Fictitious Planet Research Articles

Direct Imaging of Exoplanets beyond the Radial Velocity Limit: Application to the HD 134987 System

Abstract Future direct imaging missions will primarily observe planets that have been previously detected, mostly via the radial velocity (RV) technique, to characterize planetary atmospheres. In the meantime, direct imaging may discover new planets within existing planetary systems that have bright enough reflected flux but insufficient signals for other methods to detect. Here we investigate the parameter space within which planets are unlikely to be detected by RV in the near future due to precision limitations but could be discovered through reflected light with future direct imaging missions. We use the HD 134987 system as a working example, combine RV and direct imaging detection limit curves in the same parameter space through various assumptions, and insert a fictitious planet into the system while ensuring that it lies between the RV and imaging detection limits. Planet validity tested through dynamical simulations and retrieval tests revealed that the planet could indeed be detected by imaging while remaining hidden from RV surveys. Direct imaging retrieval was carried out using starshade simulations for two mission concepts: the Starshade Rendezvous Probe, which could be coupled with the Nancy Grace Roman Space Telescope, and the Habitable Exoplanet Observatory. This method is applicable to all other systems and high-contrast direct imaging instruments and could help inform future imaging observations and data analysis on the discovery of new exoplanets.

The stability of ultra-compact planetary systems

Aims. We investigate the dynamical stability of compact planetary systems in the CoRoT discovery space, i.e., with orbital periods of less than 50 days, including a detailed study of the stability of systems, which are spaced according to Hill’s criteria. Methods. The innermost fictitious planet was placed close to the Roche limit from the star (MStar = 1 MSun) and all other fictitious planets are lined up according to Hill’s criteria up to a distance of 0.26 AU, which corresponds to a 50 day period for a Sun-massed star. For the masses of the fictitious planets, we chose a range of 0.33–17 mEarth, where in each simulation all fictitious planets have the same mass. Additionally, we tested the influence of both the semi-major axis of the innermost planet and of the number of planets. In a next step we also included a gas giant in our calculations, which perturbs the inner ones and investigated their stability. Results. With numerous integrations of many different configurations we could show that long-time stable motion is possible for up to 10 planets with 17 mEarth within a distance of 0.26 AU. Further investigations show that the fictitious planets remain stable under certain circumstances even if a close-in gas giant is present.

Stability of terrestrial planets in the habitable zone of Gl 777 A, HD 72659, Gl 614, 47 Uma and HD 4208

Received; accepted Abstract. We have undertaken a thorough dynamical investigation of five extrasolar planetary systems using extensive numerical experiments on the supercomputer of the Max Planck Institute for Gravitational Physics (Albert Einstein Institute). This work was performed as part of the Helmholtz Institute for Supercomputational Physics Summer School on Chaos and Stability in Planetary Systems (2003). The systems Gl 777 A, HD 72659, Gl 614, 47 Uma and HD 4208 were examined concerning the question of whether they could host terrestrial like planets in their habitable zones (=HZ). First we investigated the mean motion resonances between fictitious terrestrial planets and the existing gas giants in these five extrasolar systems. Then a fine grid of initial conditions for a potential terrestrial planet within the HZ was chosen for each system, from which the stability of orbits was then assessed by direct integrations over a time interval of 1 million years. For each of the five systems the 2-dimensional grid of initial conditions contained 80 eccentricity points for the Jovian planet and up to 160 semimajor axis points for the fictitious planet. The computations were carried out using a Lie-series integration method with an adaptive step size control. This integration method achieves machine precision accuracy in a highly efficient and robust way, requiring no special adjustments when the orbits have large eccentricities. The stability of orbits was examined with a determination of the Renyi entropy, estimated from recurrence plots, and with a more straight forward method based on the maximum eccentricity achieved by the planet over the 1 million year integration. Additionally, the eccentricity is an indication of the habitability of a terrestrial planet in the HZ; any value of e > 0.2 produces a significant temperature difference on a planet's surface between apoapse and periapse. The results for possible stable orbits for terrestrial planets in habitable zones for the five systems are summarized as follows: for Gl 777 A nearly the entire HZ is stable, for 47 Uma, HD 72659 and HD 4208 terrestrial planets can survive for a sufficiently long time,while for Gl 614 our results exclude terrestrial planets moving in stable orbits within the HZ. Studies such as this one are of primary interest to future space missions dedicated to finding habitable terrestrial planets in other stellar systems. Assessing the likelihood of other habitable planets, and more generally the possibility of other life, is the central question of astrobiology today. Our investigation indicates that, from the dynamical point of view, habitable terrestrial planets seem to be quite compatible with many of the currently discovered extrasolar systems.

A study of the stable regions in the planetary system HD 74156 – Can it host earthlike planets in habitable zones?

Using numerical methods we thoroughly investigate the dynamical stability in the region between the two planets found in HD 74156. The two planets with semimajor axes 0.28 AU and 3.82 AU move on quite eccentric orbits (e=0.649 and 0.354). There is a region between 0.7 and 1.4 AU which may host additional planets which we checked via numerical integrations using different dynamical models. Besides the orbital evolution of several thousands of massless regarded planets in a three-dimensional restricted 4-body problem (host star, two planets + massless bodies) we also have undertaken test computation for the orbital evolution for fictitious planets with masses of 0.1, 0.3 and 1 MJUP in the region between HD74156b and HD74156c. For direct numerical integrations up to 10 7 years we used the Lie-integrator, a method with adaptive step-size; additionally we used the Fast Lyapunov Indicators as tool for detecting chaotic motion in this region. We emphasize the important role of the inner resonances (with the outer planet) and the outer resonances (with the inner planet) with test bodies located inside the resonances. In these two regions almost no orbits survive. The region between the 1:5 outer resonance (0.8 AU) and the 5:1 inner resonance (1.3 AU), just in the right position for habitability, is also not very likely to host planets. Our results do not strictly forbid planets to move in the habitable zone, but their existence is rare.

An Interactive Approach to Planetary Orbits at Secondary Level

I describe an interactive unit for teaching/learning planetary orbits at secondary level. The unit consists of a pre-test, activites related to the planetary orbits and a post-test. The pre-test is designed to modulate the activities to suit the needs of the students. The activities are (1) Trajectory of a particle in a gravitational field when the field is constant, and when it varies with distance (invoking a fictitious planet WonderX); (2) Plotting the orbit of a planet round the sun starting from select initial conditions and discovering Kepler’s laws; (3) Plotting orbits with force laws containing terms like 1/r3 and 1/r4 and discovering precession. Students work through graded steps. To help them, simple numerical algorithms and computer programs have been developed as they are not yet comfortable with calculus. A post-test gauges the progress of the students for a possible revisit to the topic. Such units can be very effective in teaching basic astronomy in developing countries.

Stochasticité parmi les orbites planétaires stables dans des systèmes d'étoiles doubles

In the frame of the Restricted Three-Body Problem (plane elliptic case), orbital stability of fictitious planets orbiting one of the two components of a binary star system (known the mass ratio of the two stars and the eccentricity of their orbit) is systematically explored, varying initial coordinates and velocities of the planet. For the first 5 binaries we studied, the main result is that stable planetary orbits exist at large distance from their “sun” ; for 3 of these binaries, some of these stable planetary orbits are nearly circular and, among these latter, several lie inside the habitable zone. Moreover, extrapolating to exoplanets the known dynamics of the Solar System, we may suspect that a subset of these orbits could be chaotic (and, sometimes, chaos may be bounded). A method to study this possible chaos is to compute Lyapunov Indicators; results of a numerical simulation - over Solar System age - are presented here for a fictitious exoplanet orbiting η CrB A, a planet for which the trajectory reveals bounded chaos caracteristics (during the simulation time span).

Pricing: A Normative Strategy in the Delivery of Human Services

The price of a service is an integral part of that service. Price can be viewed as a statement of value, a reflection of costs, or a marketing strategy. In human services, particularly in not-for-profit sector, price can be used as an effective means of achieving program goals. That is, price can be used to reinforce specific patterns of service utilization, to prioritize resource deployment, and to pursue other goals relevant to mission of agency. For service providers in not-for-profit sector, revenue production aspect of price may often be secondary to other organizational goals. Traditional approaches to pricing have been either or market oriented. The former is supply focused in that price is viewed as a reflection of of input used to create goods or services, whereas latter is demand focused in that price is a tool for eliciting a certain response from a potential consumer. Economic Views on Pricing Economic theory relates price to cost, competition, and elasticity of demand. In competitive markets combined forces of competition and desire of sellers to maximize profits will lead sellers to produce to point at which price, marginal cost, and average are equal (McCain, 1981). The price of health and human services paid by consumers is influenced by third-party payers and voluntary and public subsidies. Insurance policies use deductibles, copayments, and limits to make consumers conscious and to limit their financial liability (Feldstein, 1983). Paringer (1983) noted that presence of insurance leads to the large divergence between consumer out-of-pocket price and marginal cost (p. 121), which means that price paid by consumer, who is insured against a particular loss, may be far less than of producing service. Enthoven (1980) argued that price of insurance itself is, in effect, subsidized because it is purchased with pretax dollars. Therefore, consumers will tend to buy more insurance than they would if they were purchasing insurance with after-tax dollars. In short, an economic approach to pricing of services is made more complex by presence of factors external to market, as well as by social norms and values (MacRae & Wilde, 1979). The extent to which demand for a product changes when price of product changes is referred to as price elasticity. Price elasticity is said to be high when demand changes dramatically in response to a given price change and low when such a change has a minimal impact on price. When demand is relatively elastic, profits will increase when costs are adjusted downward, within certain price points, because consumers will demand a much greater volume of goods or services at lower price. When demand is inelastic, amount consumers demand is relatively constant when changes are made between price points. Profits are therefore increased when prices are increased to maximum market can bear. Gabor (1988) observed that amount a consumer may demand may actually increase when prices are raised. Consider, for example, fictitious planet Zed. On planet Zed there are very rich, middle class, and poor. Furthermore, there are only two kinds of food, rice and caviar. The poor population eats only rice, which is much cheaper than caviar; rich population eats only caviar; and middle class eats rice two meals a day and caviar one meal a day. If price of rice rises, poor population will eat same amount of rice but forgo what few luxury items they once could afford. If poor population suffers extreme poverty, they may eat less rice and go hungry, but no one is that poor on Zed. The rich population, on other hand, does not eat rice, and therefore increase in price does not affect them. The middle class, however, provides a more interesting example. Because rice is their primary food, an increase in of rice will dramatically affect their food budgets. …