scholarly journals The transfer function of the rhesus macaque oculomotor system for small-amplitude slow motion trajectories

2018 ◽  
Author(s):  
Julianne Skinner ◽  
Antimo Buonocore ◽  
Ziad M. Hafed
Keyword(s):  
Eye Movements ◽  
Transfer Function ◽  
Rhesus Macaque ◽  
Large Amplitude ◽  
Small Amplitude ◽  
Temporal Frequency ◽  
Oculomotor System ◽  
Target Motion ◽  
Velocity Gain ◽  
Eye Velocity

AbstractTwo main types of small eye movements occur during gaze fixation: microsaccades and slow ocular drifts. While microsaccade generation has been relatively well-studied, ocular drift control mechanisms are unknown. Here we explored the degree to which monkey smooth eye movements, on the velocity scale of slow ocular drifts, can be generated systematically. Two male rhesus macaque monkeys tracked a spot moving sinusoidally, but slowly, along the horizontal or vertical directions. Maximum target displacement in the motion trajectory was 30 min arc (0.5 deg), and we varied the temporal frequency of target motion from 0.2 to 5 Hz. We obtained an oculomotor “transfer function” by measuring smooth eye velocity gain (relative to target velocity) as a function of frequency, similar to past work with large-amplitude pursuit. Monkey eye velocities as slow as those observed during slow ocular drifts were clearly target-motion driven. Moreover, like with large-amplitude smooth pursuit, eye velocity gain varied with temporal frequency. However, unlike with large-amplitude pursuit, exhibiting low-pass behavior, small-amplitude motion tracking was band-pass with the best ocular movement gain occurring at ~0.8-1 Hz. When oblique directions were tested, we found that the horizontal component of pursuit gain was larger than the vertical component. Our results provide a catalogue of the control abilities of the monkey oculomotor system for slow target motions, and they also support the notion that smooth fixational ocular drifts are controllable. This has implications for neural investigations of drift control and the image-motion consequences of drifts on visual coding in early visual areas.

scholarly journals Transfer function of the rhesus macaque oculomotor system for small-amplitude slow motion trajectories

2019 ◽  
Vol 121 (2) ◽  
pp. 513-529 ◽  
Author(s):  
Julianne Skinner ◽  
Antimo Buonocore ◽  
Ziad M. Hafed
Keyword(s):  
Eye Movements ◽  
Transfer Function ◽  
Rhesus Macaque ◽  
Smooth Pursuit ◽  
Large Amplitude ◽  
Small Amplitude ◽  
Temporal Frequency ◽  
Oculomotor System ◽  
Gaze Fixation ◽  
Eye Velocity

Two main types of small eye movements occur during gaze fixation: microsaccades and slow ocular drifts. While microsaccade generation has been relatively well studied, ocular drift control mechanisms are unknown. Here we explored the degree to which monkey smooth eye movements, on the velocity scale of slow ocular drifts, can be generated systematically. Two male rhesus macaque monkeys tracked a spot moving sinusoidally, but slowly, along the horizontal or vertical direction. Maximum target displacement in the motion trajectory was 30 min arc (0.5°), and we varied the temporal frequency of target motion from 0.2 to 5 Hz. We obtained an oculomotor “transfer function” by measuring smooth eye velocity gain (relative to target velocity) as a function of frequency, similar to past work with large-amplitude pursuit. Monkey eye velocities as slow as those observed during slow ocular drifts were clearly target motion driven. Moreover, as with large-amplitude smooth pursuit, eye velocity gain varied with temporal frequency. However, unlike with large-amplitude pursuit, exhibiting low-pass behavior, small-amplitude motion tracking was band pass, with the best ocular movement gain occurring at ~0.8–1 Hz. When oblique directions were tested, we found that the horizontal component of pursuit gain was larger than the vertical component. Our results provide a catalog of the control abilities of the monkey oculomotor system for slow target motions, and they also support the notion that smooth fixational ocular drifts are controllable. This has implications for neural investigations of drift control and the image-motion consequences of drifts on visual coding in early visual areas. NEW & NOTEWORTHY We studied the efficacy of monkey smooth pursuit eye movements for very slow target velocities. Pursuit was impaired for sinusoidal motions of frequency less than ~0.8–1 Hz. Nonetheless, eye trajectory was still sinusoidally modulated, even at velocities lower than those observed during gaze fixation with slow ocular drifts. Our results characterize the slow control capabilities of the monkey oculomotor system and provide a basis for future understanding of the neural mechanisms for slow ocular drifts.

scholarly journals The vestibular-related frontal cortex and its role in smooth-pursuit eye movements and vestibular-pursuit interactions

2006 ◽  
Vol 16 (1-2) ◽  
pp. 1-22 ◽  
Author(s):  
Junko Fukushima ◽  
Teppei Akao ◽  
Sergei Kurkin ◽  
Chris R.S. Kaneko ◽  
Kikuro Fukushima
Keyword(s):  
Eye Movements ◽  
Frontal Cortex ◽  
Smooth Pursuit ◽  
Vestibular Stimulation ◽  
Target Motion ◽  
Head Movements ◽  
Image Motion ◽  
Target Velocity ◽  
Eye Velocity ◽  
Pursuit Neurons

In order to see clearly when a target is moving slowly, primates with high acuity foveae use smooth-pursuit and vergence eye movements. The former rotates both eyes in the same direction to track target motion in frontal planes, while the latter rotates left and right eyes in opposite directions to track target motion in depth. Together, these two systems pursue targets precisely and maintain their images on the foveae of both eyes. During head movements, both systems must interact with the vestibular system to minimize slip of the retinal images. The primate frontal cortex contains two pursuit-related areas; the caudal part of the frontal eye fields (FEF) and supplementary eye fields (SEF). Evoked potential studies have demonstrated vestibular projections to both areas and pursuit neurons in both areas respond to vestibular stimulation. The majority of FEF pursuit neurons code parameters of pursuit such as pursuit and vergence eye velocity, gaze velocity, and retinal image motion for target velocity in frontal and depth planes. Moreover, vestibular inputs contribute to the predictive pursuit responses of FEF neurons. In contrast, the majority of SEF pursuit neurons do not code pursuit metrics and many SEF neurons are reported to be active in more complex tasks. These results suggest that FEF- and SEF-pursuit neurons are involved in different aspects of vestibular-pursuit interactions and that eye velocity coding of SEF pursuit neurons is specialized for the task condition.

Transitions between pursuit eye movements and fixation in the monkey: dependence on context

1996 ◽  
Vol 76 (3) ◽  
pp. 1622-1638 ◽  
Author(s):  
R. J. Krauzlis ◽  
F. A. Miles
Keyword(s):  
Eye Movements ◽  
Constant Velocity ◽  
Target Position ◽  
Velocity Profiles ◽  
Target Motion ◽  
Gradual Transition ◽  
Retinal Slip ◽  
Visuomotor Processing ◽  
Backward Perturbations ◽  
Eye Velocity

1. We compared the visuomotor processing underlying the onset and offset of pursuit by recording the eye movements of three monkeys as they smoothly tracked a target that was initially at rest, started to move suddenly at a constant velocity along the horizontal meridian, and then stopped. We presented this sequence of target motions in two different contexts. In the first context the target sometimes stopped after 500 ms, but on other interleaved trials the target either continued moving at a constant velocity, slowed down, speeded up, or reversed direction. In the second context the target always stopped, but the duration of the preceding constant velocity was randomized from 500 to 700 ms. 2. The dynamics of the eye velocity during the offset of pursuit were markedly different in the two experiments. When the target stopped only sometimes, the decrease in eye velocity at the offset of pursuit often overshot zero, producing a brief, small reversal in the direction of pursuit before eye speed settled to zero. When the target always stopped, the decrease in eye velocity at the offset of pursuit followed a more gradual transition toward zero with no overshoot. Thus the eye velocity profiles obtained in the first experiment contradict, whereas those obtained in the second experiment confirm, previous characterizations of the offset of pursuit as an exponential decay toward zero eye speed. 3. To investigate the basis of the different eye velocity profiles obtained in the two experiments, we probed the state of transmission along the visuomotor pathways for pursuit with the use of small perturbations in the motion of the target. We used perturbations consisting of 1 degree step changes in target position superimposed on the constant velocity motion of the target, on the basis of previous findings that such perturbations elicit saccades during fixation but smooth changes in eye speed during maintained pursuit. Single perturbations were imposed at regularly spaced intervals on separate interleaved trials during the onset, maintenance, and offset of pursuit. 4. Perturbations imposed during the onset and maintenance of pursuit had similar effects regardless of whether the target stopped only sometimes or always. In both experiments, perturbations that stepped the target in the direction opposite to the constant velocity of the target produced decreases in eye speed; perturbations in the same direction produced negligible or inconsistent changes in eye speed. The changes in eye speed caused by perturbations were largest for those perturbations introduced within the first 100 ms after the start of target motion, before the onset of the smooth eye movement, and became progressively smaller as target motion continued. The largest changes in eye speed were therefore caused by those perturbations imposed during periods of large retinal slip and by those perturbations whose direction opposed that slip. 5. Perturbations imposed during the offset of pursuit had different effects depending on whether the target stopped only sometimes or always. When the target stopped only sometimes, forward perturbations produced large increases in eye speed, whereas backward perturbations produced negligible or inconsistent changes in eye speed. Thus the visuomotor processing underlying the offset of pursuit in this experiment strongly resembled that underlying the onset of pursuit: in both cases, those perturbations in the direction opposing large retinal slip produced the largest effects. In contrast, when the target always stopped, neither forward nor backward perturbations imposed during the offset of pursuit produced large changes in eye speed. This indicates that the visuomotor processing underlying the offset of pursuit in this experiment was different from the processing underlying the onset of pursuit. 6. Perturbations also produced changes in the frequency of saccades, although these effects were less consistent than the changes in pursuit eye speed

The Influence of Briefly Presented Randomized Target Motion on the Extraretinal Component of Ocular Pursuit

2008 ◽  
Vol 99 (2) ◽  
pp. 831-842 ◽  
Author(s):  
G. R. Barnes ◽  
C. J. S. Collins
Keyword(s):  
Eye Movements ◽  
Intertrial Interval ◽  
Peak Velocity ◽  
Target Motion ◽  
Target Velocity ◽  
Initial Exposure ◽  
Ocular Pursuit ◽  
Dependent Component ◽  
Velocity Information ◽  
Eye Velocity

We assessed the ability to extract velocity information from brief exposure of a moving target and sought evidence that this information could be used to modulate the extraretinal component of ocular pursuit. A step-ramp target motion was initially visible for a brief randomized period of 50, 100, 150, or 200 ms, but then extinguished for a randomized period of 400 or 600 ms before reappearing and continuing along its trajectory. Target speed (5–20°/s), direction (left/right), and intertrial interval (2.7–3.7 s) were also randomized. Smooth eye movements were initiated after about 130 ms and comprised an initial visually dependent component, which reached a peak velocity that increased with target velocity and initial exposure duration, followed by a sustained secondary component that actually increased throughout extinction for 50- and 100-ms initial exposures. End-extinction eye velocity, reflecting extraretinal drive, increased with initial exposure from 50 to 100 ms but remained similar for longer exposures; it was significantly scaled to target velocity for 150- and 200-ms exposures. The results suggest that extraretinal drive is based on a sample of target velocity, mostly acquired during the first 150 ms, that is stored and forms a goal for generating appropriately scaled eye movements during absence of visual input. End-extinction eye velocity was significantly higher when target reappearance was expected than when it was not, confirming the importance of expectation in generating sustained smooth movement. However, end-extinction eye displacement remained similar irrespective of expectation, suggesting that the ability to use sampled velocity information to predict future target displacement operates independently of the control of smooth eye movement.

Effect of changing feedback delay on spontaneous oscillations in smooth pursuit eye movements of monkeys

1992 ◽  
Vol 67 (3) ◽  
pp. 625-638 ◽  
Author(s):  
D. Goldreich ◽  
R. J. Krauzlis ◽  
S. G. Lisberger
Keyword(s):  
Eye Movements ◽  
Visual Feedback ◽  
Target Motion ◽  
Image Motion ◽  
Target Velocity ◽  
Feedback Delay ◽  
Total Delay ◽  
Internal Feedback ◽  
Pursuit Eye Movements ◽  
Eye Velocity

1. Our goal was to discriminate between two classes of models for pursuit eye movements. The monkey's pursuit system and both classes of model exhibit oscillations around target velocity during tracking of ramp target motion. However, the mechanisms that determine the frequency of oscillations differ in the two classes of model. In "internal feedback" models, oscillations are controlled by internal feedback loops, and the frequency of oscillation does not depend strongly on the delay in visual feedback. In "image motion" models, oscillations are controlled by visual feedback, and the frequency of oscillation does depend on the delay in visual feedback. 2. We measured the frequency of oscillation during pursuit of ramp target motion as a function of the total delay for visual feedback. For the shortest feedback delays of approximately 70 ms, the frequency of oscillation was between 6 and 7 Hz. Increases in feedback delay caused decreases in the frequency of oscillation. The effect of increasing feedback delay was similar, whether the increases were produced naturally by dimming and decreasing the size of the tracking target or artificially with the computer. We conclude that the oscillations in eye velocity during pursuit of ramp target motion are controlled by visual inputs, as suggested by the image motion class of models. 3. Previous experiments had suggested that the visuomotor pathways for pursuit are unable to respond well to frequencies as high as the 6-7 Hz at which eye velocity oscillates in monkeys. We therefore tested the response to target vibration at an amplitude of +/- 8 degrees/s and frequencies as high as 15 Hz. For target vibration at 6 Hz, the gain of pursuit, defined as the amplitude of eye velocity divided by the amplitude of target velocity, was as high as 0.65. We conclude that the visuomotor pathways for pursuit are capable of processing image motion at high temporal frequencies. 4. The gain of pursuit was much larger when the target vibrated around a constant speed of 15 degrees/s than when it vibrated around a stationary position. This suggests that the pursuit pathways contain a switch that must be closed to allow the visuomotor pathways for pursuit to operate at their full gain. The switch apparently remains open for target vibration around a stationary position. 5. The responses to target vibration revealed a frequency at which eye velocity lagged target velocity by 180 degrees and at which one monkey showed a local peak in the gain of pursuit.(ABSTRACT TRUNCATED AT 400 WORDS)

Postsaccadic Enhancement of Initiation of Smooth Pursuit Eye Movements in Monkeys

1998 ◽  
Vol 79 (4) ◽  
pp. 1918-1930 ◽  
Author(s):  
Stephen G. Lisberger
Keyword(s):  
Eye Movements ◽  
Smooth Pursuit ◽  
Visual Motion ◽  
Target Position ◽  
Target Motion ◽  
Target Velocity ◽  
Smooth Pursuit Eye Movements ◽  
Motor Pathways ◽  
Pursuit Eye Movements ◽  
Eye Velocity

Lisberger, Stephen G. Postsaccadic enhancement of initiation of smooth pursuit eye movements in monkeys. J. Neurophysiol. 79: 1918–1930, 1998. Step-ramp target motion evokes a characteristic sequence of presaccadic smooth eye movement in the direction of the target ramp, catch-up targets to bring eye position close to the position of the moving target, and postsaccadic eye velocities that nearly match target velocity. I have analyzed this sequence of eye movements in monkeys to reveal a strong postsaccadic enhancement of pursuit eye velocity and to document the conditions that lead to that enhancement. Smooth eye velocity was measured in the last 10 ms before and the first 10 ms after the first saccade evoked by step-ramp target motion. Plots of eye velocity as a function of time after the onset of the target ramp revealed that eye velocity at a given time was much higher if measured after versus before the saccade. Postsaccadic enhancement of pursuit was recorded consistently when the target stepped 3° eccentric on the horizontal axis and moved upward, downward, or away from the position of fixation. To determine whether postsaccadic enhancement of pursuit was invoked by smear of the visual scene during a saccade, I recorded the effect of simulated saccades on the presaccadic eye velocity for step-ramp target motion. The 3° simulated saccade, which consisted of motion of a textured background at 150°/s for 20 ms, failed to cause any enhancement of presaccadic eye velocity. By using a strategically selected set of oblique target steps with horizontal ramp target motion, I found clear enhancement for saccades in all directions, even those that were orthogonal to target motion. When the size of the target step was varied by up to 15° along the horizontal meridian, postsaccadic eye velocity did not depend strongly either on the initial target position or on whether the target moved toward or away from the position of fixation. In contrast, earlier studies and data in this paper show that presaccadic eye velocity is much stronger when the target is close to the center of the visual field and when the target moves toward versus away from the position of fixation. I suggest that postsaccadic enhancement of pursuit reflects activation, by saccades, of a switch that regulates the strength of transmission through the visual-motor pathways for pursuit. Targets can cause strong visual motion signals but still evoke low presaccadic eye velocities if they are ineffective at activating the pursuit system.

Participation of the caudal fastigial nucleus in smooth-pursuit eye movements. I. Neuronal activity

1994 ◽  
Vol 72 (6) ◽  
pp. 2714-2728 ◽  
Author(s):  
A. F. Fuchs ◽  
F. R. Robinson ◽  
A. Straube
Keyword(s):  
Eye Movements ◽  
Firing Rate ◽  
Smooth Pursuit ◽  
Peak Velocity ◽  
Target Motion ◽  
Fastigial Nucleus ◽  
Smooth Pursuit Eye Movements ◽  
Pursuit Eye Movements ◽  
Preferred Direction ◽  
Eye Velocity

1. We recorded single-unit activity from neurons of an output of the cerebellum, the fastigial nucleus, in two rhesus macaques while the monkeys tracked small moving targets with their eyes. Many neurons in the caudal part of the fastigial nucleus exhibited a modulation in their discharge rates when smooth-pursuit eye movements were elicited by either sinusoidal or step-ramp motions of a small target. 2. The pursuit direction that elicited the most vigorous modulation in unit firing to sinusoidal target motion could be horizontal, vertical, or oblique. Most often, the preferred direction was in the contralateral and/or downward direction (50 of 69 neurons) or in the ipsilateral and/or upward direction (13 of 69). 3. For units whose preferred smooth-pursuit directions were either contralateral/downward or ipsilateral/upward during sinusoidal pursuit, peak firing as measured by the phase shift of periodic modulation at 0.5-0.8 Hz occurred near the time of peak velocity. The discharge of 80% of the neurons with contralateral/downward preferred directions preceded eye velocity by an average of -27 degrees; thus these neurons discharged maximally during eye acceleration. In contrast, neurons with ipsilateral/upward preferred directions lagged peak velocity by an average of +10.5 degrees and therefore discharged during eye deceleration. 4. The average eye velocity sensitivity for sinusoidal pursuit between 0.5 and 0.8 Hz was 0.83 +/- 0.57 (SD) spikes/s per degrees/s. We also tested 36 units during pursuit at a variety of frequencies in their preferred directions and found that firing rates increased monotonically with peak eye velocity. However, the firing rate saturated at velocities ranging from 20 to 60 degrees/s for different units. 5. When a monkey tracked a step-ramp target motion, three discharge patterns emerged in the 27 units tested. Just over half of the units discharged a burst of spikes that preceded (average lead of 27.4 +/- 17 ms) and lasted throughout the initial third of the eye acceleration; the burst was followed by a subsequent steady firing that continued after the eye had accelerated to its steady velocity. Fewer neurons discharged a burst that began late in the acceleration and was followed by steady firing. Occasional neurons showed only a gradual increase in firing rate during acceleration followed by a steady discharge. 6. Thirty of the 31 fastigial smooth-pursuit units tested also were modulated during sinusoidal yaw and/or pitch oscillations while the animals fixated a spot that rotated with them.(ABSTRACT TRUNCATED AT 400 WORDS)

scholarly journals Learning on Multiple Timescales in Smooth Pursuit Eye Movements

2010 ◽  
Vol 104 (5) ◽  
pp. 2850-2862 ◽  
Author(s):  
Yan Yang ◽  
Stephen G. Lisberger
Keyword(s):  
Eye Movements ◽  
Target Motion ◽  
Target Velocity ◽  
Single Trial ◽  
Orthogonal Axis ◽  
Pursuit Eye Movements ◽  
Gradual Process ◽  
Eye Velocity ◽  
Consistent Direction

We commonly think of motor learning as a gradual process that makes small, adaptive steps in a consistent direction. We now report evidence that learning in pursuit eye movements could start with large, transient short-term alterations that stoke a more gradual long-term process. Monkeys tracked a target that started moving horizontally or vertically. After 250 ms of motion had produced a preinstruction eye velocity close to target velocity, an orthogonal component of target motion created an instructive change in target direction that was randomly in one of the two directions along the orthogonal axis. The preinstruction eye velocity in each trial expressed single-trial learning as a bias toward the direction of the instruction in the prior trial. The single-trial learning was forgotten within 4 to 10 s. Two observations implied that single-trial learning was not simply cognitive anticipation. First, the magnitude of the trial-over-trial change in eye velocity depended on the ongoing eye velocity at the time of the instruction in the prior trial. Single-trial learning was negligible if the prior trial had provided a well-timed cue without evoking any preinstruction eye velocity. Second, regular alternation of the direction of the instructive target motion caused reactive rather than anticipatory trial-over-trial changes in eye velocity. Humans showed very different responses that appeared to be based on cognitive anticipation rather than learning. We suggest that single-trial learning results from a low-level learning mechanism and may be a necessary prerequisite for longer-term modifications that are more permanent.

Visual tracking in monkeys: evidence for short-latency suppression of the vestibuloocular reflex

1990 ◽  
Vol 63 (4) ◽  
pp. 676-688 ◽  
Author(s):  
S. G. Lisberger
Keyword(s):  
Eye Movements ◽  
Smooth Pursuit ◽  
Initial Conditions ◽  
Short Latency ◽  
Target Motion ◽  
Head Motion ◽  
Target Velocity ◽  
Vestibuloocular Reflex ◽  
Pursuit Eye Movements ◽  
Eye Velocity

1. Monkeys normally use a combination of smooth head and eye movements to keep the eyes pointed at a slowly moving object. The visual inputs from target motion evoke smooth pursuit eye movements, whereas the vestibular inputs from head motion evoke a vestibuloocular reflex (VOR). Our study asks how the eye movements of pursuit and the VOR interact. Is there a linear addition of independent commands for pursuit and the VOR? Or does the interaction of visual and vestibular stimuli cause momentary, "parametric" modulation of transmission through VOR pathways? 2. We probed for the state of the VOR and pursuit by presenting transient perturbations of target and/or head motion under different steady-state tracking conditions. Tracking conditions included fixation at straight-ahead gaze, in which both the head and the target were stationary; "times-zero (X0) tracking," in which the target and head moved in the same direction at the same speed; and "times-two (X2) tracking," in which the target and head moved in opposite directions at the same speed. 3. Comparison of the eye velocities evoked by changes in the direction of X0 versus X2 tracking revealed two components of the tracking response. The earliest component, which we attribute to the VOR, had a latency of 14 ms and a trajectory that did not depend on initial tracking conditions. The later component had a latency of 70 ms or less and a trajectory that did depend on tracking conditions. 4. To probe the latency of pursuit eye movements, we imposed perturbations of target velocity imposed during X0 and X2 tracking. The resulting changes in eye velocity had latencies of at least 100 ms. We conclude that the effects of initial tracking conditions on eye velocity at latencies of less than 70 ms cannot be caused by visual feedback through the smooth-pursuit system. Instead, there must be another mechanism for short-latency control over the VOR; we call this component of the response "short-latency tracking." 5. Perturbations of head velocity or head and target velocity during X0 and X2 tracking showed that short-latency tracking depended only on the tracking conditions at the time the perturbation was imposed. The VOR appeared to be suppressed when the initial conditions were X0 tracking. 6. The magnitude of short-latency tracking depended on the speed of initial head and target movement. During X0 tracking at 15 deg/s, short-latency tracking was modest. When the initial speed of head and target motion was 60 deg/s, the amplitude of short-latency tracking was quite large and its latency became as short as 36 ms.(ABSTRACT TRUNCATED AT 400 WORDS)

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

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