Do orbital resonances always form naturally?

Do orbital resonances always form naturally?

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For example, if I throw two planets to orbit a star at random direction, would they form an orbital resonance?

If the question is "if I throw two planets to orbit a star at random direction, would they form an orbital resonance?" -- then in general, no. A resonance is an integral ratio (1/1, 2/1, 3/5, etc.) between the periods of motion of objects -- i.e., the ratio of their periods forms a rational number. Formally speaking the odds of getting a integral ratio (let alone a strong, low-order ratio, since those are the dynamically interesting ones) if you set the system up "randomly" should be infinitesimal, because irrational numbers are (infinitely) more abundant that rationals.

However, if the orbits of one or both of the planets can change over time, then the ratio between their periods changes, and they can end up in a resonance. (Which is maybe answering the title question.) How often this happens depends on whether the planets happen to start near a strong resonance, and on how rapidly the orbits change. (If the orbit of a planet changes slowly, then it won't encounter new resonances very often; on the other hand, rapid orbital change can overwhelm the effect of weak resonances, so that the planet passes through the resonance without being caught.)

For example, it's thought that Neptune and Pluto were originally not in resonance; but the gradual outward migrations of Neptune (due to various gravitational encounters between planetesimals and the giant planets) changed its orbital period and meant that eventually it reached 2/3 resonance with Pluto, and Pluto was "captured" by the resonance, after which it stayed in resonance with Neptune.

The vast majority of objects in the Solar System are not in resonance with anything else, which is perhaps another way of answering your question. (I.e., in practice it doesn't happen very often.)

Resonances are a key to deciphering how planetary systems form and evolve.

LET'S GIVE THEORISTS THEIR DUE. Over the years they have racked up an impressive list of prophetic prognostications, such as volcanoes on Io, neutron stars, and the cosmic microwave background. But in the realm of extrasolar planets, they have pretty much missed the boat.

Before the first planets were discovered around other solar-type stars in the mid-1990s, astrophysicists simulating planet formation on computers would churn out systems that resembled our own, with small, rocky worlds huddled close to the star and gas giants farther out. But, to their chagrin, planetary modelers completely failed to foresee the two classes that dominate the 140 or so known exoplanets: "hot Jupiters" (giant planets whirling around their host stars every one to five days) and the more numerous "eccentric planets" (gas-giant leviathans that rumble through habitable zones on elongated orbits).

"One theorist has admitted to me that he cannot think of a single prediction that he and his colleagues made about extrasolar planets that has been supported by observations," says Geoffrey W. Marcy (University of California, Berkeley), whose team has discovered the majority of known exoplanets.

In their defense, theorists have probably been correct in their ideas of how planets form in circumstellar disks from colliding planetesimals and gas accretion (S&T: April 2003, page 36). But the message from the exoplanets found so far is that the scientists were omitting a crucial ingredient, like a master chef leaving apple slices out of an apple pie. Once planets form, they don't necessarily circle serenely in their formation orbits like happy campers. They interact gravitationally with their disks, they pull on one another, and they scatter smaller bodies to and fro. In other words, planetary systems evolve, and that evolution is often violent and chaotic.

These gravitational tugs-of-war cause planets to migrate far from their formation zones. The least fortunate planets suffer fiery deaths as they plunge into their host stars or become stranded in tight hot-Jupiter orbits. Others are thrown out of systems altogether, into the frigid depths of interstellar space. Even the victors of these internecine conflicts, the eccentric planets, have their elegant circular orbits yanked into ugly, distended ovals.

But which of these processes predominate? And how often will migrating giant planets wreak havoc near their parent stars, where life-bearing planets could exist? Important clues reside in the 18 known extrasolar multiple-planet systems, each of which consists of a central star and at least two planets. Seven of these systems exhibit harmonious gravitational relationships know as resonances, which may be the missing piece to the planetary puzzle. As astrophysicist Gregory P. Laughlin (University of California, Santa Cruz) explains, "The clear presence of resonances among many of the multiple-planet systems is leading to much more insight into formation and evolution mechanisms than when only a single planet is present."

"Resonance" refers to a broad class of relationships in which one object perturbs another periodically. Astronomers were intimately familiar with resonances long before they turned up in exoplanetary systems. As early as the mid-1700s stargazers knew that the orbital periods of Jupiter's moons Io (1.77 days), Europa (3.55 days), and Ganymede (7.16 days) have a ratio of 1:2:4. In the 20th century, astronomers realized that Neptune and Pluto are in a 3:2 resonance, meaning that Neptune orbits the Sun three times as Pluto goes around twice. Various resonances abound in the rings and moons of the Saturnian system (S&T: November 2004, page 38).

These harmonious ratios by themselves don't prove that bodies are in a resonance. It's the inexorable hand of gravity that ushers planets, moons, and smaller bodies into these lockstep relationships. A dust particle could go around the Sun in one year and another in two years. But that's not a resonance because neither dust particle is massive enough to enforce the fixed 2:1 ratio. In a true resonance, at least one hefty object, such as a planet, shepherds another body gravitationally. Pluto, for example, spends most of its orbit far from Neptune but is tugged strongly by Neptune at just the one location where they come closest. Whenever Pluto starts to pull ahead or lag behind, Neptune's firm gravitational hand acts to restore the 3:2 synchronization.

The Neptune/Pluto and Jovian-moon relationships are known as mean-motion resonances. Two or more bodies in this type of resonance have orbital periods whose long-term averages can be expressed as a ratio of integers. The objects oscillate around this fixed ratio like a ball rolling back and forth in a trough. For example, Pluto orbits the Sun every 248.0 years compared to Neptune's 163.7 years. That ratio (1.512) is very close to but slightly greater than 3:2 (1.5). A few tens of thousands of years from now, the ratio might be slightly less than 3:2. But over billions of years, Pluto will go around the Sun exactly two times for every three revolutions of Neptune.

"Resonances are a weird thing because sometimes they are protective, and in other cases they are disruptive," notes Laughlin. In 1866 American astronomer Daniel Kirkwood noticed that asteroids are virtually absent in orbits that are in 2:1, 3:1, 5:2, and other integer ratios with Jupiter's 11.86-year period. In these Kirkwood gaps, repeated tugs from Jupiter add up, pumping up the eccentricity of asteroid orbits. "It's like pushing a child on a swing," explains Laughlin. "If you push the child at the high point of the swing, you build up a bigger and bigger swing." Eventually, the asteroids have such high eccentricities that they cross the orbit of either Jupiter or Mars. A close approach to Big Brother spells Big Trouble: collision or ejection from the solar system.

But Jupiter provides stability in other regions, shepherding bodies into steep valleys that are difficult to climb out of. Asteroids congregate in those orbits, which are in 3:2, 4:3, 1:1, and other resonances with Jupiter. In the 1:1 resonance, for example, thousands of Trojan asteroids share Jupiter's orbit but herd 60??ahead or behind the giant planet.

The Missing Link: Synchronicity

Astronomers didn't have to wait long before realizing that resonances play a crucial role in shaping the architecture of exoplanetary systems. In January 2001 Marcy's team announced a second planet orbiting Gliese 876, a red dwarf in Aquarius. This was just the second known extrasolar multiple-planet system around a normal star. The two planets, which have minimum masses of 0.56 and 1.89 Jupiters, orbit the star in mildly eccentric orbits with periods of 30.12 and 61.02 days, respectively. "Right away we noticed that the two planets were in a 2:1 resonance," says Marcy. Computer simulations by three groups--Laughlin and John E. Chambers (NASA/Ames Research Center), Eugenio J. Rivera (Carnegie Institution of Washington) and Jack J. Lissauer (NASA/Ames Research Center), and Man Hoi Lee and Stanton J. Peale (both at the University of California, Santa Barbara)--show that the planets are tightly locked in this 2:1 mean-motion resonance, which is stable for billions of years. Although six other exoplanetary systems appear to exhibit some type of resonant relationship, none display such beautiful harmony. Moreover, the orbital periods are so short that astronomers have watched the planets interact gravitationally in real time, a phenomenon previously observed only in the solar system and in the three planets orbiting pulsar B1257+12 (S&T: September 2003, page 18).

Although the period ratio is not exactly 2:1 today, Gliese 876's planets oscillate around that perfect ratio like a pendulum, with each swing taking nine years. Like Neptune and Pluto, if one planet's period gets slightly ahead or behind, the repeated tugs of the other planet will reel it back in. "If we could look back at the 5-billion-year history of Gliese 876, there have been exactly twice as many orbits for the inner guy as for the outer guy," says Laughlin.

There are evolutionary as well as dynamical implications. Only in the most extraordinarily fortuitous cosmic coincidence would two planets form in such a tight 2:1 resonance, and even that wouldn't explain why the two massive planets are so close to their star. It is much more likely that one or both planets migrated, and that gravitational interactions locked them into the resonance. Theorists have invoked migration to explain how hot Jupiters like 51 Pegasi's planet end up in orbits just a few million kilometers from their host stars, since it's hard to imagine them forming so close in. Computer simulations of planet formation show that a massive planet will drive a spiral wake in its circumstellar disk. The gas and dust concentrated in the wake will usually pull the planet backward, causing it to migrate steadily inward.

Although astronomers have never observed such migration in action, it offers the most straightforward and elegant explanation for how Gliese 876's planets ended up in their resonance. "The planet-disk interactions caused the outer planet to migrate inward and possibly the inner planet to migrate outward," says Lee. "The converging orbits could easily result in capture into the resonance." The planets could continue to migrate, but the resonance maintains the 2:1 ratio.

"Our models specifically predicted a 2:1 system before one was observed. Many other aspects of recent planet discoveries were not at all predicted by theory, so it's nice to get one right for a change," adds Geoffrey Bryden (NASA/Jet Propulsion Laboratory), who has developed computer simulations of planetary-system evolution with Douglas N. C. Lin (University of California, Santa Cruz) that yield the 2:1 capture mechanism. Wilhelm Kley (University of Tubingen, Germany) has achieved similar results.

Gliese 876's tight resonance argues strongly in favor of slow migration and strongly against the possibility that the two planets engaged in a fierce gravitational tug-of-war. Such violent interactions would have left them in more eccentric orbits. And even if the planets had somehow managed to settle into a 2:1 resonance, they would exhibit larger oscillations around the fixed ratio, like a pendulum gone berserk.

The two known planets orbiting the Sun-like star HD 82943 in Hydra are also locked in a 2:1 mean-motion resonance. But computer simulations indicate that HD 82943's resonance is not as tight as Gliese 876's, and the orbits are considerably more eccentric. "In Gliese 876, the oscillations around the 2:1 value are extremely small, and that system is very deeply in the resonance, whereas the HD 82943 system is just flopping around with a time scale of about a century the excursions are gigantic," says Laughlin.

Laughlin thinks the HD 82943 system resulted from three massive planets working themselves down to two. An earlier spasm of gravitational conflict tossed out one of the planets, leaving two survivors in highly eccentric resonant orbits.

The two middle planets in the 55 Cancri quadruple-planet system appear to participate in a 3:1 resonance, though the observations have yet to define the orbits precisely. If the resonance is confirmed, it will imply a slow migration-and-capture scenario as envisioned for Gliese 876.

Two recently discovered systems also exhibit mean-motion resonances. With orbital periods of 454.2 and 919.1 days, respectively, the two known planets orbiting the type-K0 star HD 128311 exhibit an obvious 2:1 relationship. The G-type star HD 37124 has planets that are probably in a 5:1 resonance, with periods of 31.085 and 154.66 days. The two planets must have migrated relatively slowly with respect to each other to be captured in such widely separated orbits.

Secular resonances also play an important role in shaping the architectures of planetary systems. "'Secular" in this context refers to several types of long-term gravitational interactions between planets that induce changes in where they come closest to their host star or where their orbital planes intersect. In some cases, a secular resonance causes the long axis of planetary orbits to precess at nearly the same rate. A classic example of a secular resonance can be found in Upsilon Andromedae, which will be forever enshrined in history books as the first known multiple-planet system around a solar-type star other than the Sun.

Debra Fischer (San Francisco State University), a member of Marcy's team, discovered Upsilon Andromedae's second and third planets in 1999--three years after Marcy and his colleague R. Paul Butler (Carnegie Institution of Washington) found the inner planet, a hot Jupiter. The outer two planets follow moderately eccentric orbits with periods of 241.27 and 1,294.4 days. Marcy's team quickly realized that the long (major) axes of both planets were nearly aligned, meaning the major axis in each elliptical orbit "points" in the same direction. "That was pretty startling, but to be honest, I didn't even know what a secular resonance was at the time," quips Marcy.

Computer simulations demonstrate that the major axes (or the lines of apsides) of the other two planets probably rock back and forth by about 30 [degrees] with respect to an imaginary line that defines the average alignment of the two elliptical orbits. The orbits themselves also precess (rotate) over long time periods.

Upsilon Andromedae's secular resonance is a good thing for the middle planet, which has about half the mass of the outer one. Without the resonance, gravitational interactions would pump up the eccentricity of both planets. Their orbits might eventually cross, and the outer planet would eject the middle planet during a close encounter. But thanks to the resonance, if the eccentricity of one planet starts to grow, the other's eccentricity will decay. The resonance keeps the planets from approaching each other too closely thus gravitational interactions are damped down, stabilizing the system for long time scales.

Like all planets, the two bodies probably started off in circular orbits. According to Eugene Chiang (University of California, Berkeley) and Norman W. Murray (University of Toronto), the outer planet interacted gravitationally with a remnant gas disk, which pumped up its eccentricity. Gravitational interactions between the two outermost planets yanked the middle one into an eccentric orbit as well, and the mutual torque forced the system into apsidal alignment. "The circumstellar disk shaped the orbit of the outer planet. The outer planet, in turn, sculpted the orbit of the middle planet," says Chiang.

As is the case with Gliese 876, Upsilon Andromedae's resonance provides a clue that young planets interact gravitationally with their disks, and that violent scattering events do not always occur. In Gliese 876, the disk interaction likely drove the outer planet inward. In Upsilon Andromedae, the interaction probably increased the outer planet's eccentricity. In each case, the resonance protects the system from utter disaster.

Gliese 876's and HD 128311's respective planetary systems appear to be in secular resonances that resemble Upsilon Andromedae's. In the double-planet system bound to the Ktype star HD 12661, Lee and Peale's research suggests a secular resonance in which the lines of apsides are anti-aligned. This means that, on average, each one's periastron is aligned with the other's apoastron, though they oscillate as much as 55 [degrees] in cycles that repeat every 12,000 years.

Resonances: Good News for Life

The other 11 known extrasolar multiplanet systems do not exhibit such strong or obvious resonances. In most of these hierarchical systems, the planets have orbital-period ratios greater than 5:1, meaning the bodies are spaced too far apart to settle into resonances. The hierarchical systems appear to be places where gravitational-scattering disasters involving three or more planets left behind reeling configurations.

In contrast, scattering did not disrupt our solar system, probably because the outer planets stayed far apart from one another. "As a conservative astronomer, I hate to say this because I like to stay close to the facts," says Marcy. "But a possible interpretation is that we wouldn't be here discussing this if our planetary system had sizable interactions among the big planets. Those big titans would have scattered the poor little terrestrial planets to kingdom come."

Clearly, many processes operate at different levels, producing a plethora of outcomes in planetary systems. Any given system's evolution depends sensitively upon initial conditions, so no single mechanism will prevail in every circumstance. But resonances demonstrate that relatively gentle processes such as migration, resonant capture, and orbital stabilization play major roles in the ongoing construction of some planetary systems. Life will have a fighting chance in certain systems because they avoid catastrophic events.

If a protoplanetary disk is to form potentially life-bearing planets, it needs to make enough planets to offer several opportunities for biology to get started, but it can't form too many Jupiter-mass behemoths. The giant planets also need to be widely spaced so they don't perturb one another too strongly, and the disk must dissipate before inbound gas giants intrude into the habitable zone.

Nobody knows how often this fortunate set of circumstances occurs, because the radial-velocity exoplanet surveys haven't been searching long enough to reveal significant numbers of Jupiter-mass planets in 10- to 15-year orbits. "What fraction of Sun-like stars have a Jupiter that resembles our own Jupiter?" asks Marcy. "We will answer this question in the next three to five years. This will tell us whether our solar system is a weirdo or not." If planet hunters like Marcy turn up relatively unperturbed carbon copies of our solar system, maybe theorists were on the right track after all.

Over the course of the next few years, planet hunters will uncover new resonant exoplanetary systems, and probably even new types of resonances. Gregory P. Laughlin's computer simulations, for example, suggest the existence of two types of systems with 1:1 mean-motion resonances.

The first type is reminiscent of the Trojan asteroids, with two planets sharing an orbit but separated by 60 [degrees], forming an equilateral triangle with the star. "The configuration is completely stable," says Laughlin. "You can distribute 3 percent of the Sun's mass in that configuration, so you can have one 30-Jupiter-mass planet and an Earth, two 15-Jupiter-mass planets, or a 20 and a 10. The 2:1 resonance is very strong, but 1:1 is even stronger." Given the stability of this configuration, Laughlin is confident that exoplanet searches will eventually turn up these types of systems.

The other type of 1:1 system is even more bizarre. One planet starts off with a circular orbit, and the other on an elliptical orbit. But over time, the two planets exchange eccentricity like a hot potato. The period ratio of the two orbits can range from 0.8 to 1.2, but it oscillates around a 1:1 average. The resonance keeps the planets from colliding. "We've done experiments where we load up systems with 10 to 12 planets, and they go crazy," says Laughlin. "Planets get tossed out and they have collisions, and they work themselves down to one or two survivors. In a non-negligible fraction of the cases, we found systems that ended up with two survivors in exactly this kind of configuration. There's a huge family of systems that will do this."

Astronomers confirm orbital details of TRAPPIST-1’s least understood planet

Scientists using NASA's Kepler Space Telescope have identified a regular pattern in the orbits of the planets in the TRAPPIST-1 system that confirmed suspected details about the orbit of its outermost and least understood planet, TRAPPIST-1h.

An international team of astronomers, including a postdoctoral research associate at the University of Central Lancashire (UCLan), used data gathered by the Kepler Space Telescope to observe and confirm details of the seventh exoplanet orbiting the star TRAPPIST-1.

They confirmed that the planet, TRAPPIST-1h, orbits its star every 18.77 days, is linked in its orbital path to its siblings and is frigidly cold. Far from its host star, the planet is likely uninhabitable — but it may not always have been so.

This most recent research follows news in February when scientists announced that the system has seven Earth-sized planets. NASA's Spitzer Space Telescope, the TRAPPIST (Transiting Planets and Planetesimals Small Telescope) in Chile and other ground-based telescopes were used to detect and characterize the planets. But the collaboration only had an estimate for the period of TRAPPIST-1h.

" The new data allowed us to measure the orbit of the outermost planet, TRAPPIST-1 h. We have determined the orbital period to be 18.77 days, which was exactly the period the team predicted. "

TRAPPIST-1 is only eight percent the mass of our sun, making it a cooler and less luminous star. It&rsquos home to seven Earth-size planets, three of which orbit in their star's habitable zone&mdashthe range of distances from a star where liquid water could pool on the surface of a rocky planet. The system is located about 40 light-years away in the constellation of Aquarius and is estimated to be between three billion and eight billion years old.

This most recent TRAPPIST-1 research has been announced in a paper published in the journal Nature Astronomy. Dr Daniel Holdsworth, a postdoctoral research associate at UCLan&lsquos Jeremiah Horrocks Institute for Mathematics, Physics and Astronomy, is one of the European scientists involved in the project and helped to determine the rotation period of the star, through the analysis of spots on the stellar surface.

He said: "We have analysed 79 days of near-continuous data of the TRAPPIST-1 system obtained with the Kepler Space Telescope in its repurposed K2 configuration. The new data allowed us to measure the orbit of the outermost planet, TRAPPIST-1 h. We have determined the orbital period to be 18.77 days, which was exactly the period the team predicted. We also found the planet to be a little smaller than Earth, with an average temperature of -100ºC.

"Furthermore, we find that all the planets form a complex chain of orbital resonances, meaning they interact with each other, effectively pulling and pushing on each other as they pass. TRAPPIST-1 is now the record for the number of planets that are interacting like this, with the previous record being four planets. This is important for the formation theory of the system as these interactions were probably present during the early stages of the system. Once the planets are in this resonant chain, it is hard to escape, and so they probably all migrated towards their host star at the same time, and remained in this intricate dance."

" We found the rotation period to be about 3.3 days which means the star is middle-aged, between three and eight Gyr old, or three to eight billion years old. "

"The K2 data also allowed us to measure the rotation period and activity of the star, results of which can be used to tell us how old the star is. We found the rotation period to be about 3.3 days which means the star is middle-aged, between three and eight Gyr old, or three to eight billion years old.&rdquo

The research was funded by the NASA Astrobiology Institute via the UW-based Virtual Planetary Laboratory as well as a National Science Foundation Graduate Student Research Fellowship, the Swiss National Science Foundation, the European Research Council and the UK Science and Technology Facilities Council, among other agencies.

Astronomy Without A Telescope – Secular Evolution

A traditional galaxy evolution model has it that you start with spiral galaxies – which might grow in size through digesting smaller dwarf galaxies – but otherwise retain their spiral form relatively undisturbed. It is only when these galaxies collide with another of similar size that you first get an irregular ‘train-wreck’ form, which eventually settles into a featureless elliptical form – full of stars following random orbital paths rather than moving in the same narrow orbital plane that we see in the flattened galactic disk of a spiral galaxy.

The concept of secular galaxy evolution challenges this notion – where ‘secular’ means separate or isolated. Theories of secular evolution propose that galaxies naturally evolve along the Hubble sequence (from spiral to elliptical), without merging or collisions necessarily driving changes in their form.

While it’s clear that galaxies do collide – and then generate many irregular galaxy forms we can observe – it is conceivable that the shape of an isolated spiral galaxy could evolve towards a more amorphously-shaped elliptical galaxy if they possess a mechanism to transfer angular momentum outwards.

The flattened disk shape of standard spiral galaxy results from spin – presumably acquired during its initial formation. Spin will naturally cause an aggregated mass to adopt a disk shape – much as pizza dough spun in the air will form a disk. Conservation of angular momentum requires that the disk shape will be sustained indefinitely unless the galaxy can somehow lose its spin. This might happen through a collision – or otherwise by transferring mass, and hence angular momentum, outwards. This is analogous to spinning skaters who fling their arms outwards to slow their spin.

Density waves may be significant here. The spiral arms commonly visible in galactic disks are not static structures, but rather density waves which cause a temporary bunching together of orbiting stars. These density waves may be the result of orbital resonances generated amongst the individual stars of the disk.

It has been suggested that a density wave represents a collisionless shock which has a damping effect on the spin of the disk. However, since the disk is only braking upon itself, angular momentum still has to be conserved within this isolated system.

A galactic disk has a corotation radius – a point where stars rotate at the same orbital velocity as the density wave (i.e. a perceived spiral arm) rotate. Within this radius, stars move faster than the density wave – while outside the radius, stars move slower than the density wave.

This may account for the spiral shape of the density wave – as well as offering a mechanism for the outward transfer of angular momentum. Within the radius of corotation, stars are giving up angular momentum to the density wave as they push through it – and hence push the wave forward. Outside the radius of corotation, the density wave is dragging through a field of slower moving stars – giving up angular momentum to them as it does so.

The result is that the outer stars are flung further outwards to regions where they could adopt more random orbits – rather than being forced to conform to the mean orbital plane of the galaxy. In this way, a tightly-bound rapidly spinning spiral galaxy could gradually evolve towards a more amorphous elliptical shape.

What are orbital resonances and how do they arise?

Orbital resonances are the orbital analogous of resonant vibrations in solids.

Imagine a metal beam oscillating, moving forward and back repeatedly. If every time it moves forward you give it a little push, the oscillation will become larger in amplitude, accumulating energy. Eventually it may no longer be able to resist the strain and undergo permanent deformation.

In an orbit the effects are very similar. If every time that an asteroid is traversing a given part of its orbit it receives a small pull, its orbital energy will increase. This will lead to a higher orbit (more distant from the central attracting body and more orbital energy).

This may happen e.g. because a planet is influencing the asteroid with its gravity. Only if their orbital periods have a simple integer ratio the perturbations will be repeated, otherwise the effect will not be cumulative (just like the metal beam, you're not consistently adding energy if you don't push it always on the same part of the oscillation).

If an orbit accumulates too much energy it may eventually reach escape speed. That's why most orbital resonances are not stable (and also in this case it's analogous to a vibration that causes a structural collapse).

There are very few cases of stable, self-correcting orbital resonances. Those happen if two or more bodies exchange energy back and forth, so that the net energy gain of a single one over an extended period of time is close to zero. One such example is the Galilean moons.

If an orbit accumulates too much energy it may eventually reach escape speed. That's why most orbital resonances are not stable (and also in this case it's analogous to a vibration that causes a structural collapse).

This is not correct. The two bodies that are resonating happily exchange energy back and forth. In the orbital context, motion near a mean-motion resonance is well approximated by a pendulum (non-forced and frictionless). Once set in motion, it will contentedly swing.

Resonances will be destabilizing only if there are multiple resonances overlapping with each other, or if over the course of a resonance cycle a body's eccentricity will get large enough to enable a close encounter with some other body.

When two things are in mean-motion resonance (MMR), the ratio of their periods is close to a small integer ratio (2/1, 3/2, 5/3, etc). (Something's "mean motion" is its mean angular speed, proportional to 1/period.) What this means physically is that the two bodies will pass each other (line up) at the same location(s) repeatedly. Watch this gif of the orbital motions of Io, Europa, and Ganymede, which are in a 4:2:1 resonance.

In reality, two bodies won't be exactly at resonance, but will oscillate around exact resonance. Take a look at this picture which shows the location of Pluto relative to Neptune's location over many orbits. If Pluto and Neptune were exactly in resonance, Pluto would come to perihelion (closest approach) at exactly two spots in that picture, exactly 90 degrees ahead of Neptune and 90 degrees behind Neptune. Instead, Pluto's perihelion oscillates around those two locations. If this oscillation of Pluto's perihelion was too big, it would experience a close encounter with Neptune. As it is, Neptune is never close by when Pluto is at perihelion. This is an example of a stable resonant interaction.

In the Asteroid Belt, resonances are typically destabilizing because the mean-motion resonances overlap with secular resonances or other mean-motion resonances , resulting in the Kirkwood gaps. (A planetary system will have several natural frequencies, the 'secular' frequencies. These are the frequencies at which the planets' eccentricities and inclinations will vary. If a small body's orbit is precessing at the same rate as one of these frequencies, this is a secular resonance.) Overlapping resonances causes chaotic variation of the orbit, and ultimately destabilization.

EDIT: As for how resonances come to be: Planets (and other bodies: moons, asteroids, etc) can migrate. In particular, their periods can change under the influence of torques from the disk in which they are forming or tides.

Orbital mechanics

Ćuk has made a career of studying the orbits of natural satellites, including asteroids and moons in the Solar System. He won a prestigious award in 2014 for his mastery of planetary dynamics, including the concept of orbital resonance, which explains how objects in orbit around another object sometimes exert a gravitational influence on one another.

This happens all over the Solar System. Most famously, Pluto and Neptune have an orbital resonance of 2:3, meaning that Pluto completes two orbits during the same amount of time that Neptune completes three. In this case, Pluto, the smaller object, has its orbit driven by Neptune. Some resonances are stable and others are unstable, even to the point where a body can be kicked out of the Solar System.

Orbital resonances becomes quite a bit more complicated with a very large planet and a sprawling system of more than five dozen moons. One might think that the moons of Saturn would, over time, fall into more elongated orbits or get knocked out of their orbital planes due to unstable resonances. But when one looks at the inner moons of Saturn, those inside the orbit of Titan, the planet’s satellites are found to be in relatively good order. They question is, should they be?

Saturn is a gas giant, of course. The gases at its surface and more exotic types of matter in its interior collectively act as a fluid. Because there are all these moons orbiting around Saturn, most of them in a plane, they exert a gravitational pull on the planet. It's therefore a bit egg shaped as it flexes, there is some friction, which causes the rotation of Saturn to slow down slightly. The tidal forces, in turn, cause the moons to slowly move away.

Spiral Galaxies Might Evolve into Elliptical Ones Naturally

A theory related to how galaxies evolve over the course of their lifespan in currently gaining increased support from the international astronomical community. Its premises challenge those of the most commonly-accepted theories of what evolutionary path spiral galaxies take.

One of the most &ldquotraditional&rdquo galaxy evolution models holds that spiral galaxies retain their trademark shape unless influenced from the outside. These influences can be vary widely.

For example, it is known that spiral galaxies grow in size and mass as they accrete dwarf galaxies. This is how the Milky Way grew to its current size. But what happens when two spiral galaxies collide?

In 3 to 4 billion years, the Milky Way will collide with Andromeda, which is a spiral galaxy of similar mass and size. Chances are high that the merger will produce a massive elliptical galaxy.

With the featureless elliptical form, many things will change. Stars and solar systems will no longer follow a narrow orbital plane as they currently do but will rather have hectic, chaotic orbits.

Therefore, according to this model, this is the only way a spiral galaxy can turn into an elliptical. But the new theory, called secular galaxy evolution, challenges this belief, Universe Today reports.

In this particular context, the term secular means separate, or isolated, and is used to denote the fact that spiral galaxies naturally evolve into ellipticals, without having to undergo mergers or collisions.

This transition is known among specialists as the Hubble sequence. What the new theory says is basically that the Hubble sequence needs no external influences to take place.

One of the basic concepts in the new proposal is the fact that spiral galaxies contain within a mechanism that enables them to transfer angular momentum outwards. What still remains to be established is whether spiral galaxies evolve or degenerate into elliptical ones.

Celestial mechanics seems to indicate that spirals develop their flattened disks and central bulge due to the fact that they spin. The rotation motion was probably imprinted to the entire structure in the earliest stages of its formation.

But the principle of conservation of angular momentum holds that spin will always be preserved. Still, experts now say, it could be that collisions, mergers and other such events might cause spiral galaxies to lose their angular momentum.

The spiral arms, which are actually density waves, may play a critical role in this process. One interesting theory holds that galactic arms are formed by the orbital resonances generated amongst the individual stars of the disk.

As enough momentum is transferred outwards, spirals might gradually come to a halt. The disk would disappear, and be replaced by an amorphous, non-defined mass of hectic star systems. In all respects, the result would be an elliptical galaxy.

Standing Waves

Maybe you've noticed or maybe you haven't. Sometimes when you vibrate a string, or cord, or chain, or cable it's possible to get it to vibrate in a manner such that you're generating a wave, but the wave doesn't propagate. It just sits there vibrating up and down in place. Such a wave is called a and must be seen to be appreciated.

I first discovered standing waves (or I first remember seeing them) while playing around with a phone cord. If you shake the phone cord in just the right manner it's possible to make a wave that appears to stand still. If you shake the phone cord in any other way you'll get a wave that behaves like all the other waves described in this chapter waves that propagate — . Traveling waves have high points called crests and low points called troughs (in the transverse case) or compressed points called compressions and stretched points called rarefactions (in the longitudinal case) that travel through the medium. Standing waves don't go anywhere, but they do have regions where the disturbance of the wave is quite small, almost zero. These locations are called . There are also regions where the disturbance is quite intense, greater than anywhere else in the medium, called .

Standing waves can form under a variety of conditions, but they are easily demonstrated in a medium which is finite or bounded. A phone cord begins at the base and ends at the handset. (Or is it the other way around?) Other simple examples of finite media are a guitar string (it runs from fret to bridge), a drum head (it's bounded by the rim), the air in a room (it's bounded by the walls), the water in Lake Michigan (it's bounded by the shores), or the surface of the Earth (although not bounded, the surface of the Earth is finite). In general, standing waves can be produced by any two identical waves traveling in opposite directions that have the right wavelength. In a bounded medium, standing waves occur when a wave with the correct wavelength meets its reflection. The interference of these two waves produces a resultant wave that does not appear to move.

Standing waves don't form under just any circumstances. They require that energy be fed into a system at an appropriate frequency. That is, when the applied to a system equals its . This condition is known as . Standing waves are always associated with resonance. Resonance can be identified by a dramatic increase in amplitude of the resultant vibrations. Compared to traveling waves with the same amplitude, producing standing waves is relatively effortless. In the case of the telephone cord, small motions in the hand result will result in much larger motions of the telephone cord.

Any system in which standing waves can form has numerous natural frequencies. The set of all possible standing waves are known as the of a system. The simplest of the harmonics is called the or first harmonic. Subsequent standing waves are called the second harmonic, third harmonic, etc. The harmonics above the fundamental, especially in music theory, are sometimes also called . What wavelengths will form standing waves in a simple, one-dimensional system? There are three simple cases.

One dimension: two fixed ends

If a medium is bounded such that its opposite ends can be considered fixed, nodes will then be found at the ends. The simplest standing wave that can form under these circumstances has one antinode in the middle. This is half a wavelength. To make the next possible standing wave, place a node in the center. We now have one whole wavelength. To make the third possible standing wave, divide the length into thirds by adding another node. This gives us one and a half wavelengths. It should become obvious that to continue all that is needed is to keep adding nodes, dividing the medium into fourths, then fifths, sixths, etc. There are an infinite number of harmonics for this system, but no matter how many times we divide the medium up, we always get a whole number of half wavelengths ( 1 2 λ, 2 2 λ, 3 2 λ, …, n 2 λ).

There are important relations among the harmonics themselves in this sequence. The wavelengths of the harmonics are simple fractions of the fundamental wavelength. If the fundamental wavelength were 1 m the wavelength of the second harmonic would be 1 2 m, the third harmonic would be 1 3 m, the fourth 1 4 m, and so on. Since frequency is inversely proportional to wavelength, the frequencies are also related. The frequencies of the harmonics are whole-number multiples of the fundamental frequency. If the fundamental frequency were 1 Hz the frequency of the second harmonic would be 2 Hz, the third harmonic would be 3 Hz, the fourth 4 Hz, and so on.

One dimension: two free ends

If a medium is bounded such that its opposite ends can be considered free, antinodes will then be found at the ends. The simplest standing wave that can form under these circumstances has one node in the middle. This is half a wavelength. To make the next possible standing wave, place another antinode in the center. We now have one whole wavelength. To make the third possible standing wave, divide the length into thirds by adding another antinode. This gives us one and a half wavelengths. It should become obvious that we will get the same relationships for the standing waves formed between two free ends that we have for two fixed ends. The only difference is that the nodes have been replaced with antinodes and vice versa. Thus when standing waves form in a linear medium that has two free ends a whole number of half wavelengths fit inside the medium and the overtones are whole number multiples of the fundamental frequency

One dimension: one fixed end — one free end

When the medium has one fixed end and one free end the situation changes in an interesting way. A node will always form at the fixed end while an antinode will always form at the free end. The simplest standing wave that can form under these circumstances is one-quarter wavelength long. To make the next possible standing wave add both a node and an antinode, dividing the drawing up into thirds. We now have three-quarters of a wavelength. Repeating this procedure we get five-quarters of a wavelength, then seven-quarters, etc. In this arrangement, there are always an odd number of quarter wavelengths present. Thus the wavelengths of the harmonics are always fractional multiples of the fundamental wavelength with an odd number in the denominator. Likewise, the frequencies of the harmonics are always odd multiples of the fundamental frequency.

The three cases above show that, although not all frequencies will result in standing waves, a simple, one-dimensional system possesses an infinite number of natural frequencies that will. It also shows that these frequencies are simple multiples of some fundamental frequency. For any real-world system, however, the higher frequency standing waves are difficult if not impossible to produce. Tuning forks, for example, vibrate strongly at the fundamental frequency, very little at the second harmonic, and effectively not at all at the higher harmonics.


The best part of a standing wave is not that it appears to stand still, but that the amplitude of a standing wave is much larger that the amplitude of the disturbance driving it. It seems like getting something for nothing. Put a little bit of energy in at the right rate and watch it accumulate into something with a lot of energy. This ability to amplify a wave of one particular frequency over those of any other frequency has numerous applications.

  • Basically, all non-digital musical instruments work directly on this principle. What gets put into a musical instrument is vibrations or waves covering a spread of frequencies (for brass, it's the buzzing of the lips for reeds, it's the raucous squawk of the reed for percussion, it's the relatively indiscriminate pounding for strings, it's plucking or scraping for flutes and organ pipes, it's blowing induced turbulence). What gets amplified is the fundamental frequency plus its multiples. These frequencies are louder than the rest and are heard. All the other frequencies keep their original amplitudes while some are even de-amplified. These other frequencies are quieter in comparison and are not heard.
  • You don't need a musical instrument to illustrate this principle. Cup your hands together loosely and hold them next to your ear forming a little chamber. You will notice that one frequency gets amplified out of the background noise in the space around you. Vary the size and shape of this chamber. The amplified pitch changes in response. This is what people hear when the hold a seashell up to their ears. It's not "the ocean" but a few select frequencies amplified out of the noise that always surrounds us.
  • During speech, human vocal cords tend to vibrate within a much smaller range that they would while singing. How is it then possible to distinguish the sound of one vowel from another? English is not a tonal language (unlike Chinese and many African languages). There is little difference in the fundamental frequency of the vocal cords for English speakers during a declarative sentence. (Interrogative sentences rise in pitch near the end. Don't they?) Vocal cords don't vibrate with just one frequency, but with all the harmonic frequencies. Different arrangements of the parts of the mouth (teeth, lips, front and back of tongue, etc.) favor different harmonics in a complicated manner. This amplifies some of the frequencies and de-amplifies others. This makes "EE" sound like "EE" and "OO" sound like "OO".
  • The filtering effect of resonance is not always useful or beneficial. People that work around machinery are exposed to a variety of frequencies. (This is what is.) Due to resonance in the ear canal, sounds near 4000 Hz are amplified and are thus louder than the other sounds entering the ear. Everyone should know that loud sounds can damage one's hearing. What everyone may not know is that exposure to loud sounds of just one frequency will damage one's hearing at that frequency. People exposed to noise are often experience 4000 Hz hearing loss. Those afflicted with this condition do not hear sounds near this frequency with the same acuity that unafflicted people do. It is often a precursor to more serious forms of hearing loss.

Two dimensions

The type of reasoning used in the discussion so far can also be applied to two-dimensional and three-dimensional systems. As you would expect, the descriptions are a bit more complex. Standing waves in two dimensions have numerous applications in music. A circular drum head is a reasonably simple system on which standing waves can be studied. Instead of having nodes at opposite ends, as was the case for guitar and piano strings, the entire rim of the drum is a node. Other nodes are straight lines and circles. The harmonic frequencies are not simple multiples of the fundamental frequency.

The diagram above shows six simple modes of vibration in a circular drum head. The plus and minus signs show the phase of the antinodes at a particular instant. The numbers follow the (D, C) naming scheme, where D is the number of nodal diameters and C is the number of nodal circumferences.

Standing waves in two dimensions have been applied extensively to the study of violin bodies. Violins manufactured by the Italian violin maker Antonio Stradivari (1644–1737) are renowned for their clarity of tone over a wide dynamic range. Acoustic physicists have been working on reproducing violins equal in quality to those produced by Stradivarius for quite some time. One technique developed by the German physicist Ernst Chladni (1756–1794) involves spreading grains of fine sand on a plate from a dismantled violin that is then clamped and set vibrating with a bow. The sand grains bounce away from the lively antinodes and accumulate at the quiet nodes. The resulting from different violins could then be compared. Presumably, the patterns from better sounding violins would be similar in some way. Through trial and error, a violin designer should be able to produce components whose behavior mimicked those of the legendary master. This is, of course, just one factor in the design of a violin.

Chladni patterns on violin plates in order of increasing frequency Source: Joe Wolf, University of New South Wales
91 Hz 145 Hz 170 Hz 384 Hz

Three dimensions

In the one-dimensional case the nodes were points (zero-dimensional). In the two-dimensional case the nodes were curves (one-dimensional). The dimension of the nodes is always one less than the dimension of the system. Thus, in a three-dimensional system the nodes would be two-dimensional surfaces. The most important example of standing waves in three dimensions are the orbitals of an electron in an atom. On the atomic scale, it is usually more appropriate to describe the electron as a wave than as a particle. The square of an electron's wave equation gives the probability function for locating the electron in any particular region. The orbitals used by chemists describe the shape of the region where there is a high probability of finding a particular electron. Electrons are confined to the space surrounding a nucleus in much the same manner that the waves in a guitar string are constrained within the string. The constraint of a string in a guitar forces the string to vibrate with specific frequencies. Likewise, an electron can only vibrate with specific frequencies. In the case of an electron, these frequencies are called and the states associated with these frequencies are called or . The set of all eigenfunctions for an electron form a mathematical set called the . There are an infinite number of these spherical harmonics, but they are specific and . That is, there are no in-between states. Thus an atomic electron can only absorb and emit energy in specific in small packets called . It does this by making a from one eigenstate to another. This term has been perverted in popular culture to mean any sudden, large change. In physics, quite the opposite is true. A quantum leap is the smallest possible change of system, not the largest.

Some probability densities for electrons in a hydrogen atom
|2,0,0⟩ |2,1,0⟩ |2,1,1⟩
|3,0,0⟩ |3,1,0⟩ |3,1,1⟩ |3,2,0⟩ |3,2,1⟩ |3,2,2⟩


In mathematics, the infinite sequence of fractions 1 1 , 1 2 , 1 3 , 1 4 , … is called the . Surprisingly, there are exactly the same number of harmonics described by the harmonic sequence as there are harmonics described by the "odds only" sequence: 1 1 , 1 3 , 1 5 , 1 7 , …. "What? Obviously there are more numbers in the harmonic sequence than there are in the 'odds only' sequence." Nope. There are exactly the same number. Here's the proof. I can set up a between the whole numbers and the odd numbers. Observe. (I will have to play with the format of the numbers to get them to line up correctly on a computer screen, however.)

0 1, 0 2, 0 3, 0 4, 0 5, 0 6, 0 7, 0 8, 0 9, …
0 1, 0 3, 0 5, 0 7, 0 9, 11, 13, 15, 17, …

This can go on forever. Which means there are exactly the same number of odd numbers as there are whole numbers. Both the whole numbers and the odd numbers are examples of sets.

There are an infinite number of possible wavelengths that can form standing waves under all of the circumstances described above, but there are an even greater number of wavelengths that can't form standing waves. "What? How can you have more than an infinite amount of something?" Well I don't want to prove that right now so you'll have to trust me, but there are more between 0 and 1 than there are whole numbers between zero and infinity. Not only do we have all the less than one ( 1 2 , 3 5 , 733 2741 , etc.) we also have all the possible (√2, 7 − √13, etc.) and the whole host of bizarre (π, e, e π , Feigenbaum's number, etc.). All of these numbers together form an set called the . The number of whole numbers is an infinity called ( ℵ0 ) the number of real numbers is an infinity called c (for ). The study of infinitely large numbers is known as . In this field, it is possible to prove that ℵ0 is less than c. There is no one-to-one correspondence between the real numbers and the whole numbers. Thus, there are more frequencies that won't form standing waves than there are frequencies that will form standing waves.

How to tell if a central element in a molecule needs to form hybridized orbitals?

Diatomic molecules will always point compatible #sigma# bonding orbital lobes along the internuclear axis, and be able to pair compatible orbitals, so there is no hybridization in molecules like #"HCl"# , #"NO"^(+)# , #"Cl"_2# , etc.

An easy way to tell when an atom has to hybridize is to count the number of surrounding atoms. I've listed examples below. Essentially:

  • Octahedral electron geometry? #sp^3d^2#
  • Trigonal bipyramidal electron geometry? #sp^3d#
  • Tetrahedral electron geometry? #sp^3#
  • Trigonal planar electron geometry? #sp^2#
  • Linear polyatomic electron geometry? #sp#


Oxygen in #"H"_2"O"# has to contribute four #sp^3# -hybridized atomic orbitals to bond because:

  1. It is bonding to more than one hydrogen. That tells you that hybridization can occur.
  2. It is bonding in a non-horizontal direction to at least one hydrogen. That tells you that hybridization should occur to orient all the orbitals correctly.
  3. It is bonding identically to each hydrogen. That tells you that hybridization has occurred to make the orbitals compatible.

It is #sp^3# because four electron groups are surrounding oxygen: two bonding to one hydrogen each, and two lone pairs not bonding at all.

Oxygen had to hybridize one #2s# and three #2p# orbitals together to generate four diagonal-oriented orbitals in three dimensions ( #x,y,z# ). Two of them could be used, but are not being used.


Boron in #"BH"_3# has to contribute three #sp^2# -hybridized atomic orbitals to bond because:

  1. It is bonding to more than one hydrogen. That tells you that hybridization can occur.
  2. It is bonding in a non-horizontal direction to at least one hydrogen. That tells you that hybridization should occur to orient all the orbitals correctly.
  3. It is bonding identically to each hydrogen. That tells you that hybridization has occurred to make the orbitals compatible.

It is #sp^2# because three electron groups are surrounding boron: three bonding to one hydrogen each, and one empty #p_z# orbital from boron that isn't compatible with hydrogen's #1s# atomic orbital. It isn't using that one to bond at the moment.

Boron had to hybridize one #2s# and two #2p# orbitals together to generate three diagonal-oriented orbitals in two dimensions ( #x,y# ).


One chosen carbon in #"H"-"C"-="C"-"H"# has to contribute two #sp# -hybridized atomic orbitals to bond because:

  1. It is bonding to more than one atom. That tells you that hybridization could occur.
  2. It is NOT bonding in a non-horizontal direction with any atoms. This doesn't tell you anything about hybridization.
  3. It is NOT bonding identically to each surrounding atom (the other #"C"# and a #"H"# ). That tells you that hybridization had to occur to make the orbitals of #"C"# and #"H"# compatible. Naturally, #"C"# is compatible with itself, so hybridization is necessary to bond with BOTH #"C"# and #"H"# .

It is #sp# because two electron groups are surrounding one chosen carbon: one bonding to one hydrogen and one bonding to the other carbon.

Carbon had to hybridize one #2s# and one #2p# orbital together to generate two horizontally-oriented #sp# hybridized orbitals in one dimension to #sigma# bond to two different atoms.

Separately, the remaining two bonds to be made to the other carbon (one triple bond has one #sigma# and two #pi# bonds) are made using the #p_x# and #p_y# atomic orbitals of carbon.

So, with acetylene, carbon is using two #sp# hybridized atomic orbitals and one #2p_x# and one #2p_y# atomic orbital to bond.

Guidance on Kepler's Laws

Teaching Guidance for 14-16

When Tycho Brahe was 17, he observed the conjunction of Jupiter and Saturn and was dismayed to find that the astronomical tables of the time were inaccurate in predicting the event by as much as a month. He decided to devote his life to making better tables, for which purpose he constructed better and better instruments.

The birth of modern planetary astronomy, with the three planetary laws discovered by Kepler, was based on the precise observations resulting from Tycho Brahe’s passion for accuracy.

Kepler: Law-giver of the heavens

In the course of his lifetime, Kepler extracted the three great planetary laws which we now call by his name.

  • The orbit of each planet is an ellipse with the Sun at one focus.
  • The arm from the Sun to a planet sweeps out equal areas in equal periods of time. If you mark the position of a planet once a month on its elliptical orbit, and draw radii from the Sun to those points, the areas of sectors between those radii are all equal.
  • If for each planet you take an average radius, R , and the time, T , the planet takes to go once round its orbit (its year) then the ratio R &thinsp3 T &thinsp2 is the same for all planets

The third law, which binds the movements of the planets together mathematically, Kepler discovered, with tremendous delight, quite late in life.

Mapping the Earth’s orbit in space and time

To map the Earth’s orbit around the Sun on a scale diagram you need many sets of measurements, each set giving the Earth’s bearings from two fixed points. Kepler took the fixed Sun for one of these and for the other he took Mars at a series of times when it was in the same position in its orbit.

Kepler proceeded thus: he marked the ‘position’ of Mars in the star pattern at one position (opposite the Sun, overhead at midnight). That gave him the direction of a base line, Sun – Earth – Mars, SE1M. Then he turned the pages of Tycho’s records to a time exactly one Martian year later. (The time of Mars’ motion around its orbit was known accurately from records over many centuries).

Kepler s Scheme to plot the Earth s orbit.

Then Kepler knew that Mars was in the same position, M, so that SM had the same direction. By now, the Earth had moved on to E2 in its orbit. Tycho’s record of the position of Mars in the star pattern gave him the new apparent direction of Mars E2 M and the Sun’s position gave him E2 S. Then he could calculate the angles of the triangle SE2M from the record thus: since he knew the directions E1 M and E2 M (marked on the celestial sphere of stars) he could calculate angle A between them. Since he knew the directions E1 S and E2 S he could calculate angle B. Then on a scale diagram he could choose two points to represent S and M and locate the Earth’s position,E2 as follows.

At the ends of the fixed base line SM, draw lines making angles A and B and mark their intersection E2 . One Martian year later he could find the directions E3 M and E3 S from the records and mark E3 on his diagram. Thus Kepler could start with the points S and M and locate E2 ,E3 ,E4 . enough points to show the orbit’s shape.

Knowing the Earth’s true orbit he could invert the investigation and plot the shape of Mars’ orbit. He found that he could treat the Earth’s orbit either as an eccentric circle or as slightly oval but Mars’ orbit was far from circular: it was definitely oval. It was an ellipse with the Sun at one focus – Kepler’s First Law of planetary motion.

Planetary data and Kepler’s Third Law

Kepler continued to brood on one of his early questions: what connection is there between the size of the planet’s orbit and the times of its year ?

Students can try and investigate the relationship between the planetary orbit radius, R , and the orbital time, T, using modern data. These are more accurate than the data available to Kepler. It will become obvious, fairly quickly, that simple proportion will not do. For example as R almost doubles in going from Mercury to Venus, T, almost triples as R grows almost 10 times from Earth to Saturn, T , grows about 30 times.

Kepler wrestled with this for a very long time, trying different combinations, until he found that R &thinsp3 T &thinsp2 was a constant. Kepler was overjoyed!

His three laws were clear, simple and powerful and they fitted the facts very accurately. He earned the title law-giver of the heavens .


Data for the planets, for Jupiter’s moons and for objects orbiting the Earth can be downloaded here.

Watch the video: Students explain Orbital Resonances (January 2023).