Download Week 2 File

Lecture 4 Gravity and the orbits of the planets Geocentric model of the universe A corner stone of the scien8fic method is the belief that the universe can be described and understood according to ra8onal principles. In other words, the universe is governed by physical laws and human beings are able to determine what those physical laws are, and can describe them using a combina8on of words and symbolic representa8on (e.g. through mathema8cs). This basic idea originated in ancient Greece and can be traced back to around 500 B.C. It was developed and expanded upon over centuries, with key contribu8ons coming from philosophers such as Pythagoras and Aristotle. In spite of this belief in ra8onal principles, the ancient Greeks (with the excep8on of Aristarchus) believed in a geocentric model of the universe – one in which the Earth lies at the centre of the celes8al sphere with the Sun, Moon, planets and stars rota8ng around the sta8onary Earth. The celes8al sphere was assumed to be real, and the stars were viewed as points of light essen8ally imbedded in its inner surface. The diurnal mo8on of the stars performing one complete revolu8on every 24 hours was understood to occur because the celes8al sphere rotated once per day. The Sun and Moon par8cipated in this daily rota8on, and to explain their monthly and yearly mo8on rela8ve to the background stars they were believed to also driO slowly around the Earth in the opposite direc8on to the East-­‐West daily rota8on of the celes8al sphere. The ancient Greeks (and others) knew of only five planets (Mercury, Venus, Mars, Jupiter and Saturn). As with the Sun and Moon, the planets follow the diurnal mo8on of the stars and also driO round the celes8al sphere. The Sun and Moon always driO from west to east, with the Sun following the path through the stars/constella8ons called the eclip8c, and the Moon following a path close to the eclip8c that is inclined by about 5 degrees. The planets also move along paths that are close to the eclip8c (because their orbit planes are only slightly inclined rela8ve to the Earth’s), but instead of showing a monotonic west-­‐east driO, they occasionally appear to move back and forth on the night sky as shown in the diagram below for Mars. The word “planet” comes from the Greek for “wanderer” – which is clearly appropriate. The usual eastward driO of a planet rela8ve to the stars is called direct mo8on. The occasional westward movement is called retrograde mo8on. All planets undergo retrograde mo8on. The Ptolemaic system Explaining this retrograde mo8on was a major challenge for ancient astronomers. One of the most successful models originated with Appollonius of Perga and Hipparchus in the 2nd century B.C. and was developed by Ptolemy in Alexandria during the 2nd century A.D. Planetary mo8on was viewed as a superposi8on of slow mo8on on a large circle called a deferent and more rapid mo8on on a smaller circle called an epicycle. Representa8on of Geocentric universe from Cosmographia (Peter Appian 1539) The Copernican Revolu8on – a heliocentric model for the Solar System Nicolaus Copernicus (1473-­‐1543) goes down in history as the person most responsible for over-­‐throwing the geocentric model of the universe and replacing it with a heliocentric one in which all the planets, including the Earth, orbit the Sun. A heliocentric model provides a natural explana8on for the apparent retrograde mo8on of the planets, as shown in the diagram. Copernicus believed that the orbits of the planets round the Sun were circular. This is not the case, so in order to make his theory agree with observa8ons that showed the planets moving with varying speed across the celes8al sphere he maintained the use of epicycles. His theory was published in a book in 1543: De revolu)onibus orbium coeles)um Copernicus noted that Venus and Mercury are always observed to be close to the Sun in the sky, and concluded that their orbits must be smaller than the Earth’s. Planets in smaller orbits are called inferior planets. Mars, Jupiter and Saturn, however, are some8mes seen high in the sky at midnight, opposite the Sun on the celes8al sphere, so their orbits must be larger than the Earth’s -­‐ such planets are called superior planets. An inferior planet observed at greatest eastern elonga8on is visible as an evening “star”. Observed at greatest western elonga8on it is a morning star. The inferior orbits of Venus & Mercury explain their cyclic occurrence as evening & morning objects. A superior planet is best observed at opposi8on when it is high in the sky at midnight and is at its closest approach to the Earth. Any planet observed at conjunc8on is above the horizon during the day and not well placed for night 8me viewing. Lecture 5 Copernicus developed a simple mathema8cal rela8onship between synodic and sidereal orbital periods (see accompanying notes on the course web page). The sidereal period of a planet is the 8me required for it to undergo one full orbit around the Sun (rela8ve to the fixed stars). This cannot be measured from observa8on on Earth and so must be calculated. The synodic period is the 8me between adjacent inferior conjunc8ons for an inferior planet, and the 8me between adjacent opposi8ons for a superior planet. The synodic period can be measured simply from observa8ons, allowing the sidereal (or orbital) period to be calculated. Using observa8ons and trigonometry, Copernicus was able to measure the rela8ve distances of the planets from the Sun in units of AU (see diagrams for measuring distance to Venus at the point of greatest eastern elonga8on and measuring distance to Jupiter by measuring the 8me between opposi8on and “quadrature” -­‐ when the Sun, Earth and Jupiter form a right angle). Copernicus did not know how large 1 AU is, and so could not measure the actual distances to the planets. In the modern age, real distances measured in metres can be determined by 8ming of radar signals between Earth and Venus. Tycho Brahe’s observa8ons Tycho Brahe (1546-­‐1601) was a Danish astronomer whose accurate observa8ons of the posi8ons of the stars and moving planets drama8cally superseded those of earlier astronomers. He built two observatories under the patronage of King Frederik II of Denmark (Uraniborg and Stjerneborg – “heavenly” and “star” castle). He did not have access to telescopes – all observa8ons were undertaken with the naked eye. He tried to measure the distance to a supernova that appeared in 1572 using parallax associated with the Earth’s rota8on about its axis. His failure to detect parallax indicated that the stars were further away than had previously been imagined. He also tried to measure the parallax of a comet in 1577, but again failed (naked eye observa8ons allow angular separa8ons on the sky of approximately 1 arcminute to be measured, a telescope is required for bejer accuracy). Tycho believed in a geocentric model of the universe because he could not detect the parallax of stars that ought to be associated with the Earth’s orbital mo8on around the Sun -­‐ no one at that 8me had any concep8on of how far the stars actually are from the Solar System, and therefore how small the parallax angles are. Kepler’s three laws of planetary mo8on Johannes Kepler (1571-­‐1630) became Tycho Brahe’s assistant one year before Tycho died, and used his observa8ons of the planets to derive three laws of planetary mo8on. His big conceptual leap was to recognise that the heliocentric system of Copernicus was essen8ally correct, but that the planetary orbits are ellipses rather than circles. 1). The orbit of a planet about the Sun is an ellipse with the Sun at one focus This law is illustrated by the figure. The long axis of the ellipse is called the major axis (and half of this is called the semimajor axis, denoted ‘a’) and the short axis is called the minor axis. The Sun sits at one focus and the other focus is empty. A larger value of the orbital eccentricity (denoted ‘e’) gives a more elongated ellipse. The closest approach of a planet to the Sun is called perihelion and its distance is Rp=a(1-­‐e). The most distant point from the Sun is called aphelion, and its distance is Ra=a(1+e). A circle is an ellipse in the limit e=0 and where the two foci merge at the centre. 2) A line joining a planet and the Sun sweeps out equal areas in equal intervals of 8me The diagram illustrates this law as both of the shaded areas are of equal area. This occurs as a result of the conserva8on of angular momentum of a planet as it orbits the Sun, causing it to move faster around its orbit when close to the Sun, and more slowly when it is more distance. Please see the supplementary notes on the course web page for a simple proof of this law (also demonstrated in the lecture). 3) The square of the sidereal period of a planet is directly propor8onal to the cube of the semimajor axis of the orbit This law states that the larger the orbit the longer the orbital period. The orbital periods and semimajor axes of two planets are related according to (P1/P2)2 = (a1/a2)3. For a single planet, if we express its period in years and semimajor axis in AU then we have P2=a3. Example: a(Mars)=1.52 AU, P=1.87 years. As we will see, Kepler’s laws are a direct consequence of Newton’s laws of mo8on and gravity. Galileo – The first telescopic astronomical observa8ons Dutch op8cians invented the telescope at the beginning of 17th century. Galileo Galilei (1564-­‐1642) was the first person to use it to perform astronomical observa8ons. Some of these provided direct evidence for the heliocentric model and were published in Siderius Nuncius (Starry Messenger). Galileo’s observa8ons of Venus showed that it displayed phases similar to the Moon (see right). These can only be observed If Venus goes around the Sun on an inferior orbit (see below), and cannot be explained by a geocentric model (below right). Venus also changes apparent size with its phases as expected if it orbits the Sun. Galileo discovered four moons when he observed Jupiter in 1610 – the “Galilean moons” Io, Europa, Ganymede and Callisto. His observa8ons over a number of nights demonstrated that the moons orbit Jupiter in the same manner that the planets orbit the Sun in the Copernican model. There was now direct evidence that some celes8al bodies orbit another celes8al body that is not the Earth -­‐ fatally undermining the geocentric model. Although Galileo had demonstrated beyond reasonable doubt that the heliocentric model provided a bejer explana8on of the available data that the geocentric model, he was not able to explain why the Earth should orbit the Sun instead of the other way round. Lecture 6 Newton’s laws of mo8on Kepler’s laws were empirical – based directly on observa8ons and data. Isaac Newton (1642-­‐1727) adopted the approach of introducing fundamental laws of nature, based on experimenta8on and observa8on, and then showed that Kepler’s laws derived logically from these laws of nature. Kepler’s laws are a direct consequence of Newton’s three laws of mo8on and his law of gravity. Newton’s first law An object remains at rest, or moves in a straight line at constant speed, unless acted on by a net external force -­‐> Circular Mo8on and Centripetal Force Newton’s second law The accelera8on (rate of change of velocity) of an object is equal to the net external force applied divided by the mass of the object: a=F/m Newton’s third law When an object exerts a force on a second object, the second object exerts an equal and opposite force on the first object. Newton’s law of universal gravita8on Two objects ajract each other with a force that is directly propor8onal to the mass of each object and inversely propor8onal to the square of the distance between them. The constant of propor8onality, G, is called the universal constant of gravita8on. G=6.67 x 10-­‐11 N m2 kg-­‐2. The gravita8onal force that the Sun exerts on the Earth can be calculated: FS-­‐E = -­‐ 6.67 x 10-­‐11 x (1.99 x 1030) x (5.98 x 1024) / (1.5 x 1011)2 = -­‐3.53 x 1022 N The force that the Earth exerts on the Sun is FE-­‐S=3.53 x 1022 N in agreement with Newton’s 3rd law. We can calculate the force that the Earth exerts on a 50 kg object sipng on the Earth’s surface: FE = -­‐ 6.67 x 10-­‐11 x (5.98 x 1024) x 50/ (6.38 x 106)2 = 490 N This is the weight of a 50 kg person standing on the surface of the Earth. Newton’s form of Kepler’s third law Newton recognised that two gravita8ng bodies in orbit exert an equal and opposite gravita8onal force on each other, causing them to orbit about their common centre of mass (COM). The size of the orbit followed by each object depends on their mass ra8o, as shown in the diagram. Using his law of gravity he derived the following form for Kepler’s 3rd law: where a=semimajor axis. Please see the notes on the course web site for a deriva8on. Earlier we discussed the fact that Galileo could not explain why the Earth orbits the Sun rather than the other way round. Newton’s laws demonstrate why this is so: the Earth is much less massive than the Sun (MEarth/Msun = 3 x 10-­‐5) so the Sun lies much closer to the COM of the Earth-­‐Sun system. In fact the COM of the Solar system lies inside the Sun for ~ 37% of the 8me and modestly above its surface ~ 63 % of the 8me, depending on the alignment of the planets. Note that the COM only moves as the Solar System orbits the Galaxy, and not as a result of the mutual orbits of the Sun and planets. Kepler’s third law can be used to calculate the combined mass of a pair of orbi8ng bodies. For example, consider, Io, one of the galilean satellites of Jupiter. It orbits at a distance of 421,600 km from the centre of Jupiter and has an orbital period of 1.77 days. From Kepler’s 3rd law d the combined masses can be wrijen as (M1+M2) = 4π2 a3/(GP2) Conver8ng km to metres and days into seconds we obtain (M1+M2) = 4π x (4.216 x 108)3/(6.67 x 10-­‐11 x 1529282 ) = 1.895 x 1027 kg This is equivalent to 9.5 x 10-­‐4 of the Solar mass. We note that Jupiter is much more massive than Io, so we can infer that Jupiter’s mass is very close to 9.5 x 10-­‐4 solar masses. Gravity and orbits The size of an orbit (the semimajor axis) depends on the total energy of the orbi8ng body – the sum of its gravita8onal poten8al and kine8c energies. Eccentricity depends on the angular momentum and energy. Consider a ball being released from point A with different horizontal veloci8es, vH. If vH is zero then the ball drops ver8cally. If vH is small then it follows trajectory B and hits the Earth. Increasing vH eventually allows the ball to go into orbit with an eccentric orbit and perigee close to the Earth’s surface (path D). Trajectory E corresponds to vH having the value required for a circular orbit. A larger vH results in an orbit with apogee far from Earth (F). Orbits can be parabolic and hyperbolic. These correspond to the sum of the kine8c and gravita8onal poten8al energies of the orbi8ng body being either zero or posi8ve. A spacecraO leaving the Earth with the escape velocity is on a parabolic orbit, and one with a velocity that exceeds the escape velocity is on a hyperbolic orbit. Comets coming in from the Oort cloud may have parabolic or hyperbolic orbits. All orbits have the shape of different ‘conic sec8ons’ as shown in the diagram. See movie at link: http://www.galacticcenter.astro.ucla.edu/animations.html
A 2.2 micron animation of the stellar orbits in the central 0.5 arcsec of our Galaxy.
Images from years 1995 through 2013.
A simple application of Kepler’s Laws, provides a mass of 4 million times the mass of the
Sun for the proposed supermassive black hole at the Center of our Galaxy.