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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

Analysis of eccentric annular incompressible seals. II - Effects of eccentricity on rotordynamic coefficients

A new analysis procedure has been presented which solves for the flow variables of an annular pressure seal in which the rotor has a large static displacement (eccentricity) from the centered position. The present paper incorporates the solutions to investigate the effect of eccentricity on the rotordynamic coefficients. The analysis begins with a set of governing equations based on a turbulent bulk-flow model and Moody's friction factor equation. Perturbations of the flow variables yields a set of zeroth- and first-order equations. After integration of the zeroth-order equations, the resulting zeroth-order flow variables are used as input in the solution of the first-order equations. Further integration of the first order pressures yields the eccentric rotordynamic coefficients. The results from this procedure compare well with available experimental and theoretical data, with accuracy just as good or slightly better than the predictions based on a finite-element model.

Nelson, C. C.↗
NASA NTRS1992

Strength After Bouts of Eccentric or Concentric Actions

This study examined the influence of an initial bout of eccentric or concentric actions and a subsequent bout of eccentric actions on muscular strength. Twenty-four healthy males, 24-45 yr old, were placed in three groups that performed eccentric actions in bouts 1 and 2 (ECC/ECC, N = 8), concentric actions in bout 1, and eccentric actions in bout 2 (CON/ECC, N = 8) or served as controls (N = 8). Bouts involved unilateral actions with the left and right quadriceps femoris. Ten sets of 10 repetitions with an initial resistance equal to 85% of the eccentric or concentric one repetition maximum (1 RM) were performed for each bout. Three minutes of rest were given between sets and 3 wk between bouts. Two weeks before bout 1 and 1, 4, 7, and 10 d after bouts 1 and 2, eccentric and concentric 1 RM were measured for the right quadriceps femoris and a speed-torque relation established for the left quadriceps femoris. Eccentric and concentric 1 RM decreased (P less than 0.05) 32% 1 d after bout 1 for group ECC/ECC. The speed-torque relation was down-shifted (P less than 0.05) 38%. Eccentric 1 RM and eccentric and isometric torque returned to normal 6 d later. Concentric 1 RM and torque at 3.14 rad-s(exp -1) had not recovered on day 10 (-7% for both, P less than 0.05). Decreases in strength after bout 2 for group ECC/ECC only occurred on day 1 (-9% for concentric 1 RM and 16% downshift of the speed- torque relation). Group CONIECC showed the opposite responses; marked decreases in strength after bout 2 but not bout 1. The results indicate that the initial decrease in strength after performance of a novel bout of eccentric exercise is comparable for eccentric, concentric and isometric muscle actions. Recovery of strength, however, appears to occur more rapidly for eccentric and isometric actions. They suggest that performance of a prior bout of eccentric but not concentric actions, as done in this study, can essentially eradicate decreases in strength after a subsequent bout of eccentric exercise. It is suggested that neural factors are, in part, responsible for adaptations to eccentric exercise.

Golden, Catherine L.↗
NASA NTRS2014

Can Eccentric Debris Disks Be Long-lived? A First Numerical Investigation and Application to Zeta(exp 2) Reticuli

Context. Imaging of debris disks has found evidence for both eccentric and offset disks. One hypothesis is that they provide evidence for massive perturbers, for example, planets or binary companions, which sculpt the observed structures. One such disk was recently observed in the far-IR by the Herschel Space Observatory around Zeta2 Reticuli. In contrast with previously reported systems, the disk is significantly eccentric, and the system is several Gyr old. Aims. We aim to investigate the long-term evolution of eccentric structures in debris disks caused by a perturber on an eccentric orbit around the star. We hypothesise that the observed eccentric disk around Zeta2 Reticuli might be evidence of such a scenario. If so, we are able to constrain the mass and orbit of a potential perturber, either a giant planet or a binary companion. Methods. Analytical techniques were used to predict the effects of a perturber on a debris disk. Numerical N-body simulations were used to verify these results and further investigate the observable structures that may be produced by eccentric perturbers. The long-term evolution of the disk geometry was examined, with particular application to the Zeta2 Reticuli system. In addition, synthetic images of the disk were produced for direct comparison with Herschel observations. Results. We show that an eccentric companion can produce both the observed offsets and eccentric disks. These effects are not immediate, and we characterise the timescale required for the disk to develop to an eccentric state (and any spirals to vanish). For Zeta2 Reticuli, we derive limits on the mass and orbit of the companion required to produce the observations. Synthetic images show that the pattern observed around Zeta2 Reticuli can be produced by an eccentric disk seen close to edge-on, and allow us to bring additional constraints on the disk parameters of our model (disk flux and extent). Conclusions. We conclude that eccentric planets or stellar companions can induce long-lived eccentric structures in debris disks. Observations of such eccentric structures thus provide potential evidence of the presence of such a companion in a planetary system. We considered the specific example of Zeta2 Reticuli, whose observed eccentric disk can be explained by a distant companion (at tens of AU) on an eccentric orbit (ep greater than approx. 0.3).

planetary systems↗
NASA NTRS1994

Eccentric exercise testing and training

Some researchers and practitioners have touted the benefits of including eccentric exercise in strength training programs. However, others have challenged its use because they believe that eccentric actions are dangerous and lead to injuries. Much of the controversy may be based on a lack of understanding of the physiology of eccentric actions. This review will present data concerning eccentric exercise in strength training, the physiological characteristics of eccentric exercise, and the possible stimulus for strength development. Also a discussion of strength needs for extended exposure to microgravity will be presented. Not only is the use of eccentric exercise controversial, but the name itself is fraught with problems. The correct pronunciation is with a hard 'c' so that the word sounds like ekscentric. The confusion in pronunciation may have been prevented if the spelling that Asmussen used in 1953, excentric, had been adopted. Another problem concerns the expressions used to describe eccentric exercise. Commonly used expressions are negatives, eccentric contractions, lengthening contractions, resisted muscle lengthenings, muscle lengthening actions, and eccentric actions. Some of these terms are cumbersome (i.e., resisted muscle lengthenings), one is slang (negatives), and another is an oxymoron (lengthening contractions). Only eccentric action is appropriate and adoption of this term has been recommended by Cavanagh. Despite the controversy that surrounds eccentric exercise, it is important to note that these types of actions play an integral role in normal daily activities. Eccentric actions are used during most forms of movement, for example, in walking when the foot touches the ground and the center of mass is decelerated and in lowering objects, such as placing a bag of groceries in the car.

Clarkson, Priscilla M.↗
NASA NTRS2008

Different Levels of Eccentric Resistance during Eight Weeks of Training Affect Muscle Strength and Lean Tissue Mass

Coupling concentric and eccentric muscle contractions appears to be important in the development of muscle strength and hypertrophy. The interim Resistive Exercise Device (iRED) currently used aboard the International Space Station does not seem to be as effective as free weight training in ambulatory subjects and has not completely protected against muscular deconditioning due to space flight. The lack of protection during space flight could be caused by iRED's proportionally lower eccentric resistance (60-70%) compared to concentric resistance. PURPOSE: To determine the effects of 8 wks of lower body resistive exercise training using five levels of eccentric resistance on muscle strength and lean tissue mass. METHODS: Forty untrained males (34.9 +/- 7 yrs, 80.9 +/- 9.8 kg, 178.2 +/- 7.1 cm; mean +/- SD) completed three 1-repetition maximum (1-RM) strength tests for both the supine leg press (LP) and supine heel raise (HR) prior to training; subjects were matched for LP strength and randomly assigned to one of five training groups. Concentric load (% 1-RM) was constant across groups during training, but each group trained with different levels of eccentric load (0%, 33%, 66%, 100%, or 138% of concentric). Subjects trained 3 d / wk for 8 wks using a periodized program for LP and HR based on percentages of the highest pre-training 1-RM. LP and HR 1-RM and leg lean mass (LLM; assessed by DEXA) were measured pre- and post-training. A two-way ANOVA was used to analyze all dependent measures. Tukey's post hoc tests were used to test significant main effects. Within group pre- to post-training changes were compared using paired t-tests with a Bonferroni adjustment. Statistical significance was set a priori at p 0.05. All data are expressed as mean +/- SE. RESULTS: LP 1-RM strength increased significantly in all groups pre- to post-training. The 138% group increase (20.1 +/- 3.7%) was significantly greater than the 0% (7.9 +/- 2.8%), 33% (7.7 +/- 4.6%), and 66% (7.5 +/- 4.3%) groups. All groups significantly increased HR strength pre- to posttraining (33%: 7.5 +/- 6.1%; 66%: 6.6 +/- 3.7%; 100%: 12.2 +/- 1.8%; 138%: 11.0 +/- 6.4%) except for the 0% (4.9 +/- 9.1%) group. There were no differences between groups. LLM increased significantly pre- to post-training in only the 138% group; there were no differences between groups. CONCLUSIONS: Eight wks of lower body resistive exercise training with eccentric overload resulted in greater increases in LP strength than training with eccentric loads of 66% or less. Post-training HR strength was not affected by eccentric training load, perhaps because of the predominance of Type I fibers typical in the gastrocnemius. Only 138% eccentric training significantly increased LLM. PRACTICAL APPLICATIONS: For athletes or others desiring to maximize muscle strength and hypertrophy gains, training with eccentric loads greater than 100% of concentric resistance will provide greater increases in muscle strength and lean tissue mass in some muscle groups. In a rehabilitation or geriatric exercise setting that places primary emphasis on program adherence and moderate strength gains, training with an eccentric underload may provide strength increases comparable to those of traditional 1:1 training but with less muscle soreness and physiologic insult to the patient, but this has yet to be proven.

English, K. L.↗
NASA NTRS2017

Effects of Retinal Eccentricity on Human Manual Control

This study investigated the effects of viewing a primary flight display at different retinal eccentricities on human manual control behavior and performance. Ten participants performed a pitch tracking task while looking at a simplified primary flight display at different horizontal and vertical retinal eccentricities, and with two different controlled dynamics. Tracking performance declined at higher eccentricity angles and participants behaved more nonlinearly. The visual error rate gain increased with eccentricity for single-integrator-like controlled dynamics, but decreased for double-integrator-like dynamics. Participants' visual time delay was up to 100 ms higher at the highest horizontal eccentricity compared to foveal viewing. Overall, vertical eccentricity had a larger impact than horizontal eccentricity on most of the human manual control parameters and performance. Results might be useful in the design of displays and procedures for critical flight conditions such as in an aerodynamic stall.

system identification↗
NASA NTRS1989

Muscle changes with eccentric exercise: Implications on earth and in space

Recent investigations of fluid pressure, morpholo gy, and enzyme activities of skeletal muscle exercised eccentrically or concentrically in normal human subjects are reviewed. Intramuscular pressures were measured before, during, and after submaximal exercise and correlated with subjective muscle soreness, fiber size, water content, and blood indices of muscle enzymes. High intensity eccentric exercise is characterized by post exercise pain, elevated intramuscular pressures, and swelling of both type 1 and 2 fibers as compared to concentric exercise. Thus, long periods of unaccustomed, high level eccentric contraction may cause muscle injury, fiber swelling, fluid accumulation, elevated intramuscular pressure, and delayed muscle soreness. Training regimens of progressively increasing eccentric exercise, however, cause less soreness and are extremely efficacious in increasing muscle mass and strength. It is proposed that on Earth, postural muscles are uniquely adapted to low levels of prolonged eccentric contraction that are absent during weightlessness. The almost complete absence of eccentric exercise in space may be an important contributor to muscle atrophy and therefore equipment should be designed to integrate eccentric contractions into exercise protocols for long-term spaceflight.

Hargens, Alan R.↗
NASA NTRS2013

The Eccentric Behavior of Nearly Frozen Orbits

Frozen orbits are orbits which have only short-period changes in their mean eccentricity and argument of periapse, so that they basically keep a fixed orientation within their plane of motion. Nearly frozen orbits are those whose eccentricity and argument of periapse have values close to those of a frozen orbit. We call them "nearly" frozen because their eccentricity vector (a vector whose polar coordinates are eccentricity and argument of periapse) will stay within a bounded distance from the frozen orbit eccentricity vector, circulating around it over time. For highly inclined orbits around the Earth, this distance is effectively constant over time. Furthermore, frozen orbit eccentricity values are low enough that these orbits are essentially eccentric (i.e., off center) circles, so that nearly frozen orbits around Earth are bounded above and below by frozen orbits.

spacecraft↗
NASA NTRS2019

Search for Eccentric Binary Black Hole Mergers with Advanced LIGO and Advanced Virgo during Their First and Second Observing Runs

When formed through dynamical interactions, stellar-mass binary black holes (BBHs) may retain eccentric orbits (e > 0.1 at 10 Hz) detectable by ground-based gravitational-wave detectors. Eccentricity can therefore be used to differentiate dynamically formed binaries from isolated BBH mergers. Current template-based gravitational-wave searches do not use waveform models associated with eccentric orbits, rendering the search less efficient for eccentric binary systems. Here we present the results of a search for BBH mergers that inspiral in eccentric orbits using data from the first and second observing runs (O1 and O2) of Advanced LIGO and Advanced Virgo. We carried out the search with the coherent WaveBurst algorithm, which uses minimal assumptions on the signal morphology and does not rely on binary waveform templates. We show that it is sensitive to binary mergers with a detection range that is weakly dependent on eccentricity for all bound systems. Our search did not identify any new binary merger candidates. We interpret these results in light of eccentric binary formation models. We rule out formation channels with rates ≳100 Gpc−3 yr−1 for e > 0.1, assuming a black hole mass spectrum with a power-law index ≲2.

B. P. Abbott↗
NASA NTRS1981

The origin of the eccentricities of the rings of Uranus

The effect of gravitational perturbations from a nearby satellite on the eccentricity e of a narrow particulate ring is considered. The perturbations near a resonance in an eccentric ring may be divided into corotation and Lindblad terms. For small e, the corotation terms damp e, whereas the Lindblad terms excite e. In the absence of saturation the corotation terms win by a small margin, and e damps. However, if the perturbations open gaps at the strongest resonances, then the Lindblad terms win, and e grows. This result offers an explanation for the existence of both circular and eccentric rings around Uranus. It is also shown that eccentricity changes induced by circular rings on eccentric satellite orbits are similar to those induced by satellites with circular orbits on eccentric rings.

Goldreich, P.↗
NASA NTRS1999

Impact of Eccentricity on East-west Stationkeeping for GPS Class of Orbits

There exists a strong relationship between eccentricity and the potential for a repeating groundtrack orbit to exhibit chaotic motion. This is true at all values of eccentricity, but, perhaps most dramatic, is that it is true even for orbits that are nearly circular. These complex motions can have a significant impact on the east-west stationkeeping process for maintaining the repeating groundtrack property of a commensurate orbit. Ely and Howell have shown that traditional stationkeeping (SK) methods are unable to maintain a repeating groundtrack in the presence of complex dynamics, such as with chaotic motion. They developed an alternate SK method that is able to maintain a repeating groundtrack for eccentric, commensurate orbits. The focus of the current study is to investigate orbits with characteristics that are similar to GPS satellites except with modestly larger eccentricities. It will be shown that at eccentricities larger than approx. .01 the chaotic regions become significant, and the need arises for a robust stationkeeping approach, such as developed in. FurtheRmore, the investigation will reveal that the influence of luni-solar perturbations contributes to the growth of eccentricity, thus increasing the probability of encountering chaotic motion during a typical satellite lifetime.

Ely, Todd A.↗
NASA NTRS2023

Exploring Climate With Obliquity in A Variable-Eccentricity Earth-Like World

Exploring planetary systems similar to our solar system can provide a means to explore a large range of possibly temperate climates on Earth-like worlds. Rather than run hundreds of simulations with different eccentricities at fixed obliquities, our variable-eccentricity approach provides a means to cover an incredibly large parameter space. Herein Jupiter's orbital radius is moved substantially inward in two different scenarios, causing a forcing on Earth's eccentricity. In one case, the eccentricity of Earth varies from 0 to 0.27 over ∼7000 yr for three different fixed obliquities (0°, 23°, and 45°). In another case, the eccentricity varies from 0 to 0.53 over ∼9400 yr in a single case with zero obliquity. In all cases, we find that the climate remains stable, but regional habitability changes through time in unique ways. At the same time, the moist greenhouse state is approached but only when at the highest eccentricities.

Exoplanet astronomy↗
NASA NTRS1988

On disk-planet interactions and orbital eccentricities

While Lindblad resonances both within and without a perturber's orbit excite its eccentricity, the present study of the eccentricity evolution due to the density wave interaction between a planetesimal and a Keplerian disk notes that coronation resonances in these regions lose their eccentricity damping effectiveness if the object is embedded in a continuous disk without a gap. Attention is given to another class of Lindblad resonances which, under these conditions, operates on disk material coorbiting with the perturber; these resonances thereby become the most important source of eccentricity damping. A model problem indicates that eccentricity ultimately undergoes decay.

Ward, William R.↗
NASA NTRS1992

Eccentric exercise training as a countermeasure to non-weight-bearing soleus muscle atrophy

This investigation tested whether eccentric resistance training could prevent soleus muscle atrophy during non-weight bearing. Adult female rats were randomly assigned to either weight bearing +/- intramuscular electrodes or non-weight bearing +/- intramuscular electrodes groups. Electrically stimulated maximal eccentric contractions were performed on anesthetized animals at 48-h intervals during the 10-day experiment. Non-weight bearing significantly reduced soleus muscle wet weight (28-31 percent) and noncollagenous protein content (30-31 percent) compared with controls. Eccentric exercise training during non-weight bearing attenuated but did not prevent the loss of soleus muscle wet weight and noncollagenous protein by 77 and 44 percent, respectively. The potential of eccentric exercise training as an effective and highly efficient counter-measure to non-weight-bearing atrophy is demonstrated in the 44 percent attenuation of soleus muscle noncollagenous protein loss by eccentric exercise during only 0.035 percent of the total non-weight-bearing time period.

Kirby, Christopher R.↗
NASA NTRS1997

Single Close Encounters do not make Eccentric Planetary Orbits

The recent discovery of a planet in an orbit with eccentricity e = 0.63 +/- 0.08 around the solar-type star 16 Cyg B, together with earlier discoveries of other planets in orbits of significant eccentricity, raises the question of the origin of these orbits, so unlike the nearly circular orbits of our solar system. In this paper I consider close encounters between two planets, each initially in a nearly circular orbit (but with sufficient eccentricity to permit the encounter). Such encounters are described by a two-body approximation, in which the effect of the attracting star is neglected, and by the approximation that their separation vector follows a nearly parabolic path. A single encounter cannot produce the present state of these systems, in which one planet is in an eccentric orbit and the other has apparently been lost. Even if the requirement that the second planet be lost is dropped, nearly circular orbits cannot scatter into eccentric ones.

Katz, J. I.↗
NASA NTRS2002

Eccentricity Evolution of Migrating Planets

We examine the eccentricity evolution of a system of two planets locked in a mean motion resonance, in which either the outer or both planets lose energy and angular momentum. The sink of energy and angular momentum could be a gas or planetesimal disk. We analytically calculate the eccentricity damping rate in the case of a single planet migrating through a planetesimal disk. When the planetesimal disk is cold (the average eccentricity is much less than 1), the circularization time is comparable to the inward migration time, as previous calculations have found for the case of a gas disk. If the planetesimal disk is hot, the migration time can be an order of magnitude shorter. We show that the eccentricity of both planetary bodies can grow to large values, particularly if the inner body does not directly exchange energy or angular momentum with the disk. We present the results of numerical integrations of two migrating resonant planets showing rapid growth of eccentricity. We also present integrations in which a Jupiter-mass planet is forced to migrate inward through a system of 5-10 roughly Earth-mass planets. The migrating planets can eject or accrete the smaller bodies; roughly 5% of the mass (averaged over all the integrations) accretes onto the central star. The results are discussed in the context of the currently known extrasolar planetary systems.

Murray, N.↗
NASA NTRS2006

Modification of Eccentric Gaze-Holding

Clear vision and accurate localization of objects in the environment are prerequisites for reliable performance of motor tasks. Space flight confronts the crewmember with a stimulus rearrangement that requires adaptation to function effectively with the new requirements of altered spatial orientation and motor coordination. Adaptation and motor learning driven by the effects of cerebellar disorders may share some of the same demands that face our astronauts. One measure of spatial localization shared by the astronauts and those suffering from cerebellar disorders that is easily quantified, and for which a neurobiological substrate has been identified, is the control of the angle of gaze (the "line of sight"). The disturbances of gaze control that have been documented to occur in astronauts and cosmonauts, both in-flight and postflight, can be directly related to changes in the extrinsic gravitational environment and intrinsic proprioceptive mechanisms thus, lending themselves to description by simple non-linear statistical models. Because of the necessity of developing robust normal response populations and normative populations against which abnormal responses can be evaluated, the basic models can be formulated using normal, non-astronaut test subjects and subsequently extended using centrifugation techniques to alter the gravitational and proprioceptive environment of these subjects. Further tests and extensions of the models can be made by studying abnormalities of gaze control in patients with cerebellar disease. A series of investigations were conducted in which a total of 62 subjects were tested to: (1) Define eccentric gaze-holding parameters in a normative population, and (2) explore the effects of linear acceleration on gaze-holding parameters. For these studies gaze-holding was evaluated with the subjects seated upright (the normative values), rolled 45 degrees to both the left and right, or pitched back 30 and 90 degrees. In a separate study the further effects of acceleration on gaze stability was examined during centrifugation (+2 G (sub x) and +2 G (sub z) using a total of 23 subjects. In all of our investigations eccentric gaze-holding was established by having the subjects acquire an eccentric target (+/-30 degrees horizontal, +/- 15 degrees vertical) that was flashed for 750 msec in an otherwise dark room. Subjects were instructed to hold gaze on the remembered position of the flashed target for 20 sec. Immediately following the 20 sec period, subjects were cued to return to the remembered center position and to hold gaze there for an additional 20 sec. Following this 20 sec period the center target was briefly flashed and the subject made any corrective eye movement back to the true center position. Conventionally, the ability to hold eccentric gaze is estimated by fitting the natural log of centripetal eye drifts by linear regression and calculating the time constant (G) of these slow phases of "gaze-evoked nystagmus". However, because our normative subjects sometimes showed essentially no drift (tau (sub c) = m), statistical estimation and inference on the effect of target direction was performed on values of the decay constant theta = 1/(tau (sub c)) which we found was well modeled by a gamma distribution. Subjects showed substantial variance of their eye drifts, which were centrifugal in approximately 20 % of cases, and > 40% for down gaze. Using the ensuing estimated gamma distributions, we were able to conclude that rightward and leftward gaze holding were not significantly different, but that upward gaze holding was significantly worse than downward (p<0.05). We also concluded that vertical gaze holding was significantly worse than horizontal (p<0.05). In the case of left and right roll, we found that both had a similar improvement to horizontal gaze holding (p<0.05), but didn't have a significant effect on vertical gaze holding. For pitch tilts, both tilt angles significantly decreased gaze-holding ility in all directions (p<0.05). Finally, we found that hyper-g centrifugation significantly decreased gaze holding ability in the vertical plane. The main findings of this study are as follows: (1) vertical gaze-holding is less stable than horizontal, (2) gaze-holding to upward targets is less stable than to downward targets, (3) tilt affects gaze holding, and (4) hyper-g affects gaze holding. This difference between horizontal and vertical gaze-holding may be ascribed to separate components of the velocity-to-position neural integrator for eye movements, and to differences in orbital mechanics. The differences between upward and downward gaze-holding may be ascribed to an inherent vertical imbalance in the vestibular system. Because whole body tilt and hyper-g affects gaze-holding, it is implied that the otolith organs have direct connections to the neural integrator and further studies of astronaut gaze-holding are warranted. Our statistical method for representing the range of normal eccentric gaze stability can be readily applied to normals who maybe exposed to environments which may modify the central integrator and require monitoring, and to evaluate patients with gaze-evoked nystagmus by comparing to the above established normative criteria.

Reschke, M. F.↗
NASA NTRS2016

Transit Timing Variations for Planets Near Eccentricity-Type Mean Motion Resonances

We derive the transit timing variations (TTVs) of two planets near a second-order mean motion resonance (MMR) on nearly circular orbits. We show that the TTVs of each planet are given by sinusoids with a frequency of jn(sub 2) - ( j - 2)n(sub 1), where j ≽ 3 is an integer characterizing the resonance and n(sub 2) and n(sub 1) are the mean motions of the outer and inner planets, respectively. The amplitude of the TTV depends on the mass of the perturbing planet, relative to the mass of the star, and on both the eccentricities and longitudes of pericenter of each planet. The TTVs of the two planets are approximated anti-correlated, with phases of ϕ and ≈ϕ + π, where the phase ϕ also depends on the eccentricities and longitudes of pericenter. Therefore, the TTVs caused by proximity to a second-order MMR do not in general uniquely determine both planet masses, eccentricities, and pericenters. This is completely analogous to the case of TTVs induced by two planets near a first-order MMR. We explore how other TTV signals, such as the short-period synodic TTV or a first-order resonant TTV, in combination with the second-order resonant TTV, can break degeneracies. Finally, we derive approximate formulae for the TTVs of planets near any order eccentricity-type MMR; this shows that the same basic sinusoidal TTV structure holds for all eccentricity-type resonances. Our general formula reduces to previously derived results near first-order MMRs.

Celestial mechanics↗