Ceres (bottom left), the Moon and Earth, shown to scale [Image NASA].
Posts Tagged 'Astronomy'
Gauss Predicts the Orbit of Ceres
Published June 24, 2021 Occasional Leave a CommentTags: Astronomy, Gauss
From Impossible Shapes to the Nobel Prize
Published October 8, 2020 Occasional 1 CommentTags: Astronomy, Geometry
Roger Penrose, British mathematical physicist, mathematician and philosopher of science has just been named as one of the winners of the 2020 Nobel Prize in Physics. Penrose has made major contributions to general relativity and cosmology.
Impossible triangle sculpture in Perth, Western Australia [image Wikimedia Commons].
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Is There Anyone Out There? The Drake Equation gives a Clue
Published August 20, 2020 Irish Times Leave a CommentTags: Astronomy
The Drake Equation is a formula for the number of developed civilizations in our galaxy, the Milky Way. This number is determined by seven factors. Some are known with good accuracy but the values of most are quite uncertain. It is a simple equation comprising seven terms multiplied together [TM193 or search for “thatsmaths” at irishtimes.com].
A plaque commemorating the first appearance of the Drake Equation at a conference at the National Radio Astronomy Observatory, Green Bank, West Virginia in 1961.
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We will describe some generic behaviour patterns of dynamical systems. In many systems, the orbits exhibit characteristic patterns called boxes and loops. We first describe orbits for a simple pendulum, and then look at some systems in higher dimensions.
Phase portrait for a simple pendulum. Each line represents a different orbit.
Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph
Published May 2, 2019 Irish Times Leave a CommentTags: Astronomy, Hamilton, Mechanics
The Greeks regarded the heavens as the epitome of perfection. All flaws and blemishes were confined to the terrestrial domain. Since the circle is perfect in its infinite symmetry, it was concluded by Aristotle that the Sun and planets move in circles around the Earth. Later, the astronomer Ptolemy accounted for deviations by means of additional circles, or epicycles. He stuck with the circular model [TM162 or search for “thatsmaths” at irishtimes.com].
Left: Elliptic orbit with velocity vectors. Right: Hodograph, with all velocity vectors plotted from a single point.
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K3 implies the Inverse Square Law.
Published April 25, 2019 Occasional Leave a CommentTags: Astronomy, Mechanics
Johannes Kepler. Stamp issued by the German Democratic Republic in 1971, the 400th anniversary of Kepler’s birth.
Kepler formulated three remarkable laws of planetary motion. He deduced them directly from observations of the planets, most particularly of the motion of Mars. The first two laws appeared in 1609 in Kepler’s Astronomia Nova. The first law (K1) describes the orbit of a planet as an ellipse with the Sun at one focus. The second law (K2) states that the radial line from Sun to planet sweeps out equal areas in equal times; we now describe this in terms of conservation of angular momentum.
The third law (K3), which appeared in 1619 in Kepler’s Harmonices Mundi, is of a different character. It does not relate to a single planet, but connects the motions of different planets. It states that the squares of the orbital periods vary in proportion to the cubes of the semi-major axes. For circular orbits, the period squared is proportional to the radius cubed.
Trappist-1 & the Age of Aquarius
Published January 3, 2019 Irish Times Leave a CommentTags: Astronomy, Mechanics
The Pythagoreans believed that the planets generate sounds as they move through the cosmos. The idea of the harmony of the spheres was brought to a high level by Johannes Kepler in his book Harmonices Mundi, where he identified many simple relationships between the orbital periods of the planets [TM154 or search for “thatsmaths” at irishtimes.com].
Artist’s impression of the Trappist-1 planetary system. Image from https://www.eso.org/public/images/eso1805b/
Kepler’s idea was not much supported by his contemporaries, but in recent times astronomers have come to realize that resonances amongst the orbits has a crucial dynamical function. Continue reading ‘Trappist-1 & the Age of Aquarius’
The 3 : 2 Resonance between Neptune and Pluto
Published December 13, 2018 Occasional 2 CommentsTags: Astronomy
For every two orbits of Pluto around the Sun, Neptune completes three orbits. This 3 : 2 resonance has profound consequences for the stability of the orbit of Pluto.
Unstable (left) and stable (right) orbital configurations.
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Gravitational Waves & Ringing Teacups
Published November 22, 2018 Occasional Leave a CommentTags: Astronomy, Relativity, Wave Motion
Newton’s law of gravitation describes how two celestial bodies orbit one another, each tracing out an elliptical path. But this is imprecise: the theory of general relativity shows that two such bodies radiate energy away in the form of gravitational waves (GWs), and spiral inwards until they eventually collide.
Warning sign, described by Thomas Moore as a “geeky insider GR joke” [image from Moore, 2013].
Johannes Kepler and the Song of the Earth
Published November 1, 2018 Irish Times 2 CommentsTags: Astronomy, Music
Johannes Kepler, German mathematician and astronomer, sought to explain the solar system in terms of divine harmony. His goal was to find a system of the world that was mathematically correct and harmonically pleasing. His methodology was scientific in that his hypotheses were inspired by and confirmed by observations. However, his theological training and astrological interests influenced his thinking [TM150, or search for “thatsmaths” at irishtimes.com].
The six planets known to Kepler [Image NASA].
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Galileo’s Book of Nature
Published February 15, 2018 Irish Times Leave a CommentTags: Astronomy, History, Mechanics
In 1971, astronaut David Scott, standing on the Moon, dropped a hammer and a feather and found that both reached the surface at the same time. This popular experiment during the Apollo 15 mission was a dramatic demonstration of a prediction made by Galileo three centuries earlier. Galileo was born in Pisa on 15 February 1564, just 454 years ago today [TM133 or search for “thatsmaths” at irishtimes.com].
Image: NASA
Darker Mornings, Brighter Evenings
Published December 21, 2017 Irish Times Leave a CommentTags: Astronomy, Time measurement
Today is the winter solstice, the shortest day of the year. We might expect that the latest sunrise and earliest sunset also occur today. In fact, the earliest sunset, the darkest day of the year, was on 13 December, over a week ago, and the latest sunrise is still more than a week away. This curious behaviour is due to the unsteady path of the Earth around the Sun. Our clocks, which run regularly at what is called mean time, move in and out of synchronization with solar time [TM129 or search for “thatsmaths” at irishtimes.com].
Sunrise in Newgrange on winter solstice [image from http://www.boynevalleytours.com/
The Star of Bethlehem … or was it a Planet?
Published December 7, 2017 Irish Times 1 CommentTags: Astronomy
People of old were more aware than we are of the night sky and took a keen interest in unusual happenings above them. The configuration of the stars was believed to be linked to human affairs and many astronomical phenomena were interpreted as signs of good or evil in the offing. The Three Wise Men or Magi were astrologers, experts in celestial matters, and would have drawn inferences from what they observed in the sky [TM128 or search for “thatsmaths” at irishtimes.com].
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Slingshot Orbit to Asteroid Bennu
Published November 16, 2017 Irish Times Leave a CommentTags: Astronomy, Mechanics
The Voyager 1 and Voyager 2 spacecraft have now left the solar system and will continue into deep space. How did we manage to send them so far? The Voyager spacecraft used gravity assists to visit Jupiter, Saturn, Uranus and Neptune in the late 1970s and 1980s. Gravity assist manoeuvres, known as slingshots, are essential for interplanetary missions. They were first used in the Soviet Luna-3 mission in 1959, when images of the far side of the Moon were obtained. Space mission planners use them because they require no fuel and the gain in speed dramatically shortens the time of missions to the outer planets.
Artist’s impression of OSIRIS-REx orbiting Bennu [Photo Credit: NASA]
Pedro Nunes and Solar Retrogression
Published October 12, 2017 Occasional Leave a CommentTags: Astronomy, Time measurement
In northern latitudes we are used to the Sun rising in the East, following a smooth and even course through the southern sky and setting in the West. The idea that the compass bearing of the Sun might reverse seems fanciful. But in 1537 Portuguese mathematician Pedro Nunes showed that the shadow cast by the gnomon of a sun dial can move backwards.
Pedro Nunes (1502–1578). Portuguese postage stamp issued in 1978.
Nunes’ prediction was counter-intuitive. It came long before Newton, Galileo and Kepler, and Copernicus’ heliocentric theory had not yet been published. The retrogression was a remarkable example of the power of mathematics to predict physical behaviour.
Quadrivium: The Noble Fourfold Way
Published July 20, 2017 Irish Times Leave a CommentTags: Arithmetic, Astronomy, Geometry, History, Music
According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato’s Republic. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6th century AD [see TM119 or search for “thatsmaths” at irishtimes.com].
Image from here.
It is said that an inscription over the entrance to Plato’s Academy read “Let None But Geometers Enter Here”. This indicated that the Quadrivium was a prerequisite for the study of philosophy in ancient Greece.
A New Window on the World
Published November 3, 2016 Irish Times Leave a CommentTags: Astronomy, Relativity
The motto of the Pythagoreans was “All is Number” and Pythagoras may have been the first person to imagine that the workings of the world might be understood in mathematical terms. This idea has now brought us to the point where, at a fundamental level, mathematics is the primary means of describing the physical world. Galileo put it this way: the book of nature is written in the language of mathematics [TM102, or search for “thatsmaths” at irishtimes.com].
Visualization of gravitational waves. Image credit MPI/Gravitational Physics/ITP Frankfurt/ZI Berlin.
Kepler’s Magnificent Mysterium Cosmographicum
Published October 13, 2016 Occasional 2 CommentsTags: Astronomy, Geometry, History
Johannes Kepler’s amazing book, Mysterium Cosmographicum, was published in 1596. Kepler’s central idea was that the distance relationships between the six planets (only six were known at that time) could be represented by six spheres separated by the five Platonic solids. For each of these regular polyhedra, there is an inner and an outer sphere. The inner sphere is tangent to the centre of each face and the outer sphere contains all the vertices of the polyhedron.
Figure generated using Mathematica Demonstration [2].
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The Search is on for Planet Nine
Published April 7, 2016 Irish Times Leave a CommentTags: Astronomy
Then felt I like some watcher of the skies
When a new planet swims into his ken.
John Keats: On First Looking into Chapman’s Homer
Pluto, discovered in 1930, orbits in the Kuiper Belt, a ring of asteroids and icy debris beyond Neptune. About ten years ago, it was reclassified as a “dwarf planet”, so now we have eight planets. Is there a ninth anywhere? A recent paper in the Astronomical Journal presents evidence of a large planet in an orbit far beyond Neptune. If confirmed, this would be a dramatic confirmation of our understanding of solar system dynamics [TM088, or search for “thatsmaths” at irishtimes.com].
Orbits of six trans-Neptunian objects aligned in one direction, suggesting a perturbing body, Planet Nine (?). Caltech/Robert Hurt.
Thomas Harriot: Mathematician, Astronomer and Navigator
Published August 27, 2015 Occasional Leave a CommentTags: Astronomy, History
Sir Walter Raleigh, adventurer, explorer and privateer, was among most colourful characters of Tudor times. He acquired extensive estates in Waterford and Cork, including Molana Abbey near Youghal, which he gave to his friend and advisor, the brilliant mathematician and astronomer Thomas Harriot.
Left: Sir Walter Raleigh (1552-1618). Right: Thomas Harriot (1560?-1621)
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The Great American Eclipse
Published August 20, 2015 Irish Times Leave a CommentTags: Astronomy, Time measurement
Just two years from now, on Monday, August 21, 2017, the Moon’s shadow will sweep across the United States at a speed of over 2,000 km/hr. The Great American Eclipse of 2017 will generate a frenzy of activity. [TM074: search for “thatsmaths” at irishtimes.com ].
Moon between NASA’s Solar Dynamics Observatory and the Sun,
giving a partial solar eclipse from space on Jan. 30, 2014. Image NASA.
An astrodynamical miracle is happening in the sky above. Our ability to launch a space-probe from the revolving Earth to reach a moving target billions of kilometres away almost ten years later with pinpoint accuracy is truly astounding. “New Horizons” promises to enhance our knowledge of the solar system and it may help us to understand our own planet too.
Four tiny moons (on left) orbit the binary system of Pluto (top centre right) and Charon [Image NASA]
The Hodograph
Published April 9, 2015 Occasional Leave a CommentTags: Astronomy, Fluid Dynamics, Geophysics
The Hodograph is a vector diagram showing how velocity changes with position or time. It was made popular by William Rowan Hamilton who, in 1847, gave an account of it in the Proceedings of the Royal Irish Academy. Hodographs are valuable in fluid dynamics, astronomy and meteorology.
Hodograph plot of wind vectors at five heights in the troposphere. This indicates vertical wind shear and also horizontal temperature gradients. Since the wind veers with height between V2 and V3, it is blowing warmer air north-eastwards to a colder region (image source: NOAA).
Falling Bodies [2]: Philae
Published November 27, 2014 Occasional Leave a CommentTags: Astronomy, Mechanics
The ESA Rosetta Mission, launched in March 2004, rendezvoused with comet 67P/C-G in August 2014. The lander Philae touched down on the comet on 12 November and came to rest after bouncing twice (the harpoon tethers and cold gas retro-jet failed to fire).
Comet 67P/Churyumov-Gerasimenko on 11 August 2014. The landing site is on the smaller knob, near the top of the image. Photo copyright ESA.
Light Weight (*)
Published October 30, 2014 Occasional Leave a CommentTags: Applied Maths, Astronomy, Mechanics, Physics
Does light have weight? Newton thought that light was influenced by gravity and, using his laws of motion, we can calculate how gravity bends a light beam. The effect is observable during a total eclipse of the sun: photographs of the sky are compared with the same region when the sun is elsewhere and a radial displacement of the star images is found. But the amount predicted by Newton’s laws is only half the observed value.
When did Hammurabi reign?
Published June 19, 2014 Irish Times Leave a CommentTags: Astronomy, History, Time measurement
The consequences of the Earth’s changing climate may be very grave. It is essential to understand past climate change so that we can anticipate future changes. This week, That’s Maths in The Irish Times ( TM047 ) is about the chronology of the Middle East. Surprisingly, this has important implications for our understanding of climate change.
Left: Image of Hammurabi in the US Congress.
Right: Part of an inscription of the Code of Hammurabi.
Solar System Perturbations
Published March 27, 2014 Occasional Leave a CommentTags: Astronomy, Mechanics
Remarkable progress in understanding the dynamics of the planets has been possible thanks to their relatively small masses and the overwhelming dominance of the Sun. The figure below shows the relative masses of the Sun, planets and some natural satellites, taking the mass of Earth to be unity.
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