoplanet searches, e.g. low-activity solar-type stars. Optical or infrared interferometric (aperture synthesis) imaging does not have this limitation, but is with current baselines (<1 km) in practice limited to giant stars and other extended objects (see Monnier et al. 2006, for a review on recent advances in stellar interferometry). Interferometry of marginally resolved stars may, however, provide some information about surface structures through the closure phase, which is sensitive to the third central moment (asymmetry) of the stellar intensity distribution (Lachaume 2003; Labeyrie et al. 2006; Monnier 2003).
Since there is limited information about surface structures on most types of stars, an interesting question is whether we can use more readily accessible photometric and spectroscopic data to infer something about possible astrometric effects. For example, dark or bright spots on a rotating star will in general cause periodic variations both in the integrated flux and in the radial velocity of the star, as well as in the photocentre and the asymmetry of the intensity distribution. Thus, we should at least expect the astrometric effect to be statistically related to the other effects.
We show that there are in fact relatively well-defined statistical relations between variations in the photocentre, total flux, closure phase and radial velocity for a wide range of possible surface phenomena. These relations are used in the following to predict the astrometric jitter in various types of stars, without any detailed knowledge of their actual surface structures.
The discovery of exoplanets by means of high-precision radial velocity measurements has triggered an interest in how
astrophysical phenomena such as magnetic activity and convective motions might affect the observed velocities
(Saar et al. 2003). Evidence for dark spots has been seen photometrically and spectroscopically for many cool
stars other than the Sun, and quantified in terms of an empirically determined spot filling factor
f, ranging from
for old, inactive stars to several percent for active stars. It is therefore natural to relate
the expected radial velocity effects to the spot filling factor. For example, Saar & Donahue (1997) used a simple model
consisting of a single black equatorial spot on a rotating solar-like star to derive the
following relation between f (in
percent), the projected rotational velocity
and the amplitude
of the resulting radial velocity
But f alone may not be a very good way to quantify the "spottiness''. For example, the photometric or astrometric effects of a large single spot are obviously very different from those of a surface scattered with many small spots, although the spot filling factor may be the same in the two cases. Therefore, more detailed (or more general) models may be required to explore the plausible ranges of the astrometric effects.
Bastian & Hefele (2005) give an assessment of the astrometric effects of starspots, and conclude that they are hard to quantify, mostly because of the insufficient statistics. Although starspots are common among cool stars with outer convective zones, data are strongly biased towards very active stars. They conclude that the effects on solar-type stars are likely to be negligible for Gaia, while much larger spots on K giants may become detectable. For supergiants and M giants, having radii of the order of (or more), the effect may reach 0.25 AU (or more), which could confuse the measurement of parallax and proper motion.
Sozzetti (2005) gives an interesting review of the astrometric methods to identify and characterise extrasolar planets. As an example of the astrophysical noise sources affecting the astrometric measurements, he considers a distribution of spots on the surface of a pre-main sequence (T Tauri) star. For a star with radius seen at a distance of 140 pc, he finds that a variation of the flux in the visual by % (rms) corresponds to an astrometric variation of as (rms), and that the two effects are roughly proportional.
While the astrometric effects cannot yet be tested observationally, it is possible to correlate the photometric and radial
velocity variations for some stars (Queloz et al. 2001; Henry et al. 2002). From a small sample of Hyades
stars, Paulson et al. (2004b) found an approximately linear relation
Svensson & Ludwig (2005) have computed hydrodynamical model atmospheres for a range of stellar types, predicting both the photometric and astrometric jitter caused by granulation. They find that the computed astrometric jitter is almost entirely determined by the surface gravity g of the atmosphere model, and is proportional to g-1 for a wide range of models. This relationship is explained by the increased granular cell size with increasing pressure scale height or decreasing g. The radius of the star does not enter the relation, except via g, since the increased leverage of a large stellar disc is compensated by the averaging over more granulation cells. For their most extreme model, a bright red giant with ( ) they find AU. Ludwig & Beckers (2005) extended this by considering the effects of granulation on interferometric observations of red supergiants. They show that both visibilities and closure phases may carry clear signatures of deviations from circular symmetry for this type of stars, and conclude that convection-related surface structures may thus be observable using interferometry.
Ludwig (2006) outlines a statistical procedure to characterise
the photometric and astrometric effects of granulation-related
micro-variability in hydrodynamical simulations of convective stars.
Based on statistical assumptions similar to our model in Appendix A,
he finds the relation
In a coordinate system
with origin at the centre of the star and away from the observer, let
be the instantaneous surface brightness
of the star at point
on the visible surface, i.e. the specific
intensity in the direction of the observer. We are interested in the integrated
properties: total flux F(t), photocentre offsets
the directions perpendicular to the line of sight, the third central moment of the
intensity distribution ,
and the radial velocity offset
These are given by the following integrals over the visible surface S (z<0):
Using a similar statistical method as Ludwig (2006), the rms variations
(dispersions) of m(t),
estimated from fairly general assumptions about the surface brightness
fluctuations (Appendix A). This calculation is approximately valid
whether the fluctuations are caused by dark or bright spots, granulation, or a
combination of all three, and whether or not the time variation is caused by the
rotation of the star or by the changing brightness distribution over the surface.
The result is a set of proportionality relations involving the radius of the star R, the limb-darkening factor a, and the centre-to-limb variation c of the
surface structure contrast (see (A.5) and (A.18) for the definition
of a and c). For a=0.6 (typical solar limb-darkening in visible light) and
c=0 (no centre-to-limb variation of contrast) we find
For the radial velocity dispersion, a similar relation can be derived under the previously mentioned conditions of a
time-independent, rigidly rotating star. Using that
As a check of the general relations in Sect. 3.1 we made numerical simulations with a very simple model, consisting of a limited number of (dark or bright) spots on the surface of a rotating star. The behaviour of the integrated properties is readily understood in this case (Fig. 1):
We assume a spherical star with N spots that are:
The star is assumed to rotate as a rigid body with period P around an axis that is tilted at an angle i to the line of sight (+z). For the present experiments we take the +y direction to coincide with the projection of the rotation vector onto the sky; thus , , and , where . Limb darkening of the form intensity is assumed, where .
To model a rotating spotted star, we place the N spots of the given size A randomly on the surface of a spherical star and tilt the axis to a certain inclination i. Letting the star rotate around its axis, we calculate the integrated quantities as functions of the rotational phase, taking into account the projection effect on the area of each spot (by the factor ) as well the limb-darkening law.
The effects of a single black spot as a function of the rotational phase are illustrated in Fig. 1. It can be noted that the effects are not unrelated, for example, the radial velocity curve mirrors the displacement in x, and both of these curves look like the derivative of the photometric curve. This is not a coincidence but can be understood from fairly general relations such as Eq. (14). With many spots the curves become quite complicated, but some of the basic relationships between them remain.
|Figure 1: The curves show the effects in magnitude, position, radial velocity and intensity skewness (third central moment) of a single dark spot located at latitude . The star is observed at inclination and the limb-darkening parameter a=0.6. The vertical scale is in arbitrary units for the different effects.|
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|Figure 2: Results of Monte Carlo simulations of rotating stars with different numbers (N) of spots, all the same size (A=0.0025). The different graphs refer to (from top to bottom) , , and , expressed on an arbitrary scale; the dots and error bars show the mean value and dispersion of the values for a set of simulations with given N. The dashed lines have slope 0.5, corresponding to .|
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The total equivalent area of the spots is AN (the spot filling factor ). As long as , all the effects are proportional to A. The dependence on N is more complex because of the random distribution of spots. For example, the photometric effect will mainly depend on the actual number of spots k visible at any time. For any random realisation of the model, k follows a binomial distribution with parameters p=0.5 and N; its dispersion is therefore . We can therefore expect the rms photometric effect to be roughly proportional to . Similar arguments (with the same result) can be made for the other effects.
Monte Carlo simulations of a large number of cases with A=0.0025(spot radius
and N in the range from 1 to 50 (assuming
random orientation of the rotation axis and a limb-darkening parameter
a=0.6) indeed show that the rms effects in magnitude,
photocentre displacements, third central moment and radial
velocity are all, in a statistical sense, proportional to
(Fig. 2). More precisely we find
The relations Eqs. (19)-(22) suggest that a measurement of any one of the four dispersions can be used to
statistically predict the other three dispersions, assuming that we know the approximate radius and rotation period of
the star, and that the different effects are indeed caused by the rotating spotted surface. An important point is that it is
not necessary to know A or N to do this. For example, expressing the other effects in terms of the photometric variation
It should also be noted that there is a considerable scatter between the different realisations reported in Eqs. (19)-(22), amounting to about 50% rms about the mean rms effect. Thus, any prediction based on either Eqs. (12)-(18) or Eqs. (19)-(22) is only valid in a statistical sense, with considerable uncertainty in any individual case. Nevertheless, the overall agreement between the results of these very different models suggests that the statistical relations among the different effects have a fairly general validity. The expressions for are the least general in this respect, as they obviously break down if the structures change on a timescale smaller than P, or if the surface structures themselves have velocity fields. Equations (12) and (13) do not depend on the assumption that the variability is caused by the rotation.
When modelling spotted stars, any brightening effect of faculae is often disregarded (for more details see Aarum-Ulvås 2005); only the darkening effect of spots is computed. For the Sun, the effect of faculae is known to be comparable and sometimes even larger than the darkening effect of sunspots (Chapman & Meyer 1986; Eker et al. 2003; Chapman 1984; Steinegger et al. 1996; ). However, since the general relationships, e.g. in Eqs. (12)-(18), are equally valid for bright and dark spots (or any mixture of them), it should still be possible to predict the astrometric effects from the photometric variations.
The (near-) proportionality between the observable effects and the spot filling factor expressed by Eqs. (1)-(3) is not supported by our spotted model, which predicts that the effects are proportional to . However, for small N and a filling factor of a few percent we have rough quantitative agreement with these earlier results. We note that Eqs. (2) and (3) can be combined to give an approximate relation similar to Eq. (17).
Equation (5) derived by Ludwig (2006) is almost identical to our Eq. (12), which is not surprising as they are based on very similar statistical models.
Both the theoretical result and the result from the simulation for
the relationship between the rms for the radial velocity
and the rms for the magnitude shows a distinct relation and this
result is confirmed by observations in the literature
(Paulson et al. 2004b) for a very limited number
of stars in the Hyades all having rotation period of days.
These are G0V-G5V stars and should therefore have approximately the
same radii as the Sun (
Equation (25) then gives
Thus the results of previous studies generally agree within a factor 2 or better with the theoretical formulae derived in this section.
In this section we use known statistics for the photometric and radial velocity variations of real stars to predict the expected astrometric jitter for different types of stars. Rather than using angular units, we consistently express the astrometric jitter in linear units, using the astronomical unit AU, mAU (10-3 AU) or AU (10-6 AU). This eliminates the dependence on the distance to the star, while providing simple conversion to angular units: AU corresponds to as at a distance of 1 pc. We also note that and km.
T Tauri stars are low-mass, pre-main sequence stars in a dynamic stage of evolution often characterised by prominent dark spots, bipolar outflows or jets, accreting matter with associated rapid brightness variations, and in many cases circumstellar discs extending to a few hundred AU (e.g., Rhode et al. 2001; Herbst et al. 2002; Sicilia-Aguilar et al. 2005). Taking the star-forming region in the Orion nebula as an example, the spectral types range from G6 to M6, with the large majority in the range K0 to M4 (Rhode et al. 2001).
Many processes may contribute to the astrometric jitter of these stars besides their surface structures, e.g. photometric irregularities of the circumstellar disc. The statistical relations derived in Sect. 3 could therefore mainly set a lower limit to the likely astrometric effects. Herbst et al. (1994) found that the photometric variability of (weak) T Tauri stars (WTTS) is of the order of 0.8 mag due to cool spots and occasional flares. Assuming a typical radius of (Rhode et al. 2001), Eq. (23) leads to an estimated astrometric variability of the order of AU.
Eyer & Grenon (1997) have used the Hipparcos photometric data to map the intrinsic variability of stars across the HR diagram. On the main sequence (luminosity class V), stars of spectral type B8-A5 and F1-F8 are among the most stable, with a mean intrinsic variability mmag and with only a few percent of the stars having amplitudes above 0.05 mag. Early B type stars are nearly all variable with a mean intrinsic variability of 10 mmag, and among the cool stars the level and frequency of variability increases from late G to early M dwarfs. In the instability strip (A6-F0) the main-sequence stars are mostly micro-variable with up to several mmag. Among F-K stars the degree of variability is probably also a strong function of age or chromospheric activity (Fekel et al. 2004); e.g., the Hyades (age 600 Myr) show variations of about 10 mmag (Radick et al. 1995).
The Sun (G2V) is located in one of the photometrically most stable parts of the main sequence, and is one of the (as yet) few stars for which the micro-variability has been studied in detail. Analysis of the VIRGO/SoHO total solar irradiance data (Lanza et al. 2003) show variability at the level mmag (relative variance , or rms) on timescales 30 days, which can largely be attributed to rotational modulation. The longer-term, solar-cycle related variations are of a similar magnitude. The optical data show a strong wavelength dependence, with mmag at 860 nm increasing to 0.4 mmag at 550 nm and 0.5 mmag at 400 nm (Lanza et al. 2004). For comparison, a single large sunspot group (equivalent area , corresponding to ) gives mmag according to Eq. (19).
The photometric variations of the Sun on short (rotation-related) timescales appears to be representative for solar-like stars of similar age and chromospheric activity (Fekel et al. 2004). Thus, we may expect mmag for "solar twins'' candidates, such as the sample studied by Meléndez et al. (2006). Inspection of the Hipparcos photometry for these stars (ESA 1997) confirm that most of them show no sign of variability at the sensitivity limit of a few mmag. Much more detailed and accurate statistics on micro-variability in solar-type stars are soon to be expected as a result of survey missions such as MOST (Walker et al. 2003), COROT (Baglin et al. 2002) and Kepler (Basri et al. 2005).
The increased frequency and amplitude of variations for late G-type and cooler dwarf stars is at least partly attributable to starspots. Aigrain et al. (2004) estimated stellar micro-variability as function of age and colour index from a scaling of the solar irradiance power spectrum based on the predicted chromospheric activity level. For example, they find mmag in white light for old (4.5 Gyr) F5-K5 stars, practically independent of spectral type, while for young stars (625 Myr) increases from 2 to 7 mmag in the same spectral range.
Variability among field M dwarfs has been studied e.g. by Rockenfeller et al. (2006), who find that a third of the stars in their sample of M2-M9 dwarfs are variable at the level of mmag. Evidence for large spots has been found for many K and M stars, yielding brightness amplitudes of up to a few tenths of a magnitude.
A large body of data on radial velocity jitter in (mainly) F, G and K stars has been assembled from the several ongoing planet search programmes and can be used to make statistical predictions as a function of colour, chromospheric activity and evolutionary stage. However, since at least part of the radial velocity jitter is caused by other effects than the rotation of an inhomogeneous surface (e.g., by atmospheric convective motions), its interpretation in terms of astrometric jitter is not straight-forward. From the observations of 450 stars in the California and Carnegie Planet Search Program, Wright (2005) finds a radial velocity jitter of 4 m s-1 for inactive dwarf stars of spectral type F5 or later, increasing to some 10 m s-1 for stars that are either active or more evolved. Saar et al. (1998), using data from the Lick planetary survey, find intrinsic radial velocity jitters of 2-100 m s-1 depending mainly on rotational velocity () and colour, with a minimum around -1.3 (spectral type K5). For a sample of Hyades F5 to M2 dwarf stars, Paulson et al. (2004a) find an average rms radial velocity jitter of 16 m s-1.
For giants of luminosity class III, Hipparcos photometry has shown a considerable range in the typical degree of variability depending on the spectral type (Eyer & Grenon 1997). The most stable giants ( mmag) are the early A and late G types. The most unstable ones are of type K8 or later, with a steadily increasing variability up to 0.1 mag for late M giants. The stars in the instability strip (roughly from A8 to F6) are typically variable at the 5-20 mmag level. As these are presumably mainly radially pulsating, the expected astrometric jitter is not necessarily higher than on either side of the instability strip. This general picture is confirmed by other studies. Jorissen et al. (1997) found that late G and early K giants are stable at the mmag level; K3 and later types have an increasing level of micro-variability with a timescale of 5 to 10 days, while b-y=1.1 (M2) marks the onset of large-amplitude variability ( mmag) typically on longer timescales (100 days). From a larger and somewhat more sensitive survey of G and K giants, Henry et al. (2000) found the smallest fraction of variables in the G6-K1 range, although even here some 20% show micro-variability at the 2-5 mmag level; giants later than K4 are all variable, half of them with mmag. The onset of large-amplitude variability coincides with the coronal dividing line (Haisch et al. 1991) separating the earlier giants with a hot corona from the later types with cool stellar winds. This suggests that the variability mechanisms may be different on either side of the dividing line, with rotational modulation of active regions producing the micro-variability seen in many giants earlier than K3 and pulsation being the main mechanism for the larger-amplitude variations in the later spectral types (Henry et al. 2000).
Several radial velocity surveys of giants (Frink et al. 2001; Hekker et al. 2006; Setiawan et al. 2004) show increasing intrinsic radial velocity variability with , with a more or less abrupt change around (K3). Most bluer giants have m s-1 while the redder ones often have variations of 40-100 m s-1.
Table 1: A summary of typical photometric and spectroscopic variability for different stellar types, and inferred levels of astrometric jitter ( ), using Eqs. (12), (18) and (27). These different estimates may represent upper or lower limits as explained in the text. References to typical observed quantities are given as footnotes. Radii and (not shown) are taken from Cox (2000).
With increasing luminosity, variability becomes increasingly common among the bright giants and supergiants (luminosity class II-Ia). The Hipparcos survey (Eyer & Grenon 1997) shows a typical intrinsic scatter of at least 10 mmag at most spectral types, and of course much more in the instability strip (including the cepheids) and among the red supergiants (including semiregular and irregular variables). Nevertheless there may be a few "islands'' in the upper part of the observational HR diagram where stable stars are to be found, in particular around G8II.
It is clear that pulsation is a dominating variability mechanism for many of these objects. However, "hotspots'' and other deviations from circular symmetry has been observed in interferometrical images of the surfaces of M supergiants and Mira variables (e.g., Tuthill et al. 1999,1997), possibly being the visible manifestations of very large convection cells, pulsation-induced shock waves, patchy circumstellar extinction, or some other mechanism. Whatever the explanation for these asymmetries may be, it is likely to produce both photometric and astrometric variations, probably on timescales of months to years. Kiss et al. (2006) find evidence of a strong 1/f noise component in the photometric power spectra of nearly all red supergiant semiregular and irregular variable stars in their sample, consistent with the picture of irregular variability caused by large convection cells analogous to the granulation-induced variability background seen for the Sun.
Table 1 summarises much of the data discussed in this section for the main-sequence, giant and supergiant stars, and gives the corresponding estimates of the astrometric jitter ( ) based on theoretical formulae. These estimates are given in three columns labelled with the corresponding equation number:
The possibility of astrometric detection of a planet depends on the angular size of the star's wobble in the sky
relative to the total noise of the measurements, including the astrophysically induced astrometric jitter discussed in
the previous section. In linear measures, the size of the wobble is approximately given by the semi-major axis of the
star's motion about the common centre of mass, or the astrometric signature
It is of interest to evaluate the astrometric signature for the already detected exoplanets. For most of them we only know from the radial velocity curve, and we use this as a proxy for . This somewhat underestimates the astrometric effect, but not by a large factor since the spectroscopic detection method is strongly biased against systems with small . Analysing the current (April 2007) data in the Extrasolar Planets Encyclopaedia (Schneider 2007) we find a median value AU; the 10th and 90th percentiles are 15 and 10 000 AU.
Future exoplanet searches using high-precision astrometric techniques may however primarily target planets with masses in the
range from 1 to 10 Earth masses (
in the habitable zone of reasonably
long-lived main-sequence stars (spectral type A5 and later, lifetime 1 Gyr). For a star of luminosity L we may take
the mean distance of the habitable zone to be
(Gould et al. 2003; Kasting et al. 1993). In this mass range
(based on data from Andersen 1991), so we find
Lopez et al. (2005) have argued that life will have time to develop also in the environments of subgiant and giant stars, during their slow phases of development. The habitable zone may extend out to 22 AU for a 1 star, with a correspondingly larger astrometric signature. However, the long period of such planets would make their detection difficult for other reasons.
The detection probability is in reality a complicated function of many factors such as the number of observations, their
temporal distribution, the period and eccentricity of the orbit, and the adopted detection threshold (or probability of false
detection). A very simplistic assumption might be that detection is only possible if the rms perturbation from the planet
exceeds the rms noise from other causes. Neglecting orbital eccentricity and assuming that the orbital plane is randomly
oriented in space, so that
the rms positional excursion of the star in a given direction on the
With a sufficiently powerful instrument, so that other error sources can be neglected, the
condition for detection then becomes
In reality, a somewhat larger ratio than
is probably required for a reliable detection, especially if the period is unknown. For example,
Sozzetti (2005) reports numerical simulations showing that
is required for detection of planetary
signatures by SIM or Gaia, where
is the single-epoch measurement error, provided that the orbital period is less
than the mission length. (For the corresponding problem of detecting a periodic signal in radial velocity data,
Marcy et al. (2005) note that a velocity precision of 3 m s-1 limits the detected velocity semi-amplitudes to greater
than 10 m s-1, implying an even higher amplitude/noise ratio of 3.3.) As a rule-of-thumb, we assume that detection
by the astrometric method is at least in principle possible if
Exoplanets of about 10 orbiting old F-K main-sequence stars in the habitable zone ( -50 AU) would generally be astrometrically detectable. This would also be the case for Earth-sized planets in similar environments ( AU), but only around stars that are unusually stable, such as the Sun.
The primary objective of high-precision astrometric measurements, apart from exoplanet detection, is the determination of stellar parallax and proper motion. We consider here only briefly the possible effects of stellar surface structures on the determination of these quantities.
Stellar parallax causes an apparent motion of the star, known as the parallax ellipse, which is an inverted image of the Earth's orbit as viewed from the star. The linear amplitude of the parallax effect is therefore very close to 1 AU. (For a space observatory at the Sun-Earth Lagrangian point L2, such as Gaia, the mean amplitude is 1.01 AU.) Thus, the size of the astrometric jitter expressed in AU can be used directly to estimate the minimum achievable relative error in parallax. For main-sequence stars this relative error is less than 10-4, for giant stars it is of the order of 10-4to 10-3, and for supergiants it may in some cases exceed 1%. We note that a 1% relative error in parallax gives a 2% (0.02 mag) error in luminosity or absolute magnitude.
If proper motions are calculated from positional data separated by T years, the random error caused by the astrometric jitter, converted to transverse velocity, is . Even for a very short temporal baseline such as T=1 yr, this error is usually very small: 0.1 m s-1 for main-sequence stars and 0.5-5 m s-1 for giants. (Note that .) In most applications of stellar proper motions this is completely negligible.
For most instruments on ground or in space, stars are still unresolved or marginally resolved objects which can only be observed by their disc-integrated properties. The total flux, astrometric position, effective radial velocity and closure phase are examples of such integrated properties. Stellar surface structures influence all of them in different ways. Our main conclusions are:
We give special thanks to Dainis Dravins, Jonas Persson and Andreas Redfors for helpful discussions and comments on the manuscript, and to Hans-Günter Ludwig for communicating his results from simulations of closure phase. We also thank Kristianstad University for funding the research of UE and thereby making this work possible.
In this appendix we derive the mean values and variances of the moments for a spherical star, where x and y are spatial coordinates normal to the line-of-sight and denotes the instantaneous flux-weighted mean. The analysis extends and generalises that of Ludwig (2006) by considering also the third moment (relevant for measurement of closure phase) and a centre-to-limb variation of the intensity contrast.
be polar coordinates on the stellar surface with
at the centre of the visible disc and
along the x axis. With
we write the instantaneous intensity across the visible stellar surface S as
and introduce the non-normalised spatial moments
It is assumed that
varies randomly both across the stellar surface (at a given instant), and as a function
of time. As a consequence, the spatial moments (A.1) and (A.2) are also random functions of time, and the goal is
to characterise them in terms of their mean values and variances. Since we are interested in quite small effects of the surface
structure it is generally true that the dispersions are small compared with the total flux and scale of the star, so that for
In this case, the variability of
is mainly produced by the numerator in (A.2), and we may use the approximations
We assume a linear limb-darkening law with coefficient a, such that
In order to compute the dispersion of Mmn we need to introduce the
second-order statistics of
Following Ludwig (2006)
we divide the visible hemisphere into N equal surface patches of size
with the centre of patch k at position
Thus the integral over the visible surface of any function
can in the limit of large N be replaced by a sum:
|Figure A.1: The scaling factor D[M10]/RD[M00] from Eq. (A.34) between the expected dispersions in photocentre position and total stellar flux, plotted as a function of the limb-darkening parameter a and the parameter for the centre-to-limb variation of surface structure contrast (c). The different curves represent, from top to bottom, c=-1, -0.5, 0, 0.5, 1. The solar symbol indicates the typical value for solar granulation in white light, a=0.6 and c=0.4.|
Closure phase is sensitive to the asymmetry of the stellar image, and the third moments (Mmn for m+n=3) are intended
to provide a statistical characterisation of this asymmetry. These moments are calculated with respect to the
geometrical centre of the disc (at x=y=0). However, intrinsic image properties such as size, shape and asymmetry are
more properly expressed with respect to the photocentre, at
y0=M01/M00, i.e., by means
of central moments (here denoted with a prime). For example, the third central moment along the x axis is given by
Since all the dispersions in (A.26) and (A.33) are proportional to C1 we obtain a set of simple scaling
relations which can be used to predict the dispersion of a certain moment from a measurement of the dispersion of a different
moment (assuming that a and c are approximately known). The most useful relations allow to predict the dispersions of the
first and third moments from that of the total flux (zeroth moment). For the first moment (photocentre) we have