Thomas J. Loredo

The Structure of the Kuiper Belt: Size Distribution and Radial Extent

The size distribution in the Kuiper Belt records physical processes operating during the formation and subsequent evolution of the solar system. This paper reports a study of the apparent magnitude distribution of faint objects in the Kuiper Belt, obtained via deep imaging on the Canada-France-Hawaii Telescope and the ESO Very Large Telescope UT1. We —nd that the entire range of observed objects (magnitudes is well represented by an unbroken power law, with the number of objects per m R D 20¨27) square degree brighter than magnitude R being of the form with a \ 0.69 and &(m R \ R) \ 10a(R~R0),

From Laplace to Supernova SN 1987A: Bayesian Inference in Astrophysics

The Bayesian approach to probability theory is presented as an alternative to the currently used long-run relative frequency approach, which does not offer clear, compelling criteria for the design of statistical methods. Bayesian probability theory offers unique and demonstrably optimal solutions to well-posed statistical problems, and is historically the original approach to statistics. The reasons for earlier rejection of Bayesian methods are discussed, and it is noted that the work of Cox, Jaynes, and others answers earlier objections, giving Bayesian inference a firm logical and mathematical foundation as the correct mathematical language for quantifying uncertainty. The Bayesian approaches to parameter estimation and model comparison are outlined and illustrated by application to a simple problem based on the gaussian distribution. As further illustrations of the Bayesian paradigm, Bayesian solutions to two interesting astrophysical problems are outlined: the measurement of weak signals in a strong background, and the analysis of the neutrinos detected from supernova SN 1987A. A brief bibliography of astrophysically interesting applications of Bayesian inference is provided.

Pencil-Beam Surveys for Faint Trans-Neptunian Objects

Motivated by a desire to understand the size distribution of objects in the Edgeworth-Kuiper belt, an observing program has been conducted at the Palomar 5 m and Canada-France-Hawaii 3.6 m telescopes. We have conducted pencil-beam searches for outer solar system objects to a limiting magnitude of R ~ 26. The fields were searched using software recombinations of many short exposures shifted at different angular rates in order to detect objects at differing heliocentric distances. Five new trans-Neptunian objects were detected in these searches. Our combined data set provides an estimate of ~90 trans-Neptunian objects deg-2 brighter than 25.9. This estimate is a factor of 3 above the expected number of objects based on an extrapolation of previous surveys with brighter limits and appears consistent with the hypothesis of a single power-law luminosity function for the entire trans-Neptunian region. Maximum-likelihood fits to all self-consistent published surveys with published efficiency functions predicts a cumulative sky density Σ(

Type Ia Supernovae, Evolution, and the Cosmological Constant

We explore the possible role of evolution in the analysis of data on Type Ia supernovae (SNe Ia) at cosmological distances. First, using a variety of simple sleuthing techniques, we find evidence that the properties of the high- and low-redshift SNe Ia observed so far differ from one another. Next, we examine the effects of allowing for an uncertain amount of evolution in the analysis, using two simple phenomenological models for evolution and prior probabilities that express a preference for no evolution but allow it to be present. One model shifts the magnitudes of the high-redshift SNe Ia relative to the low-redshift SNe Ia by a fixed amount. A second, more realistic, model introduces a continuous magnitude shift of the form δm(z) = β ln(1 + z) to the SNe Ia sample. The result is that cosmological models and evolution are highly degenerate with one another, so that the incorporation of even very simple models for evolution makes it virtually impossible to pin down the values of ΩM and ΩΛ, the density parameters for nonrelativistic matter and for the cosmological constant, respectively. The Hubble constant, H0, is unaffected by evolution. We evaluate the Bayes factor for models with evolution versus models without evolution, which, if one has no prior predilection for or against evolution, is the odds ratio for these two classes of models. The resulting values are always of order 1, in spite of the fact that the models that include evolution have additional parameters; thus, the data alone cannot discriminate between the two possibilities. Simulations show that simply acquiring more data of the same type as are available now will not alleviate the difficulty of separating evolution from cosmology in the analysis. What is needed is a better physical understanding of the SN Ia process, and the connections among the maximum luminosity, rate of decline, spectra, and initial conditions, so that physical models for evolution may be constructed, and confronted with the data. Moreover, we show that if SNe Ia evolve with time, but evolution is neglected in analyzing data, then, given enough SNe Ia, the analysis hones in on values of ΩM and ΩΛ that are incorrect. Using Bayesian methods, we show that the probability that the cosmological constant is nonzero (rather than zero) is unchanged by the SNe Ia data when one accounts for the possibility of evolution, provided that we do not discriminate among open, closed, and flat cosmologies a priori. The case for nonzero cosmological constant is stronger if the universe is presumed to be flat but still depends sensitively on the degree to which the peak luminosities of SNe Ia evolve as a function of redshift.

A new method for the detection of a periodic signal of unknown shape and period

We present a new method for the detection and measurement of a periodic signal in a data set when we have no prior knowledge of the existence of such a signal or of its characteristics. It is applicable to data consisting of the locations or times of individual events. To address the detection problem, we use Bayes’ theorem to compare a constant rate model for the signal to models with periodic structure. The periodic models describe the signal plus background rate as a stepwise distribution in m bins per period, for various values of m. The Bayesian posterior probability for a periodic model contains a term which quantifies Ockham’s razor, penalizing successively more complicated periodic models for their greater complexity even though they are assigned equal prior probabilities. The calculation thus balances model simplicity with goodness-of-fit, allowing us to determine both whether there is evidence for a periodic signal, and the optimum number of bins for describing the structure in the data. Unlike the results of traditional “frequentist” calculations, the outcome of the Bayesian calculation does not depend on the number of periods examined, but only on the range examined. Once a signal is detected, we again use Bayes’ theorem to estimate the frequency of the signal. The probability density for the frequency is inversely proportional to the multiplicity of the binned events and is thus maximized for the frequency leading to the binned event distribution with minimum combinatorial entropy. The method is capable of handling gaps in the data due to intermittent observing or dead time.

On the Orbit of Exoplanet WASP-12b

We observed two secondary eclipses of the exoplanet WASP-12b using the Infrared Array Camera on the Spitzer Space Telescope. The close proximity of WASP-12b to its G-type star results in extreme tidal forces capable of inducing apsidal precession with a period as short as a few decades. This precession would be measurable if the orbit had a significant eccentricity, leading to an estimate of the tidal Love number and an assessment of the degree of central concentration in the planetary interior. An initial ground-based secondary-eclipse phase reported by Lopez-Morales et al. (0.510 ± 0.002) implied eccentricity at the 4.5σ level. The spectroscopic orbit of Hebb et al. has eccentricity 0.049 ± 0.015, a 3σ result, implying an eclipse phase of 0.509 ± 0.007. However, there is a well-documented tendency of spectroscopic data to overestimate small eccentricities. Our eclipse phases are 0.5010 ± 0.0006 (3.6 and 5.8 μm) and 0.5006 ± 0.0007 (4.5 and 8.0 μm). An unlikely orbital precession scenario invoking an alignment of the orbit during the Spitzer observations could have explained this apparent discrepancy, but the final eclipse phase of Lopez-Morales et al. (0.510 ±+0.007 –0.006) is consistent with a circular orbit at better than 2σ. An orbit fit to all the available transit, eclipse, and radial-velocity data indicates precession at <1σ; a non-precessing solution fits better. We also comment on analysis and reporting for Spitzer exoplanet data in light of recent re-analyses.

Bayesian Adaptive Exploration

We describe a framework for adaptive astronomical exploration based on iterating an Observation-Inference-Design cycle that allows adjustment of hypotheses and observing protocols in response to the results of observation on-the-fly, as data are gathered. The framework uses a unified Bayesian methodology for the inference and design stages: Bayesian inference to quantify what we have learned from the available data; and Bayesian decision theory to identify which new observations would teach us the most. In the design stage, the utility of possible future observations is determined by how much information they are expected to add to current inferences as measured by the (negative) entropies of the probability distributions involved. Such a Bayesian approach to experimental design dates back to the 1970s, but most existing work focuses on linear models. We use a simple nonlinear problem—planning observations to best determine the orbit of an extrasolar planet—to illustrate the approach and demonstrate that it can significantly improve observing efficiency (i.e., reduce uncertainties at a rate faster than the familiar “root-N” rule) in some situations. We highlight open issues requiring further research, including dependence on model specification, generalizing the utility of an observation (e.g., to include observing “costs”), and computational issues.

State of the Field: Extreme Precision Radial Velocities*

The Second Workshop on Extreme Precision Radial Velocities defined circa 2015 the state of the art Doppler precision and identified the critical path challenges for reaching 10 cm s^(−1) measurement precision. The presentations and discussion of key issues for instrumentation and data analysis and the workshop recommendations for achieving this bold precision are summarized here. Beginning with the High Accuracy Radial Velocity Planet Searcher spectrograph, technological advances for precision radial velocity (RV) measurements have focused on building extremely stable instruments. To reach still higher precision, future spectrometers will need to improve upon the state of the art, producing even higher fidelity spectra. This should be possible with improved environmental control, greater stability in the illumination of the spectrometer optics, better detectors, more precise wavelength calibration, and broader bandwidth spectra. Key data analysis challenges for the precision RV community include distinguishing center of mass (COM) Keplerian motion from photospheric velocities (time correlated noise) and the proper treatment of telluric contamination. Success here is coupled to the instrument design, but also requires the implementation of robust statistical and modeling techniques. COM velocities produce Doppler shifts that affect every line identically, while photospheric velocities produce line profile asymmetries with wavelength and temporal dependencies that are different from Keplerian signals. Exoplanets are an important subfield of astronomy and there has been an impressive rate of discovery over the past two decades. However, higher precision RV measurements are required to serve as a discovery technique for potentially habitable worlds, to confirm and characterize detections from transit missions, and to provide mass measurements for other space-based missions. The future of exoplanet science has very different trajectories depending on the precision that can ultimately be achieved with Doppler measurements.

SPITZER SECONDARY ECLIPSES OF WASP-18b

The transiting exoplanet WASP-18b was discovered in 2008 by the Wide Angle Search for Planets project. The Spitzer Exoplanet Target of Opportunity Program observed secondary eclipses of WASP-18b using Spitzers Infrared Array Camera in the 3.6 ?m and 5.8 ?m bands on 2008 December 20, and in the 4.5 ?m and 8.0 ?m bands on 2008 December 24. We report eclipse depths of 0.30% ? 0.02%, 0.39% ? 0.02%, 0.37% ? 0.03%, 0.41% ? 0.02%, and brightness temperatures of 3100 ? 90, 3310 ? 130, 3080 ? 140, and 3120 ? 110?K in order of increasing wavelength. WASP-18b is one of the hottest planets yet discovered?as hot as an M-class star. The planets pressure-temperature profile most likely features a thermal inversion. The observations also require WASP-18b to have near-zero albedo and almost no redistribution of energy from the day side to the night side of the planet.

Thermal Emission of WASP-14b Revealed with Three Spitzer Eclipses

Exoplanet WASP-14b is a highly irradiated, transiting hot Jupiter. Joshi et al. calculate an equilibrium temperature (T eq) of 1866 K for zero albedo and reemission from the entire planet, a mass of 7.3 ± 0.5 Jupiter masses (M J), and a radius of 1.28 ± 0.08 Jupiter radii (R J). Its mean density of 4.6 g cm-3 is one of the highest known for planets with periods less than three days. We obtained three secondary eclipse light curves with the Spitzer Space Telescope. The eclipse depths from the best jointly fit model are 0.224% ± 0.018% at 4.5 μm and 0.181% ± 0.022% at 8.0 μm. The corresponding brightness temperatures are 2212 ± 94 K and 1590 ± 116 K. A slight ambiguity between systematic models suggests a conservative 3.6 μm eclipse depth of 0.19% ± 0.01% and brightness temperature of 2242 ± 55 K. Although extremely irradiated, WASP-14b does not show any distinct evidence of a thermal inversion. In addition, the present data nominally favor models with day-night energy redistribution less than ~30%. The current data are generally consistent with oxygen-rich as well as carbon-rich compositions, although an oxygen-rich composition provides a marginally better fit. We confirm a significant eccentricity of e = 0.087 ± 0.002 and refine other orbital parameters.