Eric B. Ford

SECULAR EVOLUTION OF HIERARCHICAL TRIPLE STAR SYSTEMS

We derive octupole-level secular perturbation equations for hierarchical triple systems, using classical Hamiltonian perturbation techniques. Our equations describe the secular evolution of the orbital eccentricities and inclinations over timescales that are long compared to the orbital periods. By extending previous work done to leading (quadrupole) order to octupole level (i.e., including terms of order α3, where α ≡ a1/a2 < 1 is the ratio of semimajor axes), we obtain expressions that are applicable to a much wider range of parameters. In particular, our results can be applied to high-inclination as well as coplanar systems, and our expressions are valid for almost all mass ratios for which the system is in a stable hierarchical configuration. In contrast, the standard quadrupole-level theory of Kozai gives a vanishing result in the limit of zero relative inclination. The classical planetary perturbation theory, while valid to all orders in α, applies only to orbits of low-mass objects orbiting a common central mass, with low eccentricities and low relative inclinations. For triple systems containing a close inner binary, we also discuss the possible interaction between the classical Newtonian perturbations and the general relativistic precession of the inner orbit. In some cases we show that this interaction can lead to resonances and a significant increase in the maximum amplitude of eccentricity perturbations. We establish the validity of our analytic expressions by providing detailed comparisons with the results of direct numerical integrations of the three-body problem obtained for a large number of representative cases. In addition, we show that our expressions reduce correctly to previously published analytic results obtained in various limiting regimes. We also discuss applications of the theory in the context of several observed triple systems of current interest, including the millisecond pulsar PSR B1620-26 in M4, the giant planet in 16 Cygni, and the protostellar binary TMR-1.

Quantifying the Uncertainty in the Orbits of Extrasolar Planets

Precise radial velocity measurements have led to the discovery of ~100 extrasolar planetary systems. We investigate the uncertainty in the orbital solutions that have been fitted to these observations. Understanding these uncertainties will become more and more important as the discovery space for extrasolar planets shifts to longer and longer periods. While detections of short-period planets can be rapidly refined, planets with long orbital periods will require observations spanning decades to constrain the orbital parameters precisely. Already in some cases, multiple distinct orbital solutions provide similarly good fits, particularly in multiple-planet systems. We present a method for quantifying the uncertainties in orbital fits and addressing specific questions directly from the observational data rather than relying on best-fit orbital solutions. This Markov chain Monte Carlo (MCMC) technique has the advantage that it is well suited to the high-dimensional parameter spaces necessary for the multiple-planet systems. We apply the MCMC technique to several extrasolar planetary systems, assessing the uncertainties in orbital elements for several systems. Our MCMC simulations demonstrate that for some systems there are strong correlations between orbital parameters and/or significant non-Gaussianities in parameter distributions, even though the measurement errors are nearly Gaussian. Once these effects are considered, the actual uncertainties in orbital elements can be significantly larger or smaller than the published uncertainties. We also present simple applications of our methods, such as predicting the times of possible transits for GJ 876.

Characterization of extrasolar terrestrial planets from diurnal photometric variability

The detection of massive planets orbiting nearby stars has become almost routine, but current techniques are as yet unable to detect terrestrial planets with masses comparable to the Earths. Future space-based observatories to detect Earth-like planets are being planned. Terrestrial planets orbiting in the habitable zones of stars—where planetary surface conditions are compatible with the presence of liquid water—are of enormous interest because they might have global environments similar to Earths and even harbour life. The light scattered by such a planet will vary in intensity and colour as the planet rotates; the resulting light curve will contain information about the planets surface and atmospheric properties. Here we report a model that predicts features that should be discernible in the light curve obtained by low-precision photometry. For extrasolar planets similar to Earth, we expect daily flux variations of up to hundreds of per cent, depending sensitively on ice and cloud cover as well as seasonal variations. This suggests that the meteorological variability, composition of the surface (for example, ocean versus land fraction) and rotation period of an Earth-like planet could be derived from photometric observations. Even signatures of Earth-like plant life could bexa0constrained or possibly, with further study, even uniquely determined.

Dynamical Instabilities in Extrasolar Planetary Systems Containing Two Giant Planets

Abstract Instabilities and strong dynamical interactions between several giant planets have been proposed as a possible explanation for the surprising orbital properties of extrasolar planetary systems. In particular, dynamical instabilities seem to provide a natural mechanism for producing the highly eccentric orbits seen in many systems. Here we present results from a new set of numerical integrations for the dynamical evolution of planetary systems containing two identical giant planets in nearly circular orbits very close to the dynamical stability limit. We determine the statistical properties of the three main types of systems resulting from the development of an instability: systems containing one planet, following either a collision between the two initial planets, or the ejection of one of them to infinity, and systems containing two planets in a new, quasi-stable configuration. We discuss the implications of our results for the formation and evolution of observed extrasolar planetary systems. We conclude that the distributions of eccentricities and semimajor axes for observed systems cannot be explained easily by invoking dynamical interactions between two planets initially on circular orbits. While highly eccentric orbits can be produced naturally by these interactions, collisions between the two planets, which occur frequently in the range of observed semimajor axes, would result in many more nearly circular orbits than in the observed sample.

Structure and Evolution of Nearby Stars with Planets. I. Short-Period Systems

Using the Yale stellar evolution code, we have calculated theoretical models for nearby stars with planetary-mass companions in short-period nearly circular orbits: 51 Pegasi, τ Bootis, υ Andromedae, ρ1 Cancri, and ρ Coronae Borealis. We present tables listing key stellar parameters such as mass, radius, age, and size of the convective envelope as a function of the observable parameters (luminosity, effective temperature, and metallicity), as well as the unknown helium fraction. For each star we construct best models based on recently published spectroscopic data and the present understanding of galactic chemical evolution. We discuss our results in the context of planet formation theory and, in particular, tidal dissipation effects and stellar metallicity enhancements.

Theoretical Implications of the PSR B1620–26 Triple System and Its Planet

We present a new theoretical analysis of the PSR B1620-26 triple system in the globular cluster M4, based on the latest radio pulsar timing data, which now include measurements of five time derivatives of the pulse frequency. These data allow us to determine the mass and orbital parameters of the second companion completely (up to the usual unknown orbital inclination angle i2). The current best-fit parameters correspond to a second companion of planetary mass, m2 sin i2 7 × 10-3 M☉ , in an orbit of eccentricity e2 0.45 and semimajor axis a2 60 AU. Using numerical scattering experiments, we study a possible formation scenario for the triple system, which involves a dynamical exchange interaction between the binary pulsar and a primordial star-planet system. The current orbital parameters of the triple are consistent with such a dynamical origin and suggest that the separation of the parent star-planet system was very large, 50 AU. We also examine the possible origin of the anomalously high eccentricity of the inner binary pulsar. While this eccentricity could have been induced during the same dynamical interaction that created the triple, we find that it could equally well arise from long-term secular perturbation effects in the triple, combining the general relativistic precession of the inner orbit with the Newtonian gravitational perturbation of the planet. The detection of a planet in this system may be taken as evidence that large numbers of extrasolar planetary systems, not unlike those discovered recently in the solar neighborhood, also exist in old star clusters.

Evolution of the Cluster Mass Function: Gpc3Dark Matter Simulations

High-resolution N-body simulations of four popular cold dark matter cosmologies (LCDM, OCDM, QCDM, and tilted SCDM), each containing ~105 clusters of galaxies of mass M1.5 > 5 × 1013 h-1 M☉ in a Gpc3 volume, are used to determine the evolution of the cluster mass function from z = 3 to 0. The large volume and high resolution of these simulations allow an accurate measure of the evolution of cosmologically important (but rare) massive clusters at high redshift. The simulated mass function is presented for cluster masses within several radii typically used observationally (R = 0.5, 1.0, and 1.5 h-1 Mpc, both comoving and physical) in order to enable direct comparison with current and future observations. The simulated evolution is compared with current observations of massive clusters at redshifts 0.3 z 0.8. The Ωm = 1 tilted SCDM model, which exhibits very rapid evolution of the cluster abundance, produces too few clusters at z 0.3 and no massive clusters at z 0.5, in stark contradiction to observations. The Ωm = 0.3 models—LCDM, OCDM, and QCDM—all exhibit considerably weaker evolution and are consistent with current data. Among these low-density models, OCDM evolves the least. These trends are enhanced at high redshift and can be used to discriminate between flat and open low-density models. The simulated mass functions are compared with the Press-Schechter approximation. Standard Press-Schechter predicts too many low-mass clusters at z = 0, and too few clusters at higher redshift. We modify the approximation by a simple parameterization of the density contrast threshold for collapse, which has a redshift dependence. This modified Press-Schechter approximation provides a good fit to the simulated mass functions.

Planet-Finding Prospects for the Space Interferometry Mission

The Space Interferometry Mission (SIM) will make precise astrometric measurements that can be used to detect planets around nearby stars. We have simulated SIM observations and estimated the ability of SIM to detect planets with given masses and orbital periods and measured their orbital elements. We combine these findings with an estimate of the mass and period distribution of planets determined from radial velocity surveys to predict the number and characteristics of planets SIM would likely find. Our predictions are based on extrapolating the mass distribution of known extrasolar planets by up to a factor of ~100. This extrapolation provides the best prediction we can make of the actual number of planets that SIM will detect and characterize, but may substantially over- or underestimate the frequency of Earth-mass planets, especially if these form by a different mechanism than giant planets. We find that a SIM key project is likely to detect around one to five planets with masses ≤3 M⊕ (depending on mission parameters). SIM would measure masses and orbits with 30% accuracy for around zero to two of these planets, but is unlikely to measure orbits with 10% accuracy for more than one of them. SIM is likely to detect around five to 25 planets with mass less than 20 M⊕, measure masses and orbits with 30% accuracy for around two to 12 of these, and measure masses and orbits with 10% accuracy for around two to eight such planets. SIM is likely to find ~25-160 planets of all masses, depending on the observing strategy and mission lifetime. Roughly 25%-65% of the planets detected by SIM have sufficiently large masses and short orbital periods that they can also be detected by radial velocity surveys. Radial velocity surveys could measure orbital parameters (not including inclination) for 30%-70% of the planets whose orbital parameters will be determined to within 30% by SIM.

The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2

Recent advances in deformable mirror technology for correcting wavefront errors and in pupil shapes and masks for coronagraphic suppression of diffracted starlight enable a powerful approach to detecting extrasolar planets in reflected (scattered) starlight at visible wavelengths. We discuss the planet-finding performance of Hubble-like telescopes using these technical advances. A telescope of aperture of at least 4 meters could accomplish the goals of the Terrestrial Planet Finder (TPF) mission. The 4mTPF detects an Earth around a Sun at five parsecs in about one hour of integration time. It finds molecular oxygen, ozone, water vapor, the red edge of chlorophyll-containing land-plant leaves, and the total atmospheric column density -- all in forty hours or less. The 4mTPF has a strong science program of discovery and characterization of extrasolar planets and planetary systems, including other worlds like Earth. With other astronomical instruments sharing the focal plane, the 4mTPF could also continue and expand the general program of astronomical research of the Hubble Space Telescope.

Early-Type Stars: Most Favorable Targets for Astrometrically Detectable Planets in the Habitable Zone

Early-type stars appear to be a difficult place to look for planets astrometrically. First, they are relatively heavy, and for fixed planetary mass the astrometric signal falls inversely as the stellar mass. Second, they are relatively rare (and so tend to be more distant), and for fixed orbital separation the astrometric signal falls inversely as the distance. Nevertheless, because early-type stars are relatively more luminous, their habitable zones are at larger semimajor axis. Since astrometric signal scales directly as orbital size, this gives early-type stars a strong advantage, which more than compensates for the other two factors. Using the Hipparcos Catalog, we show that F and A stars constitute the majority of viable targets for astrometric searches for planets with semimajor axes currently in the habitable zone. Thus, astrometric surveys are complementary to transit searches, which are primarily sensitive to habitable planets around late-type stars.