Posts Tagged ‘Kepler

A Historical Method Approach to Teaching Kepler’s 2nd law

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Wladimir Lyra
Kepler’s 2nd law, the law of the areas, is usually taught in passing, between the 1st and the 3rd laws, to be explained “later on” as a consequence of angular momentum conservation. The 1st and 3rd laws receive the bulk of attention; the 1st law because of the paradigm shift significance in overhauling the previous circular models with epicycles of both Ptolemy and Copernicus, the 3rd because of its convenience to the standard curriculum in having a simple mathematical statement that allows for quantitative homework assignments and exams. In this work I advance a method for teaching the 2nd law that combines the paradigm-shift significance of the 1st and the mathematical proclivity of the 3rd. The approach is rooted in the historical method, indeed, placed in its historical context, Kepler’s 2nd is as revolutionary as the 1st: as the 1st law does away with the epicycle, the 2nd law does away with the equant. This way of teaching the 2nd law also formulates the “time=area” statement quantitatively, in the way of Kepler’s equation, M = E – e sin E (relating mean anomaly M, eccentric anomaly E, and eccentricity e), where the left-hand side is time and the right-hand side is area. In doing so, it naturally paves the way to finishing the module with an active learning computational exercise, for instance, to calculate the timing and location of Mars’ next opposition. This method is partially based on Kepler’s original thought, and should thus best be applied to research-oriented students, such as junior and senior physics/astronomy undergraduates, or graduate students.

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Written by physicsgg

November 30, 2020 at 10:54 am

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Kepler’s Second Law of Motion

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Written by physicsgg

June 11, 2015 at 5:12 am


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The laws of planetary motion …

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… derived from those of a harmonic oscillator

KEPLERP. A. Horvathy
Kepler’s laws are deduced from those valid for a harmonic oscillator, following the approach of Bohlin, Levi-Civita and Arnold

1 Introduction
Kepler’s laws of planetary motion state that
1. K-I: A planet moves on an ellipse, one of whose foci being occupied by the sun;
2. K-II: The vector drawn from the sun to the planet’s position sweeps equal areas in equal
3. K-III: The squares of the periods are as the cubes of the major axes of the ellipses.
These laws can be deduced from the inverse-square force law and Newton’s equations of
…

Written by physicsgg

April 10, 2014 at 1:33 pm

Kepler 78b exoplanet is Earth-like in mass and size

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Kepler-78b is a planet that shouldn't exist. This scorching lava world, shown here in an artist's conception, circles its star every eight and a half hours at a distance of less than one million miles. According to current theories of planet formation, it couldn't have formed so close to its star, nor could it have moved there

Kepler-78b is a planet that shouldn’t exist. This scorching lava world, shown here in an artist’s conception, circles its star every eight and a half hours at a distance of less than one million miles. According to current theories of planet formation, it couldn’t have formed so close to its star, nor could it have moved there

In August, MIT researchers identified an exoplanet with an extremely brief orbital period: The team found that Kepler 78b, a small, intensely hot planet 700 light-years from Earth, circles its star in just 8.5 hours—lightning-quick, compared with our own planet’s leisurely 365-day orbit. From starlight data gathered by the Kepler Space Telescope, the scientists also determined that the exoplanet is about 1.2 times Earth’s size—making Kepler 78b one of the smallest exoplanets ever measured.

Now this same team has found that Kepler 78b shares another characteristic with Earth: its mass. By analyzing the movement of its host star, Kepler 78, the scientists determined that the exoplanet is about 1.7 times as massive as the Earth. From the same measurements, they calculated that the planet’s density is 5.3 grams per cubic centimeter, closely resembling Earth’s density (5.5 grams per cubic centimeter).
The findings make Kepler 78b the smallest exoplanet for which the mass and size are known. These new measurements provide strong evidence that Kepler 78b is composed mostly of rock and iron, similar to Earth.
However, that’s where the similarities may end: The exoplanet, due to its extreme proximity to its star, is likely blazing at temperatures too high to support life.
“It’s Earth-like in the sense that it’s about the same size and mass, but of course it’s extremely unlike the Earth in that it’s at least 2,000 degrees hotter,” says team member Josh Winn, an associate professor of physics at MIT and a member of the Kavli Institute for Astrophysics and Space Research. “It’s a step along the way of studying truly Earth-like planets.”
Winn and his colleagues, including lead author Andrew Howard, of the University of Hawaii, publish their results this week in the journal Nature. The group’s results appear in the same issue as a paper published by a separate group in Geneva, reporting similar results—scientific agreement that Winn says adds confidence to the mass measurement.

Watching for a wobble
Planets with extremely tight orbits offer scientists a wealth of data: For instance, each week Kepler 78b circles its star about 20 times, giving researchers numerous opportunities to observe its behavior.

This illustration compares our Earth with the newly confirmed lava planet Kepler-78b. Kepler-78b is about 20 percent larger than Earth, with a diameter of 9,200 miles, and weighs roughly 1.8 times as much as Earth. Credit: David A. Aguilar (CfA)

This illustration compares our Earth with the newly confirmed lava planet Kepler-78b. Kepler-78b is about 20 percent larger than Earth, with a diameter of 9,200 miles, and weighs roughly 1.8 times as much as Earth. Credit: David A. Aguilar (CfA)

The team previously determined Kepler 78b’s orbit and size by analyzing the light given off by the star as the planet passes in front of it, or transits. The researchers detected a transit each time the star’s light dipped, and measured this dimming to determine the planet’s size. (The bigger an exoplanet, the more light it blocks.)

Measuring the planet’s mass was a somewhat trickier endeavor. Instead of tracking the planet’s motion, the researchers tracked the motion of the star itself. Depending on its mass, a planet can exert a gravitational tug on its star. This stellar motion can be detected as a very slight wobble, known as a Doppler shift.
Winn and his colleagues looked to measure Kepler 78’s Doppler shift by analyzing observations from the Keck Observatory in Hawaii—one of the largest telescopes in the world. The team analyzed starlight data taken over a period of eight days. Despite the telescope’s strength, the signal from the star was incredibly faint, making a daunting task for the scientists.
“Each of the eight nights along the way, we were agonizing over it, whether it was worth continuing or not,” Winn recalls.
Out, damn starspot
In addition to the challenge of picking out such tiny signals, the researchers had to contend with an effect that initially muddled the data: starspots, dark patches on the surface of stars. Graduate student Roberto Sanchis-Ojeda, who has studied the effect of starspots on exoplanet detection, says the troublesome patches can make a star’s Doppler shift appear larger, dramatically complicating scientists’ calculations of a planet’s mass.
Sanchis-Ojeda was able to solve this puzzle by taking into account Kepler 78’s rotational period. By tracking the frequency at which certain starspots reappeared, Sanchis-Ojeda determined that the star completes a full rotation every 12.5 days—considerably longer than the planet’s orbital period of 8.5 hours. From these measurements, Sanchis-Ojeda was able to calculate the star’s true Doppler shift.
From his calculations, Sanchis-Ojeda found that the star rotates relatively slowly, at 1.5 meters per second—about the speed of a jog, or a brisk walk.
“The star is moving at the same speed as when we walk to school or go grocery shopping,” Sanchis-Ojeda notes. “The difference is that this star is 700 light-years away, so imagine how complicated it is to measure such speeds from so far away.”
From the star’s Doppler shift, the team determined that Kepler 78b’s mass is 1.7 times that of Earth—a measurement that suggests the planet is made mostly of rock and iron. Such a composition, Winn says, is not surprising, given the planet’s extremely close proximity to its star. A less massive planet, such as one made entirely of gas, would not be able to hold together in such a tight orbit.
While its similarities to Earth likely end with Kepler 78b’s size and mass, Winn says there is still more to learn about the planet, such as its surface and atmospheric composition—a goal that the group plans to pursue next.

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Written by physicsgg

October 30, 2013 at 8:53 pm


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NASA’s Kepler Discovers its Smallest ‘Habitable Zone’ Planets to Date

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elative sizes of Kepler habitable zone planets discovered as of April 18, 2013. Left to right: Kepler-22b, Kepler-69c, Kepler-62e, Kepler-62f, and Earth (except for Earth, these are artists' renditions). Image credit: NASA Ames/JPL-Caltech

Relative sizes of Kepler habitable zone planets discovered as of April 18, 2013. Left to right: Kepler-22b, Kepler-69c, Kepler-62e, Kepler-62f, and Earth (except for Earth, these are artists’ renditions). Image credit: NASA Ames/JPL-Caltech

NASA’s Kepler mission has discovered two new planetary systems that include three super-Earth-size planets in the “habitable zone,” the range of distance from a star where the surface temperature of an orbiting planet might be suitable for liquid water.

The Kepler-62 system has five planets; 62b, 62c, 62d, 62e and 62f. The Kepler-69 system has two planets; 69b and 69c. Kepler-62e, 62f and 69c are the super-Earth-sized planets.

Two of the newly discovered planets orbit a star smaller and cooler than the sun. Kepler-62f is only 40 percent larger than Earth, making it the exoplanet closest to the size of our planet known in the habitable zone of another star. Kepler-62f is likely to have a rocky composition. Kepler-62e, orbits on the inner edge of the habitable zone and is roughly 60 percent larger than Earth.

The third planet, Kepler-69c, is 70 percent larger than the size of Earth, and orbits in the habitable zone of a star similar to our sun. Astronomers are uncertain about the composition of Kepler-69c, but its orbit of 242 days around a sun-like star resembles that of our neighboring planet Venus.

Scientists do not know whether life could exist on the newfound planets, but their discovery signals we are another step closer to finding a world similar to Earth around a star like our sun….
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Written by physicsgg

April 18, 2013 at 8:24 pm

The Mystery of Tatooine Planet Formation

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How not to build Tatooine: the difficulty of in situ formation of circumbinary planets Kepler 16b, Kepler 34b and Kepler 35b

Histogram of collision velocities for Kepler 16 scenario without dust accretion, t = 250,000 binary orbits. Colours indicate different radii, with r in units of ab

Sijme-Jan Paardekooper, Zoe M. Leinhardt, Philippe Thebault, Clement Baruteau
We study planetesimal evolution in circumbinary disks, focusing on the three systems Kepler 16, 34 and 35 where planets have been discovered recently. We show that for circumbinary planetesimals, in addition to secular forcing, eccentricities evolve on a dynamical timescale, which leads to orbital crossings even in the presence of gas drag. This makes the current locations of the circumbinary Kepler planets hostile to planetesimal accretion. We then present results from simulations including planetesimal formation and dust accretion, and show that even in the most favourable case of 100% efficient dust accretion, in situ growth starting from planetesimals smaller than ~10 km is difficult for Kepler 16b, Kepler 34b and Kepler 35b. These planets were likely assembled further out in the disk, and migrated inward to their current location….
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June 19, 2012 at 10:01 am


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Kepler Catalog Adds More Planet Candidates

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The histogram summarizes the findings in the Feb. 27, 2012 Kepler Planet Candidate catalog release. The catalog contains 2,321 planet candidates identified during the first 16 months of observation conducted May 2009 to September 2010. Of the 46 planet candidates found in the habitable zone, the region in the planetary system where liquid water could exist, ten of these candidates are near-Earth-size. Credit: NASA Ames/Wendy Stenzel

Since science operations began in May 2009, the Kepler team has released two catalogs of transiting planet candidates. The first catalog (Borucki et al, 2010), released in June 2010, contains 312 candidates identified in the first 43 days of Kepler data. The second catalog (Borucki et al, 2011), released in February 2011, is a cumulative catalog containing 1,235 candidates identified in the first 13 months of data.

Now the team presents the third catalog containing 1,091 new planet candidates identified in the first 16 months of observation conducted May 2009 to September 2010. These are the same candidates that the team discussed at the Kepler Science Conference held at NASA Ames Research Center in December 2011.

Here are the highlights of the new catalog:

  •  Planet candidates smaller than twice the size of Earth increased by 197 percent, compared to 52 percent for candidates larger than twice the size of Earth.
  • Planet candidates with orbital periods longer than 50 days increased by 123 percent, compared to 85 percent for candidates with orbital periods shorter than 50 days.

Since the last catalog was released in February 2011, the number of planet candidates identified by Kepler has increased by 88 percent and now totals 2,321 transiting 1,790 stars.

The cumulative catalog now contains well over 200 Earth-size planet candidates and more than 900 that are smaller than twice Earth-size. Of the 46 planet candidates found in the habitable zone, the region in the planetary system where liquid water could exist, ten of these candidates are near-Earth-size.

The number of planetary systems found with more than one planet candidate also has increased. Last year, 17 percent, or 170 stars, had more than one transiting planet candidate. Today, 20 percent, or 365, stars have more than one.

“With each new catalog release a clear progression toward smaller planets at longer orbital periods is emerging, ” said Natalie Batalha, Kepler deputy science team lead at San Jose State University in California. “This suggests that Earth-size planets in the habitable zone are forthcoming if, indeed, such planets are abundant.”

Nearly 5,000 periodic transit-like signals were analyzed with known spacecraft instrumentation and astrophysical phenomena that could masquerade as transits, which can produce false positives. The most common false positive signatures are associated with eclipsing binary stars- a pair of orbiting stars that eclipse each other from the vantage point of the spacecraft.

The Kepler space telescope identifies planet candidates by repeatedly measuring the change in brightness of more than 150,000 stars in search of planets that pass in front, or “transit,” their host star. Kepler must record at least three transits to verify a signal as a planet.

The findings are published in the “Planetary Candidates Observed by Kepler III: Analysis of the First 16 Months of Data”. The catalog is available at the Kepler data archive at the Space Telescope Science Institute and can be downloaded from the NASA Exoplanet Archive.………
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Written by physicsgg

March 11, 2012 at 12:53 pm

NASA’s Kepler Announces 11 Planetary Systems Hosting 26 Planets

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(click image to enlarge) Kepler's Planetary Systems, Jan. 2012

NASA’s Kepler mission has discovered 11 new planetary systems hosting 26 confirmed planets. These discoveries nearly double the number of verified Kepler planets and triple the number of stars known to have more than one planet that transits, or passes in front of, its host star. Such systems will help astronomers better understand how planets form.

The planets orbit close to their host stars and range in size from 1.5 times the radius of Earth to larger than Jupiter. Fifteen of them are between Earth and Neptune in size, and further observations will be required to determine which are rocky like Earth and which have thick gaseous atmospheres like Neptune. The planets orbit their host star once every six to 143 days. All are closer to their host star than Venus is to our sun.

“Prior to the Kepler mission, we knew of perhaps 500 exoplanets across the whole sky,” said Doug Hudgins, Kepler program scientist at NASA Headquarters in Washington. “Now, in just two years staring at a patch of sky not much bigger than your fist, Kepler has discovered more than 60 planets and more than 2,300 planet candidates. This tells us that our galaxy is positively loaded with planets of all sizes and orbits.”

Kepler identifies planet candidates by repeatedly measuring the change in brightness of more than 150,000 stars to detect when a planet passes in front of the star. That passage casts a small shadow toward Earth and the Kepler spacecraft.

“Confirming that the small decrease in the star’s brightness is due to a planet requires additional observations and time-consuming analysis,” said Eric Ford, associate professor of astronomy at the University of Florida and lead author of the paper confirming Kepler-23 and Kepler-24. “We verified these planets using new techniques that dramatically accelerated their discovery.”

Each of the new confirmed planetary systems contains two to five closely spaced transiting planets. In tightly packed planetary systems, the gravitational pull of the planets among themselves causes one planet to accelerate and another planet to decelerate along its orbit. The acceleration causes the orbital period of each planet to change. Kepler detects this effect by measuring the changes, or so-called Transit Timing Variations (TTVs).

Planetary systems with TTVs can be verified without requiring extensive ground-based observations, accelerating confirmation of planet candidates. The TTV detection technique also increases Kepler’s ability to confirm planetary systems around fainter and more distant stars.

“By precisely timing when each planet transits its star, Kepler detected the gravitational tug of the planets on each other, clinching the case for ten of the newly announced planetary systems,” said Dan Fabrycky, Hubble Fellow at the University of California, Santa Cruz and lead author for a paper confirming Kepler-29, 30, 31 and 32.”

Five of the systems (Kepler-25, Kepler-27, Kepler-30, Kepler-31 and Kepler-33) contain a pair of planets where the inner planet orbits the star twice during each orbit of the outer planet. Four of the systems (Kepler-23, Kepler-24, Kepler-28 and Kepler-32) contain a pairing where the outer planet circles the star twice for every three times the inner planet orbits its star.

“These configurations help to amplify the gravitational interactions between the planets, similar to how my sons kick their legs on a swing at the right time to go higher,” said Jason Steffen, the Brinson postdoctoral fellow at Fermilab Center for Particle Astrophysics in Batavia, Ill., and lead author of a paper confirming Kepler-25, 26, 27 and 28.

The system with the most planets among these discoveries is Kepler-33, a star that is older and more massive than our sun. Kepler-33 hosts five planets, ranging in size from 1.5 to 5 times that of Earth and all located closer to their star than any planet is to the sun……
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Written by physicsgg

January 26, 2012 at 9:05 pm


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