On the relationship between instability and Lyapunov times for the 3-body problem

In this study we consider the relationship between the survival time and the Lyapunov time for 3-body systems. It is shown that the Sitnikov problem exhibits a two-part power law relationship as demonstrated previously for the general 3-body problem. Using an approximate Poincare map on an appropriate surface of section, we delineate escape regions in a domain of initial conditions and use these regions to analytically obtain a new functional relationship between the Lyapunov time and the survival time for the 3-body problem. The marginal probability distributions of the Lyapunov and survival times are discussed and we show that the probability density function of Lyapunov times for the Sitnikov problem is similar to that for the general 3-body problem.Comment: 9 pages, 19 figures, accepted for publication in MNRA

Shadowing unstable orbits of the Sitnikov elliptic 3-body problem

Errors in numerical simulations of gravitating systems can be magnified exponentially over short periods of time. Numerical shadowing provides a way of demonstrating that the dynamics represented by numerical simulations are representative of true dynamics. Using the Sitnikov Problem as an example, it is demonstrated that unstable orbits of the 3-body problem can be shadowed for long periods of time. In addition, it is shown that the stretching of phase space near escape and capture regions is a cause for the failure of the shadowing refinement procedure.Comment: 9 pages, 13 figures, accepted in MNRA

Disruption of the three-body gravitational systems: Lifetime statistics

We investigate statistics of the decay process in the equal-mass three-body problem with randomized initial conditions. Contrary to earlier expectations of similarity with "radioactive decay", the lifetime distributions obtained in our numerical experiments turn out to be heavy-tailed, i.e. the tails are not exponential, but algebraic. The computed power-law index for the differential distribution is within the narrow range, approximately from -1.7 to -1.4, depending on the virial coefficient. Possible applications of our results to studies of the dynamics of triple stars known to be at the edge of disruption are considered.Comment: 13 pages, 2 tables, 3 figure

Time matters less when outcomes differ: uni-modal versus cross-modal comparisons in intertemporal choice

Uni-modal intertemporal decisions involve comparing options of the same type (e.g. apples now versus apples later), and cross-modal decisions involve comparing options of different types (e.g. a car now versus a vacation later). As we explain, existing models of intertemporal choice do not allow time preference to depend on whether the comparisons to be made are uni-modal or cross-modal. We test this restriction in an experiment using the delayed-compensation method, a new extension of the standard method of eliciting intertemporal preferences that allows for assessment of time preference for non-monetary and discrete outcomes, as well as for both cross-modal and uni-modal comparisons. Participants were much more averse to delay for uni-modal than cross-modal decisions. We provide two potential explanations for this effect: one drawing on multi-attribute choice, the other drawing on construal level theory

Three Body Resonance Overlap in Closely Spaced Multiple Planet Systems

We compute the strengths of zero-th order (in eccentricity) three-body resonances for a co-planar and low eccentricity multiple planet system. In a numerical integration we illustrate that slowly moving Laplace angles are matched by variations in semi-major axes among three bodies with the outer two bodies moving in the same direction and the inner one moving in the opposite direction, as would be expected from the two quantities that are conserved in the three-body resonance. A resonance overlap criterion is derived for the closely and equally spaced, equal mass system with three-body resonances overlapping when interplanetary separation is less than an order unity factor times the planet mass to the one quarter power. We find that three-body resonances are sufficiently dense to account for wander in semi-major axis seen in numerical integrations of closely spaced systems and they are likely the cause of instability of these systems. For interplanetary separations outside the overlap region, stability timescales significantly increase. Crudely estimated diffusion coefficients in eccentricity and semi-major axis depend on a high power of planet mass and interplanetary spacing. An exponential dependence previously fit to stability or crossing timescales is likely due to the limited range of parameters and times possible in integration and the strong power law dependence of the diffusion rates on these quantities.Comment: submitted to MNRA

The role of causal beliefs in political identity and voting

An emerging literature in psychology and political science has identified political identity as an important driver of political decisions. However, less is known about how a person’s political identity is incorporated into their broader self-concept and why it influences some people more than others. We examined the role of political identity in representations of the self-concept as one determinant of people’s political behaviors. We tested the predictions of a recent theoretical account of self-concept representation that, inspired by work on conceptual representation, emphasizes the role of causal beliefs. This account predicts that people who believe that their political identity is causally central (linked to many other features of the self-concept) will be more likely to engage in behaviors consistent with their political identity than those who believe that the same aspect is causally peripheral (linked to fewer other features). Consistent with these predictions, in a study run when political identity was particularly salient—during the 2016 U.S. Presidential election—we found that U.S. voters who believed their political party identity was more causally central (vs those who believe it was causally peripheral) were more likely to vote for their political party’s candidate. Further, in 2 studies, we found that U.K. residents who believed that their English or British national identity was more causally central were more likely to support the U.K. leaving the European Union (Brexit) than those who believed the same identities were more causally peripheral

The narrow search effect and how broadening search promotes belief updating

Information search platforms, from Google to AI-assisted search engines, have transformed information access but may fail to promote a shared factual foundation. We demonstrate that the combination of users’ prior beliefs influencing their search terms and the narrow scope of search algorithms can limit belief updating from search. We test this “narrow search effect” across 21 studies (14 preregistered) using various topics (e.g., health, financial, societal, political) and platforms (e.g., Google, ChatGPT, AI-powered Bing, our custom-designed search engine and AI chatbot interfaces). We then test user-based and algorithm-based interventions to counter the “narrow search effect” and promote belief updating. Studies 1 to 5 show that users’ prior beliefs influence the direction of the search terms, thereby generating narrow search results that limit belief updating. This effect persists across various domains (e.g., beliefs related to coronavirus, nuclear energy, gas prices, crime rates, bitcoin, caffeine, and general food or beverage health concerns; Studies 1a to 1b, 2a to 2g, 3, 4), platforms (e.g., Google—Studies 1a to 1b, 2a to 2g, 4, 5; ChatGPT, Study 3), and extends to consequential choices (Study 5). Studies 6 and 7 demonstrate the limited efficacy of prompting users to correct for the impact of narrow searches on their beliefs themselves. Using our custom-designed search engine and AI chatbot interfaces, Studies 8 and 9 show that modifying algorithms to provide broader results can encourage belief updating. These findings highlight the need for a behaviorally informed approach to the design of search algorithms

Representations of the Self-Concept and Identity-Based Choice

We propose a novel approach to identity-based choice that focuses on peoples’ representations of the cause-effect relationships that exist among features of their self-concepts. More specifically, we propose that people who believe that a specific aspect of identity, such as a social category, is causally central (linked to many other features of the self-concept) are more likely to engage in behaviors consistent with that aspect than those who believe that the same aspect is causally peripheral (linked to fewer other features). Across three studies, we provide evidence for our approach to identity-based choice. We demonstrate that among people who belong to the same social category, those who believe that the associated identity is more causally central are more likely to engage in behaviors consistent with the social category

Is the Self-Concept like other Concepts? The Causal Structure of Identity

We investigate the age-old questions of what makes us whowe are and what features of identity, if changed, would makeus a different person. Previous approaches to identity havesuggested that there is a type of feature that is most definingof identity (e.g., autobiographical memories or moralqualities). We propose a new approach to identity thatsuggests that, like concepts in general, more causally centralfeatures are perceived as more defining of the self-concept. Inthree experiments, using both measured and manipulatedcausal centrality, we find that changes to features of identitythat are perceived as more causally central are moredisruptive to both the identity of the self and others